CHAPTERI
THE GENERALIZED PERRON INTEGRAL
Definition of the integral
-oo < a < b < +oo be given. A
pair (r, J) of a point r E R and a compact interval J C IR is
Let an interval [a, b] C 1[I,
called a tagged interval, r is the tag of J.
A finite collection Z = {(rj, Jj), j = 1, ... , k) of tagged intervals is called a system in [a, b] if rj E JJ C [a, b] for every j = 1,. .. , k and the intervals JJ are nonoverlapping, i.e.
Int(J1) fl Int(JJ) = 0 for i # j where Int(J) denotes the interior
of an interval J.
A system A = {(rj, Jj), j = 1, ... , k} is called a partition of
[a, b] if
k
UJ)=[a,b].
j-1
Given a positive function 6 : [a, b] - (0, +oo) called a gauge on
[a, b], a tagged interval (r, J) with 7- E [a, b] is said to be b -fine if
J C [r - 6(r), r + 6(r)].
A system (in particular, a partition ) A = {(rj, Jj),
j = 1, ... , k} is b-fine if the point-interval pair (rj,,Ij) is 6-fine
for every j = 1, ... , k
.
1
I. The generalized Perron integral
2
1.1 Definition. Let h(r, J) be a finite real-valued function
of point-interval pairs with r E [a, b], J C [a, b]. The function h
is called integrable over [a, b] if there is an I E IR such that given
e > 0, there is a gauge S on [a, b] such that
k
1: h(rj,Jj)-II <e
j-1
for every 6-fine partition A _ {(rj, Jj), j = 1, ... , k} of [a, b].
The number I is called the integral of the function h over [a, b].
This is a general definition of the Hens tock-Kurzweil integral
that can be found in a still growing series of books on integration
(see e.g. [40], [41], [84], [92], [93], [96], [98]) where this idea
of integration is explained from various aspects and in different
generality.
For our purposes we will consider a specific situation that has
its origin in the theory of ordinary differential equations.
Assume that U : [a, b] x [a, b] --> R is a function of two vari-
ables r, t E [a, b]. We define the integral of this function as
the Henstock-Kurzweil integral for the point-interval function
h given by
h(r,t) = U(r,/3) - U(r,a)
for r E
[a, b]
(1.1)
and J = [a,,3] C [a, b].
1.2 Definition. A function U : [a, b] x [a, b] -+ R is called
integrable over [a, b] if there is an I E lR such that given e > 0,
there is a gauge b on [a, b] such that
k
S(U, D) - II = J E[U(rj, a j) - U(rj, aj-1)] - 11 < e
j=1
for every 6-fine partition
D={(r,[aj_i,as]),j=1,...,k}_
(1.2)
Definition of the integral
3
= {ao,T1,al,...,ak-1,Tk,ak}
(1.3)
of [a, b].
The real number I E R is called the generalized Perron integral
of U over the interval [a, b] and will be denoted by fb DU(-r, t).
fb
DU(r, t) _
If fQ DU(T, t) exists then we define
- fb DU(r, t) and set fb DU(T, t) = 0 when a = b.
We denote by JC([a, b]) the set of all functions U which are
integrable over [a, b].
)]
We use the notation S(U, D) = Et=1 [U(T;, a1) - U(T;, aj_1
for the Riemann-type sum corresponding to the function U and
the partition D.
1.3 Remark. Let us mention that for a given gauge b on [a, b]
the partition D = {ao, T1i al, ... , a_ 1, Tk, ak} is S-fine if
a=ao <a1 <... <ak=b,
aj-1
2, .
. . ,
(1.4)
(1.5)
and
C [Ti - b(TI),Tj +S(T,)], 3= 1,2,...,k.
(1.6)
Looking at Definition 1.2 and the properties of a 6-fine partition
D of [a, b] it is easy to see that for the function U defined on the
square [a, b] x [a, b] only its values at points close to the diagonal
T = t play a role when defining the integral and that it is sufficient
to have U defined on a set
S = {(T, t) E R2, T E [a, b], t E [T - (S(T), T + S(T)] n [a, b]}
with some gauge S : [a, b] --j (0, +oo) on [a, b]. The gauges S in
Definition 1.2 then have to be chosen such that 6(7-) < 6(-r), r E
[a, b] and the result is the same.
I. The generalized Perron integral
4
For the integral we use the notation fa DU(r, t) introduced by
J. Kurzweil in [68]. This notation has a symbolic meaning only.
Clearly the above Definitions 1.1 and 1.2 are viable only if for
a given gauge S on [a, b] there exists at least one 6-fine partition
D of [a, b]. This fundamental question has an affirmative answer
given by the following statement.
1.4 Lemma (Cousin). Given a gauge b on [a, b], there is a
45-fine partition D = {ao, rl, al, ... , ak_1, rk, ak} of [a, b].
For the proof of this lemma see e.g. [40,Theorem 4.11 or any
of the textbooks [41], [84], [92], [93], [96], [97]. The lemma was
discovered often in various contexts. J.Mawhin [93] pointed out
that the first result of this type belongs to Cousin (1895).
1.5 Remark. Assume that f : [a, b] -- R is a real function. Let
us set U(r, t) = f (r).t for r, t E [a, b]. Then clearly by (1.1)
h(r, t) = U(r, p) - U(r, a) = f (r)(Q - a)
and
k
k
k
E h(rj, Jj) = E[U(rj, aj)-U(r1, a'J-1 )] _ E f(rj)(a7-a1-1 )
j=1
j=1
j=1
represents the classical Riemann sum for the function f and a
given partition D of [a, b].
Moreover, if g : [a, b] --> R is given then for U(r, t) = f (r).g(t),
r, t E [a, b] we obtain the Riemann-Stieltjes sum
k
k
h(rj, J1) = E f(rj)[g(aj) - g(aj-1)]
j=1
j=1
for the functions f, g and a given partition D of [a, b].
Definition of the integral
5
Using Definition 1.2 (or equivalently Definition 1.1) we obtain
in these cases certain concepts of integration based on Riemanntype sums.
If U(r, t) = f(r).t then we write fe f(s) ds instead of fb D[f(r).t]
and similarly also for the Stieltjes case U(r, t) = f (r)g(t) we use
the notation fb f (s) dg(s) instead of fb D[f (r).g(t)] provided the
generalized Perron integrals fb DU(r, t) exist in these cases.
It turns out that in the case U(r, t) = f (r)t we have
6
J1a
DU(r, t) =
j
b
D[ f (rr).t] =
rb
Ja
rb
f (s) ds = (P) J f (s) ds
a
where (P) stands for the classical Perron integral. This fact was
shown for the first time by J. Kurzweil in 1957 in his fundamental paper [68] concerning differential equations. It is also worth
mentioning that the primary definition in [68] for the integral
fa DU(r, t) is given using major and minor functions to U. These
are of course given in a more general form because of the general
form of the function U but they coincide with Perron's major
and minor functions in the known form when U(r, t) = f (r).t.
Definition 1.2 is given in [68] as a secondary one and it is shown
that the underlying idea of the integral f b DU(r, t) is the same
as for the concept based on minor and major functions.
It is clear that Definition 1.2 can be given also for functions U
with values in a general linear topological space. For the purposes
of this text we give such a definition for R"--valued functions only.
1.2n Definition. A function U : [a, b] x [a, b] -> R" is called
integrable over [a, b] if there is an I E R" such that given e > 0,
there is a gauge 8 on [a, b] such that
k
IIS(U, D) -Ill= II y:[U(r,, aj)
i=1
- U(r', a,i-i )] - III < e
(1.2)
1. The generalized Perron integral
6
for every 6-fine partition
D = {(rri, [ai-1, ai]), 7 = 1, ... , k} _
{aO,r1,ai,...,ak-l,Tk,ak}
of [a, b].
The element I E R" is called the generalized Perron integral
of U over the interval [a, b] and will be denoted by f4 DU(r, t).
If fQ DU(-r, t) exists then we define fb DU(r, t) = - fQ DU(r, t)
and set ff DU(r, t) = 0 when a = b.
We denote by IC([a, b]) the set of all functions U which are
integrable over [a, b].
Here 11.11 stands for the norm in R" (e.g. the Euclidean one).
1.6 Theorem. An R"-valued function U : [a, b] x [a, b] is integrable if and only if every comR", U = (Ui, U2,...,
ponent Un,, in = 1, 2, ... , n is integrable in the sense of Definition
1.2.
Proof. Suppose the integral f DU(r, t) = I = (Ii , ... , In) E
R" exists. For every m = 1, ... , n and any partition D of [a, b]
we evidently have S(Un D) _ (S(U, D)),n and the inequality
t.
IS(Um,D) - I,"I = I(S(U,D))m -
I(S(U,D)-I),nl
II S(U, D) - III
holds. Therefore we get the existence of fb DU .. (7-, t) and the
equality fb DU,,,, (,r, t) = I,, = (f' DU(T, t)),,,.
If we suppose that the integrals f' DU,,(r, t) = In, exist for
every m = 1, ... , n then for every E > 0 there exists a gauge 6m
Definition of the integral
7
on [a, b] such that for every Sm-fine partition Dm of [a, b] we have
I S(Um, D,,,) - Im I < e. Let us choose a gauge b on [a, b] such that
6(r) < min(Si(r), ... , S,: (r))
for r E [a, b]. Then evidently every S-fine partition D of [a, b] is
E R'° we get
also 5,,,-fine, m = 1, ... , n and for I = (11, ... ,
the inequality
n
<(nE2)2 =n2E
IIS(U,D)-III =(j
M=1
for every 6-fine partition D of [a, b].
Hence the integral fa DU(r, t) exists and its components equal
0
to fQ DUm (r, t) = I,,, for in = 1, ... , n.
1.7 Theorem. The function U : [a, b] x [a, b] -i Rn is integrable over [a, b] if and only if for every E > 0 there is a gauge b
on [a, b] such that
IIS(U,D1)-S(U,D2)II <E
for any S-fine partitions D1, D2 of [a, b], where
k
S(U, D) =
E[U(r7, aj) - U(rl, aj-1
9=1
is the Riemann-type sum corresponding to U and the partition
D = {ao, rl, al , ... , ak-l , rk, ak } of [a, b].
Proof. By the previous Theorem 1.6 it is sufficient to prove
the statement for a real valued function U : [a, b] x [a, b] --- R.
I. The generalized Perron integral
8
Assume that the Bolzano - Cauchy condition of the theorem
holds and that e > 0 is given. Then there is a gauge S on [a, b]
such that for any b-fine partitions DI, D2 of [a, b] we have
IIS(U,DI)-S(U,D2)II < z
(1.7)
Denote by M the set of all s E R such that there exists a gauge
w on [a, b] such that for every w-fine partition D we have s <
S(U, D). Assume that Do is an arbitrary 6-fine partition of [a, b].
By (1.7), for every 6-fine partition of [a, b] the inequalities
S(U, Do) - 2 < S(U, D) < S(U, Do) + z
hold.
Therefore (-oo, S(U, Do) - 2
C M and M C
i.e. the set M is nonempty and bounded
(-oo, S(U, Do) +
2)
from above. Consequently the supremum sup M of the set M
exists and by the above inclusions we have
S(U,Do) - 2 < supM < S(U,Do) + 2.
Hence IS(U,Do) -supMI < 2 and by (1.7) we have
S(U, D) - supM1 < IS(U, D) - S(U, Do) I+
+IS(U,Do) -supMI < e
for every b-fine partition D of the interval [a, b]. Therefore by definition the function U is integrable over [a, b] and fa DU(rr, t) _
sup M.
Conversely, if the integral fQ DU(T, t) exists then the BolzanoCauchy condition of the theorem can be deduced easily.
1.8 Remark. The Bolzano-Cauchy condition of Theorem 1.7
can be used also for defining integrability of a function U : [a, b] x
[a, b] -- R". In the more general case of U with values in a
linear topological space the completeness of this space is of course
necessary.
Fundamental properties of the integral
9
Fundamental properties of the integral
Now we turn our attention to some fundamental properties of
the integral fa DU(r, t).
From the evident identity
S(c1U + c2V, D) = c1S(U, D) + c2S(V, D)
for the Riemann sums of the functions U, V : [a, b] x [a, b]
c1, c2 E R and an arbitrary partition
--4R"
D = {CYO,rl,al,... iak-1,Tk,ak)
of [a, b] we immediately have the following result.
1.9 Theorem. If U, V E /C([a, b]) and c1, c2 E R then c1 U +
c2V E 1C([a, b]) and
b
b
D [cl U(r, t) + C2V (r, t)]
a
= C1
f6
Ja
DU(r, t) + C2
f
DV(r, t).
1.10 Theorem. If U E 1C([a, b]) then for every [c, d] C [a, b]
we have U E 1C([c, d]).
Proof. Assume that e > 0 is given. By the Bolzano-Cauchy
condition for the existence of the integral f6 DU(r,t) (see Theorem 1.7) there exists a gauge b on [a, b] such that
II S(U, D1) - S(U, D2)II < e
(1.8)
for every b-fine partitions D1, D2 of [a, b].
Let now D1, D2 be arbitrary b-fine partitions of [c, d]. Assume
that a < c < d < b. Let DL be a b-fine partition of [a, c] and DR
a 6-fine partition of [d, b]. They exist by the Cousin Lemma 1.4.
1. The generalized Perron integral
10
If
Dl = {ao, Tl , al , ... ak-1, rk, ak ),
DL
{ao , rl
al , ... , ai
DR=
{ao ,r1R
R
R
alR ,...,ar-1,r
,arR I
1,
TIL, aI },
thenao =a,af =c=ao,ard=ao,aR=b. Let us put
together the partitions DL, D1i DR to create a partition Dl as
follows
D1
- {ao,rl ,al ...,a(-l,T! ,ao,T1,011,...
L
L
L
L
L
R
R
R
R
,ak-1,Tk,ak,r1
,al
,...,ar-l,rr
arR
Then evidently D1 is a partition of [a, b] and it is 6-fine because
the partitions DL, D1, DR are b-fine. Similarly we put together
the partitions DL, D2i DR forming a S-fine partition D2 of [a, b].
By (1.8) we have
IIS(U,D1)-S(U,D2)II
= IIS(U,D1)-S(U,D2)II <E
because the terms in the sums S(U, D1) - S(U, D2) that correspond to the common parts DL, DR appear in each of the partitions DI, D2 and hence cancel each other. Therefore by Theorem
d
1.7, the integral f DU(r, t) exists because D1, D2 have been arbitrary b-fine partitions of [c, d].
.
1.11 Theorem. If c E (a, b) and U : [a, b] x [a, b] -+ R' is
such that U E 1C([a, c]) and U E 1C([c, b]) then U E 1C([a, b]) and
c
b
Ja
DU(r, t) =
Ja
DU(r, t) -f- f DU(-r, t).
cb
(1.9)
Fundamental properties of the integral
11
Proof. Let e > 0 be given. Denote IL = fa DU(r, t) and
IR = f b DU(r, t). By the assumptions there exists a gauge 6L on
[a, c] such that for every 6L-fine partition DL of [a, c] we have
IIS(U,DL)-ILII <e.
Similarly there exists a gauge bR on [c, b) such that for every
6R-fine partition DR of [c, b] we have
IIS(U,DR) - IRIJ < E.
Let us define
6L (T)
min(OL(c), bR(c)
for r E [a,c),
for r = c,
{
OR(T)
for r E (c,d]
and choose b : [a, b] --b (0, -boo) such that
6(r) < min(6(r), (r - cI)
if r 0 c
b(c) = 6(c).
Evidently 6 : [a, b] -* (0, +oo) is a gauge on [a, b]. Assume that
D = {ao, r] , a] , .... ak_], rk, ak } is a b-fine partition of [a, b].
Then there exists an index in such that c C [am_] , am]. Assume
that rm # c. Then we get the contradictory inequality
Irm - CI < 6(rn,) < ITm - CI
which holds because by the definition we have [am_], am] C [Tm-
b(r,,,), Tm + b(rm)] and b(r) < IT - cI for every r # c. Therefore
necessarily r,, = c. Moreover, we have
en-]
S(U,D) = ElU(Tj,aj)-U(rj,aj_])]+U(C,a,n)-U(c,am-l}+
j=]
1. The generalized Perron integral
12
k
+ E [U(rj,aj) - U(rj,aj_1 )] _
j=in+I
_
[U(7-j, aj) - U(Tj, aj_1 )]+
j=1
+U(C, C) - U(C, am-1) + U(C, am) - U(C, C)+
k
+ E [U(rj, aj) - U(Tj, aj-1 )] = S(U, DL) +S(U, DR),
j=rn+1
l DR =
where DL = {aO, T1, al, ... am-l, Tm = C, am = c),
{aR -1 = C, Tm = c, am, ... , ak-1, Tk, ak) are partitions of [a, c],
[c, b], respectively, DL is SL-fine and DR is SR-fine in virtue of
the choice of the gauge S. Hence for a S-fine partition D of [a, b]
we have
IIS(U,D)-IL - IRII
=
IIS(U,DL)+S(U,DR)-IL - IRII <
< II S(U, DL) - IL II + II S(U, DR) - IRII < 2e.
By definition this yields the existence of the integral fa DU(r, t)
0
and also the equality (1.9).
1.12 Remark. Besides the technique of joining two partitions
of two intervals having a common endpoint as used in the proof
of Theorem 1.10 we should point out the construction used in
the proof of the previous Theorem 1.11. Namely, if b is a gauge
satisfying
S(r)<Ir-clfor r#c
then it forces every S-fine partition of an interval [a, b] with c E
[a, b] to have the tag of one of its point-interval pairs at the point
c. This fact can be helpful in various constructions of special
integral sums.
Fundamental properties of the integral
13
The following statement provides an operative tool in the theory of generalized Perron integral. Its original version belongs
to S. Saks and it was formulated for generalized integrals using
Riemann-like sums by R. Henstock.
1.13 Lemma (Saks-Henstock). Let U : [a, b] x [a, b] -+ R" be
integrable over [a, b]. Given e > 0 assume that the gauge b on
[a, b] is such that
k
[
f
- U(Tj,ajDU(T,t)II < E
b
I E[U(T7,a7)
j=1
for every b -fine partition D = {ao, T1, a1 i ... , ak_1, Tk, ak) of
[a, b]. If
a <Q1 < 1 <11 <02 <'2 :5 -Y2
...:5 m:5 .:5
<...
ym <b
represents a b-fine system Wj, [$j, -yj]), j = 1, ... , m}, i.e.
b7 E [Qj, "yj] C
b(bj), Cj + b(Cj)],
j = 1, ... , m
then
m
II E[U(Cj,7) - U(e1,I3) j=1
1,j
f
DU(r,t)]II < e.
(1.10)
+91
Proof. Without any loss of generality it can be assumed that
pj < 'yj for every j = 1, ... , m. Denote Yo = a and p,,,,+1 = b. If
-yj < ,Oj+1 for some j = 0, 1, ... , m then Theorem 1.10 yields the
existence of the integral f y'+' DU(T, t) and therefore for every
> 0 there exists a gauge bj on [-y,, ij+1] such that bj(r) < b(r)
1. The generalized Perron integral
14
for r E [7j, pi+1 ] and for every Si-fine partition Di of [-yi, pi+1 ]
we have
IlS(U, Di) -
DU(T, t)II <
1,
M+1
If 'yi = pi+i then we take S(U, D') = 0.
The expression
in
m
U(ei,f.i)] + ES(U,D')
j=1
i=1
represents an integral suns which corresponds to a certain S-fine
partition and consequently
m
in
b
DU(r,t)II < e.
=1
a
)=1
Hence
in
ry,
Il >[U(i,-yi) - U(ei,lji)
DU(T, t)] II <
j=1
in
in
+ES(U,D') -
< II
7=1
i=1
in
+EIIS(U,Di)-
j=1
f
Qi+s
f
b
DU(T,t)II+
a
DU(T,t)II <e+(tn+1)m+l =e+i.
Since this inequality holds for every q > 0 we immediately obtain
0
(1.10).
Fundamental properties of the integral
1.14 Theorem. Let a function U :
15
[a, b] x [a, b] --> R" be
given such that U E k ([a, c]) for every c E (a, b) and let there
exist a finite limit
{JC
D U(r, t) - U(b, c) + U(b, b) = I.
lim
(1.11)
Then the function U is integrable over [a, b] (U E A;([a, b])) and
6
1.
DU(r, t) = I.
Proof. Assume that e > 0 is given. By (1.11) for every e >
0 we can find a B E [a, b) such that for every c E [B, b) the
inequality
11
Ja
DU(r, t) - U(b, c) + U(b, b) - ICI < e
(1.12)
is satisfied. Assume that a = co < c1 < ... is an increasing
sequence (cp)p l of points cp E [a, b) with limp_ ,- cp = b. By
the assumption we have U E JC([a, cp]) for every t. = 1, 2, ... and
therefore for every p = 1, 2,... there exists a gauge Sp : [a, cp] -(0, +oo) such that for any 6p-fine partition D of [a, cp] we have
S(U, D) -
j
Cp
DU(r, t)11 <
p = 1, 2(1.13)
For any r E [a, b) there is exactly one p(r) = 1, 2,... for which
r E [cp(r)_1i cp(r)). Given T E [a, b) let us choose b(r) > 0 such
that b(r) C bp(r)(r) and [r - b(r), r + b(r)] fl [a, b) C [a, cp(r)).
Assume that c E [a, b) is given and that
D = {ao,rl,al,...,ak-2,rk-1,ak-1}
I. The generalized Perron integral
16
is a i-fine partition of [a, c]. If p(Tj) = p then [aj-1i aj] C [Tj S(Tj ), Tj + S(T1)] C [a, cP] and also [a j-1, aj] C [rj -- 5P(Tj ), Tj +
5p(-r,)]. Let
k-1
aj
[U{Tj, aj) - U(Tj, aj-1) -
a
j=1,P(rj)=P
- DU(T, t)]
1
be the sum of those terms in the corresponding "total" sum
k-1
j- U(T, a-1) -
DU(T, t)]
t
for which the tags Tj satisfy the relation Tj E [cp-l,cp). Since
(1.13) holds we obtain by the Saks-Henstock lemma 1.13
k-1
[U(T, aj) - U(T, a1) -
II
j
- DU(T, t)]I<
'j
j=l,P(r)=P
t
and finally
k-1
C
II E[U(TI, a7) - U(7-T, a7-I )]
DU(T,t)]11 =
j=1
k-1
= II
00
Ell
P=1
E[U(Tj, aj) - U(Tj, aj_1) -
IDU(T,t)]II
j=1
k-1
E [U(Tj, aj) - U(T7, aj-1) - J
7=1,P(ri)=P
00
P=1
2p+1 = E'
DU(T, t)] II <
-1
Fundamental properties of the integral
17
Define now a gauge b on the interval (a, b] as follows. For r E [a, b)
set
0 < b(r) < min{b -7-, b(r)}
while
0<5(b) <b-B.
If D = {co, rl, al, ... , ak-1, rk, ak} is an arbitrary 6-fine partition of [a, b] then by the choice of the gauge 6 we have rk = ak = b
and ak_1 E (B, b). Using (1.12) we get
IIS(U, D)
- III =
k-1
= II
j=1
[U(rj, aj) - U(rj, aj_1)] + U(rk, ak) - U(rj, ak-l) - III <
k-1
E
<II
- U(rj,aj_1)]
j=1
-
f
Qk-1
DU(r,t)]II+
a
ak_1
DU (r, t)] - U(b, ak_1) + U(b, b) - III <
+II
fa
k-1
<e+
[U(rj, a) - U(r,
Qk
aj-1 )] -
j
1
DU(r, t)]II
Since ak_1 < b and D = {ao, 71, a], ... , ak_2, Tk-1, ak-1 } is a
b-fine partition of [a, ak_ 1 ], the second term on the right hand
side of the last inequality can be esimated by e as shown above.
In this way we obtain
IIS(U,D)-III<2e
and this inequality yields the existence of the integral f DU(r, t)
as well as the equality f b DU(r, t) = I.
1.15 Remark. The "left endpoint" analog of Theorem 1.14 can
be proved in a completely similar manner:
I. The generalized Perron integral
18
Let a function U : [a, b] x [a, b] --- R" be given such that
U E 1C([c, b]) for every c E (a, b) and let there exist a finite limit
b
lim
ca+
fc
DU (r, t) + U(a, c) - U(a, a)
I.
Then U E 1C([a, b)) and
b
DU(r, t) = I.
Theorem 1.14 and its "left endpoint" version given here represent the Cauchy extension of the generalized Perron integral. For
more details see [40, p. 67] or [41,2.10, p.115].
1.16 Theorem. Let U : [a, b] x [a, b] - R' be such that
U E 1C:([a, b]) and c E [a, b]. Then
1m
9
DU(r, t) - U(c, s) + U(c, c)J =
J
Va
Jc DU(r, t).
(1.14)
a
Proof. Let e > 0 be given and let 6 be a gauge on [a, b] which
corresponds to e by Definition 1.2n, i.e. the inequality
b
IIS(U, D) - f D U(r, t)II < e
holds for every b-fine partition D of [a, b]. If s E [c - b(c), c +
b(c)] fl [a, b] then the Saks-Henstock lemma 1.13 gives
I.9
U(c, s) - U(c, C) -
DU(r, t)II <
19
Substitution theorem
that is
II
Ja
s DU(r, t) - U(c, S) + U(c, c) -
j
DU(T, t)II
=
C
= II J DU(-r, t) - U(c, s) + U(c, c) 11 < s,
and this yields the relation (1.14).
11
1.17 Remark. Theorem 1.16 shows that the function given by
s E [a, b] ->
Ja
DU(r, t) E R",
i.e. the indefinite generalized Perron integral to U is not continuous in general. The indefinite integral is continuous at a point
c E [a, b] if and only if the function U(c, ) : [a, b] -+ R" is continuous at the point c.
We have to note that if U : [a, b] x [a, b] -> R" is such that
U E K ([a, b]) then by Theorem 1.10 the indefinite generalized
Perron integral to the function U is well defined on the whole
interval [a, bJ.
Substitution theorem
1.18 Theorem (Substitution). Assume that -oo < c < d <
+oo and that V : [c, d) -> R is a continuous strictly monotone
function on [c, d].
Let U : [p(c), V (d)] x [cp(c), cp(d)] -- R" be given.
If one of the integrals
f (C)
p(
DU(r, t),
f
c
d
GP(s))
I. The generalized Perron integral
20
exists then also the other one exists and
d
W(d)
DU(T, t)
J w(c)
=f
DU(v(a), s'(s))
(1.15)
holds.
Proof. Assume that the function tp is increasing.
Assume that fd DU(cp(o), cp(s)) exists and that e > 0 is given.
Then there is a gauge w on [c, d] such that for every w-fine partition
1i0,0'1,01,...,Nk-1,Qk,/3k}
of [c, d] we have
k
II E[U(p(oj),ta(Nj)) - U(W(Oj),WA-1))]j=1
-
j
d
DU(cp(o), (p(s)) II < e.
Since cp is increasing, cp([c, d]) = [cp(c), cp(d)], the inverse
[cp(c). cp(d)] --+
tip-1
:
[c, d] exists and is continuous and increasing on
[tp(c), cp(d)]. Hence for every r E [tp(c), cp(d)] there is exactly one
a = V -1(T) E [c, d]. For T E [cp(c), cp(d)] let us define S(r) > 0
such that
[T - 6(T), T + b(- r)] n ['(c), p(d)] c v([o - w(o), o, +w(o,)] n [c, d])
(1.16)
where v = cp-1(T) E [c, d]. This is possible since V, V-1 are
continuous.
Let (ao, T11 a1, ... , ak-1, Tk, ak} be an arbitrary 5-fine partition of [cp(c), tp(d)]. Let us set Qj = cp-1(aj), j = 0,... , k,
Substitution theorem
21
aj = cp-l (Tj), j = 1, ... , k. Then by the monotonicity of cpwe
have
po = W-1(ao)
= c < 01 < 132 < ... < ,8k-1 < Qk = w-' (ak) = d
and
/3j-1 : aj <_ / 3 j ,
j = 1, 2, ... , k.
This yields that {,Qo, al , Q1 , ... , /3k_ l , Uk,13k } is a partition of [c, d].
Since {ao, 7-1, a, .... , ak_2, Tk_1, ak_1 } is a 5--fine partition of
[cp(c), cp(d)] we have
Tj - S(Tj) C aj-1 < Tj < aj : Tj + 6(7-j)
and (1.16) implies
gyp(°j - w(°j)) : aj-1
aj C cp(cj +w(oj)).
Therefore
c - w(Qj) =
w(a, )) C
_ /3j-1 < /lj <
-1(ai-l) _
(aj) < ujRR+w(oj)
and this shows that the partition {NO, al, Nl .... , Qk-1, 0k, /3k }
w-fine.
Further, clearly
k
E[U(Tj), aj)) - U(Tj), aj-l ))] -
jd
DU(4o(o,), (p(s))II
j=1
k
= II
P(Qj )) - U((P(Oj ), VA-1))l
j=1
-
is
I. The generalized Perron integral
22
-J
d
DU(cp(a), p(s))II < e
because the partition 1,80,0'1,#1 , ... , Ak-1 , Qk, Qk} is w-fine. By
definition this yields the existence of the integral f ,(d) DU(-r, t)
and shows that (1.15) is satisfied.
Now assume that the integral f `p(d)
DU(r, t) exists and that
WI(C)
e > 0 is given. Then there exists a gauge 6 on [cp(c), go(d)]
such that f o r every S-fine partition {ao, r1i a1, ... , ak-1, rk, ak}
of [cp(c), cp(d)] we have
k
DU(r, t)II <
>[U(rj), aj)) - U(rj), aj-1 )] -
II
j=1
1W(C)
For a E [c, d] there exists a value w(a) > 0 such that
(1.17)
NOW - p(a)I <
if s E [a - w(a), a + w(a)] fl
If {Ao, a1, 91 ,
,
[c, d].
Ak-1 , ak, Ak }
is an w-fine partition of [c, d],
then we set rj = cp(aj ), j = 1,2,...,k, a j = VA),
j = 0,1, ... , k. From the monotonicity of cp and from (1.17) we
easily deduce that {ao, r1, a1, ... , ak_1, 7k7 ak} is a b-fine partition of [cp(c), cp(d)]. Hence
w(d)
/
r
E[U((aj ), P(Aj )) - U(4(aj , (Aj-1 ))] DU(T, t)II < e
k
II
JV(e)
7=1
for every w-fine partition {Ao, a1, A1,
cause
,
13k-1
, ak, Ak ) of [c, d] be-
k
E[U(cp(aj ), 4'(,8j))
j=1
- UMOj), VA-IM
Integration by parts
23
k
j:[U(Tr ), a j )) - U(r7 ), aj-1 )]
jl
where {ao, r1i a1, ... , ak-1i Tk, ak } is a 6-fine partition of
[cp(c), cp(d)]. This yields the existence of the integral
f
d
p(s))
C
0
as well as the equality (1.15).
1.19 Remark. Theorem 1.18 holds also in a more general setting when cp is continuous and monotone only, i.e. nondecreasing or nonincreasing. In this case the proof of the substitution
theorem is more technical. It can be given by generalizing the
concept of the inverse to W. If e.g. cp [c, d] --1 R is nondecrasing
:
and r E [cp(c), cp(d)] then we define
V-1(r) = {a E [c, d]; cp(a) = r}.
Clearly V-'(-r) is either a single point or a closed subinterval in
[c, d]. Using this definition of an "inverse" to cp, partitions of [c, d]
and [cp(c), cp(d)] can be mapped one to the other like in the proof
of Theorem 1.18.
Integration by parts
Now we give a general form of the integration by parts formula
for the generalized Perron integral. First of all we prove a simple
lemma.
1.20 Lemma. Assume that V : [a, b] x [a, bJ - R is such that
V(r,t) = V(t,r) for t,r E [a,b].
(1.18)
I. The generalized Perron integral
24
Let us set W (T, t) = V (t, r). Then if one of the integrals
j
DV(r, t), f. DW(T,t)
b
a
a
exists, then the other one exists as well and
I.
bDV(t,r) =
Ja
bDW(T,t) =
f DV(r,t).
a
Proof. It is easy to see that the integral sums S(V, D), S(W, D)
coincide by (1.18) for any partition of [a, b] and this yields the
result.
1.21 Theorem (Integration by parts ). Suppose that U :
[a, b] x [a, b] -1 R is given. Let us set U* (r, t) = U(t, r) and
V(t,T) = U(T,T) - U(r,t) - U(t,T) + U(t,t).
If two of the integrals fb DU(T, t), fa DU*(r, t), fQ DV (r, t)
exist then the third exists as well and the equality
b
Ja
DU(r, t) + j b DU(r, t) = U(b, b) - U(a, a) -
j
b
DV(r, t)
(1.19)
holds.
Proof. It can be checked immediately that V satisfies (1.18).
Suppose that e.g. the integrals fa DU(r, t), fQ DU*(T, t) exist.
Then there is a gauge S on [a, b] such that for any b-fine partition
D = {ao,r1ial,.... ak_1,rk,ak} of [a, b) we have
j
b
k
DU(T, t) - >[U(7-j, ai) - U(ri, a.i-1)] fI < e
j=1
Integration by parts
25
and
!b
II
Ja
k
DU*(r,t) - E[U*(TJra)) - U*(rr,aj-1)]I) < E.
j=1
Then for every S-fine partition D of [a, b] we have
II S(V, D) - U(b, b) + U(a, a) +
f
b
b
DU(-r, t) +
a
J
DU*(r, t)II =
k
IIE[(U(Tj,rj)-U(rj,aj)-U(aj,rj)+U(aj,aj))j=1
-(U(rj, r7) - U(7-j, aj-1) - U(aj_1, rj) + U(aj-1, aj-1 ))]
-U(b, b) + U(a, a) +
la
D U(r, t) + I DU*(7-, t)II <
k
C
IIE[-U(Tr>a?)-U(aj,rr)+U(TJ,aj_1)+U(aj-1,T7)]+
j=l
r6
+
Ja
DU(-r, t) +
j
b
DU*(r, t)+
a
k
+ E[U(aj, aj) - U(aj_1i aj_1)] - U(b, b) + U(a, a)II <
j=1
jb
< II
t) j=1
k
b
DU*(T, t)
+11
la
[U() - U(rj, aj-1)] II+
- [U*(rj,
aj) - U*(Tj, aj-1 )]II <
j=1
I. The generalized Perron integral
26
Hence the integral f b DV(r, t) exists and the equality (1.19) holds
by the definition of the integral.
O
The proof for the other cases is similar.
1.22 Remark. The proof of Theorem 1.21 is based on purely
algebraic manipulations of integral sums for the generalized integral. This approach to integration by parts goes back to the
paper [74] of J. Kurzweil.
Now we give a corollary to Theorem 1.21 which contains the
integration by parts formula in a more conventional form.
1.23 Corollary. If f, g : [a, b) - R are functions of bounded
variation on [a, b] then the integrals fa f (s) dg(s) and fa g(s) df(s)
exist*) and
b
b
f
f(s) dg(s) +
a
-
f
g(s)df(s) = f(b)g(b)
E 0+f(T )0+g(r) + L, 0 f(r)0 g(r),
a<r<b
where
- f(a)g(a)-
a
+f(T)
(1.20)
a<r<b
= f(T+)-f(r) = lim,_,+f (a)-f(T), 0-f(r)
f(T) - f(T-) = f(T) - lira,_r_f(a) and similarly for 0+g(r)
,
o g(r).
Proof. We postpone at this moment the proof of the fact that
both integrals fa f (s) dg(s) and fa g(s) df(s) exist. This will be
shown later (see Corollary 1.34).
Let us set U(r, t) = f(r)g(t) and U*(r, t) = g(r)f(t) for r, t E
[a, b]. Then
JDU(r,t) =
f
b
f(s) dg(s),
a
*)See Remark 1.5 for these integrals.
jDU*(r,t) = fg(s)df(s).
Integration by parts
27
and clearly also U*(r, t) = U(t, r). By Theorem 1.21 the integral
fQ DV(r,t) exists where
V(r,t) = [f(r) - f(t)][g(r) - g(t)]
Now it is clear that for proving the formula (1.20) we have to
show that
b
fa
0+f(r)D+g(r) 0 f(r)0 g(r)
E
a<r<b
a<r<b
DV(r,t) =
(1.21)
Let e > 0 be given. Set
N = {r E (a, b); If(r+) - f(r)I ? e, If(r)
-f(r-)I > e}.
Since the function f is of bounded variation, the set N is finite.
Further, for every r E [a, b] there is bl (r) > 0 such that
If(s) - f(r+)I < E, Ig(s) - g(r+)I < e for s E (r, r + bl (r))
and
If(s) - f(r-)I < e, Ig(s) -
g(r-)I <,F for s E (r - bj(r), r).
This is clear because the onesided limits for the functions f, g exist at every point in [a, b]; at the endpoints only the corresponding
ones.
Let us set b2(r) = dist(r,N) for r
N ( dist(r, N) stands
here for the distance of the point r from the set N) and b2(r) =
b1(r) for r E N. Finally, define b(r) = inin(bl(r),b2(r)) > 0 for
r E [a, b]. The function b represents a gauge on [a, b]. Let
D = {ao,rl,a],...,ak-l,rk,akI
I. The generalized Perron integral
28
be an arbitrary b-fine partition of [a, b]. If c E N then there is an
index j = 1, ... , k such that c E [a j-1 , a j]. Moreover, the form
of the gauge b forces in this case also rj = c (see Remark 1.12)
and consequently we have N C { r1, T2,. .., rk }.
For the 6-fine partition D we now get ( S(V, D) is the integral
sum corresponding to the function V and the partition D)
IS(V, D)
E 0 J (T )I
- a<r<b
E 0+f (T)L+g(T) + a<r<b
g(T)I =
k
_ E[(f(Ti) - f(aj))(g(Tj) - g(aj))j=1
-(f(7j) - f(aj-1))(g(rj) - g(aj-1))]0+f(T)A+g(T)+
a<r<b
0 f(r)A g(T)I C
r
a<r<b
k
0+ f(r)o+g(T )I +
I DU(Tj) - f (aj))(g(Tj) - g(aj ))] j=1
a<r<b
k
+I E(f(Tj)-f(aj-1 ))(g(r1)-g(aj-1 ))- Y 0-f(r)0-g(7)(.
a<r<b
j=1
(1.22)
For the first term on the right hand side of this inequality we
have
k
[(f(r) -' f(aj))(9(rj) - 9(aj)) -
0+ f(r)o+g(T)l =
a<r<b
j=1
k
(1:[A+f(Tj)0+g(Tj) - A+f(T))(g(Tj+) - g(aj))j=1
Integration by parts
29
-0+g(rj)(f (TJ +) - f (aj )) + (f (T1 +) - f (al))(g(ri+)
-
g(a7 ))]
k
0+f(ri)A+g(rj)
0+f(r)o+g(r)I :5
a<r<b
j=1
r rj
k
k
Io+f(ri)IIg(rj+)-g(aj)I +E Io+g(rj)IIf(rj+)-f(aj)I+
j=1
j=1
k
+I:If(rj+)-f(aj)IIg(rj+)-g(aj)I+ E Io+f(r)Ilo+g(r)I <
a<r<b
j=1
rVN
< e vara f + e vara g + e vara g + e vary g = e(vara f + 3 vary g).
In these cumbersome but straightforward calculations we assumed that if rj E N then aj-1 < Tj < aj. The possibility of doing
this is apparent from the fact that if for example aj-1 < rj =
a j = Tj+1 < a j+1 then
V(rj+aj) - V(rj,aj-1) + V(rj+l,aj+l) - V(rj+],aj) =
= V (rj, aj+l) - V (rj, aj-1 )
and the two intervals [aj-liaj], [aj,aj+1] of the partition D can
be replaced by a single interval [aj-1, aj+1] with aj-1 < rj < aj
without changing the integral sum S(V, D).
In a completely analogous way the second term on the right
hand side of (1.22) can be estimated which leads to the inequality
I S(V, D) -
a<r<b
&+f (r)L+g(r) +
L,
0-f (r)te-g(T )I < ce
a<r<b
where c is a constant, and this implies by the definition of the
integral that (1.21) holds.
0
I. The generalized Perron integral
30
1.24 Remark. It should be mentioned that the unusual additional terms
0+f(T)A+g(T)+ E 0 f(T)i g(T)
a<r<b
a<r<b
on the right hand side of (1.20) vanish if f and g are functions
with no common discontinuities.
The integration by parts result from Corollary 1.23 can be
strengthened for the case when one of the functions f, g is of
bounded variation and the other is regulated (i.e. has onesided
limits at every point in [a, b]). In this case the integrals in question
exist as was shown by M. Tvrdy in [164] and the proof is the same
as ours because regulated functions have only a finite number of
onesided discontinuity points at which their jumps exceed a given
positive number.
For more information on the extensive literature concerning
integration by parts see the expository papers [39] or [11].
Convergence theorems
1.25 Theorem. Let functions U, Um : [a, b] x [a, b] R", m =
1, 2, ... be given where Um E JC([a, b]) for m = 1, 2,.... Assume
that there is a gauge w on [a, b] such that
urn [Um(T, t2) - Um(T, tl )] = U(T, t2) - U(T, t1)
"a-oo
(1.23)
-
for T E [a, b] and all t1, t2 E R such that t1 < T <_ t2, [t1, t2] C
[T - w(T), T + w(r)]. Assume further that
for every ij > 0 there is a gauge S on [a, b] such that
b
IIS(Um, D) -
DU. (T, t) 11 < 17
Ja
(1.24)
for every S-fine partition D of [a, b] and every m =
1,2,....
Convergence theorems
31
Then U E 1C([a, b)] and
lim
,n
oo
I DUm(r,t)
a
DU(-r,t).
(1.25)
Proof. Let e > 0 be given. By (1.24) there is a gauge S on
[a, b], 8(r) < w(r), r E [a, b] such that for every 6-fine partition
D = {ao, rl, al , ... , ak-1, rk, ak}
of [a, b] we have
6
IIS(U,n, D) - J DUn,(r,t)II <
a
2
for m = 1, 21 .... By (1.23) for every fixed partition D of [a, b]
there exists a positive integer mo such that for m > mo the
inequality
IIS(U,n, D) - S(U, D)II =
k
II E[U,n(rj,aj)-U(rl,al)-Um(rj,al-1)+U(rj,aj-1)]II <
j=1
2
holds and this means that
D) = S(U, D).
lim
M-00
Therefore for any 6-fine partition D of [a, b] there is a positive
integer mo such that for m > mo we have
6
IIS(U,D) - j DU,n(r,t)II < e.
a
(1.26)
1. The generalized Perron integral
32
First we get from (1.26) that for all positive integers m,1 > mo
the inequality
fb
b
II
f DU,,, (r, t-
DU,(r, t)II < 2e
a
holds. This means that (fb DU,,, (r, t))n° is a Cauchy sequence
in Rn and therefore has a limit
b
DUm
lien
m-oo
(r, t) = I E Rn(1.27)
Ja
b
Ils(U, D)
- III
II S(U, D) - f DUm(r, t)II +
a
U
a
fbDU.(r,t)-III<e+ll j b
DU,,,(r,t)-I.
By (1.27) we obtain immediately from this inequality that for
every 6-fine partition D of [a, b] we have
IIS(U,D)-III <e
and this means that the integral fa D(Ui, t) exists and (1.25) is
satisfied.
0
1.26 Definition. A sequence of integrable functions U. :
[a, b] x [a, b] -- R" (U,,, E 1C([a, b])), m = 1, 2.... is called equi
integrable if the condition (1.24) of Theorem 1.25 is satisfied.
1.27 Remark. Theorem 1.25 gives a sufficient condition for a
sequence of integrable functions to tend to an integrable limit and
for the integrals of the members of the sequence to tend to the
Convergence theorems
33
integral of the limit function. The convergence of the functions
U,,, to U is given by (1.23) and the sufficient condition is the equi
- integrability (1.24) of the sequence (U,,,).
Clearly a convergence result for integrals of I8"-valued functions holds if and only if it holds for every component of the
functions (cf. Theorem 1.6 ). Therefore without loss of generality we can consider sequences of real-valued functions only.
The fundamental idea of Theorem 1.25 lies in viewing the concept of the integral as a certain limiting process and in the fact
that two limits are interchangeable provided one of them is uniform with respect to the limiting variable of the second. In our
situation equi - integrability stands for this uniformity. In this
sense Theorem 1.25 is a transparent mathematical fact, nevertheless from the practical point of view it is not easy to check that
a given sequence of integrable functions is equi - integrable. This
forces us to use another condition instead of equi - integrability
and motivates the results given in the sequel.
For a given V : [a, b] x [a, b] --> R and a tagged interval (r, J)
with r E J = [a, ,Q] C [a, b] we will use the notation
V(r, J) = V(r, a) - V(r, Q)
(1.28)
for the point-interval function which corresponds to V.
1.28 Theorem. Let U, U,,, : [a, b] x [a, b] -> R, m = 1, 2, .. .
where Um E X ([a, b]) for to = 1, 2, ... .
Assume that there is a gauge w on [a, b] such that for
every e > 0 there exist a p : [a, b] -- N*) and a positive superadditive interval function f defined for closed
intervals J C [a, b] with 4)([a, b]) < e such that for every
r E [a, b] we have
IUm(r, J) - U(r, J) I < 4)(J)
*)By N the set of positive integers is denoted.
(1.29)
1. The generalized Perron integral
34
provided m > p(r) and (T, J) is an w-fine tagged interval
with TEJC[a,b].
Let us further assume that the sequence (Um) satisfies the
following condition.
There exist a gauge 9 on [a, b] and real constants B <
C such that for all choices of functions m defined on
[a, b] taking positive integer values (m : [a, b] -- N) the
inequalities
k
B <>U.,(rj)(Tj,Jj) <C
(1.30)
j=1
hold provided
D = {(-rj,JJ), j = 1,2,...,k}
is an arbitrary 9-fine partition of [a, b].
Then the sequence (Un,) is equi - integrable, i.e. (1.24) holds.
1.28a Definition. Let U, U,.. : [a, b] x [a, b] -a R, m = 1, 2, ... ,
U,,, E K.([a, b]) f o r m = 1, 2, ... be given such that the conditions
(1.29) and (1.30) of Theorem 1.28 are satisfied.
For a given positive integer p E N let Sp be the family of all
functions V : [a, b] x [a, b] -> R such that there is a division
a=Qo <#I
<... <#1-1 <,31 =b
of [a, b] such that for any tagged interval (r, J), T E J C [a, b] we
have
V(T, J) =U1j (T, J) if T E (/lj-1, /ij),
V(T, J)
(r, J-) + Umj+, (r, J+) if r = /3j,
j =1,...,1,J- = Jn(-oo,aj],J+=Jn[1:j,oo)
(1.31)
Convergence theorems
35
where m j E N with
2
1
,
Let us give a series of statements concerning the families S,
described by this definition.
1.28b Lemma.
a) For every p E N we have Up E Sp.
b) If P1, P2 E N, p1 > p2 then Sp, C Sps
c) IfV ES1 thenV EIC([a,b]).
Proof. a) If we set 80 = a, i31 = b then clearly Up is of the
form (1.31).
b) is clear by the definition of Sp.
c) Given V E Sp then it is easy to check by (1.31) that for
every partition Dj of [/3j_1 , /3j] we have
S(V,D') =
(1.32)
for the corresponding integral sums. Since Ur,, is integrable over
[a, b], it is also integrable over
by Theorem 1.10 and
(1.32) yields the integrability of V over
This holds
for every j = 1, ... , 1 and therefore by Theorem 1.11 we obtain
V E IC([a, b]).
D
1.28c Lemma. If V E Si then
k
B < E V(Ti, Jj) = S(V, D) < C
(1.33)
j=1
f o r a n arbitrary 9-fine partition D = {(Tj, Jj), j = 1, 2, ... , k} of
[a, b].
Proof. Let {/30, . . . ,
} be the finite sequence of points used
for the definition of V in (1.31). By (1.31) we have V (r, J) =
1. The generalized Perron integral
36
if r
... , /3m } and V(r, J) = Umj (r, J- )
+Umj+,(T,J+) when r E {p0 ,...,/3m).
Umj (r, J)
{/3o,
It is easy to see that
{(rj,Ji );7 = 1,...,k,rj V 100'.. .,Am}}U
U{(rj,J,-); i = 1,...,k,rj E
0m}}U
U{(Tj,Jj+);j = 1,...,k,rj E {AO,...,Am}}
forms a 8-fine partition of [a, b] and therefore (1.33) follows immediately from (1.30).
1.28d Lemma. If (r, J) is an w-fine tagged interval, r E J C
[a, b], p > p(r), then
IV(r, J) - U(r, J)1 < -(D(J)
(1.34)
for every V E Sy.
{,(30, ... , /3m }, then V (T, J) = U,,,, (r, J) by
Proof. If Tj
(1.31) and because mj > p > p(r) (1.34) holds by (1.29).
If rj E {/3o,...,/3,,,}, then r = /3j for some j = 1,...,m and
V (T, J) = U,,,j (r, J-) + Umi+, (r, J+)
by Definition 1.28a. If J = [a, /3) then J- _ [a, pi l, J+ _ [/3j, /3]
and
U(r, J) = U(Aj, A) - U(A,, a) _
U(Aj, A)-U(Aj, Aj)+U(Aj, Aj)-U(Aj, a) = U(r, J-)+U(r, J+).
Therefore again by (1.29) we get
IV(r, J) - U(r, J)1 =
J U., (r, J-) + Umj+, (r, J+) - U(T, J-) - U(r, J+)I C
<1Umj(r,J)-U(r,J )I+IUmj+1 (r,J+)-U(r,J+)I
< 4(J-) + q)(J+) < ck(J).
0
37
Convergence theorems
1.28e Lemma. Assume that e > 0 is given. Then for every
p E N there exist Vp, VP E Sp such that
b
rb
e
DVp(r, t) - 2p < inf{J DV(r, t); V E Sp} <
a
/'b
/b
a
a
< sup{ J DV(r, t); V E Sp} < J DVp(r, t) + 2p .
(1.35)
Proof. By 1.28c the supremum and infimum exist and
rb
rb
a
a
B < inf{J DV(r, t); V E Sp} < sup{J DV(r,t);V E Sp} < C.
The existence of Vp, VP satisfying (1.35) comes immediately from
CJ
the definition of the infiinum and supremum.
1.28f Lemma. Assume that V E Sp. Let Ij
j = 1, ... , s be an arbitrary finite sequence of disjoint intervals in
[a, b]. Then for a given e > 0 we have
6
DV,(r,t) -
<
J DV(r,t) <
2p -
j=1
J=1
<
DV"(r, t) + 2p
J
f
j=1
(1.36)
where Vp, VP E Sp are the functions corresponding toe by Lemma
1.28e.
Proof. Assume for example that the second inequality in (1.36)
is not satisfied. Then there is a V* E Sp such that
9
S
r
j=1 J
r
DVp(r,t) + 2p <
DV*(r,t).
j=1
Ii
1. The generalized Perron integral
38
Define
Vo(r, J) =V*(r, J) if r E
Vo(r, J) =VP(r, J) if r E [a, b] \ Uj=1I),
Vo(r, J) =V*(r, J`) + VP(r, J+) if r = y),
j =1, ... , s, J- = J n (-oo, y;], J+ = J n [y;, oo).
Denote by Lk, k = 1, ... , r the system of closed intervals forming
the components of the closure of [a, b] \ UJ=1I;. Then evidently
Vo E Sp and we have the inequality
fbDv(t)
> DV(T,t)+f
j=1
>E J
1= I
DVP(T t)+
k=1
)
JLk
DV(r,t) >
k=1 f Lk
DVP(T t)+ 2P _
jb
DVP(T, t)+ 2P
E
which contradicts Lemma 1.28e and thus proves the second inequality in (1.36). The first inequality in (1.36) holds by a similar
argument.
Proof of Theorem 1.28. Let E > 0 be given. By (1.29) and
Lemma 1.28d for every r E [a, b] there is a p(r) E N such that for
allmEN, m p(r) we have
IVp(r)(T, J) - Um(T, J) I
IVp(r)(T, J) - U(r, J)I + JUm(T)(T, J) - U(r, J)I S 24i(J), (1.37)
and similarly also
IVP(r)(r, J) - U, (T, J)I < 211,(J)
(1.38)
Convergence theorems
39
if (r, J) is an w-fine tagged pair, T E J C [a, b], where
is a
positive superadditive interval function with (D([a, b]) < e.
For a given p E N the functions Up, Vp, VP are integrable and
therefore there is a gauge 6p on [a, b] such that
b
S(Up, D) - f DUp(T, t)I <
2P
b
S(Vp, D) -
I
S(VP, D) -
j
DVp(r, t)] < 2P ,
(1.39)
jb
DVP(T, t)I <
2p
for any 6p- fine partition D of [a, b]. For r E [a, b] let us choose
6(r) > 0 such that
6(r) < min(W(T), 61(r), 62(r), ... , SP(r)(T))
(w is the gauge from (1.29) and (1.37), p : [a, b] --+ N is introduced
in Lemma 1.28d and the gauges 61, 62, ... come from (1.39)).
Assume that D = {(ri, Jr), i = 1, ... , k} is a 6-fine partition
of [a, b] and that in E N is arbitrary. Then
k
S(Um, D) =
k
E U., (7-i, Ji) + E Um (Ti, Ji).
i=1
i=1
m<p(r,)
m>p(r;)
The first term is the suin of those
and similarly for the second term.
If m > p(ri) then by (1.38)
-24 (J) < UT.,.(ri, Ji)
(1.40)
Ji) for which in < p(Ti),
- VP(T')(Ti, Ji) < 2'(J),
I. The generalized Perron integral
40
therefore
Une(Ti, Ji) > VP(r:)(Ti, Ji) - 24 (J)
and for the second sum in (1.40) we have
k
k
E
Um(Ti,
(VP(r;)(Ti, Ji) - 2-1)(J)].
Ji) >
i=1
i=1
->P(ri)
->P(ri)
Therefore
S(U,n, D) >
k
U.,(-r,,
>
i=1
k
k
VP(r;)(ri, Ji)
Ji) + E
i=1
m<p(r;)
i=1
m>p(r;)
k
->P(ri)
m-1
k
-2
.(J) +
i=1
k
Vp(r;)(Ti, ii).
Um(Ti, Ji) +
i=1
->P(ri)
c(J) _
-2
!=1
m<P(r;)
i=1
P(r;)=1
From (1.39) and from the Saks-Henstock lemma 1.13 we obtain
k
k
Um(Ti,
i=1
i=1
m<p(r;)
k
V!(Ti,J1)-
p(r;)=1
i=1
i=1
m<p(r;)
Ji
DV!(T,t)l <
21,
DU.. (T, t)
2m
P(r:)=1
i.e.
k
DU,n(T,t)I< 2p,
m<P(r:)
k
IE
i=1
J,
k
Um(Ti,Ji) > i
i=1
,n <p(r; )
f
;
Convergence theorems
k
k
41
jDV1(r,t)_j
V!(ri, Ji) >
i
p(r1)=l
p(ri)=1
and
k
k
(J) +
S(Um,D) > -2
i=1
m<p(ri)
m>p(ri)
m-1
k
l=1
i=1
DU.(r,t)- -m+
J;
i=1
to-1
Ji
!=1
p(ri)=1
2
Since Um E Sin we have Urn E Sp for all p = 1, 2...., m - 1 by b)
from Lemma 1.28b and by Lemma 1.28f we therefore get
k
i=1
p(ri)=l
M-1
1=1
f
k
DV'(r, t) >
i
i=1
p(ri)=l
in-1
k
i=1
Ji
J i DUm(T,t) - 21
DV1(r, t) >
1=1
p(ri)=1
m-1
r
k
i=1
J
,
DU,n(r, t) 1=1
p(ri)=1
m-1
k
]i
i=1
DU.. (r, t)
J=1
21
m>p(ri)
and
m-1
k
i=1
m>p(ri )
2",
m-1
21
1=1
1=1
2t
e
21
1. The generalized Perron integral
42
k
+>
k
jDUrn(Tt)+ >
m<p(r,)
In>p(r;)
,n-1
>
b]) -
2m
-2
r=1
J
rb
2r + J DUm(T, t) >
,n-1
b
>
r
J DUm('t) >_
DUm (r, t) - e(2 +
2,n + 2
2r) >
r1
J
DU,, (T, t) - 4e.
In a completely analogous way we can use Vp instead of V1 to
show that
rb
S(U,n, D) <
DU. (r, t) + 4e,
Ja
b
IS(U,n, D) - j D Ur(r, t)I < 4e
and the sequence (Un,) is equi - integrable, the gauge 5 being
independent of m.
Using Theorems 1.28 and 1.25 the following new form of the
convergence theorem for generalized Perron integral can be given.
1.29 Theorem. Let U, Um : [a, b] x [a, b] --> R, m = 1, 2, ...
where Um E 1C([a, b]) f o r m = 1, 2, ... .
Let us further assume that the sequence (Um) satisfies the
following conditions.
There is a gauge w on [a, b] such that for every e > 0
there exist a p : [a, b] - N and a positive superadditive
interval function defined for closed intervals J C [a, b]
with qk([a, b]) < e such that for every r E [a, b] we have
IU,,,(T, J)
- U(T, J)I < t(J)
(1.29)
Convergence theorems
43
provided m > p(r) and (r, J) is an w-fine tagged interval
with rEJC[a,b].
There exist a gauge 0 on [a, b] and real constants B <
C such that for all choices of functions ni defined on
[a, b] taking positive integer values (m : [a, b] -+ N) the
inequalities
k
B < E U.,(r,)(rj, Jj) < C
(1.30)
j=1
hold provided
D = {(rj, Jj), j = 1, 2, ... , k}
is an arbitrary 0-fine partition of [a, b].
Then U E k ([a, b)] and
r6
6
lim J DU,,,(r, t) _ f DU(r, t).
m-oo
a
a
Proof. If (1.29) is satisfied then for r E [a, b] and all t1i t2 E R
such that t1 < r < t2, [t], t2] C [r - w(r), r + w(r)] we clearly
have
urn [U.. (r,t2) - U.,(r,t1)] = U(r,t2) -- U(r,tl).
By Theorem 1.28 the sequence
is equi - integrable and therefore by Theorem 1.25 the conclusion of this theorem holds.
0
1.30 Remark. It is easy to see that if f,,,
[a, b] -- 1[8, m =
1, 2,... and if g : [a, b] --> R is of bounded variation where
ttlimo fm (r) =
f(r),
:
r E [a, b]
I. The generalized Perron integral
44
then the sequence Um : [a, b] x [a, b] -+ R given by
U. (7-, t) = f,,,(T).g(t),
T, t E [a, b]
f o r m = 1, 2, ... satisfies (1.29) because
Um(T, J) - U(r, J) = (f...(T) - f(r)).(g(Q) - g(a))
for J = [a, Q] and for a given e > 0 the corresponding superadditive interval function -t can be given in the form -t(J) = e varfl g.
A result similar to Theorem 1.28 is given in the book [84,
Lemma 5.4] where the condition (1.30) is replaced by the following one.
There exists a constant K > 0 such that for every divi< 8 = b of [a, b] and every finite
sion a = go < 91 <
sequence m1, m2, ... ,1711 the inequality
1
I
0;
(T, t)J < K
holds.
A condition of this form was used for the first time in 1979 in
a common text of David Preiss and the author on elementary
Perron integration which was published in a mimeographed form
in Czech for internal use only as a preparation to a chapter of a
calculus textbook which has never been finished.
Our condition (1.30) in Theorems 1.28 and 1.29 is motivated
by the results of R. Henstock [40, Theorem 9.1]. Henstock shows
directly that a condition of type (1.30) yields the convergence
theorem for Perron integrals. Here we have shown that (1.30)
yields the more general condition of equi - integrability and the
convergence result is derived via Theorem 1.25.
Now we give a statement which plays the role of the dominated
convergence theorem for generalized Perron integrals.
45
Convergence theorems
1.31 Corollary. Let U, Um : [a,b] x [a, b] -> R, m = 1, 2, .. .
where U,n E lC([a, b]) for m = 1, 2, ... .
satisfies the
Let us assume further that the sequence
following conditions.
There is a gauge w on [a, b] such that for every e > 0 there exist
a p : [a, b] - N and a positive superadditive interval function'
defined for closed intervals J C [a, b] with 4k([a, b]) < e such that
for every r E [a, b] we have
J) - U(r, J)I < -t(J)
provided m > p(r) and (r, J) is an w-fine tagged interval with
r E J C [a, b].
Assume further that two functions V, W : [a, b] x [a, b] -> R,
V, W E 1C([a, b]) are given and that there is a gauge O on [a, b]
such that for all m E N, T E [a, b] we have
V(r, J) < U,n(r, J) < W(r, J)
(1.41)
for any 19-fine tagged interval (r, J).
Then U E JC([a, b)] and
b
lm
J
rb
DU,,(r, t) =
J
DU(r, t).
a
Proof. Assume that 9 is a gauge on [a, b] such that 9(r)
O(r), r E [a, b] and
b
JS(V, D) - J DV(r, t)J < 1, JS(W, D)
a
-
b
DW(r, t) I < 1
JQ
for every 9-fine partition D = {(rj, J,), j = 1, 2, ..., k} of [a, b].
I. The generalized Perron integral
46
Then evidently for every nt : [a, b] -+ N and any 9--fine parti-
tion D = {(ri,J1),j = 1,2,...,k} we have by (1.41)
Jj) < W (rj, Jj)
V (rj, Jj) <
and also
k
b
JfDV(r,t) - 1 < S(V,D)
a
EUnt(r;)(rj,Jj)
j=1
rb
<S(W,D)<J DW(r,t)+1.
a
Putting fa DV(r, t) - 1 = B and fQ DW(r, t) +1 = C we obtain
the condition (1.30) of Theorem 1.29 and also the statement of
0
this corollary.
Let us now show the corresponding result for Stieltjes integrals.
1.32 Corollary. Let g : [a, b] --+ R be a nondecreasing function on [a, b]. Assume that
[a, b] -- R are such functions that
the integral fa fm(s)dg(s) exists for every m E N. Suppose that
for r E [a, b] we have
l m f.. (r) = f(r)
m--.oo
and that for m E N, r E [a, b] the inequalities
V(7-):5 fm(r) < w(r)
hold where v, w : [a, b] -- R are such functions that the integrals
fa v(s)dg(s) fa w(s)dg(s) exist.
47
Convergence theorems
Then the integral fa f(s)dg(s) exists and
f f(s)dg(s) = lim fa fm(s)dg(s)
b
In- 00
a
b
Proof. If we set U,n(r,t) = fn(r).g(t), m E N, U(r,t) =
f(r).g(t), V(r,t) = v(r).g(t), W(r,t) = w(r).g(t) for r E [a,b], t E
[a, b] then (1.41) is satisfied and by Corollary 1.31 we obtain the
result.
Definition. A function f : [a, b] -+ R is called a finite step
< 3,n =
function if there is a finite division a = 3o < 01 <
i = 1,... , m the
b such that in every open interval
function f is identically equal to a constant c; E R.
1.33 Proposition. If f : [a, b] -+ R is a finite step function
(of the form given by the previous definition) and g : [a, b] -+ R is
of bounded variation on [a, b] then the integral fQ f (s)dg(s) exists
and
b
f(s)dg(s) =
Cin
_ f(a)(g(a+) - g(a)) + L, f (Qj)(g(f3 +) - g(fi -))+
i=1
m
c.i(g(/31-) -g(/3,_1+)).
+f(b)(g(b) -g(b-)) +
(1.42)
j=1
(By g(s+), g(s-) the onesided right and left limits of g at the
point s E [a, b] are denoted.)
Proof. The function g is of bounded variation and therefore the
onesided limits of this function exist at every point in [a, b]. For
a given e > 0 let us choose a gauge 6 on [a, b] such that b(fi) <
48
I. The generalized Perron integral
$m }) for r
p(e), j = 0,1,-..,m, b(r) < dist(T, {QOM QI
/3,, j = 0, 1, ... , m. For a gauge 6 satisfying these assumptions
we know that every 6-fine partition D of [a, b] contains all the
points /3a, ,01 i ... , Nm as tags of its 6-fine point-interval pairs. If
we choose p(e) sufficiently small we obtain the existence of the
integral fa f(s)dg(s) as well as the equality (1.42) for its value.
0
Definition. A function f : [a, b] - R is called regulated on the
interval [a, b] if the onesided limits f (s-), f(s+) exist for every
s E (a, b], s E [a, b), respectively.
It is well known that every function which is regulated on [a, b]
is bounded on this interval and is the uniform limit of finite step
functions (See e.g. [43] or [6].) Clearly also every function of
bounded variation on [a, b] is regulated on [a, b].
1.34 Corollary. If f : [a, b] -+ R is a regulated function on
the interval [a, b] and g : [a, b] -3 R is of bounded variation on
[a, b] then the integral fa f (s)dg(s) exists.
Proof. Since f is bounded on [a, b] and there is a sequence of
finite step functions f,n which converges to f uniformly on [a, b],
there is an index mo E N such that for rn > ino we have I f,n(r)I <
K where K > 0 is a constant. Since the integrals b f fm(s)dg(s)
exist, in = 1,2,... by Proposition 1.33 and the assumptions of
Corollary 1.31 are satisfied with v(r) = -K, w(r) = K, Corollary
1.31 yields the result.
Inequalites for the integral
1.35 Theorem. Assume that a function U : [a, b] x [a, b] -->
a
1R' is given for which the integral f DU(r, t) exists. If V : [a, b] x
[a, b] --> R is such that the integral f 6 DV (r, t) exists and if there
49
Inequalites for the integral
is a gauge 8 on [a, b] such that
t) - U(T, T)II < (t - r).(V(,r, t) - V(T, r)) (1.43)
It-T
for every t E IT - 8(T), T + 8(r)] then the inequality
b
rb
DU(T, t)
< j DV(T, t)
(1.44)
a
holds.
Proof. Assume that e > 0 is given. Since the integrals
f DU(T, t), f a DV(r, t) exist there is a gauge 6 on [a, b] with
b
b(s) < B(s) for s E [a, b] such that for every b-fine partition
D = {ao,T1,a1,...ak-1,Tk,ak)
of [a, b] we have
k
[U(ri, ai) - U(T, ai_)] i=1
j
b
DU(r, t) < e,
(1.45)
k
1: [V(Tr,ai) - (7-i,ai-1)] - j DV(T,t) < e.
j=1
(1.46)
a
It is easy to see that (1.43) implies IIU(Ti,ai) - U(Ti,Ti)II <
V(ri,ai) - V(Ti,Ti) when ai > Ti and IIU(ri,ai) - U(Ti,r;)II
V(Ti, Ti) - V(Ti, ai) when ai < Ti. Hence for i = 1, 2, ... , k we
have
IIU(Ti, ai) - U(ri, ai-1)II < IIU(Ti, a,) - U(Ti, r )II+
+II U(Ti, Ti) - U(r1, ai-1 Al :5 V (r , ai) - V(Ti, ai-1).
1. The generalized Perron iniegral
50
By (1.45) and (1.46) we get
b
j DU(-r, t)
k
b
< E[U(Tj, aj)
//
j=1
- U(Tj, aj-1 )] - 1. DU(r,/ t)
/
a
+ [U(Tj,aj) -UITj,aj-1 )]
<
J=1
k
<
E+E[V(Tj,aj) -V(Tj,aj_1)] _
j=1
k
b
=E+1:[V(Tj,aj)-V(Tj,aj_1)]-f
J=1
DV(T,t)+
a
f
b
DV(T,t) <
a
b
< 2e + 1. DV(T, t).
a
Since e > 0 was arbitrary, the inequality (1.44) is satisfied.
0
This theorem gives an estimate of the integral fQ DU(-r, t) by
another integral of a real valued function. In some cases it is useful to have an estimate of this type with a Stieltjes type integral.
This is given in the following statement.
1.36 Corollary. Assume that a function U : [a, b] x [a, b] R" is given for which the integral fQ DU(-r, t) exists. If V : [a, b] x
[a, b] -+ 1R is such that the integral fa f (s) dg(s) exists for f, g
[a, b] -> R and if there is a gauge 0 on [a, b] such that
It -
rI.IIU(T,t)
- U(T,T)II
(t - T).f(r)(g(t) - g(r))
Inequalites for the integral
51
for every t E IT - 0(T), r + 0(r)] then
jDU(rt)
6
< f f(s) dg(s).
(1.47)
a
Proof. The result immediately follows from Theorem 1.34 for
the case when V(r,t) = f(T)g(t).
1.37 Remark. Theorem 1.34 can be also used to deduce the
following known result.
If f : [a, b] -4R, If(s) I _< c for s E [a, b] where c is a constant,
g : [a, b] -+ JR is of bounded variation on [a, b] and the integral
fa f (s) dg(s) exists then
b
f(s)dg(s)
c varQ g.
Indeed, it is easy to see that
It -,r 1. If (r).g(t) - g(r)l < (t - r).c.(vara - vary)
for every t, r E [a, b] and the statement follows immediately from
the known fact that the integral fa c d(vara) = c. varQ exists and
from Theorem 1.34.
1.38 Lemma. Let h : [a, b] - JR be a nonnegative nondecreasing function which is continuous from the left on (a, b]. Assume
that f : [0, +oo) --> [0, +oo) is a continuos nondecreasing function
with the primitive F : [0, +oo) -- R, i.e. F(s) =
for s E [0, +oo).
F(s) = f (s)
I. The generalized Perron integral
52
Then the integral fQ f(h(s)) dh(s) exists and
j
b
f (h(s)) dh(s) < F(h(b)) - F(h(a)).
(1.48)
Proof. The composition of functions f and h given by f (h(s))
for s E [a, b] is evidently nondecreasing on [a, b]. Therefore the
integral fQ f (h (s)) dh(s) exists by Corollary 1.34.
For proving (1.48) Theorem 1.34 will be used. Assume that
e > 0 is given. By the definition of the primitive F to f, for
every s E [0, +oo) there exists O(s) > 0 such that for every i
with 0 < 1ij < O(s) we have
+F(s+rl)-F(s)-f(s)77I SET/.
(1.49)
Since limf_,r+ h(t) = h(r+) for r E [a, b), there is a S+(r) > 0,
S+(b) = 1 such that for t E (r, r + b+(r)] fl [a, b] we have
0 < h(t) - h(r+) 5 O(h(r+)).
Putting s = h(r+) and '1 = h(t) - h(r+) we obtain
f(h(r+))(h(t) - h(r+)) < F(h(t)) - F(h(r+)) + e(h(t) - h(r+))
for t E (r, r + b+(r)] fl [a, b). Further, we have
f(h(T+))(h(t) - h(r+)) - [F(h(t)) - F(h(r+))] _
h(r+)
J
[.f(h(T)) - f(s)] ds < 0
h(r)
because f(h(r)) < f(s) for s E [h(r),h(r+)]. Therefore
If(h(r))(h(t) - h(r))j _
Inequalites for the integral
53
= f(h(r))(h(t) - h(r+)) + f(h(r))(h(r+) - h(r)) <
< f(h(r+))(h(t) - h(r+)) + F(h(r+)) - F(h(r))
< F(h(t)) - F(h(r)) + e(h(t) - h(r))
provided t E (r, r + 6+ (-r)] fl [a, b]. From the inequality (1.49)
and from the continuity from the left of the function h at the
point r E (a, b] there is b-(r) > 0, 6-(a) = 1 such that for
t E [T - b-(T), r] fl [a, b] the inequality
If (h(r))(h(t) - h(r))I < F(h(r)) - F(h(t)) + e(h(T) - h(t))
is satisfied. Take b(r) = inin(b-(r), 6+(r)) for r E [a, b]. Then
for T E [a, b] and t E IT - b(T), r + b(r) fl [a, b] we obtain by the
above inequalities the relation
- f(h(r))h(r))l <
it - rl If
(t - r) (F(h(t)) + eh(t) - F(h(r)) - eh(r)).
By Corollary 1.35 this inequality implies
b
b
J.
f(h(s)) dh(s) < 1. d[F(h(s)) + eh(s)]
a
= F(h(b)) - F(h(a)) + e(h(b) - h(a)).
This inequality yields (1.44) because e > 0 can be chosen arbi-
0
trarily small.
1.39 Remark. Using essentially the same technique as above it
can be shown that if in Lemma 1.38 the function h is continuous
from the right instead of being continuous from the left then the
inequality
F(h(b)) - F(h(a)) <
r
b
f(h(s)) dh(s)
1. The generalized Perron integral
54
holds, where the integral f4 f(h(s)) dh(s) exists for the same reason as was shown in the setting of Lemma 1.38.
If ry > 0 and f (s) = s" for s E [0, +oo) then F(s) = 1 s-Y+'
1
is the primitive to f on 10, +oo). Using Lemma 1.38 for this
is
special case ve obtain the inequality
f
+
/m (s) dh(s) <
[h''+'(b)
1
ti
- h'+'(a)]
(1.50)
for an arbitrary nonnegative nondecreasing function h : [a, b] -- R
which is continuous from the left.
It can be also shown that if h : [a, b] - R is continuous from
the right on [a, b] then
6
1
1
[V+'(b)
- h''+1(a)] < J V(s) dh(s).
(1.51)
a
1.40 Theorem. Let 0 : [a, b] - [0, +oo), h :
[a, b] ---> [a, +oo)
be given where 0 is bounded and h is non decreasing and continuous from the left on the interval [a, b]. Suppose that the function
w : [0, +oo) -> R is continuous, nondecreasing, w(0) = 0, w(r) > 0
for r > 0.
For u > 0 let us set
u
1(u)_
1
uo
4'(r)
dr
(1.52)
with some uo > 0.
The function S : (0, +oo) - R is increasing, 1Z(uo) = 0 and
liin._.o+ 11(u) = a _> -oo, limu._.,+,12(u) = 8 < +oo.
Assume that for 1; E [a, b] the inequality
k+
j
w(b(r)) dh(r)
(1.53)
Inequalites for the integral
55
holds, where k > 0 is a constant.
If 1Z(k) + h(b) - h(a) < Q then for E [a, b] we have
Q-' (Sl(k) +
h(a)),
(1.54)
where 1Z-' : (a, Q) - R is the inverse function to the function fl
given by (1.52).
Proof. If we have 11(1) + h(b) - h(a) < 0 for some 1 > 0 then
for all r E [a, b] we have
a < SZ(1) + h(r) - h(a) < 0.
Therefore the value of
SZ(l) + h(r) - h(a)
belongs to the domain of 1Z-' provided r E [a, b], and for these r
we can define
w!(r) = c-1(11(1) + h(r) - h(a)).
Define further
V(s) = w(11-1(i(1) + s)) for s E [0, Q - R(1)].
(1.55)
At Q-1 (Q(1) + s) there exists a derivative St' of the function 12
and
s)) =
0.
w(Q-' (I(1) + s)) #
The well known formula for the derivative of the inverse function
leads to
WS-
M(fl-1(110)
+ s))] = SZ'(11-1(fl(1)+S))
.56)
I. The generalized Perron integral
56
= w(1-,(Il(1) + s)) = cp(s)
for s E [0, /3 - Sl(l)]. If now E [a, b] is given then using the
definition of the function cp from (1.55) we obtain
I
Ja
w(w1(r)) dh(r) =
=
j
Ja
w(S2
(SI(1) + h(r) - h(a))) dh(T) _
(h(T) - h(a)) d(h(r) - h(a)).
This together with (1.56) and Lemma 1.38 implies
J
{ w(w,(r))
dh(r) < fr'(st(l) +
h(a))
-12-'(Q(1))
_
= w,(e) - 1,
and consequently for l E [a, b] we have the inequality
1+
J
w(wi(T)) dh(r) < w,(e).
Assume that eo > 0 is such that 92(k + co) + h(b) - h(a) < /i. Let
us take an arbitrary e E (0,,-o) and set 1 = k + e. For this case
the last inequality reads
k+e+
j
(wk+(r)) dh(r) < wk+(e),
and taking into account the relation (1.53) for every e E [a, b] we
get
(
k+
Ja
)-
w(O(r)) dh(r) - k - e - j w(wk+,, (r)) dh(r) _
Inequaliles for the integral
57
t
-e + la [w(0(T ))
- w(Wk+e(T ))] dh(r).
(1.57)
Hence 0(a)-wk+f(a) < -e and also w(0(a))-w(wk+E(a)) < 0
because the function w is assumed to be nondecreasing. The functions , and Wk+f are bounded and therefore there is a constant
K > 0 such that
I w(O(T ))
- w(wk+E(r))J <
K
for r E [a, b].
Using Theorem 1.16 and the estimate given in Remark 1.37 we
get from the last two displayed inequalities
OW
- Wk+E(C) : -e + [w(&(a)) - w(wk+E(a))](h(a+) - h(a))+
+ b-'0+
lim
Ja+b
[w(0(r)) - w(Wk+c(T))] dh(r) <
< -E + K lim [h(C) - h(a + a)] = -e + K[h(C) - h(a+)].
Because
h(C) = h(a+), an , > 0 can be found such that
for C E (a, a +,) the inequality h(C) - h(a+) <
1 holds and
2K
therefore also
+
for eE(a,a+rq).
Let us set
T = sup It E [a, b]; 't'(e) - wk+c(C) < 0 for l; E [a, t]}.
As we have shown above, we have T > a and for C E [a, T)
the inequality O(C) - wk+e(C) < 0 and therefore also w(0(C)) -
I. The generalized Perron integral
58
0 holds. The last conclusion is a consequence of
the assumption that w is nondecreasing. By (1.57) and Theorem
1.16 we have
O(T) - wk+e(T) <
T-6
< -e + 60+
urn
J
[w(
(r)) -w(wk+e(r))] dh(r)+
+[w(b(T)) - W(wk+£(T))](h(T) - h(T-)):5
<0
because h(T) - h(T-) = h(T) - limr.T_ h(r) = 0 and
/T-6
lim
J
6--0+ a
[w(a(r)) - w(Wk+£(T))] dh(r) < 0.
If we assume that T < b then we can repeatthis procedure for
C > T by virtue of the inequality
Wk+e(C) < -e +
T
IT
thus obtaining
z > 0. Hence T = b and
W(t(r)) - W(wk+a(r))] dh(r),
0 for l; E [T,T + 17] for some
wk+c(e) = Q-' (1Z(k + e) + h(C) - h(a))
for l; E [a, b]. Since the function Q is continuous and the last
inequality holds for every sufficiently small e > 0, we obtain the
0
inequality (1.53).
1.41 Remark. Theorem 1.40 represents a Bellman-BihariGyori type inequality (see e.g. [34]). Results of this type are
especially useful for deriving uniqueness results for equations like
ordinary differential equations.
Inequalites for the integral
1.42 Corollary. Assume that the function h
59
:
[a, b] --*
[0, +oo) is nondecreasing and continuous from the right. If the
inequality (1.53) in Theorem 1.40 is replaced by
k+
f
b
w(t,b(r)) dh(T)
(1.58)
where k > 0 is a constant and if S2(k) + h(b) - h(a) < /3 then for
e E [a, b] we have
Q-' (12(k) + h(b) - h(e))
(1.59)
Proof. For o E [-b, -a] define
h(a) = h(b) - h(-a).
It is easy to see that h : [-b, -a] - [0, +oo) is continuous from
the left and nondecreasing.
By the Substitution Theorem 1.18 we have
-b
jb
J
w(t)b(r)) dh(T) =
J
w(O(-rr)) dh(-r) =
= - / w(b(-r)) dh(-r) _
b
- w(ti(-T)) dh(T)
b
and we can write the inequality (1.58) in the form
k+
f
t
w(t,b(-r)) dh(r)
b
for e E [a, b], i.e.
k+
f
w(ib(r)) dh(r)
6
t. The generalized Perron integral
60
for e E [-b, -a], where Ri(o) = t&(-c) for o E [-b, -a] . Therefore Theorem 1.40 can be applied to obtain the inequality
+ h(Q) - h(-b)) = Sl-'(cl(k) + h(b) - h(-o-))
for o E [-b, -a]. Hence for C E [a, b] we obtain the inequality
cr' (11(k) + h(b) - h(C))
and the corollary is proved.
1.43 Corollary. If 0, h and k satisfy the assumptions of Theorem 1.40 and if for 1; E [a, b] the inequality
k+L
rt
&(r) dh(-r),
Ja
l; E [a, b]
holds with a constant L > 0 instead of (1.53), then for every
E [a, b] the inequality
OW <
keL(h(t)--h(a))
is satisfied.
Proof. This result is an immediate consequence of Theorem
1.40 if we set w(r) = Lr for r > 0. In this case we have
SZ(u)=
Lju
r = L(lnu-lnuo)=
In
a
a nd 11 '(u) = uoeL". The final inequality of the statement now
easily follows from (1.54).
In a completely analogous way also the following statement
can be proved using Corollary 1.42.
A ToneUi -type theorem
61
1.44 Corollary. If h : [a, b] -- [0, +oo) is nondecreasing and
continuous from the right and if, k satisfy the assumptions of
Theorem 1.40 where for e E [a, b] the inequality
b(l;) < k + L
Jitf
+/'(r) dh(-r),
e E [a, b]
holds with a constant L _> 0 instead of (1.53), then for every
e E [a, b] the inequality
&(s) :5
keL(h(b)-h(f))
is satisfied.
A Tonelli -type theorem
1.45 Theorem. Let [a, b], [c, d] C R, -oo < a < b <
00, -oo < c < d < oo be given intervals. Assume that cp :
[a, b] -- R is a function of bounded variation on [a, b] and w
[a, b] x [c, d] x [c, d) -> R is a given function satisfying
(A) for every (a, s) E [c, d] x [c, d] the function cp(t)w(r, a, s)
is integrable over [a, b], i.e. the integral
f
Ja
b
a, s)]
exists for every (a, s) E [c, d] x [c, d],
(B) the system of functions w(r, , ), r E [a, b] is equi - integrable over [c, d], i.e. for every q > 0 there is a gauge
bo : [c, d] --+ (0, oo) on [c, d] such that
m
d
[Dw(r,aj,0j)-w(r,ajaNj-1)]7=1
D.w(r,a,s)I <
c
I. The generalized Perron integral
62
for every bo-fine partition
Do :c=Qo :5 ai C#1 !5 ... GN,n-1 Co'm </3m =d
of [c, d] and any r E [a, b].
Then both the integrals f' Dt(f d
a, s)]) and
f d(Da f b De [yo(t)w(r, a, s)]) exist and have the same value, i.e.
f Dt(f
b
a
d
b
D,[,p(t)w(r, a, s)]) =
c
f d(D8 f
c
a, s)])
a
(1.60)
Remark. By Dt, D, the integration with respect to the couple
(r, t), (o, s) is marked in the formulation of the theorem. This
means that the symbol D3 corresponds to the integration over
[c, d] with s, a E [c, d] where for tagged intervals (a, K), K C [c, d]
the integral sums are determined by terms h(a, K) = V(a,-y)
V (o, J3) for a function V : [c, d] x [c, d] - R and K
and similarly for Dt and integration over [a, b].
-
C [c, d],
Proof. Let e > 0 be given. By the assumption (B) there is a
gauge bo on [c, d] such that
m
d
[w(T, aj, 0j) - w(T, a1, Nj-1 )] - IC
//
j=1
Dew(T, Q, s)I
(1.61)
2(1 + varb
for any r E [a, b] and for every bo-fine partition
Do =
{Qo,Q1,01,...
7 fln:-17 ormIPm}
of [c, d] with aj being the tags in [Qj-1, /3j].
63
A ToneUi -type theorem
First let us prove the existence of the integral
jb
D1(
D.[(t)w(T, a, s)]) =
z
1.
D((t)
D,w(r, a, s)).
e
Assume that the bo-fine partition Do of [c, d] is fixed. By (A) we
find a gauge b on [a, b] such that for every b-fine partition
D=
jaogrl,al,...
ap-l,rp,ap)
of [a, b] we have
m
p
E I E(W(ak) - Sp(ak-1))w(Tk, Qj, Nj )j=1 k=1
and
m
f
a
b
4
(1.62)
p
I J>(ak) - V(ak-1))w(Tk, O'j, flj-1
j=1 k=1
-
f
b
D,[(p(t)w(T,aj,Qj-1)]I < 4.
(1.63)
a
Denote U(r, t) = W(t) f ' D,w(r, a, s) for r, t E [a, b]. We show
that for every b-fine partition D of the interval [a, b] the corresponding integral sum S(U, D) is e-close to the value of
m
E
j=1 J
rb
a
aj, $) - w(r, aj, #j-IM-
I. The generalized Perron integral
64
Using the inequalities (1.61), (1.62) and (1.63) we obtain for every
6-fine partition D of [a, b] the following estimate.
M
rb
/ Dt [ (t)(w(r, aj, Yj) -' w(r, uj, fij-1 ))] I =
I S(U, D) i=1
a
P
j((a) - (ak-1))
Dew(Tk, or, s)-
k=1
M
jb
-EDt[W(t)(w(r,aj,f)-w(r,aj,/3j-1))]I :!5
j=1
d
P
I L(V(ak) - s'(ak-1 ))
J
k=1
Dew(Tk, o,, s)-
M
P
- E E(cP(ak) - W(ak-1))[W(rk, O'j, Aj) - w(Tk, O'j, #j-1 )]I +
k=1 j=1
m
P
+1 L r (s'(ak) - c'(ak-1))[w(rk,
Qj) - w(rk, O'j,
j=1
M
b
- j-1
1 Ja
Dt [V(t)(w(r, aj, Qj) - w(r, aj, Oj-1 ))]I
P
d
<- L I V(ak) - W(ak-1)I
k=1
I
J D9w(rk, a, s)-
m
-
))[w(Tk, a1, Qj) - w(rk, aj,
q
j=1
m
+EI
p
j=1 k=1
(V(ak) - V(ak-1))w(rk, a'j, Qj )-
A Tonelli -type theorem
65
b
- f Dt[4o(t)w(T,crj,aj)]I+
a
M
+
p
I E(V(ak) - (P(a"k-i ))w(Tk, aj, Qj-1 )j=1 k=1
b
a
P
E
E
E
LWak)-V(«k-1)1+4+4 <
2(l+vara ) k=1
<
varab
E
< E.
cp +
2(1 + var
n
Therefore for every two S-fine partitions D1i D2 of the interval
[a, b] we clearly have
IS(U,DI) - S(U,D2)I < 2e
and Theorem 1.7 yields the existence of the integral
b
DtU(T, t) =
D,w(T,
b
s)]
Ia
The inequality (1.61) further implies
m
It-TI
a , Nj) - w(T, aj, Aj-1 )]
j=1
-d
D,w(T,v,s)I <
< It - T IIP(t) - (P(T)I2(1
+ varQ ca
1. The generalized Perron integral
66
<It-Tllvaracp - varacpl
6
b
2(1+varaV)
=
E
= (t - r)(vara <p - vara Cp)2(1 + varb
a (P)
d
Since the integral fa Dt[cp(t) f D,w(r, 0., s)] exists and by the
assumption (A) also the integral
In
b
Dt (V (t) E(w(r) aj, Nj)
j=1
a
z
- w(r, aj, Nj-1 ))] =
b
Dt [sv(t)w(T, aj, Oj )] j=1
a
j=1
f
b
0j, Nj-1)]
a
exists we have by Corollary 1.36
m
J
Dt [fi(t) (w(T, j, Qj) - w(r, j, j-1 ) )]j=1
fbDE(i)jdD]I
b
fa
d(var'r )
b
E vara cp
E
2(1 + vara W)
2
E
" 2(1 + vara cp)
Hence
+n
I
b
b
Y, [f Dt[V (t)w(r, 0j, Qj )] - fa D t
j=1
/
qq
0.j, Nj-1
a
Id
-I'Dt[,p(t)
a
D.w(r,o,,s)]I <E
for every c > 0 and every 60-fine partition
Do = I&al,01, ...
N m-1,01m,Qm}
of [c, d]. Consequently, the integral f d(D, f b Dt[cp(t)w(r, a, s)])
exists and by definition equals to fb Di[V(t) f d D,w(r, a, s)].
A Tonelli -type theorem
67
1.46 Corollary. Let [a, b], [c, d] C R, -oo < a < b <
oo, -oo < c < d < oo be given
intervals. Assume that cp
:
[a, b) -- R, b : [c, d) - R are functions of bounded variation on
[a, b], [c, d], respectively, and that f : [a, b] x [c, d] --+ IR is a given
function such that
t
for every a E [c, d] the integral f f (r, a)dcp(r) exists,
r E [a, b] is equi the system of functions
integrable over [c, d], i.e. for every q > 0 there is a gauge
bo : [c, d] - (0, oo) on [c, d] such that
m
,/'
r,
E[f(r,o'j)(i'(aj) - ,b(Q,-1)] -
I
j=1
J
d
f(r,o,)dY'(a)I < i
c
for every bo-fine partition
Do :c=Qo :5 al :5 al <... <Qm-I <am :5 Nm =d
of [c, d] and any r E [a, b].
Then both the integrals fa (f d f (r, o) dt(o,)) dye(r) and
fd(fa f(r, a) dcp(r))di(o) exist and have the same value, i.e.
d
f (f f(r, a) db(a))
b
a
c
j (f
d
J.
b
f(r, a) dcp(r))db(a).
Proof. The convention described in Remark 1.5 for (Perron- )
Stieltjes integrals is used here. Let us set w(r, or, s) = A r, a)t(s).
For this function w : [a, b] x [c, d] x [c, d] -* R the assumptions of
Theorem 1.45 are satisfied and this proves the corollary.
0
1.47 Remark. The results of Theorem 1.45 remain valid if
we assume that cp : [a, b] -- L(R") is an n x n-matrix valued function of bounded variation on [a, b] and the function w :
I. The generalized Perron integral
68
[a, b] x [c, d] x [c, d] -* R" is n-vector valued. This can be
shown using Theorem 1.45 for the components of the product
cp(t)w(r, a, s) where Theorem 1.6 for k"- valued functions has to
be taken into account.
1.48 Remark. Theorem 1.45 and Corollary 1.46 represent Tonelli - type results on the reversal of order of double integrals.
More sophisticated results of this kind as well as Fubini - type
theorems for the Perron integral can be found e.g. in [40], [46],
[76].
It should be mentioned that in general it is not easy to check
the equi - integrability condition (B) in Theorem 1.45. We will use
the result in Chap. VI for generalized linear differential equations
where fortunately it is not difficult to show that the condition (B)
is satisfied.
CHAPTER II
ORDINARY DIFFERENTIAL EQUATIONS
AND THE PERRON INTEGRAL
The present approach to the concept of an ordinary differential
equation goes back to C. Caratheodory, in particular to his book
[14] published in 1918. In this work C. Caratheodory accomplished the construction of an analysis (calculus) course based
purely on the concept of the Lebesgue integral.
Given an ordinary differential equation of the form
x = f(x,t)
(2.1)
with f : B x [a, b] -+ R" where B C R" is an open set (e.
g.
B = B, = {x E R"; jjxIJ < c}).To solve it, in the classical setting
means:
find (if possible, all) functions x : J --> R' defined on a nonde-
generate interval J C [a, b] such that
x(t) E B for t E J
(2.2)
x is differentiable everywhere in J,
(2.3)
i.e. the derivative i(t) exists for every t E J and
i(t) = f (x(t), t) for every t E J.
(2.4)
A function x : J - R" satisfying (2.2), (2.3) and (2.4) is
called a solution of (2.1) and of course the properties are satisfied
69
70
II. ODE'S and the Perron integral
componentwise, i.e. if x = (X 1, ... , xn) then (1.3) means that all
xk, k = 1, ... , n are differentiable and (1.4) reads
xk(t) = fm(xl(t),...,x"(t),t) fort E J and k
where f," are the components of f = (fl, ... , fn).
Given a solution x : J - R" then
x(t) = x(a) +
j
f(x(s), s) ds
(2.5)
provided a, t E J. On the other hand, if x : J - R" is a function
satisfying (2.2) such that for some a E J the equality (2.5) holds
for t E J, then x is a solution of (2.1) provided f : B x [a, b] -Ht" satisfies some additional conditions. If e.g. f is continuous
then this conclusion holds, provided the integral in (2.5) is the
Riemann integral.
The starting point for Caratheodory's generalized approach to
ordinary differential equations of the form (2.1) is the integral
equation (2.5) where the Lebesgue integral is involved in (2.5).
The fundamental question of existence of a solution of the ordinary differential equation (1.1) is treated by C. Caratheodory as
the question of existence of a solution of the integral equation
(2.5) with the Lebesgue integral on the right hand side. By the
properties of the Lebesgue integral a function x : J --+ R" satisfying (2.5) is necessarily absolutely continuous in its interval of
definition because the indefinite Lebesgue integral has this property. Therefore it cannot be expected that a solution of (2.1)
in the sense of Caratheodory possesses a derivative everywhere
in its domain of definition. Generalized solutions to (2.1) are
absolutely continuous functions for which their derivative exists
almost everywhere with respect to the Lebesgue measure.
More precisely, a function x : J --> R", J C [a, b] being an
interval, is called a solution of (2.1) in the Caratheodory sense if
x(t) EBforalmostalltEJ,
If. ODE'S and the Perron integral
x is absolutely continuous on J
71
(x E AC(J))
and
th(t) = f (x(t), t) for almost all t E J,
or equivalently, if x(t) E B for almost all t E J and
t2
x(t2) - x(t1) _ (L)
J
f (x (s), s) ds
t,
for all ti,t2 E J.*
Caratheodory's proof of existence of a solution to the initial
value problem
x=f(x,t),x(a)=vEB
makes use of successive approximations. Assume that J = [a, a+
i7] C [a, b] where rl > 0. Define the approximations as follows:
cpi(t)=vfortE [a,a+
cpi(t)=v+J
' f(cpj(s),s)dsfortE(a+17,a+r1]
(2.6)
j=1,2, ...
Using this definition a sequence of continuous functions cps
:
[a, a + ,] -+ R" is defined. The idea of C. Caratheodory is to
take a subsequence of (cps) (denote it again by (pi)) which for
j -> oo tends to a limit y(t) for every t E [a, a + 9]. This limit
function y is a solution of (2.5) provided also
slim
00
J
f((s)s) ds = J f(y(s), s) ds
*(L) in front of the integral indicates that the Lebesgue integral is
considered.
72
II. ODE'S and the Perron integral
is satisfied. To this end a convergence result for the Lebesgue
integral has to be used and the Lebesgue dominated convergence
theorem is the right and sufficiently powerful tool for this case .
The Caratheodory assumptions are the following.
(Cl) f t) is continuous for almost all t E [a, b],
(C2) f (x, ) is measurable for x E B,
(C3) there is a Lebesgue integrable function rn(t) > 0, such
that for x E B and almost all t E [a, a + 77] the estimate
Ilf(x, t)II < rn(t)
holds.
The local version of the existence theorem for a solution of (2.1)
or (2.5) in the Caratheodory setting can be found in [18], [86] or
[23]. For local existence theorems the requirement x(t) E B for t
in the domain of definition of a solution x plays a role.
In this context a very natural problem arises. It is connected
with the possibility of using Perron's concept of the nonabsolutely
convergent integral in the integral equation (2.5). The question is
what concept of a generalized solution of (2.1) will be the result
of this approach. The fundamental problem of existence of a
solution of
x (t)
= x(a) + (P)
f
f (x(s), s) ds,
fort E[a,a+ri]C[a,b],11>0
should give the first information about the properties of a solution
and about the possibly most general right hand sides of (2.1).*
*We write (P) in front of the integral to emphasize that the Perron integration is used.
!!. ODE'S and the Perron integral
73
First, it is clear that when we are looking for a solution of (2.5)
then the Perron integral (P) fa f (x(s), s) ds should exist for every
t E [a, a +,q] and therefore any function satisfying (2.5) behaves
like the indefinite integral of a Perron integrable function. Hence
a solution of (2.5) (or of (2.1) in the Perron - Henstock sense)
is a function x : J - R" defined on some interval J on which it
is ACG* and has almost everywhere in J a derivative for which
x(t) = f(x(t),t) a.e. in J, because this are the properties of the
indefinite Perron integral.
In the book [40, Chap. 7, Sec. 19] R. Henstock is following the
lines of C. Caratheodory in deriving an existence result for the
integral equation (2.5). For the same reason as mentioned above
(for the case studied by C. Caratheodory with the Lebesgue integral), convergence results giving conditions for the possibility of
interchanging the limit and integral play an essential role. Therefore let us quote a convergence result from [40, Theorem 9.1] in
a simplified version suitable for the purposes of differential equations presented here.
2.1 Theorem. Let gj : [a, Q] - K, j = 1, 2.... be a sequence
of functions Perron integrable over [a, l3]. Let 6 : [a, /3] - (0, oo)
be a gauge on [a, /3] and
D = `ao,T1,a1,... ,ak-1,Tk,ak}
an arbitrary b-fine partition of [a, /3].
If
lim gj(t) = g(t)
j-+00
almost everywhere in [a,,8] and
k
a:-1 } < C
B<
i=1
H. ODE's and the Perron integral
74
for some real numbers B < C, all 6-fine partitions D of [a, ,Q] and
for all choices of positive integer-valued functions At), t E [a, Q]
then g is Perron integrable over [a, /3] and
(P)
r#
Ja
a
g(s) ds = lim (P)
I
gi(s) ds.
This convergence theorem for Perron integral is the basis for
the following existence result for the equation (2.5) given by R.
Henstock in [40, Theorem 19.1].
2.2 Theorem. Assume that f : R" x [a, b] ---* R" satisfies the
following conditions:
(H1) f t) is continuous for almost all t E [a, b],
(H2) the Perron integral(P) fQ f (z, s) ds exists for every z E
Rn
(H3) there is a compact set S C R" and a gauge b on [a, b] such
that f o r all S-fine partitions D = tao, -r1, a1, ... ,. rk, ak}
and all functions w : [a, b] -- R" we have
k
E f(w(r ), r)(ai - ai-1) E S
i=1
Then for every v E R", a E [a, b] there is a function y : [a, b] -+ R'
such that
y(t) = v + (P)
for t E [a, b).
j
f (y(s), s) ds
The proof of this theorem is postponed at this moment. It
should be mentioned that we assume here the function f (x, t)
being defined for all t E [a, b] and x E R". Therefore the existence
of a solution on the whole interval [a, b] is asserted in the theorem.
H. ODE's and the Perron integral
75
Let us pay attention to Henstock's conditions (H1) - (H3) for
the existence of a solution of (2.5). Clearly (H1) is the same as
Caratheodory's condition (Cl). Condition (H2) is not mentioned
explicitly in [40], nevertheless it is evidently used in the proof of
the result and its role is similar to that of Caratheodory's (C2). In
view of Theorem 2.1 (H3) is the condition that makes it possible
to interchange the order of limit and integral in the form
slim
(P)
00
R
Ja
f (w (s), s) ds = (P) IQ a f (slim c 3(s), s) ds
-00
fA
_ (P)J
provided the sequence of functions (p1(s)) converges with j ---> 00
pointwise to w(s) for s E [a, fl] and (H1) _ (Cl) holds. This
means that the role of (H3) in Theorem 2.2 is the same as that
of Caratheodory's condition (C3) that guarantees (2.6) with the
Lebesgue integral by the Lebesgue Dominated Convergence Theorem.
Now we will have a closer look at Henstock's conditions (H1)
- (H3) and at the functions f : [a, b] x R' --> R' for which these
conditions are satisfied.
2.3 Lemma. If f : R" x [a, b] -+ R' satisfies (H3) from Theorem 2.2 and v E R" is given then there exists a constant A > 0
such that
k
IIf(w(ri),ri)-f(v,r=)II(at -ai-i) <4nA
i=1
for every w : [a, b] --f R" and all 6-fine partitions
D = {ao,r11a1,...,ak-l,rk,ak}
(2.7)
11. ODE'S and the Perron integral
76
of [a, b] where the gauge b is given in (H3).
Proof. Since S from (H3) is a compact set in R" there is an
A > 0 such that S C [-A, A]" C R" where [-A, A]" is the
n-dimensional cube centered at the origin in R" with the edge
length 2A. Let
D = {ao, 71 , a1 , ... , ak-1, Tk, ak}
be an arbitrary b-fine partition of [a, b]. By (H3) we have
k
-A <
fn: (w(ri ), r1)(ai - a2-1) < A
i=1
for every function to : [a, b] --> R" and every m = 1, ... , n where
f," is the m-th component of the function f. For the special case
of the constant function w(t) = v, t E [a, b] we have
k
-A < > fm(v,Ti)(ai
A
i=1
and
k
-2A <E [f. (W(Tj), Ti)
i=1
- fm (VI Ti)](ai - ai-1) < 2A.
Denote
[fm(w(ri),r2)
- fm(v,ri)]+ = maX{fm(w(Ti),r) - fm(v,Ti),O}
and
[fm(w(ri),ri) - fm(v,Ti)] = min{fm(w(ri),Ti) - fm(v,Ti),0}
H. ODE'S and the Perron integral
77
If fm(w(Ti), Ti) - fm(v, r) > 0 then define z(T) = w(r) for T E
[ai-l,ai] and if fm(w(T;),T;)- fm.(v,Ti) < 0 then we set z(r) = v
for r E [ai_l, ai]. Evidently z((: [a(, b]( -- Rn and
fm(z(Ti), Ti) - fm(v, Ti) = [fm(W(Ti), ri) - fm(v, 7-01' >- 0.
Therefore
0<
E[fm(z(Ti),Ti) -
f-,(v,Ti)](ai - ai-1) =
i=1
k
E[fm(w(Ti ), Ti) - fm (v, Ti )]+(ai - ai-1) < 2A.
i=1
Similarly it can be shown that
k
-2A < E[fm(W(Ti), r) - fm(v, Ti)] (ai - ai-0< 0.
i=1
Since evidently
I fm(w(Ti),Ti) - fm(v,Ti)I =
[fm(w(ri), rs)
we obtain
- f. (V, TO) + - [fm(W(Ti),TO - fm(v,ri)]
k
Ifm(w(Ti), Ti) - fm(v, Ti)I(ai - ai-1) _
i=1
k
E[fm(w(Ti),Ti)
i=1
- fm(V,Ti)]+(ai - ai-0-
k
- E[fm(w(r
), Ti) i=1
fm(v,Ti)]-(ai
- ai-1) < 4A
for every m = 1, . . . , n and therefore (2.7) is satisfied.
0
II. ODE's and the Perron integral
78
2.4 Theorem. A function f : R" x (a, b] -+ R" fulfils (H1) (H3) from Theorem 2.2 if and only if
f(x) t) = g(t) + h(x,t)
(2.8)
for (x, t) E R" x [a, b] where
g : [a, b] --> at" is Perron integrable over [a, b]
(2.9)
and h : R" x [a, b] --> R" fulfils
t) is continuous for almost all t E [a, b],
(Cl)
(C2) h(x, ) is measurable for x E R",
(C3) there exists m : [a, b] -+ [0, oo) Lebesgue integrable over
[a, b] such that
Ilh(x,t)II < m(t)
for x E R" and almost all t E [a, b].
Proof. If f is of the form (2.8) with g and h satisfying (2.8)
and (Cl) - (C3) then evidently f satisfies (H1) and (H2) from
Theorem 2.2.
Since the integrals (P) f' g(s)ds, (L) fa m(s)ds exist, there is
a gauge b : [a, b] --> (0, oo) on [a, b] such that
k
rb
9(ri)(ai - ai-1) - (P)
k
m(r)(a- -) - (L)
9(s) ds'l < 1,
(2.10)
J m(s) dsj) < 1
(2.11)
J
II. ODE'S and the Perron integral
79
provided D = {ao, r1 i a1,... , ak_1, Tk, ak) is an arbitrary 6-fine
partition of [a, b]. For such a partition and an arbitrary w
[a, b] -- R" we have by (2.10), (2.11) and (C3) the inequality
k
f(w(Ti),T,)(ai - ai-1)II <
II
i=1
k
g(Ti)(ai - a;-1)Ii+
k
+ II E h(w(T;), Ti)(ai - ai-1 )II <
i=1
k
<
g(Ti)(ai
II
i
=1
II(P)
f
-
ai-1) - (P)
f
b
g(s) dsll+
a
k
b
g(s)dsll +
a
IIh(w(Ti),T;)II(ai
k
rb
1 + 11(P)
- ai-1) <
i=1
Ja
m(Ti)(ai - ai-1) <
g(s) dsll +
i=1
11(P) j b g(s) ds11 + j b
and this means that the sum
to the compact ball
Ek
m(s) ds + 2
f (w(r;),1Ti)(a; -ai_1) belongs
b
S = {x E R"; IIxII < 11(P) f g(s) dsll + f bm(s) ds + 2)
a
a
in R", i.e. (H3) is satisfied.
Let now f : R" x [a, b] -i R" fulfil (H1) - (H3) from Theorem
2.2. Assume that u E R" and set
g(t) = f (V, t) fort E [a, b],
II. ODE'S and the Perron integral
80
h(x,t) = f (x, t) - g(t) = f (x, t) - f (v, t) fort E [a,b], x E
Rn.
Then evidently (2.8) holds and by (H2) g is Perron integrable
over [a, b], i. e. (2.9) holds.
By Lemma 2.3 we have
k
(2.12)
JIh(w(Ti),T*)II(ai - ai-1) < C
i=1
with a constant C > 0 where w :
[a, b] -->
Rn and the 8-fine
partition D = {ao, T1, al, ... , ak-1 i rk, ak } of [a, b] are arbitrary.
The gauge 8 is given in (H3).
Put M(0) = 0 and
k
M(s) = sup{> 11h(w(ai),ai)II(Qi - fi-1)},
s E (a,b]
i=1
where the supremum is taken over all b-fine partitions
{$o,a],01,...,13 m-1,am,Qm}
of [a, s] and all functions w : [a, b] - R".
Let us show that if a < s 1 < s2 < b then
M(s1) < M($2)-
Assume that
{Yo, Pl,'Y1, ... ,71-1, PI,'ri}
is an arbitrary 6-fine partition of [Si, s2] and that
{oo,a1, 01,.,/9m-1,a,n,fm}
(2.13)
II. ODE'S and the Perron integral
81
is a 6-fine partition of [a, s1 ]. Then
00,al,01,.../3tn-1,QmiAn) U {-'0,PI,yl,... 71-1,P1771)
is a 6-fine partition of [a, 82 ] and
k
Il h(w(o=), ai)II()- Q=-1 )+
r
i=1
I
+
Il h(w(Pi),Pi)ll (7i - 7j-1) < M(s2)
j=1
Passing to the supremum over all b--fine partitions of [a, s 1 ] we
get
I
M(s1)+EIlh(w(P,),pj)ll(7i -7,i-1) <- M(s2).
.i=1
Further evidently
I
M(Ss) C M(s1) + E Ilh(w(P,i), Pj)Il (7,j
- 7,i-1) < M(s2)
j=1
and therefore (2.13) holds.
By (2.12) clearly
0 < M(s) < C,
S E [a, b].
(2.14)
Assume now that t E [a, b] and a < s1 < s2 < b where
t - b(t) < s1 < t < s2 < t + b(t).
(2.15)
Having an arbitrary 6-fine partition of [a, sl ] we add to it the
triple
{sl,t,s21
82
II. ODE'S and the Perron integral
and we obtain by (2.15) a 6-fine partition of [a, s21. Therefore
M(sl) + jjh(x,t)II(s2 - sl) < M(s2) for any x E R"
and consequently
jjh(x, t)jj(s2 - SO < M(s2) - M(sl ),
x E R.
(2.16)
Since M is monotone by (2.13) the derivative M of M exists
almost everywhere in [a, b]. Putting
"I(t) = M(t)
if the derivative k(t) exists and
m(t) = 0
otherwise, we get by (2.16) the inequality
1Ih(x, t)II < m(t)
for x E R" and almost all t E [a, b]. Except a set of Lebesgue
measure zero the function m is the derivative of the absolutely
continuous part of the nondecreasing function M and is therefore
Lebesgue integrable over [a, b] with (L) fa m(s) ds < M(b) < C.
Hence h satisfies (C3) and this concludes the proof.
0
2.5 Remark. Theorem 2.4 represents a surprising result which
states that Henstock's existence theorem 2.2 covers "only" the
case of Caratheodory functions perturbed by a Perron integrable
function, i.e. differential equations of the form
th = h(x, t) + g(t)
(2.17)
where h is a Caratheodory function satisfying (Cl) - (C3) and g
is Perron integrable over the interval at which we are looking for
solutions to (2.17) in the sense of the integral equation (2.5) , i.e.
for functions x : J -- 1R" such that
x(t) = x(a) + (P) jt h(x(s), s) ds + (P)
a
Theorem 2.4 was proved in [87].
r
Ja
t
g(s) ds.
II. ODE'S and the Perron integral
83
2.6 Lemma. If f : R" x [a, b] --; ]R satisfies conditions (HI)(1-13) from Theorem 2.2 and if cp : [a, b] - Rn is continuous then
(P) f f (cp(s), s) ds exists for every a < a < Q < b.
Proof. The function So is continuous on [a, b] and therefore it
is the uniform limit of a sequence t,ik of step functions. By (H2)
the Perron integral (P) fQ f (k (s), s) ds exists for every k E N
and the continuity condition (H1) yields
im
s) = f(W(s), s),
sE[a,b].
Using (H3) makes it apparent that the conditions of Theorem 2.1
are satisfied for every component of the function f and therefore
the
integral
(P) fa f (cpk(s), s) ds
and
also
(P) fa f (cp(s), s) ds exists for every a < a < Q < b.
Before proceeding with the proof of the existence result stated
in Theorem 2.2 we will prove one more lemma.
We define the Caratheodory approximations by
cpj(t)=vfortE[a,a+77
rrcpj(t) = v+(P)J
«
1f(pj(s),s)ds fort E (a+ l,a+y)]
.7
j = 1, 2, .. .
(2.18)
where f is the function from Theorem 2.2, a E [a, b) and y =
b - a. For t E [a, a + 41 the function cpj is evidently continuous.
Therefore by Lemma 2.6 the integral
(P) 1.
f(cp(s), s) ds
If. ODE'S and the Perron integral
84
exists and is continuous as a function of the variable t for t E
[a + ., a + 34], i.e. Vj is continuous on [a, a + ]. In this way
for every j = 1, 2.... the function cps is defined inductively on
the whole interval [a, a + q] = [a, b], and is continuous in this
interval.
2.7 Lemma. If f : [a, b] x R" --> R" satisfies conditions (HI)(H3) from Theorem 2.2 then the sequence of functions (Soj)J° 1
defined on [a, b), a < a < b contains a pointwise convergent subsequence.
Proof. Assume that
a+
is an arbitrary division of [a + i , b]. Using the formulas (2.18)
for every m = 1, ... , I we have
CM-4.
f(v, s) dsjl =
IIVJ(Cm) - Oj(Cm-1) - (P)
Cm_1f
Cm-1
= II(P)
f
Cm -
((s), s) ds - (P)
a
j
f (co (s), s) ds-
Cm
(P)
f(v, s) dsll =
Cm_1-
Cm-4
= II(P)
(f(v;(s), s) - f(v, s)) dsII.
(2.19)
Let 6 : [a, b] -- (0, oo) be the gauge on [a, b] given by the condition
(H3) in Theorem 2.2. Since the integral
Cm- L
(P)
Cm-1-
(f(c' (s),s) - f(v,s))ds
II. ODE's and the Perron integral
85
exists, there is a gauge b* with S*(t) < S(t) such that if
m
m
m
ff ,r1
m ,a1
mm
Dm= la0
,...,akm-1,Tkm,akm
km1
is a b* -fine partition of [cii-1 - 4, Cm - 4 ] then
Cm-
(f (V;(s), s) - f (v, s)) ds-
11(P)
Cm_1-
-
E(f(
Tm,WJTm)) - f(v,Tm)](ai - ai= 1)II
I.
i=1
Hence
Cm-'
11(P)
Cm_1-
(f(V,(s), s) - f(v, s)) dsll <
CmjT
< II(P)
C
m_1-il
(f(svs(s), s) - f(v, s)) ds-
km
- ,E[f(v7(Ti =),Tm) - f(v,Ti 1)](ain -aim-1)II+
i=1
km
[f (co,(Tm), Ti-) -
+II
f(v, Ti l)](ai l - am 1)II
i=1
and using (2.19) we get
Cm-
Ilvj(Cm) - V,(cm-1) - (P)
f(v, s) dsll <
Cm _
1
km
< 1 + II >[f(co,(Tm),Tm) - f(v,Tm)](am - am 1)II
i=1
11. ODE'S and the Perron integral
86
for every m = 1, ... ,1. Therefore
Csn-
(Pj(Cm-1) - (P)
m=1
C
I
f(v,s)dsll <
.-
k,,,
IIff(wj(Ti `),T;") - AV, T'
<1+
ai `
m=1 i=1
Since the union of the above mentioned partitions Dm of the
intervals [c,n_1 - 41 Cm - ] is evidently a 6-fine partition of [a, b-
Lemma 2.3 can be used for w = p and by the last inequality
we obtain the estimate
f
CM-1-'
E II (C,n) -Vl(Cm-I) `(P)
f(v,s)ds11 < 1 + 4nA
M=1
(2.20)
< ct of the interval [a +
for every division co < c1 <
Hence
vary+ (cpj(t) - (P) J
f (v, s) ds) < 1 + 4nA
for every j. Put
T/,j(t)=v fort E [a,a+'],
J
It ?
O, (t) = (.oj(t) - (P)
f (v, s) ds for t E (a + 77,b]
b]
j = 1, 2, .. .
Then
vary b, = var« ' b + vary+ zfij < 1 + 4nA
?
4,
b].
87
11. ODE'S and the Perron integral
by (2.20) and evidently also
IIl)(t)II < IIO)(a)II + II')(t) -')(a)II < IIvII + var, i) <
IIvII + 1 + 4nA.
Since the sequence (t)) is equibounded with uniformly bounded
variation Helly' s Choice Theorem ( see e.g. [104]) yields that
there exists a subsequence of (0j), we denote it again by (0j),
which is pointwise convergent, i.e.
slim O (t)
00
= 'li m(cpj(t) - (P) J
f(v,s)ds),
t>a
lim Oi(a) = lien cp)(a) = v.
)- 00
i- 00
Since the indefinite Perron integral is continuous, we have
lim(P)J
a
f(v,s)ds = (P) J f(v,s)ds
a
and therefore lim yp)(t) exists and equals some value V(t) for
)-00
0
every t E [a, b).
Proof of Theorem 2.2. Let a E [a, b) be given and assume that
the sequence of functions c p ) : [a, b] -- R" , j = 1, 2, ... is defined
by the relations (2.18). By Lemma 2.7 there is a subsequence of
this sequence, which we denote again by (W)), such that
lim cp)(t) = W(t)
3-00
for every t E [a, b].
Since by (H1) f (t, .) is continuous for almost all t E [a, b], we have
lim f (cp)(t), t) = f (w (t), t)
00
almost everywhere in [a, b].
H. ODE'S and the Perron integral
88
Further, the integral
rb
(P)J f(W1(s),s)ds
exists for every j = 1, 2, ... by Lemma 2.6 because the functions
cpj : [a, b] -> I8" are continuous. Finally, (H3) implies that for
every 1 = 1, ... , n we have
k
-A <
),r))(ai - a;-1)
A
:=1
for an arbitrary j : [a, b] --> N and an arbitrary b-fine partition
{ao,7j,al,...,0k-1,Tk,ak}
!jlim
of [a, t].
It remains to show that
t(P)
00
Z
f (s, cp (s)) ds = (P)
J.
f (cp(s), s) ds.
By Henstock' s Theorem 2.1 we have
j-.
t
ft
lim (P)
Ja
f (cp(s), s) ds = (P)
Ja
f (cp(s), s) ds.
Since
t
(P)1a
_ (P)
f(v1(s), s) ds =
Ja f(cpj(s),s)ds - (P) J
t
t
f(wj(s),s)ds
(2.21)
89
II. ODE'S and the Perron integral
it is now sufficient to show that
lim00 (P)
f-
f (cp;(s), s) ds = 0.
(2.22)
At this moment we use the result of Theorem 2.4 which states that
under the given condition we have f(Vi(s), s) = g(s)+h(cpj(s), s)
where g is Perron integrable and h is a Caratheodory function
with IIh(x,s)II < m(s). Hence
t
(P)
(P)
t?
_
Jt?
h(Vi(s), s) ds.
For this expression we have
rt
jlim (P)
00
Jt-?
g(s) ds = 0
because the indefinite integral of a Perron integrable function is
continuous, and
II (P)
t
Jt-
h(Vj(s), s) dsll
(P)
t
Jc-
II h(Vj(s),s)II ds <
t
< (P)
m(s) ds - 0
for j --- oo for the same reason. Therefore (2.22) holds and (2.21)
is true.
H. ODE'S and the Perron integral
90
Now, taking the limit for j - oo of the Caratheodory approximations given by (2.18) we obtain
t
Si(t) = v+(P)J f(V (s),s)ds,
t E [a,b],
a
i.e. cp is by definition a solution of (2.1) in the Perron sense on
[a, b].
It is straightforward to define the Caratheodory approximations for the interval [a, a] similarly as in (2.18) and repeat the
above procedure for examining the existence of the desired solution on [a, a] . Since both solutions coincide at a Theorem 2.2 is
0
proved.
2.8 Remark. The proof of Theorem 2.2 follows the lines of
Henstock's proof from [40]. It should be pointed out that the
relation (2.22) is used but not mentioned explicitly in Henstock's
proof and it is not very clear how to derive it without using
Theorem 2.4 to this end as we have done above.
2.9 Remark. Let us consider again the situation in which we
are after having proved Theorem 2.2.
Denote by Car the class of functions f : [a, b] x R" - Rn
satisfying conditions (Cl), (C2) and (C3) and let 'Hen be the
class of functions f : [a, b] x R' -+ It" for which (H1), (H2) and
(H3) are satisfied.
A function x : [c, d] -> IR" is said to be a solution of the
ordinary differential equation
x = f(x,t)
in the Caratheodory sense on the interval [c, d] C [a, b] if
x(t) = x(a) + (L)
j
f (x(s), s) ds
(2.1)
H. ODE'S and the Perron integral
91
for every a, t E [c, d] and x : [c, d] --4R" is called a solution of
the ordinary differential equation (2.1) in the Perron - Henstock
sense if
e
x(t) = x(a) + (P) J f(x(s), s) ds
for every a, t E [c, d].*)
By the well known relation between the Lebesgue and Perron
integrals it is clear that if x : [c, d] -+ Rn is a solution of (2.1) in
the Caratheodory sense then it is a solution of (2.1) in the Perron
- Henstock sense as well.
Moreover, Theorem 2.4 shows not only that Car C Hen but it
gives the formal relation
Hen = P + Car
where by P the set of Perron integrable functions g : [a, b] --> Rn
is denoted. This relation means that f E Hen if and only if f =
g + h where g E P and h E Car, and it contains the information
that the Henstock - Perron concept of a solution of the ordinary
differential equation (2.1) is more general than the concept of
Caratheodory provided (H1) - (H3) is fulfilled. It shows also how
far the generalization goes with respect to the possible choices of
right hand sides of an ordinary differential equation.
It is worth mentioning here that Henstock's conditions (H1),
(H2) and (H3) are not the only ones under which the Perron Henstock notion of a solution of the ordinary differential equation
(2.1) is studied. For example in [12] a result of P. Bullen and R.
Vyborny [13] is mentioned where the following conditions are used
for f :Rx[a,b]-+ R:
(B1) f
t) is continuous for almost all t E [a, b],
*)We write (L) , (P) in front of the integral to emphasize that the Lebesgue or Perron integration is used.
H. ODE'S and the Perron integral
92
(B2) for all continuous ACG* functions Sp
:
[a, b] -- R the
function f (ap(t), t) : [a, b] --+ R is measurable ,
(B3) there exist continuous M, p : [a, b] --4R, M(a) = µ(a) _
0 for some a E [a, b] such that for all continuous and
ACG* functions cp : [a, b] --+ R we have
DM(t) > APO), t) > Dp(t)
(D, D stand for the lower and upper derivative, respectively.)
It is clear that the conditions (B1) - (B3) have to be modified
for R"- valued functions f : R' x [a, b] -+ R" in assuming them
for the components of the function f.
It turns out that if a function f satisfies (B1) - (B3) then
f E P + Car = 1-len. Indeed, define g(t) = f (v, t) for some v E R
and t E [a, b] and put h(x, t) = f (x, t) - g(t) for x E R, t E [a, b].
The function g is measurable by (B1) and since by (B3) we have
DM(t) > g(t) > Dp(t)
the function g is Perron integrable by P. Bullen's result given in
[12].
By (B3) we further have
Dp(t) - DM(t) < f(x, t) - f (v, t) = h(x, t) < DM(t) -
y(t),
i.e.
0 < IIh(x, t)I1< DM(t) - Dp(t) < D[M(t) - p(t)J,
t E [a, b].
(2.23)
Since M and p are major and minor functions to g, respectively,
the difference M - p is nondecreasing in [a, b] and therefore the
derivative de (M - p) exists almost everywhere. Defining
m(t) = dt (M - µ)
IL ODE'S and the Perron integral
93
for t E [a, b] at which the derivative exists and
m(t) = 0
elsewhere, we have by (2.23) the inequality
h(x, t) < m(t) for x E R and almost all t E [a, b].
The function m is nonnegative and integrable and therefore h
satisfies Caratheodory's condition (C3). It is also clear that (B1)
and (B2) imply respectively (Cl) and (C2) for the function h and
this yields that if f satisfies Bullen's conditions (B1) - (B3) then
f E P + Car = ?-fen, i.e. Bullen's case includes the same class
of ordinary differential equations as Henstock's. On the other
hand, it is a matter of routine to show that f satisfies (B1) (B3) whenever f E P + Car. This means that for the class 13u1
of functions f satisfying (B1) - (B3) we have
13u1= P + Car =Hen,
i.e. all the aproaches mentioned above concern the same class of
equations.
Finally, let us turn our attention to another concept described
by T.S. Chew and F. Flordeliza in [16]. They consider the class
of functions f : R x [a, b] --> R such that
(Cl) f t) is continuous for almost all t E [a, b],
(C2) f (x, ) is measurable for every fixed x,
(C-F) there exist two Perron integrable functions g, r on [a, b]
such that for
g(t) < f (x, t) < r(t),
t E [a, b], x E R.
11. ODE'S and the Perron integral
94
Put in this case h(x, t) = f (x, t) - g(t). Then h evidently satisfies
(Cl) and (C2) and by (C-F) we have
0 < f(x,t) - g(t) < r(t) - g(t)
0 < h(x, t) < m(t)
(2.24)
where m(t) = r(t) - g(t) is a nonnegative Perron integrable function and therefore m is also Lebesgue integrable. h is nonnegative
and this yields by (2.24) that h satisfies Caratheodory's condition
(C3). In this way we have again that conditions (Cl), (C2) and
(C-F) for f imply f E P + Car, i.e. the case described by T.S.
Chew and F. Flordeliza in [16] leads to equations of the form
(2.17).
The existence theorem 2.2 for Perron - Henstock solutions of
(2.1) was presented in a global form by assuming that the right
hand side f is defined for t E [a, b] and x E R". Now we present
the local version of the existence theorem.
2.10 Theorem. Assume that the function f is defined on a
set
x [a-A,a+A]
Q
with some P > 0 and that f(x, t) = g(t) + h(x, t) for (x, t) E Q
where
g :[a - A, a + A] ---* R'
is Perron integrable,
and
t) is continuous for almost all t E [a - A, a + A],
(2) h(x, ) is measurable for
(1)
x E B(v, e) = {y E R"; Iiy
- vjj < e}
If. ODE'S and the Perron integral
95
(3) there exists m : [a - A, a + A] -- R Lebesgue integrable
over [a - A, a + A] such that
IIh(x,t)II < m(t)
for x E B(v, o) and almost all t E [a - A, a + A].
Then there exists a S > 0, S < a such that on the interval
[a - S, a + 6] there exists a Perron - Henstock solution x(t) of
(2.1) such that x(a) = v.
Proof. The indefinite integral
G(t) _ (P)
rt
J
g(s)ds,
t E [a -A, a+ A]
being an ACG* function is continuous at t = a and the same
is true for the indefinite integral M(t) = fa m(s)ds because it is
absolutely continuous. Hence there is a S > 0 such that IIG(t)II +
M(t)I < P for every t E [a - S, a + S].
For t E [a, a + S] define the Caratheodory approximations
b
cpj(t) =v fort E [a,a+-],
=v+(P)g.(s)ds+
(2.25)
rt
+ (P)
J
h(cpi (s), s) ds
fort E (a +
,a+
S]
j=1,2, ...
as above in (2.18) for the proof of Theorem 2.2. Since by Theorem
2.4 the assumptions of Theorem 2.2 are satisfied and
Ilcp.i(t)-vII S
=
IIG(t-bG(a)II+IM(t-bM(a)I =
<P
(2.26)
11. ODE'S and the Perron integral
96
holds for every t E [a, a + b] we conclude from Theorem 2.2 and
Lemma 2.7 that there exists a subsequence of (Spy) (we denote
it again by (Vi)) which converges pointwise to a certain function x(t) on [a, a + 6]. Using the continuity of the function h
in the second variable and the Lebesgue Dominated Convergence
Theorem we obtain
1 rn(P)
rt-
t
s) ds = (P)
J«
Ja
s) ds.
(The functions h(s, cp(s)) are dominated by m on [a, a + b].)
Passing to the limit for j -' oo in (2.25) we obtain
x(t) = v + (P)
J
t
g(s) ds + (P)
J
t
h(x(s), s) ds
for t E [a, a + b] and this means that x : [a, a + b] -- R" is a
solution of (2.1) in the Perron - Henstock sense because by (2.26)
evidently
IIx(t) - vfl < Lo,
t E [a, a + b],
i.e. x(t) E B(v, e) for every t E [a, a + b].
For t E [a - b, a] the same can be proved by defining the
appropriate Caratheodory approximations and this completes the
proof.
2.11 Remark. Theorem 2.10 gives the local existence of a Perron - Henstock solution of the ordinary differential equation (2.1)
with the initial condition x(a) = v. This result has the same form
and essentially also the same proof as the well-known local existence theorem for Caratheodory solutions when the right hand
side f of (2.1) satisfies conditions (C1),(C2) and (C3) locally (see
e.g. Theorem 2.1 of Chap. II in [18]). The proof of Theorem
2.10 differs only slightly from the proof of the Caratheodory local
II. ODE'S and the Perron integral
97
existence theorem because of the special form g(t) + h(x, t) of the
right hand side of (2.1). Since we have the benefit of the result
given in Theorem 2.4 we can claim that the local existence of a
Perron - Henstock solution of an initial value problem for the
equation (2.1) is guaranteed also in the general case of equations
with right hand sides satisfying Henstock's conditions (H1), (H2)
and (H3) given in Theorem 2.2.
If f : G --> R" where G C R" x R = R"+' is an open set then
for every (a, v) E G there exist A, ,o > 0 such that
RA,e= {yER";IIy-vjj <to} x [a-A,a+A]CG.
Assuming that the assumptions of Theorem 2.10 are satisfied for
every such RA,,, we can use Theorem 2.10 to state that for every
initial point (to, x) E G there exists locally a Perron-Henstock
solution of (2.1).
Example. A typical equation which cannot be treated in the
frame of Caratheodory's approach to ordinary differential equations is the following.
Define F(t) = t2 sin
_ for t 54 0, F(0) = 0. Then the derivative
F(t)= f(t)=2tsin
-
cos=21
t#0,
F(0) = f(0) = 0
exists. Since f (t) is not Lebesgue integrable, for the differential
equation
x = f(t),
t E [-1, 1]
the Caratheodory existence theorem does not hold. Nevertheless,
x(t) = F(t) + c, where c is a constant, is evidently a Perron Hensto'k solution of this equation. Even without knowing this,
Theorem 2.10 applies to this case and gives the local existence of
such a solution.
98
H. ODE's and the Perron integral
Another simple example based on this classical Perron integrable function f is the linear ordinary differential equation
x= ax + f(t),
t E [-1, 1],xElit
Its general solution has the form
fe
x(t) = C eat +
eat
e-a' f (s) ds,
t E [-1, 1]
J0
where the integral on the right hand side exists when treated as
the Perron integral.
2.12 Remark. In the book [23] the notion of a solution of the
ordinary differential equation (2.1) was weakened to the following:
find a function x : J -+ Rn defined on a nondegenerate interval
J C R such that (x(t), t) belongs to the domain G of the function
f, x is continuous on J, almost everywhere differentiable in J
and x(t) = f (x(t), t) for almost all t E J.
This concept is more general than the concept of Caratheodory; absolute continuity is not required for the solution. In [23]
it is pointed out that if x is a function satisfying
x (t)
= x(a) + (P)
j
f (x(s), s) ds
for a, t E J then it is a solution of (2.1) in the above mentioned
weakened sense. The fundamental existence and unicity results
for such solutions of (2.1) are given in [23] for the case when f
satisfies (H1) and (H2) from Theorem 2.2 provided the Lipschitz
condition
- f(y, t)II < L(t)fIx - yll
is satisfied locally in the domain of f with L integrable in the
Lebesgue sense. The results in [23] concerning solutions in the
11f (x, t)
weakened sense are reduced to the above mentioned case of solutions in the Perron-Henstock sense.
CHAPTER III
GENERALIZED ORDINARY
DIFFERENTIAL EQUATIONS
Let an open set G C R"+' be given. Assume that F : G --,
R"+' is a given R"+1-valued function F(x, t) defined for (x, t)
G,xER",tER.
3.1 Definition. A function x : [a,,0] - R" is called a solution
of the generalized ordinary differential equation
dx = DF(x,t)
dr
(3.1)
on the interval [a, /3] C R if (x(t), t) E G for all t E [a, /3] and if
32
x(s2) - x(s1) =
J
DF(x(r),t)
(3.2)
holds for every sl, s2 E [a, /3] the identity.
The integral on the right hand side of (3.2) has to be understood as the generalized Perron integral introduced by Definition
1.2n in Chapter I.
3.2 Remark. Let us mention that the notation (3.1) is symbolical only. The letter D indicates that (3.1) is a generalized
differential equation, this concept being defined via its solutions.
Even the symbol dx does not mean that the solution has a derivative.
99
III. Generalized ODE'S
100
For example if r : [0, 1] - R is a continuous function for which
its derivative does not exist at any point of the interval [0, 1], then
we can define F(x, t) = r(t). In this case we evidently have
9,
JSi
82
DF(x(r), t) =
Jt
Dr(t) = r(s2) - r(si)
by the definition of the integral, and this means that the function
x : [0, 1] --+ R given by x(s) = r(s) for s E [0, 1] is a solution
of the generalized differential equation dT = DF(x, t) = Dr(t)
though it has no derivative at any point in [0,1].
From this point of view a generalized ordinary differential
equation is a formal equation-like object for which we have defined its solutions. The definition of the solution given by the
integral relation (3.2) reminds us of the concept of a solution of
an ordinary differential equation given for example in Chap. II .
3.3 Remark. Assuming that F(x, r, t) is defined for (x, T, t) E
H C R"+2, where H C R"+2 is such that if (x, s, s) E H then
there is a 9 > 0 such that (x, s, t) E H whenever It - sI < 0, we
can introduce the solution. of the generalized differential equation
dx
dr
= DF(x, r, t)
(3.3)
similarly as in Definition 3.1.
3.4 Definition. A function x : [a,#] -+ R' is called a solution of the generalized ordinary differential equation (3.3) on the
interval [a, /3] C R if (x(t), t, t) E H for all t E [a, /3] and if for
every sl, s2 E [a,#] the identity
32
x(s2) - x(s1) =
holds.
J
DF(x(r), T, t)
101
III. Generalized ODE'S
3.5 Proposition. If a function x : [a, /3] --, 1R" is a solution of
the generalized ordinary differential equation (3.1) on [a, /3] then
for every fixed y E [a, /3] we have
X(S) = x(-y) + J DF(x(r),t),
s E [a,
(3.4)
7
If a function x : [a, /3] -> 1R" satisfies the integral equation
(3.4) for some -y E [a, Q] and all as E [a, 0] then x is a solution of
the generalized ordinary differential equation (3.1).
Proof. The first statement follows directly from the definition
of a solution of (3.1) when we put Si = y and 32 = s.
If x : [a,#] --* R' satisfies the integral equation (3.4) then by
the additivity of the integral we have
x(s2) - x(sl) =
/ 82
= x(y) + J
12
DF(x(r), t) - x(y) -
7
J
DF(x(r), t) _
'Y
(32
=
J
DF(x(r), t)
81
for every S1, s2 E [a, /3], and x is a solution of (3.1).
0
Definition 3.1 itself does not provide any information about
the properties of the function x [a, 0] -* R' which is a solution
of (3.1) in the sense of Definition 3.1. The only fact we know
is that in this case the integral f DF(x(r), t) exists for every
s1, s2 E [a, /3]. By Proposition 3.5 we know that a solution x of
(3.1) behaves like the indefinite generalized Perron integral of a
function integrable in this sense. Using Theorem 1.16 from Chap.
I the following result can be deduced.
III. Generalized ODE'S
102
3.6 Proposition. If x : [a, /3] -' R" is a solution of the generalized ordinary differential equation (3.1) on [a, /3] then
lim [x(s) - F(x(u), s) + F(x(a), a)] = x(a)
S -.O
(3.5)
for every a E [a, /3].
Proof. Let a E [a, /3] be fixed. Then by Proposition 3.5 we
have
rS
x(s) - J DF(x(r), t) = x(a)
0
and therefore also
x(s) - F(x(a), s) + F(x(a), a)/'s
- J DF(x(rr), t) + F(x(a), s) - F(x(a), a) - x(a) = 0 (3.6)
s
for every s E [a, $].
By Theorem 1.16 we get
[j5
li m
DF(x(rr), t) - F(x(a), s) + F(x(a), a)J = 0
J
and this together with (3.6) yields the existence of the limit
lim,., [x(s) - F(x(a), s) + F(x(a), a) - x(a)] as well as the relation
m [x(s) - F(x(a), s) + F(x(a), a) - x(a)] = 0
which is equivalent to (3.5).
0
3.7 Remark. By virtue of Proposition 3.6 the following assertion is true.
III. Generalized ODE'S
103
If x : [a, /3] - R" is a solution of (3.1) then for every v E [a, /3]
the value of x(s) can be approximated by x(a) + F(x(o), s) F(x(a), a) provided s E [a,#] is sufficiently close to or.
Now we introduce a class of functions F : G - R" for which
it is possible to get more specific information about the solutions
of (3.1).
Given c > 0, we denote
Bc = {x E R"; IIxII < C}.
Let (a, b) C I8 be an interval with -oo < a < b < oo and let us
set
G = B, x (a, b).
We will use the set G C R"+1 in our subsequent study of generalized differential equations (3.1). Assume that h : [a, b] -- R is a
nondecreasing function defined on [a, b] and that w : [0, +oo) -> R
is a continuous, increasing function with w(0) = 0.
3.8 Definition. A function F : G -p R" belongs to the class
,F(G, h, w) if
II F(x, t2) - F(x, tl )II < I h(t2)
- h(t1)I
(3.7)
for all (x, t2 ), (x, tl) E G and
IIF(x,t2)-F(x,tl)-F(y,t2)+F(y,tl)II <
< w(II x - yiI)Ih(t2) - h(tl )I
for all (x,t2), (x,tl), (y,t2), (y,t1) E G.
(3.8)
III. Generalized ODE'S
104
3.9 Lemma. Assume that F : G -+ K" satisfies the condition
(3.7). If [a, /3] C [a, b] and x : [a, /3] - K" is such that (x(t), t) E
G for every t E [a,,81 and if the integral fp DF(x(r), t) exists,
then for every sl, s2 E [a, /3] the inequality
82
j
DF(x(r), t) < I h(s2) - h(si )I
(3.9)
81
is satisfied.
Proof. Using (3.7) we obtain
it - rI.II F(x(r), t) - F(x(r), 7-)II : (t - 7-)(h(t) - h(r))
for any r, t E [a, /3]. The integral fq dh(t) exists and
82
j
dh(t) = h(s2) - h(si)
for every s1, s2 E [a, /3]. Therefore (3.9) is an immediate consequence of Corollary 1.35.
3.10 Lemma. Assume that F : G --> IR" satisfies the condition (3.7). If [a, /3] C [a, b] and x : [a, l3] - R" is a solution of
(3.1), then the inequality
IIx(sl) - x(s2)II < I h(s2) - h(sl)I
(3.10)
holds for every sl , s2 E [a,#] .
Proof. The result follows directly from Lemma 3.9 if we take
into account that by definition we have
f82
x(s2) - x(sl) =
J
91
for every Si, s2 E [a, /3].
DF(x(r), t)
III. Generalized ODE'S
105
3.11 Corollary. Assume that F : G ---' R' satisfies the condition (3.7). If [a, /3] C [a, b] and x : [a, /3] - R" is a solution of
(3.1) then x is of bounded variation on [a,,8] and
vara < h(/3) - h(a) < +oo.
(3.11)
Moreover, every point in [a, /3] at which the function h is continuous is a continuity point of the solution x : [a, /3] -> R".
Proof. Let a = so < 51 <
< sk = /3 be an arbitrary
division of the interval [a,#]. By (3.10) we then have
k
k
I h(si) - h(sj-1I < h(/3) - h(a).
JIx(sj) - x(sj-111
.i=1
i=1
Passing to the supremum over all divisions of [a,#] we obtain
(3.11). The second statement is a consequence of the inequality
(3.10).
3.12 Lemma. If x : [a, l3] -i R" is a solution of (3.1) and
F : G - R" satisfies the condition (3.7) then
x(s+) - x(s) = of m x(a) - x(s) = F(x(s), s+) - F(x(s), s)
(3.12)
forsE[a,/3)and
x(s) - x(s-) = x(s) - of m x(a) = F(x(s), s) - F(x(s), s-)
(3.13)
for s E (a, /3] where F(x, s+) = limo,S+ F(x, v) for s E [a, /3)
and F(x, s-) = limo.._ F(x, o) for s E (a, /3].
Proof. First of all note that the limits F(x, s+) and F(x, s-)
exist because the function h, being a function of bounded variation, possesses onesided limits at every point of [a, /3] C [a, b] and
111. Generalized ODE'S
106
(3.7) quarantees the Bolzano-Cauchy condition for the existence
of the corresponding onesided limits of the function F(x, a)
[a, ,D] -> IR" for every x c- B,.
For a<s<c </9wehave
x(0) - x(s) =
J
DF(x(r); t)
a
by the definition of a solution of (3.1). By Theorem 1.16 we have
ff1 m
x(v) - x(s) = 1 m J DF(x(r),t)
$
= lim [F(x(s), a) - F(x(s), s)]
c-s+
and (3.12) holds. The relation (3.13) can be proved analogously.
0
3.13 Remark. The above statements describe the properties of
solutions of the generalized differential equation (3.1) whenever
the "right hand side" F satisfies (3.7) from the definition of the
class F(G, h, w). Of course this are merely preliminary results
because at this moment we have not the fundamental information about the existence of a solution. Nevertheless, we have a
description of the "candidates" for a solution.
Now we present a result concerning the existence of the integral involved in the definition of the solution of the generalized
differential equation (3.1).
3.14 Theorem. Assume that F E .F(G, h, w) is given and
that x : [a, /9] - R", [a, /3] C (a, b) is the pointwise limit of
a sequence (xk )kEN of functions xk : [a, /31 --+ R" such that
(x(s), s) E G ,(xk(s), s) E G for every k E N and s E [a,#] and
III. Generalized ODE'S
107
that f ! DF(xk(r), t) exists for every k E N. Then the integral
fa DF(x(r), t) exists and
DF(x(r), t) = hm 1.9iDF(xk(r), t).
Proof. According to Theorem 1.6 it can be assumed without
any loss of generality that F is a real valued function. Assume
that E > 0 is given. By (3.8) we have
I F(xk(r), t2) - F(xk(r), t1) - F(x(r), t2) + F(x(r), tl )I <
< w(Ilxk(r) - x(r)II)(h(t2) - h(t1))
(3.14)
for every r E [a, /3], t1 < r < t2, [t1, t2] C [a, /3]. Let us set
µ(t) = h(A) - h(a) + 1 h(t) for it E [a,#] .
The function µ : [a, /3] -- R is nondecreasing and µ(/3)-µ(a) < E.
Since
kllmoxk(r) = x(r)
for every r E [a, /3] and the function w is continuous at 0, there
is a p(r) E N such that for k > p(r) we have
E
w(Ilxk('r) - x(r)II) < h(/3) - h(a) + 1'
for k > p(r) the inequality (3.14) can be rewritten to the
i.e.
form
IF(xk(r),t2)
- F(xk(r),tl) - F(x(r),t2) + F(x(r),t1)I <
III. Generalized ODE'S
108
< µ(t2) - µ(tl )By (3.7) we have
-h(t2) + h(tl)
F(xk(r),t2) - F(xk(r),tl) <
< h(t2) - h(t1)
for every r E [a Q], k E N and r E [a, /3}, tl < r < t2, [t1, t2] C
The integrals f D[h(t)] = h(/3) - h(a), f' D[-h(t)] _
h(a) - h(P) obviously exist. Therefore by Corollary 1.31 the
integral fa DF(x(rr), t) exists and the conclusion of the theorem
holds.
3.15 Corollary. Assume that F E .F(G, h, w) is given and
that x : [a,,Q] -* R", [a,,Q] C (a, b) is the pointwise limit of a
sequence of finite step functions such that (x(s), s) E G for every
s E [a,,8]. Then the integral f DF(x(r), t) exists.
Proof. By the previous Theorem 3.14 it is sufficient to prove
that the integral f'8 DF(cp(r), t) exists for every finite step function cp : [a,#] -- Rn. If cp is a finite step function then there
is a division a = so < s1 <
< sk = ,Q of [a,,61 such
that W(s) = cj E R' for s E (sj_l, sj), j = 1, ... , k where
cj, j = 1, ... , k are constants. Assume that (w(s), s) E G for
every s E
By the definition of the integral it is easy to see that if s,_, <
t) exists and
O'1 < 02 < sj then the integral
aZ
ZI
DF(cp(r), t) = F(ci, u2) - F(c,, a1).
Assume that oo E (3,-i, sj) is given. We have
rso
lim
[J
S
DF(cp(r), t)+
III. Generalized ODE'S
109
+F(co(s.i-1), s) - F(cp(si-1), si-1)] _
lim
S-sj_1+
[F(c,, ao) - F(cj, s) +
s)-
-F(cp(s.i-1), s'-1)] =
= F(cj, oo) - F(ci, si-i +)+
+F((o(s1-1), sa-1 +) - F(cp(si-1), si-1).
Hence by Theorem 1.14 (see Remark 1.15) the integral
DF(V(r), t) exists and equals the above computed limit.
Similarly it can be shown that also the integral fro DF(cp(r), t)
exists and
Si
DF(V(r), t) = F(c,i, s; -) - F(c;, oo )£0
f °p
9J
1
-F(cp(si), si-) + F(cp(sj ), si)
Therefore by Theorem 1.11 we obtain the existence of the integral
DF(p(r), t) and
DF((p(r), t) _
aj_1
= F(c.i, sj -) - F(c,i, co) - F(V(si ), sj -) + F(co(s.i ), s1)+
+F(cj, ao) - F(cj, s,i-1 +) +
+F(V(s,i-i ), si-1 +) - F(cp(si_i ), s,i-i) _
= F(cl, s,-) - F(c,, s,i-i+) + F(p(si-i ), s'-i+)-F(a(s,-i ), sj-1) - F(V(s.i ), sj-) + F(V(si), si ).
III. Generalized ODE's
110
Repeating this argument for every interval [sj_1, sj], j = 1, ... , k
and using again Theorem 1.11 we obtain the existence of the
integral f' DF(cp(r), t) as well as the identity
k
DF(cp(r),t) = E[F(cj,sj-) - F(cj,sj_1+)]+
fa0
j=1
k
si-1+) - F((p(sj-1), sj-1)-
+
j=1
-F(cp(sj), sj-) + F(co(sj), sj)]
This completes the proof.
3.16 Corollary. If F E .F(G, h, w) is given and x : [a,#] ->
Rn, [a, /3] C (a, b) is a regulated function (in particular a function
of bounded variation) on [a, /3] such that (x(s), s) E G for every
s E [a, p] then the integral fa DF(x(r), t) exists.
Proof. The result follows from Corollary 3.15 because every
regulated function is the uniform limit of finite step functions
(see e.g. [6]).
3.17 Remark. Corollary 3.16 implies that the class of functions x : [a,#] -> R", [a, Q] C (a, b) for which the integral
fa DF(x(r), t) exists is sufficiently large provided
F E .F(G,h,w). At least it is clear that ff DF(x(r),t) exists
for every function which can be expected to be a solution of the
generalized differential equation (3.1) with F E .T(G, h,w) (see
Corollary 3.11).
3.18 Remark. In a more general setting it can be assumed
that G C R"+1 is an open set and that the function F : G -i Rn
locally satisfies the requirements given by the properties of the
III. Generalized ODE'S
111
class .F(G, h, w). More precisely, if (x, t) E G then there exists
A > 0 such that
G= {xER";((x-il'<L) x(t-A,i+O) C G
and F satisfies on G the conditons (3.7) and (3.8) from Definition
3.8 with some functions h and w, and of course these functions
can be specific for the set G.
CHAPTER IV
EXISTENCE AND UNIQUENESS OF SOLUTIONS
OF GENERALIZED DIFFERENTIAL EQUATIONS
Let us consider the generalized ordinary differential equation
dx
dT
= DF(x, t)
described in Definition 3.1 in the previous chapter for the case
when F : G -+ R" belongs to the class F(G, h, w) (see Definition
3.8).
We assume here that G = B, x (a, b) where B, _ {x E
Rn; IIx II < c}, c > 0, (a, b) C R -oo < a < b < +oo and
h : [a, bJ - R is a nondecreasing function defined and continuous
from the left on [a, bJ, w : [0, +oo) -* R is a continuous, increasing
function with w(O) = 0. Let us mention that in addition to the
assumptions from Chap. III we now assume that the function h is
continuous from the left. This assumption is made to avoid some
technicalities which are not substantial for the problem itself.
Existence of solutions
Some of the fundamental properties of a solution of (4.1) are
established in Lemma 3.10, Corollary 3.11 and Lemma 3.12. A
solution of (4.1) has to be a function of bounded variation which
in our case is continuous from the left and has discontinuities of
the first kind given by Lemma 3.12, i.e. if for some to E (a, b) the
112
Existence of solutions
113
value of the solution x of (4.1) is x(to) = x then the right limit
at the point to fulfils
x(to+) = x(to) + F(x(to ), to+) - F(x(to ), to) =
= x + F(x, to +) - F(x, to).
Because of the possible discontinuities of a solution it can happen
that for some x E B, i.e. for some (x, to) E G, the value
x+ = x + F(x, to+) - F(x, to)
does not belong to B, and this means that the corresponding
solution x with x(to) = x jumps off the open set B, at the moment
to and cannot be continued for t > to.
Therefore to prove a local existence theorem for a solution of
(4.1) satisfying the initial condition x(to) = x we make the quite
natural assumption
x+ = x + F(x, to+) - F(x, to) E B,
(4.2)
4.1 Remark. The proof of the main result will be based on
the Schauder-Tichonov fixed point theorem. We use the usual
notation BV [a, I 1 = BV for the space of functions of bounded
variation on the interval [a, /3] with the usual norm given by
IIXIIBV = IIx(a)II + var,6 x
for x : [a,#] -- R. The classical monograph [21] is the basic
reference for the space BV as well as for all fundamental facts
from functional analysis used in this text.
IV. Existence and uniqueness for GDE's
114
4.2 Theorem. Let F : G -> R" belongs to the class
F(G, h, w) and let (Y, to) E G be such that (4.2) is satisfied. Then
there exist A-, 0+ > 0 such that on the interval [to-A-, to +A+]
there exists a solution x : [to - 0-, to + A+] --> R" of the generalized ordinary differential equation (4.2) for which x(to) = Y.
Proof. The Schauder-Tichonov fixed point theorem will be
used for proving this result (see [21]).
Since B, is open there is a A- > 0 such that if t E [to - A-, to]
and x E R" is such that llx - ill < lh(t) - h(to)l then (x, t) E
G = B, x (a, b), and similarly there exists a A+ > 0 such that if
t E [to, to +A+] and x E R" is such that llx-x+ll < l h(t)-h(to+)l
then (x, t) E G = B, x (a, b). This follows easily from (4.2) and
from the properties of the function h.
Denote by A the set of all functions z : [to -A-, to +A+] --+ R"
such that z E BV [to - A-, to + A+], llz(t) - ill < l h(t) - h(to)l
for t E [to - A-, to) and llz(t) - i + () < l h(t) - h(to+)l for
t E [to, to +A+].
It is easy to see that if z E A then z(to) = Y and z(to+) = i+
by virtue of the inequalities in the definition of A. It is also easy
to check that the set A C BV [to - A-, to + A+] is convex, i.e. if
z1, z2 E A, a E [0, 1] then azl + (1 - a)z2 E A.
Next, let us show that A is a closed subset of BV [to A-, to +
A+] = By. Let zk E A, k E N be a sequence which converges in
BV to a function z. Since
-
llzk(t) - z(t)ll < Ilzk - zll BV
for every t E [to - A-, to + A+], we have
km lizk(t) - z(t)ll = 0
uniformly for t E [to - A-, to + A+]. Therefore we have
Ilz(t)-x+11 5
llzk(t)-z(t)II+llzk(t)-x+II < e+[h(t)-h(to+)[
Existence of solutions
115
for any e > 0 whenever k E N is sufficiently large and t E [to, to +
A+]. This yields
II z(t) - x + II < h(t) - h(to+)
for t E [to, to + A+]. Similarly we can show that
II z(t) - ill < h(to) -- h(t)
for t E [to - A-, to] and therefore for the limit z we have z E A
and A is closed.
For a given z E A define the map
Tz(s) = x + jDF(z(r)t),
s E [to - A, to +
].
(4.4)
o
The map T is well-defined because by Corollary 3.16 the integral
feo DF(z(r), t) exists for every s E [to - 0-, to + A+].
Using Lemma 3.9 and Theorem 1.14 (Remark 1.15) we get
IITz(s) - x + II = IIx + j DF(z(r), t) - ill =
eo
= II f
DF(z(r), t) - [F(i, to+) - F(x, to)]II =
:o
r9
= II
Jto
DF(z(r), t)
- [F(z(to), to+) - F(z(to), to+)] II =
9
e rlim+ II Ir D F(z(r),t)II
lim+ Ih(s) - h(r)I = h(s) - h(to+)
-to
for s E [to, to + A+]. Analogously we have also
IITz(s) - xII < h(to) - h(s)
IV. Existence and uniqueness for GDE's
116
for every s E [to - A-, to]. Hence Tz C A for z E A, i.e. T maps
the set A into itself.
Let us show that T : A -- A is continuous. If z, v E A then
IITz - TvII By =
= IITz(to - A-) - Tv(to -A-)II + varto+,a+
II Tz(to) - Tv(to)II + 2 var'o±Q±(Tz
z - Tv) <
- Tv) _
= 2 varto+A+
to -O (Tz - Tv).
(4.5 )
Further, by (3.8) from Definition 3.8 and by Theorem 1.35 we
also obtain
IITz(t2) - Tv(t2) - Tz(tj) + Tv(ti )II =
r12
= II
D[F(z(T),t) - F(v(T),t)]II <
J t,
<
j
t2
tl
for tl, t2 E [to
D[w(II z(T) - v(T)II)h(t)]
- A-, to + A+] and therefore
to+o+
varto+A+(Tz - Tv) <
to-0-
D[w(IIz(T) - v(T)II)h(t)].
(4.6)
Assume that z, Zk E A, k E N and limk-. Ilzk - ZIIBV. Then by
(4.3)
kliM
114(t) - z(t)II = 0
uniformly for t E f to - A-, to + A+] and by Corollary 1.32 of the
convergence theorem 1.29 we obtain
to+G1+
lim
k-oo to-A- D[w(II zk(r) - z(T)II)h(t)] = 0.
Existence of solutions
117
Therefore by (4.6)
lim varto+o+(Tzk
- Tz) = 0
to
k-+oo
and (4.5) yields limk-.. IITzk -TzIIBV = 0, i.e. T is a continuous
map.
Finally we show that T(A) C A is sequentially compact in
the Banach space BV [to - A-, to + A+]. Let zk E A, k E N
be an arbitrary sequence in A. The sequence (zk) consists of
equally bounded functions of equibounded variation and therefore
Helly's Choice Theorem (see e.g. [104] or [21]) yields that this
sequence contains a pointwise convergent subsequence which we
again denote by (zk). Hence we have limk.,,. zk(T) = z(T) for
every T E [to - A-, to + A+], the values of z belong to B, and
z E BV[to - A-, to + A+]. Put
r9
y(s) = Tz(s) = i +
J:o DF(z(T), t)
for s E [to - A-, to + A+]. By Lemma 3.9 we have y E BV [to
A-, to + A+] and it is not difficult to show that
-
l imp iITzk - Y IBV[ta-0-,to+0+j = 0.
This immediately leads to the conclusion that every sequence in
T(A) contains a subsequence which converges in
BV[to -A-, to +A+] and consequently, T(A) is sequentially compact.
All assumptions of the Schauder-Tichonov fixed point theorem
being satisfied we can conclude that there exists at least one x E
A such that x = Tx, i.e.
!9
x(s) = Tx(s) = x +
Jto DF(x(T ), t),
s E [to - A-, to + A+],
118
IV. Existence and uniqueness for GDE's
and this implies (by the definition) that x is a solution of (4.1)
on [to - 0-, to +,A+], which concludes the proof.
0
4.3 Remark. Theorem 4.2 represents the fundamental local existence result of a solution of the generalized differential equation
(4.1) with a given initial value x. It is easy to see that the proof
can be repeated also for the general case when the left continuity
of the function h is not assumed. In this case we define
x- =x+F(i,to-)-F(i,to) E Be
and set 0- > 0 such that for t E [to - 0-, to] and x E R" with
lix - x - II < I h(t) - h(to-)I we have (x, t) E G = B, x (a, b).
Then also the set A in the proof Theorem 4.2 has to be modified
as the set of all functions z : [to - 0-, to + 0+] -- R" such that
z E BV[to-0-,to+0+], z(to) = x IIz(t) - x- II < Ih(t)-h(to-)I
for t E [to - 0-, to) and 11z(t) - x + II < Ih(t) - h(to+)l for
t E (to, to + L1+].
A theorem similar to our Theorem 4.2 was proved in [69
Theorem 2.1] and [70] by J. Kurzweil for increasing values of t, i.e.
for t E [to, to + 0+] by the method of successive approximations
with shifted arguments (see also Chap. II).
As we have mentioned in Remark 3.18 it can be assumed that
G C R"+' is an open set and that the function F : G --> R"
locally satisfies the requirements following from the properties of
the class F(G, h, w), i.e. if (x, t) E G then there exists 0 > 0
such that
G= {x E R";Ilx - xH < A} x (t-n,t+o)C G,
F satisfies on G (3.7) and (3.8) from Definition 3.8 with some
functions h and w and these functions can be specific for the set
G. It is clear that if we are looking for a local existence result
and if we require (4.2) then we can use Theorem 4.2 without any
Existence of solutions
119
essential changes for showing that a solution of (4.2) locally exists.
It is evident that if x : [to - 0-, to + 0+] -- R" is a solution of
(4.1) on [to - 0-, to + A+] given by Theorem 4.2 and if its value
at the right endpoint of this interval satisfies the condition (4.2),
i.e. if
x((to + 0+)+) =
= x(to+o+)+F(x(to+o+),
(to+o+)+)-F(x(to+0+), to+o+)
and (x((to + 0+)+), to + 0+) E G then the local existence result
can be applied to this point for continuing the solution to the
right.
For this reason it is useful to introduce the set
GF = {(x, t) E G; (x + F(x, t+) - F(x, t), t) E G}.
Then we can say e.g. that if (x(to + A+), to + A+) E GF then
the solution x can be continued to the right.
The background for the continuation of a solution of (4.1) is
given by the following result.
4.4 Lemma. If x : [a, /3] -- K" and y
-y] -+ R" are solutions of the equation (4.1) on [a, /3] and [/3, 7], respectively, where
[a, -y] C (a, b) and x(/3) = y(/3) then the function z : [a, -y] --- II8"
defined by the relations z(s) = x(s) for s E [a,,61, z(s) = y(s) for
s E [l3, -y] is a solution of the equation (4.1).
Proof For s1, s2 E [a, -y], s1 < /3 < s2 we have by Theorem
1.11
z(s2) - z(s]) = z(32) - z(/3) + z(/3)
- z(sl) _
= y(s2) - y(f3) + x(fl) - x(si) =
f
DF(y(r), t) +
j
1
DF(x(r), t) _
IV. Existence and uniqueness for GDE's
120
(32
=
J
DF(z(T), t)
and this evidently yields the result.
0
4.5 Remark. If x : [to, t1] --> R" is a solution of (4.1) on [to,t1]
such that (x(s), s) E GF for every s E [to, t1], then it can be
continued to the right (for values s > t1) to a solution of (4.1).
This procedure is well-known for the case of ordinary differential
equations. In the present situation we can say that if x : [to, t1] -R" is a solution of (4.1) which cannot be continued to the right,
then either (x(t1), t I ) GF or t1 = b.
A different situation occurs when we wish to continue a solution to the left. Given a solution x : (ti, to] ---i R" of (4.1) on the
interval (ti,to] C (a, b), t1 < to, it can be continued to the left to
the value t1 if and only if there exists x* such that
x(tl +) = x* + F(x*, ti +) - F(x*, tl ),
i.e. if the value x(tl+) = lim,_..i,+x(s) belongs to the range of
the map
x E B, H x + F(x,tl+) - F(x,t1) E R".
It is easy to see that this is true when the function h is continuous
at the point t1.
4.6 Example. Let us set
G={(x,t); jxI<1,Itl<1)CR2
and define
F(x,t)=xforIxj <1,-1<t<0,
F(x,t)=0 forjxf<1,0<t<1.
121
Uniqueness of solutions
Then F E .F(G, h,w) where h(t) = 0 for -1 < t < 0, h(t) = 1 for
0 < t < 1, w(r) = r, r > 0.
For t # 0 we have F(x, t+) - F(x, t) = 0 and F(x, 0+) F(x, 0) _ -x. Therefore x+F(x, 0+) - F(x, 0) = x - x = 0 E B1,
i.e. GF = G and every solution of the equation
dx
dT
= DF(x, t)
(4.1)
with x(O) = x jumps to the point 0 and satisfies x(t) = 0 for
t E (0,1).
It is not difficult to verify that if (x*, to) E G, to _< 0 then
the solution of (4.1) with x(to) = x* is given by x(t) = x* for
- 1 < t < 0, x(t) = 0 for 0 < t < 1. If (x*, to) E G, to > 0 and
x+ # 0 then the function x(t) = x* is a solution of (4.1) with
x(to) = x* on (0, 1) and cannot be continued for t < 0 because
the range of the mapping
x E B1 H x + F(x, 0+) - F(x, 0) = 0
consists of the point 0 E R and x* =
x(t) is beyond of
this range.
If x* = 0, to > 0 then x(t) = 0 is a solution of (4.1) on the
whole interval (-1, 1).
Uniqueness of solutions
4.7 Definition. A solution x : [to, to+rl] -- R" of the generalized differential equation (4.1) is called locally unique for increas-
ing values of t (locally unique in the future) if for any solution
y : [to, to + a] --> R' of (4.1) with y(to) = x(to) there exists 771 > 0
such that x(t) = y(t) for t E [to, to +,q] n [to, to + a] n [to, to + i1].
A point (x, to) E G is called a point of local uniqueness in the
future for the equation (4.1) if every solution x of (4.1) such that
x(to) = x is locally unique in the future.
IV. Existence and uniqueness for GDE's
122
We say that the equation (4.1) has the local uniqueness prop-
erty in the future if every point (x, to) E G is a point of local
uniqueness in the future for the equation (4.1).
Similarly the local uniqueness for decreasing values of t (local
uniqueness in the past ) can be defined for a solution of (4.1).
4.8 Theorem. Let us assume that F E .F'(G, h, w) where the
function h is non decreasing and continuous from the left, w :
[0, +oo) --+ R is continuous, nondecreasing, w(r) > 0 for r > 0,
w(O) = 0 and
lim /
u
V
1
w(r)
dr = -boo
(4.7)
for every u > 0. Then every solution x of (4.1) such that x(to) = x
where (x, to) E GF is locally unique for increasing values of t.
Proof. Assume that x, y : [to, to + q] -+ ]R" are solutions of
(4.1) such that x(to) = y(to) = x. Then
11x(s)
- y(s)II = II
f
D[F(x(T), t) - F(y(r), t)] 11 <
to
9
<
to
to+b
jo
w(II
w(II x(T)
y(r) II) dh(r) _
x(r) - y(T)II) dh(r) +
j
w(II x(T) - y(r)II) dh(r)
o +b
where 0 < 6 < s - to. We have
t o+b
Lo
w(IIx(r)
y(r)II)dh(r) =
= w(IIx(to) - y(to)II)[h(to+) - h(to)]+
rto+b
+ttlim+J,
w(IIx(r) - y(r)II)dh(r <
Uniqueness of solutions
<
123
w(II x(r) - y(T)11)[h(to + S) - h(to+)] = A(b)
sup
rE(to,to+6]
because w(IIx(to) - y(to)II) = w(0) = 0. Since the limit h(to+)
exists we have also limo-.o+ A(b) = 0. Therefore
11xs
y(s)II <Ab
J
xT
T
dhT
o+6
for s E [to + b, to + 77].
Take uo > 0 and set
1l(u) =
ru
1.0
w(r) dr.
Using Theorem 1.40 we get
jix(s) - y(s)II < 92-'(Q(A(b)) + h(s) - h(to + b))
(4.8)
for s E [to + 6, to + y] provided 11(A(b)) + h(to + y) - h(to + b) < Q
where # = limu-+oo 11(u) < +oo.
Evidently, we have
1l(A(b)) + h(to +,q) - h(to + 6) < 1Z(A(b)) + h(to +'1) - h(to+)
and because lim6_.o+ A(b) = 0 and
tain
lim 11(A(b)) + h(to + tj)
6- 0+
1Z(u) = -oo we ob-
- h(to+) = -oo.
Hence there is a bo > 0 such that for b E (0, bo) the inequality
11(A(b)) + h(to + 77)
- h(to+) < ,8
holds. Applying now the map 11 to both sides of (4.8) we obtain
11(lix(s) - y(s)II) < 11(A(6)) + h(s) - h(to + b),
IV. Existence and uniqueness for GDE's
124
and this yields
c(x(s) - y(s)II) - Z(A(b)) < h(s) - h(to + b) < h(s) - h(to+).
From the definition of fl we therefore have
II=(s)-.Y(s)II
fA(b)
1
W(r)
..!!
dv < h(to + il) - h(to+)
for s E [to + b, to + 71] and 6 E (0.bo).
Assume now that IIx(s*) - y(s*)Il = k > 0 for some s* E
(to, to + r7). Then
1
IA'(
6) w( r)
-boo
dr < h(to + r!) -
for every b E (0, bo) such that b < s* - to (i.e. s* > to + b). Now
it is possible to take b --> 0+ for obtaining the inequality
1k
lim
1
JA(b) w(r)
dr < h(to + rt) - h(to+) < +oo,
-
which contradicts the assumption on the function w. Therefore
x(s) - y(s)I1 = 0 for s E (to, to + q] and the result is proved. 0
4.9 Remark. Theorem 4.8 represents an Osgood type uniqueness theorem for the case of local uniqueness for increasing values
of t.
4.10 Corollary. If F E .F(G,h,w) with w(r) = Lr, r >
0, L > 0 then every solution of the generalized differential equation (4.1) starting from a point (x, to) E GF is locally unique for
increasing values oft.
Proof. For u > 0 we have evidently
ra 1
lim
dr = lim
v_o+ Jv w(r)
1
in
-u
v
= +oo
Uniqueness of solutions
125
and the condition of Theorem 4.8 is satisfied.
0
The local uniqueness for increasing values of t can be extended
to the global uniqueness for increasing values of t in the same
manner as this is done for the case of classical ordinary differential
equations.
4.11 Theorem. Assume that F E .F(G, h, w) and that x
[a1, 01 ] --> R", y : [a2, 02] - R" are two solutions of the generalized differential equation (4.1).
If the condition (4.7) is satisfied and if x(s) = y(s) for some
s E [al, fl1 ]n[a2, /32], then x(t) = y(t) for all t E [a1, l3,]n[a2, /32]n
Is, b).
Proof. The intersection [a,,
N1 ]
n [a2, /32 ] n [s, b) is a closed
interval of the form Is, c], where c < b. Denote
M = it E Is, c]; x(6) = y(a) fore E Is, t]}.
If s = c then there is nothing to prove. Assume therefore that
s < c and put /3 = sup M. We evidently have /3 < c.
Because the solutions x nd y are continuous from the left in
virtue of the assumption that the function h is continuous, we
have Is, /3] C M and our goal is to show that /3 = c. If we
had /3 < c then Theorem 4.8 could be used to show that there
is q > 0 such that x(a) = y(v) for all v E [#,,3 + q] because
/3 E M and x(/3) = y(/3). This contradicts the definition of /3 and
consequently M = Is, c]. The theorem is proved.
0
Theorems 4.8 and 4.11 give conditions for local, and consequently also global uniqueness in the future. A completely different situation arises when we consider the question of uniqueness
in the past, i.e. for decreasing values of the independent variable.
4.12 Example. Let B = {x E R; Ix1 < 11, [a, b] _ [-1, 1]. For
(x, t) E G = B x [-1,1] define
F(x, t) = a.g(t).x
126
IV. Existence and uniqueness for GDE's
where a>0,g(t)=tfort<0,g(t)=t-lfort>0.
Then x(t) = 0, t E [-1, 1] is a solution of the generalized
differential equation
dx
r= DF(x, t),
d
and also the function
y(t) = e°t,
y(t) = 0,
t E [-1, 01
t E (0,1]
is a solution of this equation on the interval [-1, 1]. We have
x(1) = y(l) but for t E [-1,0] the solutions x,y are different in
spite of the fact that the assumptions of Corollary 4.10 are evidently satisfied. The problem of nonuniqueness for decreasing
values of the independent variable is specific for generalized differential equations wich can have solutions with right hand side
discontinuities.
Maximal solutions
Let us assume that F E .F(G, h, w). In the situation considered in this chapter where the function h describing the class
,F(G, h, w) is assumed to be continuous from the left, a solution of
the generalized differential equation (4.1) can be in general continued - if at all - only to the right,that is for increasing values
of the independent variable.
If the local uniqueness of a solution for increasing values of the
independent variable is ensured (for example by the conditions
on the function w given in Theorem 4.8) then a unique "forward"
maximal solution of (4.1) can be defined when an initial condition
x(to) = i is prescribed for some to E (a, b) and x E B,.
Maximal solutions
127
It is clear that a maximal forward solution can be defined only
if (x, to) E GF, i.e. if
x + F(x, to+) - F(x, to) E Bc
because otherwise for the possible solution x it can happen that
x(t) V Bc for t > to and this would contradict the definition of a
solution.
Assume therefore that G f = G, i.e.
x + F(x, t+) - F(x, t) E Bc
for every x E B, t E (a, b). This means that there are no points
in G from which the solution of (4.1) can jump off the set Bc.
Let x : [to, to + 77] -, Rn, q > 0 be a solution of (4.1) on
[to, to +,q]. The solution y : I - Rn of (4.1) where I = [to, to + a]
or I = [to, to+o) a > 0 is called a prolongation of x if [to, to+rl] C
I and x(t) = y(t) fort E [to, to + rt]. If [to, to + 77] I, i.e. o > q
then I is called a proper prolongation of x to the right.
If (x, to) E G then a solution x of (4.1) with x(to) = x defined
for t > to is maximal if there is a value b(x,to) > to such that x
exists on [to, b(x, to)) and cannot be prolonged to a larger interval
of the form [to, /3] where Q > b(x, to), or in other words there is no
proper prolongation to the right of the solution x : [to, b(x, to)) ->
Rn of (4.1).
Let us give a few results concerning maximal solutions of (4.1).
4.13 Proposition. Let F E .F(G, h, w) and (x, to) E G. If
the equation (4.1) has the local uniqueness property in the future
then there exists an interval J with the left endpoint to and a
function x : J --- Rn such that to E J, x(to) = x and x : J -* R'
is a maximal solution of (4.1). The interval J and the function
x are uniquely defined by the initial condition x(to) = x and the
maximality property of the solution.
Proof. Assume that x' J' -. Rn, x2 : J2 -- Rn are two
maximal solutions of (4.1) with x'(to) = x2(to) = x. The local
:
128
IV. Existence and uniqueness for GDE's
uniqueness property implies xI(t) = x2(t) for every t E J1 fl J2 fl
[to, +oo). Define x(t) = x' (t) for t E J1 and x(t) = x2(t) for
t E J2. Then it is easy to show that x : J1 U J2 - R" is a
solution of (4.1) on x j1 U J2. Since the solutions x1, x2 are
assumed to be maximal, we have J' = J2 = J and x1(t) = x2(t)
for t E J. Hence the maximal solution x is unique.
Now let us show that the solution x : J -- 1Rn exists. Denote
by S the set of all solutions x : Jx - R' of (4.1) with x(to) = x
and the interval of definition Jx for which to is the left endpoint
of Jx. The set S is nonempty by the local existence of a solution
given in Theorem 4.2. Define
J = n Jx.
zES
If t E J,z fl J. where y, z E S then z(t) = y(t) by the assumption
of the uniqueness. Hence if we define x : J -- R" by the relation
x(t) = y(t) where y E S and t E Jy we obtain a solution of
(4.1) defined on J which satisfies the initial condition x(to) = Y.
Looking at the definition of J we can immediately see that x
J -* R" is a solution of (4.1).
4.14 Proposition. Let F E ..'(G, h, w) and (x, to) E G. Assume that the equation (4.1) has the local uniqueness property
in the future. Let x : J --> R" be th maximal solution of (4.1)
with x(to) = x, where to E J is the left endpoint of the interval
J. Then J = [to, /3) fl (a, b), to <,3:5 +oo.
Proof. It is clear that for the maximal interval J we have J E
(a, b). Let t* E J. Take y = x(t*) E Bc. Theorem 4.2 yields the
existence of a b > 0 such that on [t*, t* + b] there is a solution
v : [t*, t* + 6] --- R" of (4.1) such that v(t*) = y. The point
(y,t*) is a point of local uniqueness in the future and therefore
x is a prolongation of v and [t*, t* + S] C J. This means that
Maximal solutions
129
relatively to (a, b) the interval J is open at its right endpoint and
the statement holds true.
0
4.15 Proposition. Let F E ,F(G, h, w) and (x, to) E G. Assume that the equation (4.1) has the local uniqueness property
in the future. Let x : [to, /3) -* R" be a maximal solution of (4.1)
and let M C G = B, x (a, b) C R"+1 be a compact set. Then
there exists c E [to, /3) such that (x(t), t) M fort E (c,,3).
Proof. Assume to the contrary that the statement does not
hold. Then there is a sequence tk E [to, /3), k E N such that
limk-0tk = /3 and (x(tk),tk) E M, k E N. Since M is assumed
to be compact and b < +oo, the sequence (x(tk ), tk )kEN contains a
convergent subsequence which we denote again by (x(tk), tk)kEN.
Then limk._.o. x(tk) = y and (y, /3) E M C G.
By Theorem 4.2 there exists a b > 0 such that on [/3, /3+a] there
is a solution v of (4.1) with v(/3) = y. Define u :
S] -> R"
by
u(t) = x(t), t E [to, $),
u(t) = v(t), t E [,3, ,8 + E].
Now assume that s1 E [to, /3) and s2 E [/3, /3 + S]. Then for k E N
sufficiently large we have tk E (s1, /3) and
J
82
DF(u(r), t) =
rtk
az
Q
Js,
DF(u(r), t) +
DF(u(r), t) +
J#
DF(u(r), t) _
DF(u(r), t) + U(52) - u(/3)
Jtk
= x(tk) - u(s1) + u(s2) - y + j DF(u(r), t).
tk
(4.9)
130
IV. Existence and uniqueness for GDE's
By Lemma 3.9 we have
tk
DF(u(r),t)I( < h(13)
- h(tk)
Since h is continuous from the left and limk-oo tk
we have
k
lim
koo
=0,tk <,o
t) = 0.
it
Using this and limk_.0 x(tk) = y we take k --' oo in (4.9) and
obtain
DF(u(r), t) = u(s2) - U(81)82
L
For all other possible positions of sl, s2 E [to, /3+S) we obtain the
same relation directly from the definition of u. In this way we
obtain that u : [t0, ,Q + b] --, Rn is a solution of (4.1) on [t0, ,3 + 6]
which is evidently a proper prolongation of the solution x which
is assumed to be maximal. This contradiction proves the result.
0
CHAPTER V
GENERALIZED DIFFERENTIAL
EQUATIONS AND OTHER CONCEPTS
OF DIFFERENTIAL SYSTEMS
Assume that
W= {x E R"; DDxji < c}
for some c > 0 and that [a, b] C R is a bounded closed interval.
Let w : [0, +oo) -> R be a continuous increasing function with
w(0) = 0 (the function w has the character of a modulus of continuity).
Let y be a finite positive regular measure on [a, b] (see [21]).
5.1 Definition. A function g : B x [a, b] -- R' belongs to the
class C(B x [a, b], u, w) if
g(x, ) is measurable with respect to the measure y,
(5.1)
there exists a p-measurable function m : [a, b] --+ R such
that f' m(s) dp < +oo and
Ilg(s, xjj < m(s)
(5.2)
for (x, s) E B x [a, b],
there exists a p-measurable function 1 : [a, b] - R such
that fQ 1(s) dp < +oo and
IIg(s, x) - g(s, 011 :5 1(s)w(l ix - yJI)
for (x, s), (y, s) E B x (a, b].
131
(5.3)
V. GDE's and other concepts
132
5.2 Remark. Integrability here, and also in the subsequent
parts of this chapter, has to be understood as the LebesgueStieltjes integrability with respect to the finite positive regular
measure fi.
Define
G(x, t) =
g(x, s) dp
(5.4)
it o
for (x, t) E B x [a, b] where to E [a, b] and g E C(B x [a, b], p, w).
By (5.1) and (5.2) it is clear that the function G : B x [a, b]
]R" is well defined by (5.4).
5.3 Lemma. If a function g : B x [a, b] - R' satisfies (5.1)
and (5.2) then for'the function G given by (5.4) we have
t2
II G(x, t2) - G(x, t1 )II < f m(s) dp
(5.5)
t,
for every x E B and tl, t2 E [a, b].
Proof. Since (5.2) holds, we have
IIG(x,t2)-G(x,ti)II =
IIf
t2
g(x,s)dµ
11
,
rt2
Jt,
!t2
Ilg(x, s)II dµ <
J,
m(s) dp
0
for every x E B and ti, t2 E [a, b].
5.4 Lemma. If g E C(B x [a, b], µ,w) then for the function G
given by (5.4) we have
IIG(x,t2)-G(x,ti)-G(y,t2)+G(y,ti)II <_ w(IIx-yJI)
f
t2
l(s)d,u
I,
(5.6)
V. GDE's and other concepts
133
for every x, y E B and ti , t2 E [a, b].
Proof. By the definition of the function G and by (5.3) we have
IIG(x,t2) - G(x,t1) - G(y, t2) + G(y, ti)II =
t2
t2
= II
Jt,
[g(x, s) - g(y, s)] dell <- it
<w(IIx-ylI)J
IIg(x, s) - g(y, s)II dy
,
12
1(s)dp
t,
for every x, y E B and ti , t2 E [a, b].
Our next statement shows how the class C(B x [a, b], µ, w) is
related to the class F(B x [a, b], h,w) which was introduced in
Definition 3.8.
5.5 Proposition. If g E C(B x [a, b], p, w) then the function
G given by (5.4) belongs to the class T(B x [a, b], h, w) where
h(t) =
t
m(s) dµ +
o
f
1(s) dp
to
with to E [a, b] is a nondecreasing function.
Proof. By Lemma 5.3 we have
12
II G(x, t2) - G(x, ti )II <
J
m(s)dµ < I h(t2) - h(ti )I
for every x E B and t1 , t2 E [a, b], and therefore (3.7) from Definition 3.8 is satisfied. From Lemma 5.4 we get
IIG(x,t2) - G(x,t1) - G(y,t2) +G(y,ti)II <
< w(IIx
- yIt)it
12
1(s) dy <w(Ilx - yIl)Ih(t2) - h(tl)I
1
for every x, y E B and t1, t2 E [a, b], and this shows that (3.8)
from Definition 3.8 holds true.
0
134
V. GDE's and other concepts
5.6 Definition. A function g : B x [a, b] - Rn belongs to the
class Car(B x [a, b], p) if
g(x, ) is measurable with respect to the measure p,
(5.1)
there exists a p-measurable function in : [a, b] -4 R such
that f' m(s) dp < +oo and
IIg(s, xII < m(s)
(5.2)
for (x, s) E B x [a, b],
g(., s) is continuous for every s E [a, b].
(5.7)
5.7 Remark. This definition of the class Car(B x [a, b], p) of
functions g : B x [a, b] -> R" is closely related to the class of functions satisfying the Caratheodory conditions as they are described
in Chap. II. Indeed, if p is the Lebesgue measure on [a, b] then
they are the same except that (5.1) and (5.2) here are required
to hold everywhere instead of p-almost everywhere. Clearly this
difference is not essential from the point of view of Caratheodory
differential equations.
In the definition of Car(B x [a, b], p), (5.3) from Definition 5.1
of C(B x [a, b], p, w) is replaced by (5.7). The condition expressed
by (5.3) explicitly requires that the continuity from (5.7) has a
given modulus w. It is evident that
C(B x [a, b], p, w) C Car(B x [a, b], p).
The assumption (5.7) was replaced here by (5.3) only for the sake
of simplicity. In fact the following can be shown.
V. ODE'S and other concepts
135
5.8 Proposition. If g E Car(B x [a, b], p) then there exist an
increasing continuous function w : [0, 2c] --> R, w(0) = 0 and a
nonnegative p-integrable function p : [a, b] - R such that for the
function G given by (5.4) we have
t2
p(s)dp
JIG(x,t2)-G(x,ti)-G(y,t2)+G(y,ti)J(<w(jjx-yfl) it
,
for every x, y E B and t1 , t2 E [a, b].
The proof of this result is rather technical, it uses the so called
Scorza-Dragoni property of functions which belong to Car(B x
[a, b], p). The detailed proof of this result is given in [150]
Therefore we actually can use the proof of Proposition 5.5 to
show the following.
5.9 Proposition. If g E Car(B x [a, b], p) then the function G
given by (5.4) belongs to F(B x [a, b], h, w) with a nondecreasing
function h : [a, b] -- R and a modulus of continuity w.
5.10 Remark. Let us mention that the previous results hold
for a positive regular measure p on [a, b].
If p is a general "signed" measure on [a, b] then it can be
represented in the form of the difference of two positive regular
measures p+ and p-, i.e. p = p+ - p-. In this case for an
arbitrary p-measurable M C [a, b] and f : [a, b] -- R" we have
JMJMJM
provided the right hand side in this equality is defined. The integral fm f dp converges if and only if the integrals
fm f dp+, fm f dp- converge and this is equivalent to the convergence of fm f dp* where p* = p+ + p- is the total variation
V. GDE's and other concepts
136
of the measure it. Clearly also the p-measurability of a function
is equivalent to its µ+- and y--measurability.
If the measure y for the case introduced in Definition 5.1 is
not assumed to be positive then we have
t
G( x,t)
=
g(x,s)dy
11.0
=
L
g(x, s) dµ-
j g(x, s) dp(5.8)
o
where µ+ and y- are positive measures and the results of Lemmas
5.3 and 5.4 apply to each term on the right hand side of (5.8).
Let us now assume that u : [a, bJ -' R is of bounded variation
on [a, b]. Let p be the Legesgue-Stieltjes measure on [a, b] which
corresponds to the function u : [a, b] - R. The function u can
be written in the form u = u+ - u- where u+, u- :
[a, b] --> R
are bounded increasing functions, and if for a function g : B x
[a, b] -p R" the integral fto g(x, s) du exists then we write as usual
fto g(x, s) du(s) for this integral.
5.11 Proposition. If g : B x [a, b] --+ R" is such that g E
C(B x [a, b], p, w) where t is the Lebesgue-Stieltjes measure given
by the function u : [a, b] - R which is of bounded variation, then
for the function
G(x, t) =
it,o
g(x, s) du(s),
x E B, t, to E [a, b]
there is a nondecreasing function h : [a, b] -+ R such that
IIG(x, t2) - G(x, t, )II
I h(t2) - h(t, )I
and
IIG(x, t2) - G(x, tl) - G(y, t2) + G(y, tl )II <
(5.9)
V. GDE's and other concepts
137
< w(Iix - yll)Ih(t2) - h(t1)I
fortl,t2 E [a,b], x,y E B, i.e. G E.F(B x [a,b],h,w).
Proof. Let u = u+ - u- be the Jordan decomposition of the
function u on [a, b], the functions u+, u- being bounded and increasing on [a, b].
Let us consider the function
G(x, t)jY(x,8)d(3)
=x E B, t, to E [a, b].
(5.10)
o
By Lemma 5.3 we have
G+ (X, t2)
-
t2
G+ (x, t1)II <-
J t,
m(s) du+(s)
for every x c B, t1, t2 E [a, b].
Similarly also for the function
G- (X, t) =
J
g(x, s) du-(s),
x E B, t, to E [a, b]
eo
we have
t2
1I G (x, t2) - G- (x, t1)II <
J
m(s) du+(s)
,
for every x E B, t1i t2 E [a, b].
Hence
[[G(x, t2) - G(x, ti )II =
= JIG+(x, t2) - G+(x, t1) - G-(x, t2) + G-(x, t1)11 :
2
Jt ,
m(s) du+(s) +
Jt,
2
m(s) du-(s) _
(5.11)
V. GDE's and other concepts
138
=
J!l
If we set
t
hi (t) =
then hi
m(s) d(var u).
Ja
m(s) d(varQ u),
t E [a, b]
[a, b] -* R is nondecreasing since m is nonnegative on
[a, b] and the function s E [a, b] H vara u is nondecreasing. Hence
we have
:
IIG(x,t2) - G(x,ti)II s Ihi(t2) - hi(ti)I
(5.12)
for x E B, t1, t2 E [a, b].
Similarly, Lemma 5.4 implies
- G+(x,ti) - G+(y,t2) + G+(y,ti)II :
IIG+(x, t2)
t2
W(II x - VII)
J t,
l(s) du+(s)
if x, y E B and t1, t2 E [a, b], and a similar inequality holds for
the function G- which is given in (5.11). Hence the function G
from (5.9) satisfies
II G(x, t2) - G(x, ti) - G(y, t2) + G(y, ti) II <
12
< W(II x - VII) j
1(s) du(s) + w(II x - VII)
l
for x, y E
J
t2
1(s) du-(s) <
l
< w(II x - yII)Ih2(t2) - h2(ti )I
and ti, t2 E [a, b] where
h2(t) =
Ja
t
1(s) du-(s) +
Ja
(5.13)
t 1(s) du+(s)
for t E [a, b].
The function h2 is evidently nondecreasing on [a, b]. If we take
h(t) = hi (t) + h2(t) for t E [a, b] then (5.12) and (5.13) imply the
statement.
0
V. GDE's and other concepts
139
5.12 Proposition. Assume that g : B x [a, b] - R' belongs
to C(B x [a, b], p, w) where p is the Lebesgue-Stieltjes measure
given by the function u : [a, b] -+ R which is of bounded variation
on [a, b], let G : B x [a, b] --+ R' be defined by (5.9).
If x : [a,#] -- B, [a,#] C [a, b] is the pointwise limit of
finite step functions then both the generalized Perron integral
fa DG(x(r), t) and the Lebesgue-Stieltjes integral
f Q g(x(s), s) du(s) exist and have the same value.
Proof. By Proposition 5.12 we have G E F(B x [a, b], h, w)
where h is nondecreasing and w has the character of a modulus of
continuity. Therefore the existence of the integral f DG(x(r), t)
is guaranteed by Corollary 3.15. It is a matter of routine to show
that for every finite step function cp : [a,#] -' R" the integral
f,'# g(cp(s), s) du(s) exists and its value is the same as the value of
ff DG(cp(-r), t) (the technique of the proof of Corollary 3.15 can
_
be used ).
Assume now that cpk : [a, /3] -p B, k E N is a sequence of finite
step functions such that
lim cpk(s) = x(s),
00
sE
k
Then (5.3) yields
kTo
g(cpk(s), s) = g(x(s), s)
s E [a,#]
and (5.2) enables us to use the Lebesgue dominated convergence
theorem for showing that the integral fa g(x(s), s) du(s) exists
and
DG(x(-r), t)
lim
J
k-'oo a
= im
r
J
9(cpk(s), s) du (s) =
DG(Vk(7-),t) _
R
Ja
g(x(s), s) du(s)
V. GDE's and other concepts
140
where Theorem 3.14 is used to establish the first equality.
0
5.13 Remark. The results given above will be used in the sequel for the representation of some more conventional concepts of
ordinary differential equations within the framework of the generalized ordinary differential equations. The way how to do it is
based on the construction of the function G given in (5.4) for a
function g.
Caratheodory equations
Assume that a function f : B x [a, b] -- ]R" satisfies the following conditions:
f (x, ) is Lebesgue measurable on [a, b],
(5.14)
there exists a Lebesgue measurable function m : [a, b] ->
R such that f4 m(s) ds < +oo and
jjg(s, xjj < m(s)
(5.15)
for (x, s) E B x [a, b],
there exists a Lebesgue measurable function 1 : [a, b] -4R
such that f b l(s) ds < -boo and
II9(s, x) - 9(s, y)JJ : 1(s)w(II x - YII)
(5.16)
for (x, s), (y, s) E B x [a, b].
Let us associate f with the function
F(x, t) =
f
f (x, s) ds
x E B, t, to E [a, b].
(5.17)
a
Looking at (5.14) and (5.15) it is easy to see that the function
F : 11 x [a, b] - R" is well defined and because all assumptions
141
Caratheodory equations
of Proposition 5.11 are satisfied with u(t) = t, t E [a, b] we know
from Proposition 5.11 that F E F(B x [a, b], h, w) where h and
w have all the properties required in Definition 3.8 of the class
.''(B x [a, b], h, w).
Let us recall the Caratheodory concept of a solution of the
differential equation
th = f(x,t).
(5.18)
A function x : [a,,31 - R' is called a solution of (5.18) on [a,,0] C
[a, b] if x is absolutely continuous on [a, /3], x(s) E B for almost
all (in the sense of Lebesgue measure) s E [a, p] and if for almost
all t E [a,,31 the equality
= f(x(t),t)
(5.19)
is satisfied. These properties of the solution are equivalent to the
requirement that for every s1, s2 E [a, i3] we have
r82
X(S2) - x(sl) =
J8,
f (x(s), s) ds.
(5.20)
The following result connects the Caratheodory theory with
the theory of generalized differential equations.
5.14 Theorem. A function x : [a,#] --ti R", [a,#] C [a, b] is
a solution of (5.18) (in the sense of Caratheodory) on [a,,13] if
and only if x is a solution of the generalized differential equation
dx
dr
= DF(x, t)
(5.21)
on [a, a] in the sense of Definition 3.1, where F is given by (5.17).
Proof. Assume that x : [a, /3] --- R' is a solution of (5.18). By
Proposition 5.12 the integral f' DF(x(r), t) exists and
r32
X(S2) - x(sI) =
J
81
82
f (x(s), s) ds =
J
Su
DF(x(r), t)
142
V. GDE's and other concepts
for all s1, $2 E [a, p]. Hence x is a solution of (5.21). If, conversely, x is a solution of (5.21) then again Proposition 5.12 shows
that x : [a,#] --> Rn satisfies the equality (5.20) and x is absolutely continuous because by Lemnia 3.10 we have
IIx(s2) - x(s1)II < Ih(S2) - h(S1)I
for every Si, s2 E [a, /j] and h can be chosen as an absolutely
continuous function on [a, b].
0
5.15 Remark. Theorem 5.14 justifies the term generalized dif-
ferential equation in the sense that for an ordinary differential
equation we can find a generalized differential equation such that
the two equations have the same set of solutions.
It has to be mentioned here that the condition (5.16) is stronger
than the condition used for defining the class of Caratheodory
functions.
If we replace (5.16) by
f(., s) is continuous for every s E [a, b],
(5.22)
if we assume that f E Car(B x [a, b]) = Car(B x [a, b], p)
where y is the Lebesgue measure on [a, b], then Proposition 5.9
yields that the function F given by (5.17) belongs to a certain
class F(B x [a, b], h, w) and all the above constructions can be
i.e.
repeated without any changes.
5.16 Historical comments. The generalized Perron integral described in Chapter I was introduced for the first time in 1957 in
the paper [68] by J. Kurzweil and its introduction and motivation comes from the theory of ordinary differential equations. The
problem goes back to the early fifties when the averaging principle
for differential equations was substantiated by I.I. Gichman in his
paper [31] I.I. Gichman showed that the averaging principle can
Caratheodory equations
143
be proved correctly by using some new results on continuous dependence of solutions of differential equations on a parameter. In
1955 the paper [66] of M.A. Krasnoselskij and S.G. Krejn on this
problem appeared, being followed in a short time by the paper
[88] of J. Kurzweil and Z. Vorel.
From these results it became clear that for continuous dependence of solutions on a parameter of a nonautonomous ordinary
differential equation of the form
x = f(x,t)
(5.23)
the indefinite integral
F(x, t) =
rt
J
f (x, s) ds
(5.24)
eQ
to the right hand side f of (5.23) is significant, i.e. if these independent integrals for two ordinary differential equations are close
to each other, then the solutions starting e.g. from the same point
are close to each other.
One of the problems motivating the introduction of generalized
ordinary differential equations can be stated as follows:
Describe the solution of (5.23) in terms of F from (5.24).
The idea how to deal with this problem is the following. The
initial value problem
x = f(x,t),
x(to) = x*
(5.25)
is (e.g. in the Caratheodory setting ) equivalent to the integral
equation
x(s) = x* + j f(x(Q), o) do,
eo
with the Lebesgue integral on the right hand side.
(5.26)
V. GDE's and other concepts
144
If x is a solution of (5.25) on [to, t] then x is absolutely continuous on [to, t] and it can be approximated by a finite step function
XI which is constant on intervals of the form (a, _ 1, a,) where
to =ao <al
<...<ak, =t
and which on (ai _ 1 , aj) assumes the value x(Tj) where T3 E
[aj_1, a,], j = 1,.. . , kt. Moreover, this can be done in such
a way that
1lim
xt(s) = x(s)
00
(5.27)
uniformly on [to, t]. If the function f (x, t) satisfies the current
assumption of continuity in the variable x then
lim
too f (xt(s), s) = f (X (3), s)
on [to, t]. If now e.g. the assumptions of the Lebesgue dominated
convergence theorem are satisfied for the sequence of functions
f (xI(s), s), I = 1, 2, ... (of this kind is (5.15)) then it can be
concluded that
lim it.,
too
f (x1(s), s) ds
t
ji,
f (x(s), s) ds.
However, for a fixed 1 E N we have
kt
t
o
Jx',8
=
f
ai
j=1 a,
f(x(r),s)ds =
4
1
k1
[F(x(rr),ai)-F(x(rj),ai-1)]
_
1=1
(5.28)
145
Caratheodory equations
and this together with (5.28) shows that the integral
can be approximated by sums of the form
ft' f (x(s), s) ds
ki
L [F(x(Tj ), aj) - F(x(Tj ), aj-1)],
j=1
i.e. using (5.26) the value x(t) can be approximated by
ki
x* + >[F(x(Tj), aj) - F(x(Tj), aj_1)].
(5.29)
j=1
The only thing we have to do now is to fix the process of constructing a sufficiently fine division to = ao < al < . . < ak = t
and the choice of Tj E [a j_1, a j], j = 1, ... , k in order to obtain
the uniform convergence (5.27).
Looking at the sum in (5.29) it is easy to recognize that it
is formally related to the integral sums treated in Chapter I.
Moreover, if the problem is to describe the solutions of (5.25)
with f fastly oscillating in the variable t then it is clear that we
have to take the intervals [aj_1, aj] of the division in dependence
on the choice of the point Tj because otherwise we would ignore
the instantaneous influence of the fastly changing vector field of
the right hand sides of the equation. This leads to the concept
of a 6-fine partition for a given gauge as it was described in the
introduction to Chapter I.
These few vague remarks are given here to describe the essence
of the intuitive idea which was (probably) behind the theory of
generalized ordinary differential equations created by J. Kurzweil
in 1957 and, of course, also of the Kurzweil branch of the HenstockKurzweil approach to integration.
For the sake of historical truth it is necessary to intimate that
the sum approach to the generalized Perron integral in Kurzweil's
146
V. GDE's and other concepts
work [68] is secondary. The primary one is based on the concept
of major and minor functions and is very close to the concept used
in the theory of the Denjoy-Perron integral. Since it is shown in
[68] that these two concepts are equivalent, the equivalence of the
new sum integral and the Denjoy-Perron integral was clear from
the very beginning of Kurzweil's work even if this fact was not
sufficiently emphasized as concerns the theory of integration.
Measure differential equations
Measure differential equations have been investigated by many
authors, e.g. W.W. Schmaedeke [124], P.C. Das and R.R. Sharma
[19], M. Rama Mohana Rao and Sree Hari Rao [114], S.G. Pandit
[P] and others. A survey of this approach to differential systems
is given in the monograph [107] written by S.G. Pandit and S.G.
Deo.
The main purpose of the concept of measure differential equations is the description of systems exhibiting discontinuous solutions caused by the impulsive behaviour of the differential system.
The solutions of a measure differential equation are discontinuous
functions of bounded variation, i.e. they have the same properties
as generalized ordinary differential equations described in Chapter III.
Assume as before that
B = {x E R"; IIxlI < c}
andG=Bx[a,b],-oo<a<b-f-oo. Let f :G->R"be a
function which satisfies (5.14) - (5.16) with some wl in the role
of w. Further assume that u : [a, b] --4R is of bounded variation
on [a, b] and continuous from the left. Let g : G -> R" be such
that g E C(G, p, w2) where It is the Lebesgue-Stieltjes measure
generated by the function u (see Proposition 5.11 ).
Measure differential equations
147
The measure differential equation is formally written in the
form
Dx = f (x, t) + g(x, t)Du
(5.30)
where Dx and Du stand for the distributional derivatives of the
functions x and u in the sense of distributions of L. Schwartz.
In the paper [19] Das and Sharma have shown that the concept
of a solution of (5.30) satisfying the initial condition x(to) = Y,
to E [a, b], x E B, is equivalent to the concept of a solution of the
integral equation
x(t) = Y +
J
t
f(x(s), s) ds +
tp
J
t
g(x(s), s) du(s)
(5.31)
tp
for t E [a, b]. In other words, a function x : [a,13) -> Rn is a
solution of the measure differential equation (5.30) if and only if
(x(s), s) E G for s E [a, /3]and
32
x(82) - x(s1) =
Jsl
f (x(s), s) ds +
sz
g(x(s), s) du (s) (5.32)
sl
for every s1, s2 E [a, /3].
Since we have got some experience with the Caratheodory
equation in the previous part it is now evident that for x E B,
t, to E [a, b] we have to define
t
Fl (x, t) = f f(x(s), s) ds
to
and
F2 (x, t) =
rt
Jto g(x(s), s) du(s).
The function F1 : G -' R" is the same as in the previous section
on Caratheodory equations because it corresponds to f satisfying
V. GDE's and other concepts
148
(5.14) - (5.16) and we have F1 E ,F(G, hl, w,) where h1 is absolutely continuous on [a, b]. To the function F2 Proposition 5.11
can be applied for concluding that F2 E ,F(G, h2, w2) where h2
is nondecreasing and continuous from the left because u is continuous from the left. (Using Remark 5.7 and Propositions 5.8
and 5.9 it can be assumed that f is a Caratheodory function and
g E Car(G, p) where µ is the Lebesgue-Stieltjes measure generated by u on [a, b] and the results are the same.)
Let us set
F(x,t) = Fi(x,t) + F2(x,t)
(5.33)
for (x, t) E G. It is a matter of routine to show by Definition 3.8)
that F E .F(G, h, w) where h = hl + h2 and w = w1 + w2. The
functions h and w have the properties required in Chapters III
and IV for the generalized differential equation
dx
d = DF(x, t).
(5.34)
Proposition 5.12 can now be used to show that for every function
x : [a, ,Q] -* R', [a, Q] C [a, b] which is the pointwise limit of a
sequence of finite step functions the integrals
f (x (s), s) ds,
g(x(s), s) du(s),
J
Q
1
DFi (x(7-), t), j D F2(x(T), t)
exist and
/Q
J«
a
g(x(s), s) du (s) _
DF2(x(r), t),
«
!#
1
f (x(s), s) ds =
J«
DF2 (x(T), t)
(5.35)
149
Measure differential equations
Looking at the integral form (5.32) of the measure differential
equation (5.30) it is easy to observe that every solution is a func-
tion of bounded variation. Hence by (5.35) the relation (5.31)
can be writen in the form
s,
/ -92
X(S2) - x(sl) =
f (x(s), ds +
8y
112
g(x(s), du(s) _
!82
DF2(x(r), t) = /
DFi (x(r), t) +
DF(x(r), t)
.Jlsi
81
1
J
81
J81
for every solution x : [a, p] -> R" of (5.30) and every sl, s2 E
[a, /3]. Hence x : [a, /3] -+ R" is a solution of (5.43). The converse
statement, namely that every solution x : [a,#] --+ R" of (5.34)
with F given by (5.33) is also a solution of (5.30), can be shown
quite analogously. In this way we obtain the following result.
5.17 Theorem. A function x : [a, /3] ---> R", [a, /3] C [a, b] is
a solution of the measure differential equation (5.30) on [a,,0] if
and only if x is a solution of the generalized differential equation
(5.34) on [a, /3] with the function F given by (5.33).
5.18 Remark. In [107] the function u involved in the measure
differential equation (5.30) is assumed to be continuous from the
right. There is no essential difference between our setting of left
continuous u. One case can be easily transformed into the other.
5.19 Example. Let us consider the measure differential equatipu
Dx = 2(t + 1)-'x Du,
t E [0, 2]
where u(t) = t fort E [0,1], u(t) = t-1 for t E (1, 2] (see Example
2.1 in [107]). Let us define
t
F(x, t) =
j2(s + 1)x du(s).
V. GDE's and other concepts
150
Then
F(x,t) = A(t)x,
where
A(t) = J t 2(s + 1)-'du(s) = J t 2(s + 1)-1 ds = 21n(t + 1)
for t E [0, 1] and
A(t) = J 2(s + 1)-1 ds + J t 2(s + 1)-1 du(s) _
1
0
1
ft
21n2+21(u(1+)-u(1))+J 2(s+1)-1 ds = -1+21n(t+1)
i
fort E (1, 2].
The generalized differential equation which is associated to the
given measure differential equation has the form
dx
dr = D[A(t)x].
If we consider the initial value problem x(0) = 0 for this equation
then
x(t)=0fortE[0,1],
x(1+)
- x(1) = x(1)[A(1+) - A(1)] = -x(1), i.e. x(1+) = 0
and for t E (1, 2] we get
x(t) = x(1)+ Jt x(r) dA(r) = x(1)+x(1)+ lim J tx(r) dA(r) _
b
1
=
2
Jl
t
x(r) dln(1 + r) = 2 1t x(r)(1 + r)`1 dr.
1
Measure differential equations
151
Therefore for t E (1, 2] the function x is the solution of the initial
value problem
y(1) = 0
y = 2(1 + t)-ly,
and x(t) = y(t) = 0 for t > 1. The situation is different when we
consider the initial value problem with x(2) = 0 for this equation.
Then we have as in the previous case
x(t) = 0 for t E (1,2]
because for t E (1, 2] we have
t
2
x(t) = f x(r) dA(r) _ -2 it x(r)(1 +,r)-' dr.
t
2
Further
and
0 = x(1+) = x(1) + x(1)[A(1+)
- A(1)] = x(1) - x(1).
This means that x(1+) = 0 for all possible values of x(1) E R,
and therefore every function x : [0, 1] -> R for which
t
t
x(t) = x(1) + 2
J
x(r) dA(r) = x(1) - 2
J
x(r)(1 + r)-1 dr
holds is a solution of the equation. Hence
x(t) = 2(1 +t)-'x(t), x(1) = c, c E R
and therefore
x(t) _ (t
where c E R is arbitrary.
41)z c,
t E [0,1]
152
V. GDE's and other concepts
Differential equations with impulses
In this part we describe shortly another approach to systems
exhibiting impulsive behaviour, namely ordinary differential equa-
tions with impulses. The theory of these equations was extensively studied e.g. by A.D. Myshkis, A.M. Samojlenko, N.A.
Perestjuk and others (see [99],[103], [116], [117], [118], [119], [108]
etc. ). A survey of the state of art is given in the comprehensive
monograph [120].
Assume again that B = {x E R"; lixil _< c} and G = B x [a, b],
-oo < a < b + oo. Let f : G -> R" be a function which satisfies
(5.14) - (5.16).
Further let a finite set of points t; E [a, b], i = 1, ... , k be given
with ti < ti+1 f o r i = 1, 2, ... , k - 1 and a system of k continuous
maps
Ii :B-"R", i=1,2,...,k.
The system with impulsive action at the fixed instants
t 1 , t2, ... , tk is usually written in the form
i = f (x, t),
t j4 ti,
z xI ti = x(ti+) - x(ti) = Ii(x(ti)),
i = 1, ... , k.
(5.36)
(5.37)
The equation (5.36) describes the behaviour of the state at instants different from ti, i = 1,. .. , k and (5.37) represents the
discontinuity from the right of the solution for t = ti.
We describe the differential system with impulses (5.36), (5.37)
more exactly by describing its solutions. A function x : [a, Q]
Rn, [a, /3] C [a, b] is called a solution of the differential system
with impulses (5.36), (5.37) if (x(t), t) E G for t E [a,,61, the
function x is absolutely continuous on every interval [a, tl ] fl [a,
(ti, ti+l] fl [a,,61, i = 1, ... , k - 1, (tk, b) n [a, fl]
x(t) = f (x(t), t) for almost all t E [a,#],
153
Equations with impulses
and
x(ti+) = lun x(t) = x(ti) + Ii(x(ti))
if ti E [a, (3].
For a given d E [a, b) define
Hd(t)=Ofort<d,Hd(t)=1fort >d
and define
F(x, t) =
j
f (x, s) ds + > I(x)H(t).
(5.38)
i=1
Since B is compact and the mappings Ii are continuous, there
exists a constant K > 0 such that IIII(x)II < K for all x E B and
i = 1, ... , k. Therefore if x E B and s1i 82 E [a, b] we obtain
IIF(x,s2) -F(x,s1)II < II
f
82
f(x,s)dsII+
it
k
+KI k(Ht;(s2) - Ht,(s1))I <
i=1
< Ih1(s2) - h1(s1)I + KIh2(s2) - h2(s1)I
where h1 : [a, b] --+ R is the nondecreasing, absolutely continuous
function corresponding to f in the same way as in the part on
Caratheodory equations (i.e. the function F1 (x, t) = f t P X, s) ds
belongs to the class F(G, h1, w1)) and
k
h2(t) = E(Ht; (t),
i=1
t E [a, b].
V. GDE's and oilier concepts
154
Clearly, h2 is nondecreasing and continuous from the left on [a, b].
If w2 is the common modulus of continuity of the finite system of
mappings Ii, i = 1, ... , k then
IIIi(x) - Ii(Y)II < w2(IIx - Y11)
for x, y E B. Using the information from the part on Caratheodory equations we obtain easily
IIF(x,s2) - F(x,s1) - F(y,s2) + F(y,s1)II <
< w1()Ix - y11)Ih1 (S2) - h1 (Si )I + w2(IIx - yll)lh2(S2) - h2(SI )I
for x, y E B and s1, s2 E [a, b]. The first term corresponds to f
and for the second term in (5.38) we have the following estimate:
k
ll E(Ii(x) - Ii(y))(Htj (s2) - Ht: (s1))il <
i=1
k
< w2(IIx - YII)l k(Ht,(S2) - Hti(Si))I <
i=1
< W2(IIx - yii)lh2(s2) - h2(SI)I
If we take h(t) = h1(t) + h2(t) for t E [a, b] and w(r) = w1(r) +
w2(r) then we obtain that the function F defined by (5.38) belongs to the class F(g, h, w) and the result of Lemma 3.12 states
x(ti+) - x(ti) = F(x(ti),ti+) - F(x(ti),ti) =
k
k
lim E Ii(x(ti))Ht; (t) - E Ii(x(ti))Ht; (ti) =
i=1
i=1
= Ii(x(ti))(Ht, (ti+) - Ht; (ti )) = Ii(x(ti ))
for a solution of the generalized differential equation
dx
d = DF(x, t).
After this we summarize the results as follows.
(5.39)
V. GDE's and other concepts
155
5.20 Theorem. A function x : [a, /9] -> R'2, [a,#] C [a, b] is
a solution of the differential equation (5.36) with impulses (5.37)
on [a, ,0] if and only if x is a solution of the generalized differential
equation (5.39) on [a, Q] with the function F given by (5.38).
By the results given in the above sections in this chapter the
Caratheodory equations, measure differential equations and differential equations with impulses are embedded in a quite natural
way into the class of generalized ordinary differential equations
with an appropriately chosen right hand side F which belongs to
the class .F(G, h, w).
5.21 Remark. From Chap. III we know that a solution of a
generalized ordinary differential equation with F E F(G, h, w) is a
function of bounded variation and therefore it cannot be expected
that this equation would be equivalent in general with a classical
differential equation. Nevertheless the following problem can be
posed.
Given a generalized ordinary differential equation
dx
d = DF(x, t).
(5.40)
under what condition is this equation equivalent to a classical
differential equation?
Let us consider the case when F E F(G, h, w) where B =
{x E Rn; Ix11 < c} and G = B x [a, b], -oo < a < b + oo,
h : [a, b] --> Rn is nondecreasing and continuous from the left and
w has the properties of a modulus of continuity as given in Chap.
III.
Assume that n = 1, i.e. F : G --> R. By Definition 3.8 of
F(G, h, w) we have
IF(x,t2) - F(x,tl)j < Ih(t2) - h(tl)j
for every x E B and t1, t2 E [a, b].
(5.41)
V. GDE's and other concepts
156
In the usual way the function h defines a Lebesgue-Stieltjes
measure on [a, b] via the additive interval function given by
p([c, d)) = h(d) - h(c) for an interval [c, d) with a < c < d < b.
For a fixed x E B we define
px([c, d)) = F(x, d) - F(x, c)
for a < c < d < b. Clearly µx induces a measure on [a, b] and
by the inequality (5.41) we can see immediately that the measure µx is absolutely continuous with respect to the measure p.
Therefore by the Radon-Nikodym Theorem (see e.g. [21, 111.10.7.
Theorem] or [25, (8.19) Theorem]) there exists a p-measurable
function gx on [a, b] such that for each p-measurable set E C [a, b]
we have
µx(E)
= JE gc
This means that
F(x, t2)
- F(x, ti) =
F(x, t) - F(x, a) =
j
(5.42)
dµ.
f
gdp =
t2
gx dµ,
f
g(s) dh(s).
a
Let us set f (x, t) = g5(t). Then f (x, ) : [a, b] -* R is the RadonNikodym derivative of µx with respect to the measure it it is
p-integrable and has a finite integral (= µ5(E)) over every measurable set E C [a, b]. Moreover,
F(x, t) = F(x, a) +
Ja
g5 dp = F(x, a) + J tf (x, s) dh(s).
a
Denote by Dp (t) the general derivative of the set function µ5
given by (5.42). Then we have
Dpx(t) = g-- (t) = f(x, t)
157
V. GDE's and other concepts
almost everywhere with respect to the measure it. (See e.g. [25,
(8.29) Theorem].)
Hence for. p-almost all t E [a, b] we have by (5.41)
t ,t) I -<
m F(
If(x,t)I = I9x(t)I = IDpx(t)I < lis-t
h(s)
h,) - h(t)
I
h(t)I = 1.
< lim Ih(s) -
(5.43)
- s-t Ih(s) - h(t)J
Since F E..F(G,h,w) we have also
IF(x,t2)-F(x,ti)-F(y,t2)+F(y,tl)I <w(Ix-yi)Ih(t2)-h(t1)I
(5.44)
for x, y E B and t1i t2 E [a, b]. Essentially in the same way as
above we can deduce from (5.44) that
If(x, t) - f(y, t)I = g=(t) - 9y(t)I < w(Ix - yl)
(5.45)
holds for x, y E B and p-almost all t E [a, b]. From (5.43) and
(5.45) we can see that (5.2) and (5.3) from the definition of the
class C(G, p, w) is satisfied with m(s) = 1(s) = 1 p-almost everywhere in [a, b].
Assume now that x : [a, p] - R is a solution of (5.40). Using
the method of the proof of Proposition 5.12 it is easy to show
that the integral fa f (x(a), a) dp = = f8 f (x(a), tr) dh(a) exists
and equals to f« DF(x(r), t). Therefore
x(s) = x(a) +
/s
Ja
s
DF(x(r), t) = x(a) +
Ja
f(x(a), u) dh(u)
and x : [a, ,0] -+ R is a solution of the measure differential equation
Dx = Ax, t)Dh
(5.46)
V. GDE's and other concepts
158
(see (5.30) and (5.31)). Taking into account the result stated in
Theorem 5.17 we obtain the following.
Every generalized ordinary differential equation (5.40) with
F E .F(G, h,w) is equivalent to a certain measure differential
equation of the form (5.46).
Assume in addition that the function h is absolutely continuous (with respect to the Lebesgue measure) on [a, b]. Then we
have h(t) = fa m(s) ds where m is Lebesgue integrable over [a, b].
Using the construction of the function f given in the more general
case above we have
x(s) = x(a) +
Ja
s DF(x(r),
t) = x(a) + J f(x(a),
a) do,
s
a
for an arbitrary solution x : [a, Q] --4R of the generalized differ-
ential equation (5.40) and all s E (a,#]. This means that x is a
solution of the Caratheodory differential equation
i = f(x,t)
and it can be shown that the right hand side of this equation
satisfies (5.14) - (5.16). In this way we arrived at the following
result.
Every generalized ordinary differential equation (5.40) with
F E .F(G, h, w) where his absolutely continuous on [a, b] is equivalent to a certain classical Caratheodory differential equation.
We have restricted our consideration of (5.40) to the case
n = 1. It is clear that this restriction is not essential. All the reasoning can be used for the components of a general n-dimensional
function F for obtaining the same results for an arbitrary n E N.
Finally it is worth to mention here that in a more general
setting the question given at the beginning of this remark was
treated by T.S. Chew in the paper [15]. Chew gives a necessary and sufficient condition for the right hand side of the equation (5.40) for being equivalent to a classical ordinary differential
V. GDE's and other concepts
159
equation with the right hand side f E Car(G, p) where p is the
Lebesgue measure on [a, b].
Remark. It should be mentioned that D. Frankova [26], [30]
developed another approach to generalized ordinary differential
equations, the so called generalized ordinary differential equations with a substitution. Generalized ordinary differential equations with a substitution represent a powerful technical tool. The
method of D. Frankova was used e.g. in [27] and [145] for treating
second order linear equations with impulses (see also the paper
[67] of K. Kreith).
CHAPTER VI
GENERALIZED LINEAR
DIFFERENTIAL EQUATIONS
Assume that J C R is an interval. Denote by L(R") the set of
all n x n-matrices with real components and assume that A : J -L(R') is an n x n-matrix valued function defined on the interval
J. We assume further that A is locally of bounded variation,
i.e. varQ A < oo for every compact interval [a, b] E J where
the variation is defined using the norm of the matrix which is
the norm of the corresponding linear operator on R", i.e. the
operator norm in L(R" ).
Let us note that if A(t) = (a;j(t));j=1,,.,,,, then every component aid : J - R, i, j = 1,... , n is locally of bounded variation
provided A has this property.
For x E Rn, t E J we set
F(x, t) = A(t)x.
If (Y, to) E I[8" x J is an arbitrary point then for every closed ball
B,(x) = {x E R"; fix - ill < c} in R" with radius c > 1 centered
at i and [a, b] C J, to E [a, b] we have
II F(x, t2) - F(x, tl )II <_ II A(t2) - A(t1)II IIxII <_ Ih(t2)
- h(t1)I
for x E B,(x), t1 i t2 E [a, b] where h : [a, b] -> R is given by
h(t) = (c+ IIxfi) varQ A fort E [a, b], and for x, y E Bo(x), tl, t2 E
160
VI. Generalized linear ODE'S
161
[a, b] we have
II F(x, t2) - F(x, tl) - F(y, t2) + F(y, tl )II <
< II A(t2) - A(tl )ll llx
- yll < II x - yll lh(t2) - h(tl )l.
The function h defined on [a, b] is evidently nondecreasing because
A is assumed to be of locally bounded variation.
The above inequalities show that the function F : Rn X J
Rn locally satisfies the requirements of Definition 3.8, i.e. F E
.F(G, h,w) where G = B,(x) x [a, b], w(r) = r, r > 0.
If g : J -- Rn is a function of locally bounded variation on J
then for x E Rn, t E J we can set
F(x, t) = A(t)x + g(t)
and show in the same way as above that F E .F(G, h, w) for
G = B,(x) x [a, b] where h(t) = (c + IIxil)var' A + vara g for
t E [a, b] and w(r) = r, r > 0.
In this chapter we will study the generalized linear differential
equation
aT D[A(t)x + g(t)]
where A, g have the properties given above.
(6.1)
Using Definition 3.1 of a solution we can see that a function
x : [a, fl] - R' is a solution of (6.1) on [a,#] if for any s1, 32 E
[a,,3] we have
82
X(S2)
=
- x(sl) = J D[A(t)x(T) + g(t)] _
S,
f
92
al
D[A(t)x(T)] +g(32) -g(sl)
(6.2)
VI. Generalized linear ODE'S
162
Using the more conventional notation f 2 d[A(r)]x(r) for
fs12 D[A(t)x(r)] we can rewrite (6.2) in the form
82
X(S2) - x(S1) =
fs,
d[A(r)]x(r) + 9(S2) - 9($1)
where the integral here is the Perron-Stieltjes integral (cf. Remark 1.5) with the usual convention on integration of R"-valued
functions.
Remark. Sometimes instead of the notation given in (6.1) the
notation
dx = d[A]x + dg
is used.
6.1 Lemma. If x
:
[a, /3] - R" is a solution of (6.1) on
[a,./3] C J then x is of bounded variation on
Proof. From the existence of the solution x of (6.1) we have
x(t) = x(to) + f d[A(r)]x(r) + g(t) - g(to)
eo
for every t, to E [a, /3] and the integral fto d[A(r)]x(r) exists for
t, to E [a, /3]. Hence by Theorem 1.16 the limits
t
lim- ft., d[A(r)]x(r) and t lim
t
ft"
d[A(r)]x(r)
exist for to E (a, /3], to E [a, /3), respectively. Hence the solution
x has onesided limits at every point in [a, /3] because also the
function g evidently has this property. Therefore for every to E
[a, /3] there exists a b > 0 and a constant M such that 11x(t)I I < M
for t E (to - b, to + b) fl [a, Q]. By the Heine - Borel Covering
V1. Generalized linear ODE'S
163
Theorem there is a finite set of intervals of the type (to - 6, to + 6)
covering the interval [a, /3]. Therefore there is a constant K > 0
such that 11x(t)II < K for every t E [a, 3]. If now a = so < sl <
< sk = b is an arbitrary division of [a, /3] then by Corollary
1.6 (see also Remark 1.37 ) we obtain
Ilx(s;) - x(si
for every i = 1,
)If < K vars_, A + IIg(si) - g(si
,
)II
k and
k
]Ix(si) - x(si-I) II < K var,8 A + var13 y.
i=l
Hence vary x _< K varq A+vara g by the definition of the variation
because a = so < sl <
< sk = b was an arbitrary division of
0
[a, Q]
Assume that [a, b] C J is a compact interval. Let us denote
by BV([a, b]) = BV the set of all functions x : [a. b] --+ JRn of
bounded variation on [a, b] and set
II xII By = IIx(a)II + var6 X.
(6.3)
It is known that II II By is a norm on BV and that BV with the
-
norm I I- II B y is a Banach space.
Assume further that to E [a, b] is fixed.
Given x E BV([a, b]) we define
Tx(t)
J d[A(s)]x(s), t E [a, b].
(6.4)
a
By Corollary 1.34 the integral on the right hand side of (6.4)
exists for every t E [a, b] because A is of bounded variation on
[a, b] and x is evidently regular because we assume x E By.
V1. Generalized linear ODE'S
164
6.2 Proposition. If x E BV then Tx : [a, b] - R" given
by (6.4) is a function of bounded variation on [a, b], i.e. Tx E
BV([a, b]).
Moreover, the mapping T : BV - BV given by (6.4) is a
bounded compact linear operator on BV.
Proof. Assume that s1, s2 C [a, b], sl < s2. Then by Corollary
1.36 we have
IITx(s2)-Tx(sl )II
= II
f
82
,
d[A(s)]x(s)II <
f
82
IIx(s)II d(varta A).
S1
Therefore for an arbitrary division a = so < si <
< sk = b of
[a, b] we have
k
k
IITx(s;) - Tx(s;-,)II <
=
j
9
i=1
i=1
rb
b
IIx()II d(varo A) < II xII By
Ja
j-1
IIx(s)II d(varto A) _
d(varo A) = II xII By varA
because for every s E [a, b] we evidently have
11x(s)II < 11x(s)
- x(a)II + 11x(a)II -< IIx(a)II +varb x = II xIIBV
Hence passing to the supremum over all divisions a = so < sl <
<sk=bof [a, b] we obtain
vary Tx < vary AIIxIIBV
(6.5)
and Tx : [a, b] -* R" is of bounded variation on [a, b].
Clearly, the mapping T : BV -+ BV is linear. Using the same
argument as above we have
to
a
HTx(a)II = II f d[A(s)]x(s)II <
to
Ja
IIx(s)II d(varto A) S
VI. Generalized linear ODE's
165
IIxIIBVvara° A <
and this together with (6.5) yields
IITxIIBV = IITx(a)II +varQTx <2varQAIIxIIBV.
(6.6)
Therefore the operator T : BV - BV is bounded.
Finally, we have to show that T : BV -- BV is compact.
Assume that xk E BV, k = 1, 2,... is a bounded sequence in
By, i.e. Ilxk II $ C, k = 1, 2, ... for some constant C > 0. By
Helly's Choice Theorem the sequence (xk) contains a subsequence
(xk,) which converges pointwise to a function x E By, i.e
slim xk, (s) = i(s), s E [a, b].
00
(6.7)
Define
y(t) =
d[A(s)]x(s), t E [a, b];
it
then evidently y E BV. Let us set further
z, (s) = xk, (s) - x(s), s E [a, b].
Evidently zi E BV and by (6.7) we get
lim z, (s) = 0, s E [a, b].
(6.8)
Moreover,
Ilz,(s)II < IIz,(a)II + IIz,(s) - z1(a)II < IIz1IIBV =
= IIXk, -LIIBV < Ilxk,IIBV+IIxIIBV <C+IIxIIBV
(6.9)
for every s E [a, b]. We have also
a
ItTxki(a) - y(a)II = II
Jto
d[A(s)]zl(s)II <
l
a
to
Ilzi(s)II d(vart0 A).
VI. Generalized linear ODE's
166
From (6.8) and (6.9) we obtain by Corollary 1.32 the relation
slim IITXk,(a) - y(a)II = 0.
(6.10)
00
Further, for s1, s2 E [a, b], s1 < s2 we obtain
IITxk, (s2) - y(s2) - (Txk, (si) - y(si ))II = IITz!(s2) - Tz,(sl )II =
=
II
f
az
$z
d[A(s)]zi(s)II <-
9,
J
Ilzi(s)II d(var=o A)
91
and for an arbitrary division a = so < s1 <
< sk = b of [a, b]
also
k
II T xk, (si) - y(si) - (Txk1(si-1)
i=1
k
<-
b
Jr
Ilzi(s)II d(varto A) _
:.1 s
This yields as usual
IIz1(s)II d(var 0 A).
a
1
var' (TX k, - y) <
and
- Y(si-1))II <
jb
IIzl(s)II d(var,o A)
again (6.8), (6.9) and Corollary 1.32 imply
b
lm
1
00
IIzi(s)II d(varto A) = 0
a
and consequently
lim vara(Txk, - y) = 0.
r-00
This relation together with (6.10) yields
lim IITxk,
1- 00
- YIIBV = 0,
i.e. the sequence (Txk) contains a subsequence which converges
in BV (to y E BV) and the operator T is compact.
0
VI. Generalized linear ODE'S
167
6.3 Proposition. If [a, b] C J is a compact interval, to E [a, b]
then either
(A) the equation
x(t) =
Jto
d[A(s)]x(s) + f (t), t E [a, b]
(6.11)
admits a unique solution in BV([a, b]) for any f E BV
or
(B) the homogeneous equation
rr
x(t) =
J
d[A(s)]x(s), t E [a, b]
(6.12)
eo
admits at least one nontrivial solution in BV([a, b]).
Proof. The equations (6.11), (6.12) can be written in the form
x-Tx=f, x-Tx=O,
respectively, where T : BV -+ BV is the operator defined by
(6.4). This operator is compact by Proposition 6.2 and therefore
the alternative given in the statement follows immediately from
the well known Fredholm alternative for equations of this form
(see e.g. [21], [121] or any other standard textbook on linear
operator equations in Banach spaces).
Let us now consider the case when in Proposition 6.3 (A) occurs. This is equivalent to the situation when the homogeneous
equation (6.12) has only the trivial solution x = 0 in By.
6.4 Proposition. If [a, b] C J is a compact interval, to E [a, b]
then (6.12) has only the trivial solution x = 0 in BV if and only
if
(C)
I - [A(t) - A(t-)] = I - 0-A(t)
is regular for any t E (to, b]
VI. Generalized linear ODE'S
168
and
I + [A(t+) - A(t)] = I - A+A(t)
is regular for any t E [a, to).
Here I denotes the n x n identity matrix.
In other words the condition (C) is equivalent to (A) from
Proposition 6.3.
Proof. Define
e(t) = var'o A, t E [a, b].
[a, b] --> R is a nondecreasing function.
Assume that the condition (C) is satisfied and that x : [a, b] 1[8n, x E BV is a solution of (6.12). Then evidently x(to) _
0.
Suppose that to < b; then there is a c E (to, b) such that
(c) - S(to) < Z
For arbitrary sI, s2 E [to, c], s1 < 82 we have
II x(S2) - X(SI )II = II
f
92
d[A(s)]x(s)II <
91
f
92
IIx(s)IIde(s)
9
and, as usual, this yields
varto
x<
10c
IIx(s)IIde(s) =
c
= II x(to )II [e(to+) - 00)] + ali0+1
IIx(S)II df(S) <
o+6
< II xII BV([to,c1) Elio
(to + a)) _
= IIxIIBV([ta,C1)(e(c) - Wo+)).
VI. Generalized linear ODE's
169
Hence by the choice of c we obtain
IIx11BVc[eo,c]) <- 111x11BVc[eo,c]),
i.e. 11x11BV([eo,c]) = 0 and x(t) = 0 for every t E [to, c].
Let us assume that t* E (to, b] is the supremum of all c E (to, b]
such that x(t) = 0 on [to, c]. Then evidently x(t) = 0 for all
t E [to, t*). Using Theorem 1.16 we get
x(t*) - x(t*-) = lim J
d[A(s)]x(s)
lim [A(t*) - A(t* - 6)]x(t*) = A-A(t*)x(t*),
6-»0+
0 = x(t*-) = [I - 0-A(t*)]x(t*),
and the regularity of the matrix I - A-A(t*) yields x(t*) = 0.
If < b then in the same way as above we can show that there is
a c > t* in [a, b] such that x(t) = 0 on [to, c], but this contradicts
the assumption that t* is the supremum of all c E [to, b] for which
x(t) = 0 on [to, c]. Hence necessarily t* = b and x(t) = 0 on [to, b].
In a completely analogous way it can be shown that x(t) = 0 on
[a, to] and therefore x(t) = 0, t E [a, b] is the only solution of
(6.12) on [a, b].
To show the other implication let us assume that the condition
(C) is not satisfied.
Since A is of bounded variation on [a, b], there can be only a
finite set of points t* > to at which the matrix I - 0-A(t*) is not
regular because 110-A(t)11 > 1 only for a finite set of points in
(to, bJ. At all other points in this interval we have 110 A(t)11 < 2
and consequently the matrix I - i-A(t) has an inverse [I o'A(t)]-' at these points and I - A-A(t) is regular there.
VI. Generalized linear ODE's
170
Assume e.g. that there is a point t* E (to, b] such that the
matrix I - A-A(t*) is not regular and I - A-A(t) is regular
for every t E (to, t*). Then there exists y E R" such that the
algebraic system
[I - A-A(t*)]z = y
has no solution in R' .
Define f (t) = 0 for t E [a, b], t # t* and f(t*) = y. Then
evidently f E BV. Suppose that x is a solution of the nonhomogeneous equation (6.11) with this f on the right hand side. Then
we can show as above that x(t) = 0 for t E [to, t*) and we have
x(t*) = x(t*) - x(t*-) =
=
0-A(t*)x(t*) + f(t*) - f(t*-)
= A-A(t*)x(t*) + y,
and this means that for the value x(t*) we obtain
[I - 0-A(t*)]x(t*) = y.
By the assumptions given above such a value cannot exist in
R" and therefore also the equation (6.11) cannot have a solution
in BV for the given choice of f. Consequently, by Proposition
6.3 there is at least one nontrivial solution of the homogeneous
equation (6.12) and this proves the desired implication as well as
the statement of our proposition.
0
After this short excursion into the functional analysis of the
operator T : BV --> BV we turn our attention again to the
generalized linear differential equation (6.1).
The result given in Proposition 6.4 motivates the following
additional assumption on the matrix valued function A : J --+
L(R"):
I - [A(t) - A(t-)] = I - 0-A(t)
is regular for any t E J
VI. Generalized linear ODE's
171
and
I + [A(t+) - A(t)] = I - A+A(t)
(6.13)
is regular for any t E J.
By Propositions 6.4 and 6.3 the conditions (6.13) (or equiv-
alently (C)) assure that the equation (6.11) has a unique solution of bounded variation on [a, b] for an arbitrary choice of
f E BV([a, b] ), i.e. that the case (A) from Proposition 6.3 occurs.
6.5 Theorem. If J C R is an interval (finite or infinite),
A : J --+ L(R"), g : J --+ R" are functions of locally bounded
variation in J and (6.13) holds, then for every (i,to) E R' x J
there exists a unique solution x(t) of
dx
_
dT
D[A(t)x + g(t)]
(6.1)
satisfying the initial condition x(to) = x. This solution exists on
the whole interval J and is locally of bounded variation on J.
Proof. By definition we have
x(t) = i+
d[A(s)]x(s) + g(t) -- g(to)
(6.14)
to
for the solution of the equation (6.1). Put f (t) = Y+ g(t) - g(to).
Then f : J -- R" is locally of bounded variation on J and on
every compact interval [a, b] C J with to E [a, b] the equation
(6.14) is equivalent to the equation (6.11). Using Proposition 6.3
and 6.4 we obtain that there is a unique solution x E BV([a, b])
of (6.11) with this function f and therefore we have also the
existence of a solution of (6.14) (or equivalently of (6.1)) on [a, b].
This applies for any compact interval [a, b] C J and consequently
VI. Generalized linear ODE's
172
our statement holds because evidently the whole interval J can
be exhausted by intervals of the form [a, b] C J and the resulting
prolongation of the solution is unique by Propositions 6.3 and
6.4.
6.6 Corollary. If J C R is an interval (finite or infinite),
A : J -; L(R") a function of locally bounded variation in J and
(6.13) holds, then the homogeneous generalized linear differential
equation
dx = D[A(t)x]
(6.15)
dT
with the initial condition x(to) = 0, to E J has only the trivial
solution x(t) = 0 for t E J.
Proof. The result easily follows from Theorem 6.5.
6.7 Theorem. Assume that J C R is an interval, A : J -->
L(R"), g : J --> R" are of locally bounded variation in J and
(6.13) holds.
If x, y : J --> R" are solutions of (6.15) and a, /i E R, then
ax + fly is also a solution of (6.15).
If x : J -4 R" is a solution of (6.15) and z : J -+ R" is a
solution of (6.1) then x + z : J --4 R" is a solution of (6.1).
Proof. For every s1, s2 E J we have by definition
x(52) - x(51) =
f
32
d[A(s)]x(s)
and similarly for y : J - R". Hence by the linearity of the
integral we get
ax(s2) + IQy(s2) - ax(s1) - Ny(51) _
=a
f
S2
at
d[A(s)]x(s) + 0
f
a2
a1
d[A(s))x(s) _
VI. Generalized linear ODE's
=
raz
Ja,
173
d[A(s)](ax(s) + Qy(s)),
and ax + fly is a solution of (6.15) by the definition. For z : J ->
R" we have
82
Z(S2)
-
z(sl) =
f
d[A(s)]z(s) + 9(s2) - 9(31)
al
for every s 1, s2 E J. Hence
z(s2) + x(s2) - z(s1) - x(sl) =
=
J
d[A(s)](z(s) + x(s)) + g(s2) - g(s1)
l
f or every s1, s2 E J, and x + z is a solution of (6.1).
0
6.8 Theorem. If J C R is an interval (finite or infinite),
A : J - L(R") a function of locally bounded variation in J
and (6.13) holds, then the set of all solutions of the homogeneous
generalized linear differential equation (6.15) is an n-dimensional
subspace in the space BVio,(J) of all R"- valued functions on J
which are locally of bounded variation in J.
Proof. The linearity of the set of all solutions of the homogeneous generalized linear differential equation (6.15) is established
by Theorem 6.7. Every solution of (6.15) belongs to the space
BV10C(J) by Theorem 6.5.
If to E J then to a solution x of (6.15) we assign its value
x(to) E R". It is easy to see (by the uniqueness of the solutions
of (6.15)) that in this way a one-to-one map between R" and
the set of solutions of (6.15) is defined and this implies that the
solutions of (6.15) form an n-dimensional subspace in the space
BV10 (J).
0
174
VI. Generalized linear ODE'S
6.9 Theorem. Assume that J C R is an interval, A : J L(R"), g : J --> R" are of locally bounded variation in J and
(6.13) holds. Let z : J - R" be a fixed solution of (6.1).
Then every solution of (6.1) can be written in the form x + z
where x : J --> R" is a solution of (6.15).
Proof. The difference of any two solutions of (6.1) is a solution
of the homogeneous equation (6.15). The result is a consequence
of Theorem 6.7.
6.10 Remark. The results of Theorems 6.7 - 6.9 show that the
set of all solutions of the generalized linear differential equation
(6.1) has the usual linear structure which is known for the case
of classical linear differential equations.
The fundamental matrix
6.11 Theorem. Assume that A : J - L(R") is of locally
bounded variation in J and satisfies (6.13).If to E J then for
every n x n-matrix X E L(R") there exists a uniquely determined
n x n-matrix valued function X : J --+ L(R") such that
X(t)
X+
j
d[A(s)]X(s)
(6.16)
o
for t E J.
Proof. It is easy to see that X : J -- L(R") satisfies (6.16)
if and only if any column of X satisfies (6.15), i.e. if for every
k = 1, . . . , n the k-th column Xk of X satisfies
_
Xk(t) = Xk +
rt
Jto
d[A(s)]Xk(s).
The result now easily follows from Theorem 6.5.
The fundamental matrix
175
Let us introduce some notions analogous to the case of classical
linear ordinary differential equations.
A matrix valued function X : J - L(R") is called a solution
of the matrix equation
dX T
dr
= D[A(t)X]
(6.17)
if for every s1, $2 E J the identity
82
X(s2) - X(sl) = J d[A(s)]X(s)
91
holds.
A matrix valued function X : J -> L(R") is called a fundamen-
tal matrix of the equation (6.15) if X is a solution of the matrix
equation (6.17) and if the matrix X (t) is regular for at least one
value t E J.
6.12 Theorem. Assume that A : J -> L(]R") is of locally
bounded variation in J and satisfies (6.13). Then every fundamental matrix X : J - L(R") of the equation (6.15) is regular
for alit E J.
Proof. By definition the fundamental matrix X is a solution
of the matrix equation (6.17) and there is a to E J duch that
the matrix j C'
X(to) is regular. Assume that for some tl E
J the matrix X(t1) is not regular. Then there exist constants
cl , ... , c" E R, ck # 0 for at least one k = 1,...,n such that
n
E CkXk(tl) = 0
k=1
where X k denotes the k-th column of the matrix X. Since every
column of X is a solution of the homogeneous equation (6.15), by
176
VI. Generalized linear ODE'S
Theorem 6.7 the linear combination x(t) = Fk=l CkXk(t) is also
a solution of (6.15) where x(t2) = Ek_1CkXk(tl) = 0. Hence
by the uniqueness of solutions we have x(t) = 0 for t E J and
therefore x(to) = Ek= CkXk(to) = 0 . Consequently, we get
Ck = 0 f o r all k = 1, ... , n because the columns Xk(to) of the
regular matrix X(to) are linearly independent. This contradiction
0
proves that the matrix X (t) is regular for all t E J.
Now we reformulate the result given in Theorem 6.11.
6.13 Theorem. Assume that A : J -- L(IIfn) is of locally
bounded variation in J and satisfies (6.13). Then there exists a
uniquely determined n x n-matrix valued function U: J x J -+
L(Rn) such that
U(t, s) = I+
rt
Js
d[A(r)]U(r,s)
(6.18)
for t, s E J. For every fixed s E J the n x n-matrix valued
function U(., s) is locally of bounded variation in J.
Proof. For a given s E J the matrix U is a solution of the
equation
X (t) = 1+
rt
J
d[A(r)]X(r).
By Theorem 6.11 this solution exists for all t E J and is uniquely
determined for every fixed s E J. The local boundedness of the
variation of this solution on J is guaranteed by Theorem 6.5. 0
6.14 Theorem. Suppose that A : J --> L(R') is of locally
bounded variation in J and satisfies (6.13). Then the unique solution x : J --> Rn of the initial value problem for the generalized
linear differential equation
dx
)]
dr =DA(t
j
x
(6.15)
The fundamental matrix
177
with the initial condition x(s) = Y E R", s E J is given by the
relation
x(t) = U(t, 3)x,
tEJ
(6.19)
where U : J x J -> L(R") is given by Theorem 6.13.
Proof. The function x : J -> R" given by (6.19) is evidently of
locally bounded variation on J by Theorem 6.5. Hence for every
t E J the integral f ' d[A(r)]x(r) exists and we have
t
:
d[A(r)]x(r) = J d[A(r)] U(r, s)i =
= (U(t, S) - I)x = x(t) - x
for t E J and x(s) = U(s, s)x = Y. This means that x is a
solution of the initial value problem from the statement, and it
0
is of course unique by Theorem 6.5.
Now we give a survey of basic properties of the n x n-matrix
valued function U : J x J - L(R') introduced in Theorem 6.13.
6.15 Theorem. Suppose that A : J - L(R") is of locally
bounded variation in J and satisfies (6.13). Then the n x nmatrix valued function U : J x J -> L(R'), which is uniquely
determined by (6.18) has the following properties.
(a) U(t, t) = I for t E J,
(b) for every compact interval [a, b] C J there is a constant
M > 0 such that
IIU(t,s)II < M for all t,s E [a,b],
vary U(t, ) < M for t E [a, b],
vara
S) < M for S E [a, b],
VI. Generalized linear ODE'S
178
(c) for r, s, t E J the relation
U(t, s) = U(t, r)U(r, s)
holds,
(d) U(t, s) E L(R") is regular for every t, s E J,
(e)
U(t+, s) = [I + 0+A(t)]U(t, s),
U(t-, s) = [I - 0-A(t)]U(t, s),
U(t, s+) = U(t, s)[I + A+A(s)] -',
U(t, s-) = U(t, s)[I - 0-A(s)]-'
for t, s E J whenever the limits involved make sense,
(f) for t, s E J the relation
[U(t, s)]-' = U(s, t)
holds.
Proof. The property (a) is an immediate consequence of the
definition given by (6.18).
Denoting by Uk the k-th column of the matrix U we can write
(6.18) in the form
/'e
Uk(t, s) = ek + J d[A(r)]Uk(r, s)
for k = 1,... , n where ek is the k-th column of the identity matrix
I E L(R-).
Assume that s E [a, b) is fixed and s < t < b. For r E [s, b]
define A(r) by the relations A(s) = A(s) and A(r) = A(r-) when
179
The fundamental matrix
r E (s, b]. It is not difficult to check that for any y E BV([s, b])
and t E (s, b] we have
t d[A(r)]y(r) = J d[A(r)]y(r) + [A(t) - A(t-)]y(t) _
J.
s
= J t d[A(r)]y(r) + i A(t)y(t).
s
Therefore
t
Uk(t, s) = ek +
f
_
d[A(r)]Uk(r, s) + A-A(t)Uk(t, s)
s
and also
[
I - AA(t)]Uk(t, s) = e+
j
d[A(r)]Uk(r, s).
Hence
_
e
Uk(t, s) = [I - A-A(t)]`' (ek +
J
d[A(r)]Uk(r, s))
and also
II Uk(t, s)II : II[I - A-A(t)]-' IK(ileklf +
f
l
11Uk(r, s))11dvar'A
for every t E (s, b]. Since A is continuous from the left in (s, b]
the function var, A, r E (s, b] is also continuous from the left
in its domain of definition. Since the matrix valued function
[I - 0-A(t)] is assumed to be regular on J and A is of bounded
VI. Generalized linear ODE'S
180
variation on [a, b] there exists a constant C > 0 such that II [I
0-A(t)J-' 11 < C for all t E [s, bJ. Therefore we obtain
IIUk(t, s)II < CIlekIi +
f
t
-
_
IIUk(r, s))11dvarr A
and Corollary 1.43 yields the estimate
IIUk(t,s)II <
< <'
ClleklleCvar: A <
IleklleCvar; A
= Lk
if s < t < b. If s E (a, bJ is fixed and a < t < s then for
r E [a, s] we can define A(r) by the relations A(s) = A(s) and
A(r) = A(r+) if r E [a, s]. Essentially the same reasoning as
above makes it possible to use Corollary 1.44 for showing that
IIUk(t,s)II < ClleklleCvara A = Lk
for a < t < s with some constant C. Since this can be done for
every column of U we have the estimate
IIU(t, s)ll < M1,
t, s E [a, bJ.
Using this estimate we obtain for arbitrary t1, t2 with a < t1 <
t2 < b the inequality
rt2
IIU(t2, s) - U(tl, $)II =11
t,
d[A(r)]U(r, s)ll <
t2
<
Jt
lIU(r, s)Ildvar; A < M varti A
The fundamental matrix
181
and therefore
MlvarQA=M2
for S E [a, b].
If a<sl <S2 <bthen
U(t, S2) - U(t, S1) =
t
J92
d[A(r)]U(r, S2) - 1.2` d[A(r)]U(r, Sl )92
32
J
d[A(r)]U(r, sl) = -
d[A(r)]U(r, sl )+
J
91
31
+
j
d[A(r)](U(r, s2) - U(r, sl )),
2
and this means that the difference X (t) = U(t, S2) - U(t, sl ),
t E [a, b] satisfies on [a, b] the matrix equation (6.17) with the
initial condition
$2
X (S2) = - J
d[A(r)]U(r, S1),
91
and by Theorem 6.14 (when applied to the columns of X) we get
r 82
X(t) = U(t, S2) - U(t, sl) _ -U(t, S2)
J
d[A(r)]U(r, sl )
91
for t E [a, b]. Hence
(82
II U(t, $2) - U(t, S1 )II C IIU(t, S2)II
J
911
< Mi var;I A
IIU(r, sl )Ildvarr A <
VI. Generalized linear ODE'S
182
and therefore
varI U(t, ) < Mi vary A = M3
for every t E [a, b]. Putting M = max(M1, M2, M3) we obtain
(b).
By (6.18) which defines U we have
t
U(t, S) = I+ J d[A(T)JU(T, s)
4
=I+J
r d[A(T)]U(T, s) + J t d[A(T)]U(T, s) _
r
s
= U(r, S) +
f
d[A(T)]U(T, s).
By this relation the k-th column (k = 1, ... , n) Uk(t, s) of U(t, s)
is a solution od the initial value problem
dT
D[A(t)xJ,
x(r) = Uk(r, s)
and therefore by Theorem 6.14 we have
Uk(t, s) = U(t, r)Uk(r, s),
k = 1'...,n
and (c) evidently holds.
The property (d) is the content of Theorem 6.12.
The property (e) is a consequence of Theorem 1.16. Let us
prove for example the second identity in (e). By definition we
have
t
d[A(r)]U(r, s)
U(t, S) - U(t - b, s) =
t-a
The variation of constants formula
183
whenever b > 0, t, t - b E J. Theorem 1.16 now gives
U(t, S) - U(t-, s) = lien [U(t, S) - U(t - b, s)] _
Jim
j
t
d[A(r)] U(r, s)
and therefore
U(t-, s) = [I - 0-A(t)]U(t, s).
The other identities can be proved in the same way.
The identity in (f) is an easy consequence of the identity
U(t, s)U(s, t) = U(t, t) = I which holds by (c) and (a).
The variation of constants formula
O
6.16 Lemma. Assume that A : J -> L(R') is of locally
bounded variation in J and that (6.13) is satisfied. Let V : J -+
R' be of locally bounded variation in J and let K E L(R",Rm)
be an m x n-matrix valued function which is of bounded variation
on [a, b].
Then for U : J x J --> L(R") given by (6.18) in Theorem 6.13
the equality
t
r
d[h (r)](J d,[U(r, s)]cp(s)) _
to
to
t
= intt d,[K(s)]cp(s) +
J
d,[J dr[K(r)]U(r, s&(s)
tot
$
(6.20)
V1. Generalized linear ODE'S
184
holds for to, t E J.
Proof. Assume that t > to and define
V(r, s) = U(r, s) for to < s < r,
V(r, s) = U(r, r) = I for to < r < s.
If to < r < t then
L
and
d,[V(r, s)]cp(s) =
f
Jr
r
d,[U(r, r)]cp(s) = 0
r
d9[V(r,s)](p(s) =
to
t
f ds[U(r,s)]p(s)
to
and therefore for to < r < t we get
fds[U(r,s)](s) =
o
it
(6.21)
o
For p, s, a E [to, t] now define
W (p, o, s) = v (p, s)'p(a)
By (6.21) we have
f
r
to
ds[U(p, s)] p(s) =
and also
/t
Jto
f tot Ds[W (p, o, s)]
r
d[K(r)](J d,[U(r, s)]cp(s)) _
:o
= J eDr[ ftotD3[h(r)V(p,s)]4p(a)]
o
The variation of constants formula
185
t
=
J
Dr[J D3[K(r)W(P,o,s)]]
to
I.
to
Our goal is to interchange the order of integration in this double
integral. We will use the Tonelli-type Theorem 1.45 to this end.
Theorem 1.45 is here used in fact for the components of a vector
valued function.
is of bounded variation
By Theorem 6.15 the function
on [to, t].By Corollary 1.34 this yields the existence of the integral
fto D[li(r)V(p,s)] for any s E [to,t].
Therefore the integral
f t Dr[h (r)V (P, s)V(a)] = f t Dr[K (r)W(P, a, s)]
to
to
exists for every (a, s) E [to, t] x [to, t].
The next thing we have to do is to prove that the system of
is equiintegrable over [to, t] for
functions W (p, , ) = V (p,
p E [to, t]. Using (c) from Theorem 6.15 we obtain by (6.21) the
equality
t
J to
D,W(P, tr, s) =
f
p
d,[U(P,
to
P
= U(P, to) 10 d,
(6.22)
o
for p E [to, t]. Let 71 > 0 be arbitrary.
The function U(to, ) is of bounded variation in the interval
[to, t] (see (b) in Theorem 6.15), the integral fco d8[U(to, s)]cp(s)
exists and therefore by definition there is a gauge t on [to, t] such
that
,n
II EUto,Qi) - U(to,,3i-i))v(aj) - J
i=1
ds[U(to,s)](p(s)II < q
0,
(6.23)
VI. Generalized linear ODE'S
186
for every 6-fine partition
D = {,3o,Ql,Q1......
of [to, t] with tags oj in [,8j-1,13j]. Let P E [to, t] be given and
let D be a b-fine partition of [to, t] of the form described above.
Then there is an index k E { 1, ... , m} such that p E [/3k-1,1 k]
and
t
II (W(P,vj,/33) - W(p,aj,/3J-1 )) - J D,W(P,o,,3)II =
j=1
°
M
= II E(V(P,Q;)
-
V(P,/3;-1))V(oj)-
j=1
P
-U(P, to) f ds[U(to, s)]sv(s)II =
to
k-1
= II E(U(P, Q$) - U(P, Q,i-1))V(Oi )+
.1=1
P
+( U(P, P) - U(P, /3k-l))p(Qk) - U(P, to)
J"o
ds[U(to, s)]cp(s)II S
k-l
II U(P, to) II.II E(U(to, f3) - U(to, a;-l ))cP(Oj )j=1
_J
o
P) - U(to,/k-1))cp(ak)-
+II
/
"
k-1
d,[U(to, s)]V(s)II
(6.24)
The variation of constants formula
187
The Saks-Henstock Lemma 1.13 yields by (6.23) the estimate
k-1
II
$j -i
>(U(to, /3j) - U(to, P.i-1))v(o,) -
ds[U(to, s)]V(s)II :5,q.
J to
i=1
(6.25)
If Ok _< p then [Pk-1, PI C [Pk-1, Pk] C (Ok - b(Ok), Ok + b(Ok))
and again by the Saks-Henstock Lemma we get
p
ds[U(t0,
II(U(to, P) - U(to, Pk-i ))'(Ok) - J
77.
(6.26)
If P < Ok then the Saks-Henstock Lemma similarly yields
II(U(to,Qk) - U(to,P))V(Ok) -
f
ak
d9[U(to,sP(s)II
<i
P
and
p
II(U(to, P) - U(to,
ds[U(to, s)]cp(s)II <
Pk-1))w(Ok) - f
Ok
< II(U(to, Pk) - U(to, Pk-1))'(Ok) - J
ds[U(to, s)](s)II+
Qk-l
/k
de[U(to, s)] p(s)II < 2q.
+II(U(to, ,3k) - U(to, P))W(Ok) lp
(6.27)
Taking into account the inequalities (6.24) - (6.27) we arrive at
"I
t
IIE(W(P,O;,P;)-W(P,u,Pj-1))-Jo
i=1
< 377IIU(P,to)II < 317M
D.W(P,O,s)II <
V1. Generalized linear ODE'S
188
where M is the bound of U on [to, tJ x [to, t] given by (b) from
Theorem 6.15. Since the partition D was independent of p and
r, > 0 was arbitrary, this last inequality shows the required equiintegrability of the system of functions W (p, , ) = V (p, )I V(.)
This enables us to utilize Theorem 1.45 to state that
J t dr[K(r)](ftor
d5[U(r, s)lV (s))
o
= j Dr[ f tot Dr[K(r)V (p, s)Jp(o)J _
o
f
Dr
to
t
D. [K(r) W(p, a, s)]]
to
t
Jto Ds[JtoDr[K(r)W(p, a, s))j
By the definition of the function W we have
f tot Dr[K(r)W (p, a, s)] = J t Dr[K(r)V (p, s)V(a)) _
o
= J Dr[K(r)V (p, s)(a)] + f Dr[K(r)V (p, s)(a)) _
S
S
o
= j Dr[K(r)]cp(a) + f t Dr[K(r)U(p, s)(a)] _
S
S
o
=
f
S
to
dr[K(r)]cp(u) + f t dr[K(r)]U(r,
S
_ [K(s) - K(t2&(a)] +
J8
dr[K(r)]U(r, s)V(a)
The variation of constants formula
189
This yields
t
J to
Ds [J t Dr[K(r)W (p, a, s)]] =
to
ds[I.
L d3[K(s)] p(s) +
dr[K(r)]U(r, s)J(p(s),
t
s
t
i.e.
dr[K(r)]((r
J
t
to
it
o
ds[K(s)] p(s) +
ds[U(r,
o
f
t d8[
to
f t dr[K(r)]U(r, s)Jsv(s)
s
and (6.20) is proved for the case t > to. The case t < to can be
treated similarly.
O
6.16a Corollary. Assume that A : J -> L(R") is of locally
bounded variation in J and that is satisfied (6.13) is satisfied.
Let V : J --+ R" be of locally bounded variation in J.
Then for U : J x J - L(R") given by (6.18) in Theorem 6.13
the equality
t
J to
d[A(r)](for ds[U(r,
t
J t d[A(s)] p(s) + ittot d[U(t, s)] p(s))
o
holds for to, t E J.
Proof. If we set K = A in Lemma 6.16 then by (6.18) we have
J
is
t dr[K(r)]U(r, s) =
Js
t
dr[A(r)]U(r, s) = U(t, s)
-I
VI. Generalized linear ODE's
190
and
f ` d,
to
[1,
t
dr[h (r)]U(r, s)]ip(s)
= f t d9[U(t, s) - I]y(s) =
to
t
J to
ds[U(t, s)]ip(s)
This proves the formula given in the corollary.
0
6.17 Theorem (variation of constants formula). Assume that
A : J --a L(R") is of locally bounded variation in J and that (6.13)
is satisfied. Then for every to E J, x E R" and g : J --> Rn of
locally bounded variation in J the unique solution of the initial
value problem
dx
= D[A(t)x + g(t)],
d7
x(to) = x
(6.28)
can be written in the form
x(t) = U(t,to)x+g(t)-g(to)- f d9[U(t,s)](g(s)-g(to)) (6.29)
to
for t E J where U : J x J -p L(R') is given by (6.18) in Theorem
6.13.
Proof. The function x(t) given in (6.29) is well defined by the
properties of the function U described in Theorem 6.15. Assume
that t > to, t E J. Using (6.18) and (6.29) we obtain
t
f to/
d[A(r)]x(r) =
J
t
d[A(r)]U(r, to)x+J d[A(r)](g(r)-g(io))to
t o
-J
r
t
to
d[A(r)] f
d,[U(r, s)](g(s) - g(to)) _
to
The variation of constants formula
_ (U(t, to) - I)+
-J
d[A(r)]
J to
j
191
d[A(r)]cp(r)o
d,[U(r, s)]sp(s)
with cp(s) = g(s) - g(to) for s E J. The function cp evidently
preserves the property of g being of locally bounded variation on
J. Therefore Corollary 6.16a can be used for the last integral in
this relation to obtain
t
Jto
d[A(r)]x(r) = U(t,to)x - Y +
ft
d[A(r)]cp(r)-
to
rt
rt
- J d[A(r)]cp(r) - J d[U(t, s)]cp(s)) _
to
to
= U (t,
to )- -
L d[U(t, s&(s))
= -x - (g(t) - g(to)) + x(t),
and this means that
at
x(t) = +
j
d[A(r)]x(r) + g(t) - g(to)
holds for t E J, t > to and the function x given by (6.29) is a
solution of (6.28) for these values of t. For the remaining case
t E J, t < to the result can be proved in a completely analogous
way.
0
VI. Generalized linear ODE'S
192
6.18 Lemma. If A : J -p L(R") is of locally bounded variation in J , (6.13) is satisfied and X : J -4L(R") is an arbitrary
fundamental matrix of (6.15) then
U(t,s) = X(t)X-'(s)
(6.30)
for every t, s E J where U : J x J -> L(R") is given by (6.18) in
Theorem 6.13.
Proof. By Theorem 6.13 the matrix X (s) is regular for every
s E J and therefore the product X(t)X' (s) is well defined and
regular for t, s E J. Since X is a solution of the matrix equation
(6.17) we have
d[A(r)]X(r) = X(t) - X(s)
and this means that
a
I
d[A(r)]X (r)X -' (s) = X(t)X-1(s) - X(s)X-1(s) _
= X(t)X-'(s)
- I.
Hence the product X (t)X -' (s) satisfies the equation (6.18) and
the uniqueness of U stated in Theorem 6.13 yields the result.
6.19 Corollary. If A : J - L(R') is of locally bounded
variation in J, (6.13) is satisfied, to E J, Y E R", g : J R" is of locally bounded variation in J and X : J -- L(R')
is an arbitrary fundamental matrix of the equation (6.15) then
the unique solution of the initial value problem (6.28) can be
represented in the form
x(t) = g(t) - g(to)+
The variation of constants formula
+X(t) (X-1(to)i -
d9[X-'(s)](9(s)
to
- 9(to)))
193
(6.31)
Proof. The result follows immediately from the variation of
constants formula (6.29) if the equality (6.30) from Lemma 6.18
is taken into account.
6.20 Example. Consider the linear differential system with impulses
x = F(t)x,
(6.32)
Oxjt; = x(ti+) - x(ti) = Bix(ti)
(6.33)
on an interval J C R where F : J -- L(R") is an n x n matrixvalued function which has locally integrable (in the sense of Le-
besgue) components on J. We assume that t; E J, ti < ti+j
for i = 1, 2,... and that Bi E L(Rn) are n x n matrices for
i = 112.... such that I + Bi are regular matrices for i = 1, 2, ... .
For the concept of a differential system with impulses see Chapter
V. Linear differential systems with impulses of the form (6.32),
(6.33) have been considered by A.M. Samojlenko and others (see
e.g. [116], [117], [118], [120] etc.).
Given a fixed a E J define
t
-
A(t) _ f F(s) ds + > B;Hg; (t)
a
(6.34)
i=I
f o r t E J where Ht; (t) = O fort < ti and Ht; (t) = 1 fort > ti.
The integral in (6.34) is the integral in the Lebesgue sense.
If it is supposed that in every compact interval [a, b] C J the
set of points ti such that ti E [a, b] is finite, then the matrix valued
function A given by (6.34) is locally of bounded variation on J,
continuous from the left (in spite of the definition of the function
Ht;), and we have
I+O+A(t)=Iift54 ti,
VI. Generalized linear ODE's
194
I+ A+A(t) =I+ Bi ift=ti.
Hence the matrix I + A+A(t) is regular for every t E J and of
course also I - 0-A(t) = I is regular for every t E J because A
is continuous from the left. Therefore A satisfies (6.13).
By the results stated in Chapter V we know that the linear differential system with impulses of the form (6.32), (6.33) is equivalent to the homogeneous generalized linear differential equation
dx
dr = D[A(t)x].
(6.35)
Let us consider the fundamental matrix U : J x J --. L(IR") of
(6.35) which satisfies
U(t, s) = I+
J
d[A(r)]U(r, s)
9
(see Theorem 6.13)).
If t,r E (ti,ti+I] then
U (t, s)
U(T, s) +
j
d[A(r)]U(r, s) _
t
= U(r,s) +1, F(r)U(r,s)dr,
r
U(t,s)
(6.36)
: J x J -4 L(R") is the classical fundamental matrix of
the linear system of ordinary differential equations
where Sl
x = F(t)x
The variation of constants formula
195
which is defined by the relation
dt
r)) = F(t)4(t, -r),
t) = I.
(6.36) and the property (e) from Theorem 6.15 imply for t, r E
(ti,ti+1] the relation
U(t, S) = urn T -> ti+4k(t, r)U(T, S) = 4(t, ti)U(ti+, S) =
=
ti )[I + 0+A(ti)]U(ti, S) = .4(t, ti )[I + Bi]U(ti, s).
This procedure can be repeated for U(ti, s) to obtain
U(t, s) = (P(t, ti)[I + Bi]U(ti, s) =
= .4(t, ti)[I + Bi]c(ti, ti-1)[I + Bi-1 ]U(ti-1, S).
A continuation of this procedure leads to the relation
U(t, S) =
ti )[I + Bi]lb(ti, ti-1)[I + Bi-1 ] ... [I + Bj]4(tj, s) =
_ (1(t, ti) II [I + Bk]4'(tk, tk-1 )[I + Bj]4)(tj, S)
k=j+1
whenever s E J, s < t, s E (tj-1i tj] for some j or j = 1 ifs < t1.
If t < s, s E [4, 4+11, t E (tj_1i tj] or t < tl then by (f) from
Theorem 6.15 we obtain
U(t,s) = [U(s,t)]-' =
_ [(k(s, ti )[I + Bi]lt(ti, ti-1)[I + Bi-1] ... [I + Bj]4(tj, t)]-' =
[I+ Bi]-'[(I(s,ti)]-' _
I(t, tj )[I + Bj]-' ... [I + Bi]-14(ti, s).
VI. Generalized linear ODE'S
196
Now by Theorem 6.14 the solution of the initial value problem
(6.35) with the initial condition x(s) = i can be expressed in the
form
x(t) = U(t, s)x =
i
_ ob(t, ti) 11 [1+ Bk]Ctk, tk-1)[I + B;]t(tj, s)x
k=j+1
fort>_s,tE(ti,ti+i],SE(tj_1iti]orj=1ands<t1.
In this way we obtain the formula for solutions of the initial
value problem for the linear differential systems with impulses
(6.32), (6.33) which is known from the work of A.M. Samojlenko
and N.A. Perestjuk [120] and can be also found in a slightly differ-
ent setting in Chap. 4 of [107]. It should be noted that if Bi = 0
for all i then we have an ordinary linear differential equation and
the fundamental matrix in this "impulse" setting coincides with
the classical one.
It is clear that the results presented in this chapter apply also
to the case of a nonhomogeneous differential system with impulses
of the form
i = F(t)x + h(t),
Oxlti = x(ti+) - x(ti) = Bix(ti) + ai
where h : J -> R' is locally integrable in J and ai E R", i =
112
... .
Boundary value problems
Let us consider the following problem:
Given a generalized linear ordinary differential equation
dx = D[A(t)x
dr
+ g(t)]
(6.1)
Boundary value problems
197
with A : [a, b] -> L(R") satisfying the condition (6.13) find a
solution of (6.1) which satisfies the side condition
b
1.
d[K(s)]x(s) = r
(6.37)
where K : [a, b] -+ L(R", R"`) is an in x n-matrix valued function
of bounded variation on the interval [a, b] and r E R'"
This is a general form of a boundary value problem for generalized linear ordinary differential equations.
It should be mentioned that a side condition of the form
b
Mx(a) + Nx(b) +
J
d[L(s)]x(s) = r
(6.38)
with M, N E L(R", R' ), L : [a, b] -+ L(R", R"') of bounded
variation on the interval [a, b] and r E H8"` assumes the form
(6.37) if we take
K(t)=-M+L(a) fort=a
K(t) =L(t)
K(t) =N + L(b)
for a < t < b
for t = b.
This follows from an easy computation. Moreover, if L is a constant function then (6.38) has the conventional form
Mx(a) + Nx(b) = r.
Assume that ' : [a, b] --> L(R") is the fundamental matrix of
the equation
dx = DA(t)x
dT
such that %P (a) = I.
VI. Generalized linear ODE'S
198
6.21 Lemma. The boundary value problem (6.1), (6.37) has
a solution if and only if
7T {j6 d[K(t)]9(t) -
f ab d[K(t)](t)a f d[((s))'](s)
=
,YT?.
(6.39)
holds for every -y E R,n such that
b
7T f d[K(t)]W(t) =
(6.40)
0
a
where by 7T the transpose to 7 E R"' is denoted.
Proof. By the variation of constants formula (6.31)
x : [a, b] --> Rn is a solution of (6.1) if and only if
x(t) _ W(t)c + 9(t) - 9(a) -
(t)
j
d[{{s))']{9{s) - 9(a))
(6.41)
for some c E Rn and all t E [a, b]. Inserting (6.41) into the left
hand side of the side condition (6.37) we obtain
f
b
(f d[K(t)]`'(t))c + f d(K(t)](9(t) - 9(a))b
d[K(t)]x(t) _
a
j
b
b
a
d[K(t)]4'(t)
a
f
t
9(a))
a
_ (f b d[K(t)]WY(t))c + f b d[K (t)](9(t) - g(a))a
b
a
j ad[K(t)]W(t) fd[(,P(s))-']g(s)+
Boundary value problems
199
b
+jd[K(t)j4,(t)[(IP(t))-' - I]g(a) _
b
b
Ja
d[K (t)]`I'(t))c +
- f ab d[K(t)]%'(t)
ft
f d[K(t)]g(t)a
-
d[(`f'(s))-' ]g(s)
a
f
b
d[K(t)]`I'(t)g(a).
a
This implies that x is a solution to the boundary value problem
(6.1), (6.37) if and only if x is given by (6.41) where c E R" is
such that
(jb
d[K(t)]`I'(t) c = r +
+
- f ab d[K(t)]g(t)
f
a
f d[K(t)](t)g(a)a
ft
b
d[K
a
d[(`4(s))-' ]g(s).
(6.42)
Using the well known facts about linear systems of algebraic equations we conclude in particular that the boundary value problem
(6.1), (6.37) possesses a solution if and only if the right hand
side of (6.42) is orthogonal to the null space of the adjoint linear
system (6.39), i.e. if and only if
'yT r
-
= 'yT
-f
b
a
f
b
d[K(t)]`I'(t)g(a) +
a
f
a
d[K(t)]W(t) f
b
d[K(t)]g(t)-
t d[(I'(s))-']g(s)
a
holds for every y E R" for which (6.40) is satisfied. Since
-tT
-
jbd[K(t)]I1(t)g(a)}
=
VI. Generalized linear ODE'S
200
by (6.40), the assertion of the lemma follows readily.
Now we modify slightly the condition of solvability of the
boundary value problem (6.1), (6.37) given in Lemma 6.21.
6.22 Lemma. The boundary value problem (6.1), (6.37) has
a solution if and only if
f ab
ds[f b dt['YT K(t)'J(t)('F(s))-' l9 (s) +'YT r= 0
s
for every solution -y E I8" of the equation (6.40)
Proof. Let us turn our attention to the relation (6.39). By the
identity (6.20) from Lemma 6.16 we have
f
a
b
d[K(t)]W(t) fa t d[(`F(s))^')g(s) =
6
b
b
= f d[K(s))9(s) + f d9[f
a
a
s
Inserting this into the left hand side of (6.39) we obtain
,?,T
- fb
11b
d[K(t)J9(t)
a
'YT
d[K(t)l W'(t) f
t
a
-1 da[f b dt[_'T
b
a
9
and this proves the statement.
The results given in Lemmas 6.21 and 6.22 represent solvability conditions for the boundary value problem (6.1), (6.37). They
form the basis for more detailed studies of boundary value problems of the type (6.1), (6.37) from the point of view of functional
Boundary value problems
201
analysis. The reader interested in a more detailed theory of such
boundary value problems (the adjoint problem, Green's function,
etc.) should consult e.g. [155], [154]. It is worth to mention that
the method of proving Lemma 6.21 can be used for the construction of a solution of the boundary value problem provided the
condition of Lemina 6.21 or that of Lemma 6.22 is satisfied.
CHAPTER VII
PRODUCT INTEGRATION AND GENERALIZED
LINEAR DIFFERENTIAL EQUATIONS
Multiplicative integration has a long history. Vito Volterra
[169] invented product integration at the end of the last century.
Product integration was studied by L. Schlesinger [122], [123] in
connection with linear differential equations. The present state
of art is well described in the book [20] of J.D. Dollard and C.N.
Friedman and in a good survey [32] of R.D. Gill and S. Johansen
which tends to applications in statistics and Markov processes.
This was also one of the points in the book of V. Volterra and B.
Hostinsky [170] from 1938.
The original approach to product integrals is based on partial
integral products which are similar to Riemann integral sums.
This way leads also to a "Lebesgue type" product integration by
the extension of the integral product of "step functions", see e.g.
[125] or [20].
Here we present the theory when the classical Riemann approach is replaced by the Kurzweil - Henstock concept of b-fine
partitions which are fine with respect to gauges. This idea was
used for the first time by J. Jarnik and J. Kurzweil in [53] and we
present here a certain refinement of their approach on the basis
of [148].
We denote by L(R") the set of all linear operators from R1 to
is the
R' (the set of all n x n-matrices) and assume that 11
202
11
VII. Product integration
203
operator norm on L(R") which corresponds to the norm used in
the space R".
Let an interval [a, b] C R, -oo < a < b < +oo be given. Let 3
be the set of all compact subintervals in [a, b], i.e. of intervals of
the form [a, 0], where a < a < 3:5 b.
Assume that a function V : [a, b] x 3 - L(R") is given; V is
an n x n-matrix valued point-interval function. Let us assume
that a partition
D={(ri,J,),j=1,...,k}={(ri,[ai-i,a.i]),j,...,k}=
= {ao,7,J,ai,...,ak-1,Tk,akI
is given where
a=ao <ai
ai-i <ri <ai, j=1,2,...,k
and
[ai-i,ai] C [ri - 5(ri),ri +b(ri)], j = 1,2,...,k.
For the function V : [a, b] x 3 - L(R") and a given partition
D of the interval [a, b] we denote
P(V, D) = V (7-k, Jk)V (rk-1, Jk-l)
... V(rl, Jl) _
= V(rk, [ak-1, ak])V(rk-1, [ak-2, ak-1]) ... V(ri, [ao, a1]),
the ordered product of elements of L(R").
204
VII. Product integration
7.1 Definition. A function V : [a, b] x 3 --> L(R") is called
Perron product integrable if there exists Q E L(R") such that for
every e > 0 there is a gauge b : [a, b] --> (0, +oo) on [a, b] such
that
IIP(V, D) - Q1I <
(7.1)
for every b-fine partition D of [a, b).
Q E L(R") is called the Perron product integral of V over [a, b]
and we use the notation Q = fa V(t, dt).
The following statement is based on the fact that L(R") with
the operator norm is complete because it is a Banach space. The
proof of the result follows exactly the lines of the proof of Theorem
1.7.
7.2 Proposition. Let V : [a, b] x 3 - L(R") be given. The
following two conditions are equivalent.
(i) There is a Q E L(R z) such that for every e > 0 there is a
gauge b : [a, b] -- (0, +oo) on [a, b] such that II P(V, D) - Q11 < e
for every b-fine partition D of [a, b).
(ii) For every s > 0 there is a gauge b : [a, b] -- (0, +oo)
on [a, b] such that IIP(V, D1) - P(V, D2)CI < e for every b-fine
partitions D1i D2 of [a, b).
7.3 Remark. It is easy to see that the concept of the Perron
product integral is defined in a manner very similar to Definition
1.2 (or 1.2n) where the generalized Perron integral is described.
The role of summation is replaced by multiplication and because
the product of two matrices is not commutative in general, the
order of multiplication in P(V, D) is fixed.
Now we turn our attention to some more specific requirements
concerning the n x n-matrix valued point-interval function V :
[a, b] x 3 -* L(R"). These conditions play an essential role in our
205
VII. Product integration
subsequent studies:
(7.2)
V(r, [7-,,r]) = I for every r E [a, b],
where I E L(R") is the identity operator in L(R"),
for every r E [a, b] and ( > 0 there exists or > 0 such that
11V(T, [a,,8]) - V(r,
[a,T])II < C
(7.3)
for alla,/3 E [a,b], r-o,<a<T<Q<r+o;
for every r E [a, b) there is an invertible V+(r) E L(R")
such that
llm
(i- r+
IIV(r,[r,/3])-V+(T)II =0, i.e.
lim V(r,[r,$])=V+(T),
#-r+
(7.4+)
and
for every r E (a, b] there is an invertible V_(r) E L(R") such
that
lien II V(T, [a, r]) - V_(r)II = 0, i.e.
ar-
lim V(r, [a, r]) = V_(T).
(7.4-)
7.4 Definition. If the function V : [a, b] x 3 --+ L(lR') satisfies (7.2), (7.3),(7.4-) and (7.4+) then we say that V satisfies
condition C.
7.5 Lemma. Assume that the function V : [a, b] x,1 -+ L(R")
satisfies the condition
for every r E [a, b] and C > 0 there exists o, > 0 such that
IIV(r,[a,/1)-III <C
for all a,/0E[a,b],r-a<a<T<Q<T+a.
(7.5)
VII. Product integration
206
Then the function V satisfies condition C.
Proof. Since we have
V(T, [a, /3]) - V(T, [T,Q])V(T, [a, T]) =
=V(T,[a,Q])-V(T,Fr,NI)-V(T,[a,T])+I-(V(T, [T,,3]) - I)(V(T, [a, Tl) - I) =
= V (r, (a, al) - I + I - V (T, [T, Q]) + I - V (T, [a, T l )---(V(T, [T, /3]) - I)(V(T, [a, T]) - I)
we have also
IIV (T, [a, a]) - V (T, [T, Q] )V (T, [a, T]) II <
s IIV(T,[a,pl)-III +IIV(T,[r,01)-III +IIV(T,[a,r])-III+
+IIV(T,[r,01)-IIIIIV(T,[a,r])-III
This inequality implies that if the function V : [a, b] x 3 - L(R)
satisfies the condition (7.5) then
IIV (T, [a, N]) - V (T, [T, /3])V(T, [a, T])II < 3( +
(2
for all a, 0 E [a, b], r - a < a < r < fj < T + a and this implies
that (7.3) is fulfilled.
Moreover, (7.5) evidently yields limp-,+ V(T, [r,,3]) = I for
T E [a, b) and lima-,_ V (r, [a, T]) = I for r E (a, b] and therefore
(7.4+) and (7.4-) hold with V+(T) = V_(T) = I and it is easy to
see that also (7.2) is satisfied. This means that the condition C
is satisfied and the lemma is proved.
0
VII. Product integration
207
7.6 Lemma. Assume that for the function V : [a, b] x 3 -4
L(Rn) the condition C is satisfied. Then for every r E [a, b]
there exists a al = al (T-) > 0 such that V(T,[a,/3]) E L(Rn) is
invertible ( the inverse matrix (V(T, [a,/3]))-' E L(Rn) exists )
provided a,/3E [a,b],T-al <a<r</3<r+al.
Proof. Let r E [a, b] be given. For a given C > 0 let a, (T) > 0
be such that we have
II V(7, [a, N]) - V (T, [T, #])V(7, [a, T ] )II < C
(7.6)
and
IV(r, [T, Q]) - V+(r)II < C, IIV(T, [a, T])
provided
- V (,r) 11 < C
(7.7)
E [a, b], r - al < a < r _< /3 < 'r + al. The
possibility of choosing such a al > 0 is ensured by (7.3), (7.4-)
and (7.4+).
Since V_(r) and V+(T) are invertible operators (we define
V_(a) = I and V+(b) = I ) their product V+(T)V_(T) is also
invertible with (V+(T)V_(T))-1 = (V_(r))-1(V+(T))-'.
It is easy to verify that
V (-r, [a, l3])
- V+(T )V (T) =
= V (,r, [a, /3]) -(V (T, [T, /3] )V (T, [a, r] )+
+(V (T, [r, fl]) - V (r))(V (T, [a,
r]) - V (T)) + (V(T, [r, /]) - V+(r))V (T)
Hence
II V (T, [a,,3]) - V+(T)V_(T)II <
<- 11 V (r, [a, 0]) - V (,r, [T, l3])V(T, [a, T J)II+
208
VII. Product integration
+IIV(T, [T, p]) - V+(T)IIIIV(r, [a, T]) - V -(T)11+
+IIV+(T)IIIIV(T, [a, r]) - V(0II+IIV(T,[T,/3])-V+(T)IIIIV (T)II,
and if a,/lE[a,b], r - aY < a<r_</3<r+a, then by (7.6)
and (7.7) we obtain
IIV(T, [a,Q]) - V+(T)V_(T)II :
<_ C + CZ + C(IIV+(T)II + IIV (T)II) =
_ ((1 + C + 11 V+ (7) 11 + 11 V- (7) 11).
Since t; > 0 can be chosen arbitrarily small, the operator
V(r, [a, /3]) is invertible. (It is sufficient to take ( > 0 such that
C(1+(+IIV+(T)II+IIV_(r)II) >
II(V+(T))-'(V_(T))-'11-'.)
If e.g.
a = T < /3 then the result comes immediately from the second
inequality in (7.7) for a sufficiently small C. The case a < r =
0 is a consequence of the first relation in (7.7), and finally for
a = r = /3 we have V (-r, [a, /3]) = I and V (,r, [a, /3]) is evidently
invertible.
7.7 Lemma. Assume that for the function V : [a, b] x 3
L(R") the condition C is satisfied. Then for every T E [a, b] there
exists a a2 = 0'2(r) > 0 such that
IIV(T,Ice, T])II < 11 V- (T)II+ 2I1 (V-(7))
II,
II (V (T, [a, r]))-' II < 2II(V (T))-' N
(7.8)
for all a E [a, b] such that r - a2 < a < r and
IIV(T, [r,l3])II < IIV+(r)II + 11[(V+(T))`
II (V(T, [T, /3]))-' 11 < 2I1(V+(T))-' 11
II,
209
VII. Product integration
for all /3 E [a, b] such that r < /3 < T + o2.
Proof. We prove the result stated in (7.8), the proof of (7.9)
is analogous. If r = a then there is nothing to prove because
there is no a E [a, b] such that a < r. Assume therefore that
r E (a, b]. The matrix V-(7) E L(Rn) is invertible by (7.4-). If
now B E L(R") and
JIB - V-(T)II < 1 ll(V (T))-' II
then B-' E L(R") exists (see the general result given in [D S
,VII.6.1] and
00
B-' = (V (T))-' E[(V (T) - B)(V (T))-']k
k=0
Therefore
00
IIB-
II = II (V (T ))-' II [II(V (T) - B)IIII(V(T))-'
Illk
k=O
II(V(r))-'II
__
1 - II(V-(r)
- B)IIII(V(r))-1II
Since in this case IJ(V-(r) - B)IIII(V-(r))-'Il < 2 we have
1 - li(V(T) - B)IlII(V(T))-' II >
2
and therefore
IIB-'it
< 211(V-(r))-111.
(7.10)
VII. Product integration
210
By (7.4-) there is a o,2 (r) > 0 such that if a E [a, b], r - Or2 <
a < r, then
IIV(r,[a,r])-V (T)II <
II1(V(r))-1II.
(7.11)
Hence (7.10) yields
II (V (r,
[a,T]))-1II < 211(V (T))-1II
and (7.11) implies also
11V (r, [a, r])11 <_ IIV(r, [a, r]) - V_(r)11 + IIV(r)11 <
< 1 II(V (T))-111 + IIV (r)II
provided r - or2 < a < r, i.e. (7.8) holds for such a E [a, b].
For the case r E [a, b) we can find a Q2 (r) > 0 such that (7.9)
holds if ,0 E [a, b], r < Q < r +
Taking Q2 = min(o2 , Q2) we
obtain the statement of the lemma.
0
0'2.
7.8 Theorem. Let V : [a, b] x 3 -- L(Rn) be Perron product
integrable over [a, b] with fa V (t, dt) = Q where Q E L(Rn) is
invertible and assume that the condition C is satisfied.
Then there exists a constant K > 0 such that for every s E
[a, b] the Perron product integrals Ja V(t, dt), f9 V (t, dt) exist,
the equality
6
s
6
jjV(t,dt)jIV(t,dt) = 11V(t,dt)
a
a
holds and
s
s
11 jj V (t, dt)II < K, 11([J V(t, dt))-' II < K.
a
a
VII. Product integration
211
Proof. Let C > 0 be arbitrary. Let bo : [a, b] --> (0, oo) be a
gauge on [a, b] such that ba(t) < min(ol(t), C2 (t)) for t E [a, b]
where al (t), U2 (t) are given in Lemma 7.6 and 7.7, respectively,
and such that
II P(V, D) - QII <
211Q-1 II-l
(7.12)
holds for every bo-fine partition D of the interval [a, b]. Assume
further (by (7.3)) that
(7.13)
II V(7-, [a, 9]) - V (T, [r,13])V(r, [a, r])II < C
for all y,a,/3E[a,b],r-bo(r)<a<r</3<r+bo(r).
The proof of the theorem will be divided into several steps.
First we prove the following assertion.
For every r E [a, b] there is a Ki(r) > 0 such that
ifs E (r - bo (r ), r] fl [a, b] and Dl is a bo fine partition
of [a, s] then
max{IIP(V,D1)II,II(P(V,D1))-1II) <- K1(r),
(7.14-)
and
ifs E (,r, ,r + bo(r)] fl [a, b] and D2 is a bo fine partition of [s, b]
then
max{IIP(V,D2)II,II(P(V,D2))-'II} 5 K1(r).
(7.14+)
For proving this statement let us first mention that because we
have 60 (-r) < Ql (r), Lemma 7.6 implies that V(r, [a,,8]) E L(R")
is invertible for every r, a,,8 E [a, b] such that r - bo(r) < a <
r <,8 < r + bo(r). In order to prove (7.14-) let D3 be a bo-fine
partition of the interval [r, b]. Let
Dl = {ao,rl,al,...,a,-,,rl,a,}
212
VII. Product integration
be a So-fine partition of [a, s] and let
D3 = 101+1, T1+2, a'1+2, ... , ak-1, Tk, ak }
be a bo-fine partition of [t, b]. Set
D = {ao, rl, al, ... , at-1, 7,1, a, = s, rt-1 = T,
a1+1 = T, 'rt+2, al+2, ... , ak-1, Tk, ak }.
In the sequel we will use the notation D = D1 o (r, [s, r]) o D3 for
this construction of a partition of the interval [a, b]. This partition
D is in fact the union of ordered finite sets in which the ordering
preserves the ordering of the components DI, {s, T, r}, D3; o denotes the union of ordered sets. It is evident that D is a bo-fine
partition of [a, b] and that V (ri, [ai-1 i a,]) E L(R") is invertible
f o r every i = 1, ... , k. Therefore
P(V, D1) = V(r,, [al-1, 011)V(r1-1, [at-2, at-1]) ...
V (r1, [ao, a1]) E L(]R")
and
P(V, D3) = V(Tk, [ak-1, ak])V (Tk-1, [ak-2, ak_1]) ...
V(Tt+2, [a1+1, a1+2]) E L(R")
are invertible and also the inequality (7.12) holds where by definition we have
P(V, D) = P(V, D3)V (rt+1, [at, a1+1])P(V, D1) =
= P(V, D3 )V (r, [s, r])P(V, D1)
213
VII. Product integration
and
IIP(V, D,) - (V(r, [s, rJ))-'(P(V, D3))-'QII =
= II (V (r, [s, r]))-' (P(V, D3 ))-'
[(P(V, D3 )V (r, [s, r])P(V, DI) - Q] 11 <
<
II(V(r,[s,r]))_'IIII(P(V,D3))-'II.2IIQ-'II-'
Consequently, by Lemma 7.7 (see also (7.11)) we obtain
IIP(V,D1)II < IIP(V,DI) - (V(r, [s,TI))-'(P(V,D3))-'QII+
+II(V(r,[s,r]))-'IIII(P(V D3))-'1111QII <
II(V(r,[s,r]))-'IIII(P(V,D3))-'II(1IIQ-'II-' +IIQII) <
1
211 (V- (r))"' IIII(P(V, D3))-111( IIQ-' II-' + IIQII)
= Ko(r) > 0.
(7.15)
On the other hand, we have
II(P(V,Dl))-' -Q-'P(V,D3)V(r,[s,rJ)II =
= IIQ-' (Q - P(V, D3 )V (r, [s, r])P(V, D,))(P(V, DI ))-' II
< IIQ-' IIII(Q - P(V, D)IIII (P(V, DI))-'11 <
< IIQ-111111Q-'
II
II(P(V, D, ))
11= 2 H(P(V, D, ))-' II
and by Lemma 7.7 we therefore obtain
II(P(V,D1))-'II < II(P(V,Di))-' -Q-'P(V,D3)V(r,[s,r])II
+1IQ-' II II P(V, D3)I1IIV(r, [s, r1)II <
VII. Product integration
214
1II(P(V,D1))-'II + IIQ"'IIIIP(V,D3)II(IIV (r)II+
+1ll(V (r))-'ID ,
i.e. we obtain the inequality
II (P(V, Dl ))-' II <_
< 211Q-' IIIIP(V,D3)II(IIV (T) 11 + 1II(V (r))-'II) = K°(r) > 0.
(7.16)
Taking K_(r) = max(K°(r), K°(r)) > 0 we can conclude by
(7.15) and (7.16) that
max{IIP(V, D1)II, II(P(V, DI ))-' II)
< K_(r)
holds. A fully analogous reasoning gives also that if s E [r, ,r +
6°(r)) fl [a, b] and D2 is a 60-fine partition of the interval [s, b]
then
max{IIP(V, D2)II, II(P(V, D2))-111):5 K+(r)
where K+(r) > 0. Putting Ki(r) = max(K_(r),K+(r)) we
obtain (7.14-) and (7.141-).
Now we show that the following is holds.
For every r E [a, b] there is a K2(r) > 0 such that
max{IIP(V, D,)II, II (P(V,D,))-'II,
IIP(V, D2)II, II (P(V, D2))`' II} < K2(r)
(7.17)
if s E (r-60(r), r+6°(r))fl[a, b] and D1, D2 are arbitrary
bo -fine partitions of [a, s], [s, b], respectively.
215
VII. Product integration
Let us take e.g. s E IT, r + bo(r)) and set D = D1 o D2. Then
P(V, D) = P(V, D2)P(V, D1)
and P(V, D1), P(V, D2) E L(R') are invertible by Lemma 7.6.
Since (7.12) is assumed to be satisfied we have
2IIQ-111-1
II P(V,D2)P(V,DI) - Qll <
and
IIP(V, DI) - (P(V, D2))-'QII =
= II (P(V, D2 ))-' (P(V, D2)P(V, DI)
- Q)II <
< II(P(V,D2))-'II2IIQ-III-'
Therefore
IIP(V,D1)II <
< II P(V, DI) - (P(V, D2))-'Qll + II(P(V, D2))-' IIIIQII <
< II(P(V,D2))-'II(2IIQ-'II-1 +
IIQII).
On the other hand, we have
II (P(V, DI ))-' - Q-1 P(V, D2 )II =
= IIQ-' (Q - P(V, D2)P(V, D1))(P(V, D1))-' II <
< IIQ-' IIIIQ - P(V, D2)P(V, D1)II II (P(V, D1))-' II <
< 1 II(P(V, DI ))-' Il,
henceforth
II(P(V,DI))-'II <
(7.18)
VII. Product integration
216
<- II(P(V, D1))-' - Q-' P(V, D2)II + IIQ-' II IIP(V, D2)II <
< 2 II(P(V, Di ))-' II + IIQ-' II IIP(V, D2)II
and therefore we have
(7.19)
II(P(V,D1))-'II <- 2IIQ-'
Using (7.14+), (7.18) and (7.19) we get the estimate
IIIIP(V,D2)II.
max{IIP(V,D1)II,II(P(V,D1))-'II} <
< K1(r)[211Q-' II +
2111Q-' II-' + IIQII]
= KL(T) >0.
Similarly it can be also shown that
max{IIP(V,D2)II, II(P(V,D2))-' II} < KR(T)
where KR(T) > 0. Putting now K2(r) = max{KL(r), KR(T)} we
obtain (7.17) from the two inequalities given above.
The intervals of the form (r - So(r), r + bo(T)) with r E [a, b]
form an open covering of the compact interval [a, b]. Therefore
there is a finite set {t1, ... , t1} C [a, b] such that
bo(t,),tj +bo(t,))
Define K = max{1, K2(t1), K2(t2), ... , K2(ti)} where K2(r) is
[a,b] C
given by (7.17). Then (7.17) implies the following statement.
There exists a constant K > 1 such that
max{IIP(V,Di)II,II(P(V,D1))-'II} <K
(7.20-)
if s E (a, b] and D1 is an arbitrary bo -fine partition of
[a, s]
and
max{IIP(V,D2)JI,
II(P(V,D2))-' II} < K
(7.20+)
if s E [a, b) and D2, is an arbitrary bo-fine partitions of
IS, b]
217
VII. Product integration
Now let us prove the following statement.
Let C E (0, 2 IIQ-' II-') be given and let b be a gauge on
[a, b] such that b(r) < So(r) for r E [a, b] and
IIP(V,D)-QII <e
for every b -fine partition D of [a, b].
If s E (a, b] and D1, D3 are arbitrary 6 -fine partitions
of [a, s], then
(7.21-)
II P(V, Di) - P(V, D3 )II < 2Ke.
If s E [a, b) and D2, D4 are arbitrary b -fine partitions
of [s, b], then
(7.21+)
II P(V, D2) - P(V, D4 )II < 2Ke.
K is the constant given in (7.20).
Let us prove (7.21+) only; the proof of (7.21-) is similar. Let
s E [a, b). Denote by D1 an arbitrary S--fine partition of [a, s] and
let us put D5 = Dl o D2 and D6 = D1 o D4. Evidently D5 and
D6 are S-fine partitions of the interval [a, b]. Hence
II P(V, D2)P(V, Dl) - P(V, D4)P(V, D1)II <
< IIP(V,D5) -QII + IIP(V,D6) -QII
2e
and
II P(V, D2) - P(V, D4) 11 =
= II [P(V, D2)P(V, Dl) - P(V, D4)P(V, Dl)](P(V, D1))-1 II
II P(V, D2)P(V, Di) - P(V, D4)P(V, D1)II II (P(V, Di ))-' II < 2Ke
VII. Product integration
218
by (7.20-). This yields (7.21+), and (7.21-) can be shown similarly.
Using (7.21) and Proposition 7.2 we have the following result.
If s E (a, b) then there exist Q-, Q+ E L(R") such that
for every c E (0, IIQ-1 II-1) there is a gauge b1 : [a, b] ->
2
(0, +oo) on the interval [a, b] such that
I I P(V, D1) - Q -1 1 < e
(7.22-)
for every b1 -fine partition D1 of [a, s] and
IIP(V,D2) - Q+II < c
(7.22+)
for every 61 -fine partition D2 of [s, b].
Now we are able to complete the proof of the theorem.
Assume that s E (a, b). Let us choose a gauge S2 on [a, b]
such that b2(r) _< min(b(r), bo(r), bl (r), I r - sI) for r # s and
b2 (s) < bl (s). By this choice every b2-fine partition
D = {(r1, J1), j = 1,...,k} _ {(r3,[a,i-1,a,]), I = 1,...,k)
has the property that there is a j E 11, 2, ... , k } such that ri = s.
For a b2-fine partition D of [a, b] and b2-fine partitions D1, D2 of
[a, s], [s, b], respectively, we have by (7.20) the inequality
II P(V, D) - Q+Q II <- II P(V, D) - P(V, D2)P(V, D1)II+
+II P(V, D2)P(V, D1)
- Q+Q II <
: II P(V, D) - P(V, D2) P(V, D1)II +
+IIP(V, D2)P(V, D1) - Q+P(V, D1) + Q+(P(V, D1) - Q-)II S
II P(V, D) - P(V, D2)P(V, D1)II+
VII. Product integration
219
+II P(V, D2) - Q+IIIIP(V, D1)II+
+IIQ+ - P(V, D2II II P(V, Dl) - Q H+
+IIP(V,D2IIIIP(V,DI)-Q II <
< IIP(V, D) - P(V, D2)P(V, D1)II + E(2K + e).
(7.23)
For a given b2-fine partition
D = {ao,Tl,a,,...,aj-1,Tj = s,aj,Tj+l,aj+1,
.... ak-1, Tk, ak }
we put
D- = {ao, Tl, a1, ... , rj-1, aj-1 }
and
D+ = {aj,Tj+l,aj+1,...,ak-1,Tk,ak}
and also
D1 = D- o {aj_1,tj = s,aj = s},
D2 = {«j_1 = s, Ti = s,aj} o D+.
It is easy to see that D1, D2 are S2-fine partitions of [a, s], [s, b],
respectively, and that
P(V, D) = P(V, D+)V(s, [aj-1, aj])P(V, D-),
P(V, D1) = V (s, [aj-1, s])P(V, D-),
P(V, D2) = P(V, D+)V(s, [s, ajJ).
Moreover,
II P(V, D) - P(V, D2)P(V, D1)II =
= 11 P(V, D+)V (s, [aj-1, aj])P(V, D-)-
V11. Product integration
220
-P(V,D+)V(s,[s,a.i])V(s,[a.i-l,sl)P(V,D-)II =
= II P(V, D+)[V (s, [a.i-i, ail )-V(s, [s, a.i])V (s, [a.i-1, sl )l P(V, D-)11,,5 K
2C
by (7.20) and (7.13) because we have aj_1i aj E [a, bl and
s - 6o(s) < s - 62(s) < aj_1 < s < aj < s + 62(s) < s + 6o(s).
By the inequality (7.23) we therefore obtain
II P(V, D) - Q+Q- II < K2( + e(2K + e).
If we take for example t; = K22 and use the inequality
IIP(V,D) -QII < e
then for every 62-fine partition D of [a, bl we get (see (7.21))
IIQ - Q+QII<- IIQ-P(V,D)II +IIP(V,D)-Q+QII <
<e+e+e(2K+e)=e(2+2K+e)
and because e > 0 can be chosen arbitrarily small we finally
obtain
Q=Q+Q-.
(7.24)
Since Q E L(R") is invertible, we have by (7.24) the identity
Q-1 Q+Q- = I
and this means that Q`' Q+ E L(R") is the inverse to Q- (in
fact this shows that Q-1Q+ is the left inverse to Q- but we have
also Q-Q-1Q+Q- = Q- and consequently Q-Q-1Q+ = I; this
means that Q-1 Q+ is also the right inverse to Q-). Similarly
VII. Product integration
221
it can be also shown that Q+ E L(Rn) is also invertible with
(Q+)-' = Q-Q-'
This yields by (7.21) that the Perron product integrals
Ila V(t, dt) = Q-, H; V(t, dt) = Q+ exist and have invertible
"values" Q-, Q+, respectively. Moreover, (7.24) is in fact the
equality
b
a
b
11 V(t, dt) fJ V(t, dt) _ 11 V(t, dt)
(7.25)
a
a
a
given in the statement of the theorem.
The estimates
s
II 1 f V(t, dt)II < K, II(fl V(t, dt)) -' II < K
a
a
are simple consequences of (7.20) and of (7.25).
7.9 Lemma. Let V : [a, b] x 3 -- L(R") be Perron product
integrable over [a, b] with 11a V(t, dt) = Q where Q E L(R") is
invertible and assume that the condition C is satisfied.
Let us define 4 : [a, b] -+ L(R") by the relations
fi(a) = I,
fi(s) =
ft
V(t, dt) for s E (a, b].
(7.26)
a
The function fi is well defined and its values are invertible
elements of L(R"), fi(b) = Q.
For a given e > 0 let 6 : [a, b] -* (0, +oo) be a gauge on [a, b]
such that
b
II P(V, D) -P(b)II = II P(V, D) - Ij V(t, dt)II <
a
(7.27)
V11. Product integration
222
holds for every b-fine partition D of the interval [a, b]. If
a<81:5 1 <7'1 C$2:5 2C72C...
... <Qn& <61n <7'in <b
represents a 6-fine system {(6i, [/3,, yi]), j
6i E [ , 8 j ,
= 1, ... , mi.e.
C [e.i - b(ei),ei +b(ei)],
i = 1,...,m
then
Nm, [Nm,7m])4(I3n).
V (Sm-1 [Nm-l 7n:-1
(1(y' })-' V(b1, [Nl,
DCOM-1)..
.
-Ills II(oP(b))-' Ile.
(7.28)
Proof. The function 4 : [a, b] - L(R") given by (7.26) is well
defined by Theorem 7.8 and the same theorem yields also the
invertibility of ,D(s) for every s E [a, b]. By Theorem 7.8 also
the Perron product integral f1d V (t, dt) exists over every interval
[c, d] C [a, b]
Let us denote yo = a and Q,,,+1 = b.
Since the product integral fl' V(t, dt) exists for every j =
0, 1, ... , m we have by definition the following assertion: For every i > 0 there is a gauge bi : [yi, pi+1 ] -+ (0, oo) such that
bi(t) < b(t) fort E [yi, Qi+1 ] and
Aj+i
IIP(V,Di) - 11 V(t,dt)II =
-Yj
= II P(V, Di) - (Qi+1)( (y; ))-' II <'l
(7.29)
for every bi fine partition Di of [yi,/ii+l] and for every j =
0,1'...,m.
VII. Product integration
223
For bj-fine partitions Dj of [yj, /3j+1 ], j = 0,1,..., m let us set
D = Do o(C1,[Q1,yl])oDl 0(C2,[/32,y2])oD30...
oD,,,,-1 0 (Gi, [Qm,7,n]) o Din.
D given in this way evidently forms a 6-fine partition of [a, b]
and therefore (7.27) holds for this partition. Hence
II (,t (b)) -' P(V, D) - III = II(,k (b)) -' [P(V, D) - (b)]II <
(7.30)
< II(-,b(b))-'II--.
Further, we have
P(V, D) = P(V, Dm)V (em, [/3m, y,n])P(V, Din-1) ...
P(V, D1)V(e1, [/31,y1])P(V, Do)
and
(4(b))-'P(V,D) _
_
(`(b))-'P(V,Dm.)V(Cm,[l3m,ym])P(V,Dm-1)...
P(V,DI )V(C1,[131,y1])P(V,Do) _
_ (4)(b))-' P(V, D7 n )V (CnL,
})-1
Pin,
V(ttSm, [Nm,
P(V,
))-1 ... .
4b(i32)(1(/32))-'P(V,D1)4(yl)( (yi))-'V(f1,[Qi,y1])x(/31)
(`x(/31))-'P(V, Do)t(yo)
Denoting
(,b(aj+1))-' P(V, Dj),t(y3) = Aj + I
VII. Product integration
224
forj=0,1,...,mand
(p(y,))-'V(c,,[Q,,-j])lb(p,) = Z, + I
for j = 1, 2, ... , m we obtain
(4)(b))-' P(V, D) _
_ (I + Am)(I + Zm)(I + Am-1)(I + Zm-1) ...
(I + A1)(I + Zi)(I + Ao )
and we can write (7.30) in the form
11(I + Ar)(I + Zm)(I + A»:-1)(I +
(I + A1)(I + Z1)(I + Ao) - III <
(7.31)
II(4(b))-' IIe.
By (7.29) we have
IIAill
= II
=
II(,D(Q,+1))-'P(V,D,)t('Y,)-III
=
K2ri
3 + ))-' [P(V, D,) - 4(fl,+i )(4'('y ))-'
(7.32)
where K is the constant given by Theorem 7.8 and
j = 0,1,...,m.
Using the estimate (7.32) we obtain the following estimate:
for every 9 > 0 there is an 7 > 0 such that
11(I +Am)(I +Zm)(I +Am-2)(I
+Zm-1)...(I+A1)(I +Z1).
(I+Ao)-(I+Zm)(I+Z--1)...(I+ZI)Il
Hence by (7.31) we have
II(I + Zm)(I + Z,n_1) ... (I + ZO - III <
<11
225
VII. Product integration
s 11 (1 + A,n)(I + Zr)(I + A,«-j)(I +
Zm-,)..
.
(I+AI)(I+ZI)(I+Ao)-(I+Z":)(I+Zm-,)...(I+Z1)II+
+11(I + Am)(I + Zm)(I + Am-1)(I +
Zm-1)..
.
(I+A,)(I+Z2)(I+Ao)-III <
< 9 + II((1(b))`' IIe
where 0 > 0 is arbitrary, and consequently we arrive at
II(I+Zm)(I+Zm-,)...(I+Zj)-III < II(4,(b))-'IIE
and by the definition of Zi, j = 1,2,. .. , m we obtain (7.28).
Remark.. The result given in the second part of Lemma 7.9
plays a similar role for the Perron product integral as the SaksHenstock Lemma 1.13 does for the generalized Perron integral.
7.10 Corollary. Let V : [a, b] x3 --- L(R') be Perron product
integrable over [a, b] with I]a V(t, dt) = Q where Q E L(R") is
invertible and assume that the condition C is satisfied.
Then for every i > 0, r E [a, b] there exists a b > 0 such that
[Qm,7m])4(Q) - III < ri
(7.33)
and
II V (Tm, [Nm, rym]) - (7)(4P(/3))-' II
K277
(7.34)
whenever 0, 7 E [a, b], T - b < Q _< r < 7 < r + b, where
: [a, b] -+ L(R") is given by (7.26) and K is the constant from
Theorem 7.8.
Proof. Taking e =
II- > 0 we obtain (7.33) di-
rectly from (7.28) in Lemma 7.9 when b : [a, b] ---* (0, +oo) is the
gauge on [a, b] corresponding to this choice of e by the definition
of the Perron product integral (see (7.27) ).
VII. Product integration
226
Since we have
qq
11V(Tm, [[m, Ym]) -
=
II
-
V(Tm, [[3m,7'm])4)(N) - I](4)(/3))-' II
-III,
5 II-(7)IIII(1D(a))-' IIII
we obtain the inequality (7.34) from (7.33) and from the inequalities
II < K
II,k(T)II 5 K,
0
which hold for every r E [a, b] by Theorem 7.8.
7.11 Lemma. Assume that A, Ak E L(Rn), k = 1, 2, ... are
invertible and such that
lim Ak = A.
(7.35)
lim (Ak)-' = A-'.
(7.36)
k-'oo
Then
k-oo
Proof. By (7.35) there is a ko E N such that for k > ko we
have IIA
- AkIl < IIA-' II-' and therefore
III - AkA-' II = II(A -
Ak)A-'
II 5 II(A - Ak)II
IIA-'II < 1.
Hence AkA-' has an inverse given by
00
(AkA-')-l = > (I-AkA-')t =
t=o
00
E((A-Ak)A-1)1
t=a
Consequently
(Ak)-' = A-' E00 ((A t=o
Ak)A-')t
=
= A(Ak)-i
227
VII. Product integration
00
= A-' + A-' >`((A -
Ak)A-')t,
1=11
00
(Ak)-1
-
A-1
=
A-'
E((A -
Ak)A-1)t
1=1
and
00
II(Ak)-'
- A-'11:5 IIA-'II E(IIA -
AkIIIIA-' 11'
<
1=1
< IIA-' II
1
IIAA AAIkIIAAII1
II
fork>ko.
-* 0 for k -> oo we obtain from this estimate
- AkII
A-1
II -+ 0 for k --> oo, i.e. (7.36) holds.
that II(Ak)-1 Since IIA
7.12 Lemma. Let V : [a, b] x 3 - L(R") be Perron product
integrable over [a, b] with fl' V(t, dt) = Q where Q E L(R") is
invertible and assume that thea condition C is satisfied.
Then
(7.37)
forrE(a,b] and
lim '1('y) _
(7.38)
for T E [a, b).
Proof. It follows immediately from Corollary 7.10 that
urn II ('(T ))-' V (T, [Q, T])I(Q)
- III = 0
(7.39)
VII. Product integration
228
for r E (a, b) and
-1
V(r, [r, y])4 (r) - III = 0
fin,
r+
(7.40)
for r E [a, b).
By (7.4-) and (7.4+) from the condition C we also have
,6
-y
lim IIV(r,[/3,r])-V (r)II = 0
(7.41)
- V+(T)II = 0
(7.42)
limn
r+ IIV(r, [r, 7])
for r E [a, b) where V-(r), V+(r) E L(1R") are invertible. Since
II _< K, we get
by Theorem 7.8 we have 114P(r)II < K,
for r E (a, b], ,3 < r the inequality
II (P(,Q))-' - (41 (r))-' V-(r)II =
((t(r))-' V (,r, [/3, r]) + (4 (r))-' V (r, [a, r])-
= II
-(4(r))-'V (r)II =
= II[I - (
+(It(r))-'V(r, [/3,r]) - (t(7))-'V (r)II <
+IIV(r,[/3,r])-V (r)II].
This inequality together with (7.39) and (7.41) implies
lim (gy(p))-' _ (1(r))-' V-(r)
,6
r-
and Lemma 7.11 immediately yields (7.37).
VII. Product integration
229
Similarly for r E [a, b), -y > r we have
114(7) - V+(r)'(r)II =
V(r, [r, y)A(r) + V (r,
J14(y)[I -
('(y))-'V(r,
V+(r)F(r)II :5
[r,7[A(r)]11+
+11[V(r, [r, -YD - V+ (01 C-011 !5
< K[III -(I(-Y)) -' V (T-, [r,
11 V(r, IT, y]) - V+(r)II
and (7.40) together with (7.42) directly imply (7.38).
7.13 Lemma. Assume that Yl, Y2i ... , Yk E L(R") where
k111YiII<1.Letusset
X = (I + Yk)(I + Yk-1) ... (I + Yi) - I
and
k
Z -X - EY.
i=i
Then
k
IIXII < 2 E IIY=1I
i=i
and
k
1IZ11 < (E Will)
2.
i=1
Proof. Putai=IIYi11fori=1,2,...,kandA=EkIAi<1.
We have
(I+Ak)(I +Ak-l)...(I+A,)=
VII. Product integration
230
k
A 3A,2A,1 + ... +
`A`
Ai + E Aj21 A,1 +
1+
i=1
j3>j2>il
)2>31
+AkAk-1 ... Al < eAkeAk-t
...
e.A.
Hence
k
Ai + E Aj2Aj1 + ... + AkAk-1 ... Al <
eA
- 1 < 2A
j2>jl
i=1
and
'\j2 Aj1 +
j2>31
Aj3 Aj2 Aj1
+ ... + Ak \k-1 ... A <
13>32>31
A < 1. We evidently have
k
Yi+
X=
yj2Yj1
+...+YkYk-I...Y1
j2>31
==1
and
Z = E Yj2Yjl + > Yj3Yj2Yj1 + ... + Ykyk-I ... YI .
72>ll
J3>32>31
Hence
k
IIXII < > III" I{ + E I1yj21111yjl II + ... + IIykIIIIYk-1 II ... IIYI II =
i=1
j2>31
k
Aj2 Aj1 + ... + AkAk-1 ... AI
Ai +
i=1
J2>31
VII. Product integration
231
k
<e"-1 <2A=2EIIY=II,
i=1
and similarly also
IIzII <- i IIYi2II IIYiI II +
i3>i2>il
i2>il
IIYi3IIIIYi21IIIYi1 II +
+
+IIYkIIIIYk-111... IIY1II =
Ai2Ail + E Aj3Ai2Ail +
72>jl
+AkAk-1
... Al <
.73>32>)l
k
A2 =(EIIYjII)2.
i=1
7.14 Theorem. Assume that V : [a, b] x 3 -> L(Rn) satisfies
the condition C and that for every c E [a, b) the Perron product
integral 1a V(t, dt) exists and has invertible values.
Let the limit
C
lim V(b, [c, b]) fj V(t, dt) = Q
c-+b-
(7.43)
a
exist, where Q E L(Rn).
Then V : [a, b] x 3 -1 L(Rn) is Perron product integrable over
[a, b] and
b
II V (t, dt) = Q.
(7.44)
a
Proof. Let e E (0, 1) be given. Since the limit (7.43) exists,
there is a B E [a, b) such that for every c E [B, b) we have
c
IIV(b,[c,b])[JV(t,dt)-QII <e.
a
(7.45)
VII. Product integration
232
Let us have a sequence a = co < c1 < ... , limp...., cp = b. Over
every [a, cp], p = 1, 2.... the function V is Perron product integrable and therefore there exists a gauge by : [a, cp] - (0, +oo),
p = 1, 2, ... such that for every by-fine partition D of [a, cp] we
have
Cy
IIP(V, D)
- fj V (t, dt)I j
a
!5
2p+1(r(1ap V(t, dt))-1 II
(7.46)
forp= 1,2,....
For every r E [a, b) there is exactly one p(r) E N such that
T E [cp(r)_1i cp(r)). For T E [a, b) let us choose b°(r) > 0 such
that 60 (r) bp(r)(T) and [T - 60 (T), r +S°(T)] n [a, b) C [a, cp(r)).
If c E [a, b) and
D = l010,71 , 01, ... 01k-2, Tk-1, ak-1 }
is a b°-fine partition of [a, c], then if p(rj) = p, we have
[aj-1, aj] C (TJ
- b°(Tj), Tj + b°(Tj)) C [a, c p]
and also
[aj-1, a,] C (Tj -6p(r1),Tj +bp(TJ))
For the partition D- we have
(7.47)
P(V, D-) = V(7-k-1, [ak-2, ak-1])V (Tk-2, [ak-3, ak-2]) ... X
xV(rl, [ao, a1]) = AmAm-1
... Al
where Aj, j = 1, 2, ... m is the ordered product of all factors
V (T,, [a1_1 i a,]),
1 < 1 < k - 1 with ri E [Cpl _1, cp1 ], i.e.
Aj =V(Trj+aj,[ar,+aj-i,ari+aj])x
233
V11. Product integration
X V (Trj +,j -1 , [arj +-,j -2, arj +sj -1 ]) ... V(Trj , [arj -1, ar, ] )
and T.,7Trj+17...,Trj+sj E [cp,-1,cpj] with 1 < rj rj +sj
k - 1. Since the partition D- satisfies (7.47) we also have
[ai-1, ail C (Ti
(Ti),
i + bp1 (Ti) ),
i = rj, rj + 1, ... , rj + s j. Using (7.46) and Lemma 7.9 we obtain
rrj+, j
[j
a
Frl
...V(Trj, [ar;-1,ar1])11V(t,dt) - III =
a
rrj +,j
rr!
= II( IT V(t, dt))-' Aj [J V(t, dt) -III <
a
a
EII(fIa°' v(t, dt)) II
E
=
2pj+1
V(t,dt))-111
C 2pi+1
for
II(fI?aDj
every j = 1, 2, ... , m. Hence
in
rrj
rrj +,j
EII( rj
j=1
III
V(t,dt))-'Aj11V(t,dt)-1115 E
a
E
2p1+1 <
a
(7.48)
Denoting
rri +ej
Yj =
II( jj
a
rr,
V(t, dt))-' Aj
jI V(t, dt) - III
a
VII. Product integration
234
for j = 1, 2, ... , m we have by (7.48) and by the choice of e the
inequality
m
EIIY;II<e<1,
j=1
and for
x =(I+Ym)(I+Ym-i)...(I+Y1)-I =
a
Q'k-1
_ (H V(t, dt))-' Am.Am-1 ... Al 11 V(t, dt) - I =
a
a
Qk-1
]I V(t, dt))-' A,An1-1 ... Al
-I=
a
_ ([I V(t, dt))-' P(V, D-) - I
a
we obtain by Lemma 7.13 the estimate
m
c
IIXII = 11(11 V(t, dt))-'P(V, D-) - III
a
2> IIYjII < 2e, (7.49)
;-1
and this estimate does not depend on c E [a, b).
Define now a gauge b on [a, b] such that
D11
0 < b(T) < min(b - T,b°(T))
for r E [a, b) and
If
0<b(b)<b-B.
= {a0, TI ,al, ... ak-21 Tk-1 1 ak-1, Tk, ak }
VII. Product integration
235
is an arbitrary b-fine partition of [a, b] then by the choice of the
gauge b we necessarily have rk = ak = b and ak_1 E (B, b). We
have also D = D o (b, [ak-1, b]) where
D = {ao, r1, al, ... ak-2, rk - I, a k-1 )
and
P(V,D) = V(b,[ak-1ib])P(V,D-).
Hence
IIP(V,D)-QII = IIV(b,[ak-1,b])P(V,D )-QII =
IIV(b,[ak_1,b]) fl V(t,dt)( JJ V(t,dt))-1P(V,D-)-QII =
a
a
ak-1
Crk-I
= IIV(b, [ak-1, b]) JJ V(t, dt)[( 11 V(t, dt))-'P(V, D-)
a
- I]+
a
ak-1
V(t, dt) - QII <
+V(b, [ak-1, b])
a
ak-1
< [II V(b, [ak-1, b]) Ij V(t, dt) - QII + IIQIIJ.
a
II( [V(t, dt))-'P(V, D-) - III+
ak-I
+IIV(b, [ak-1, b]) 11 V(t, dt) - QII
a
Since B <ak-1 <b we have by (7.45)
ak-t
IIV(b, [ak-1, b]) H V(t, dt) - QII <
a
(7.50)
V11. Product integration
236
and by (7.49) we get
Ak-L
fj V(t, dt)) -'P(V, D-) - III < 2e.
a
Therefore (7.50) yields
II P(V, D) - QII < (e + IIQII)2e + e = e(2e + 1 + 211QII)
for an arbitrary 6-fine partition D od [a, b), and this means by
definition that the Perron product integral fa V (t, dt) exists and
its value is Q.
0
In a completely analogous way also the following result can be
proved.
7.15 Theorem. Assume that V : [a, b] x J - L(R") satisfies
the condition C and that for every c E (a, b] the Perron product
integral f6 V(t, dt) exists and has invertible values.
Let the limit
b
lim II V(t, dt)V(a, [a, c]) = Q
c- a+
C
exist, where Q E L(R").
Then V : [a, b] x J -> L(IR") is Perron product integrable over
[a, b] and
b
II V(t, dt) = Q.
a
Remark. It is not difficult to check that if V : [a, b] x 3 -r
L(R') is Perron product integrable over [a, b], then for every d E
(a, b] we have
c
d
cd - [J V(t, dt) _ (V_(d))-' fJ V(t, dt).
lim
237
VII. Product integration
and similarly for d E [a, b)
6
b
lim
c-.d+
V(t, dt) _ [I V (t, dt)(V+(d))-' .
d
c
If d E (a, b) then
6
c
6
II V (t, dt) =clam 11 V(t, dt)V+(d)V- (d) cllim fl V(t, dt).
c
a
a
The proof of the following theorem was given in the paper [65].
7.16 Theorem. Assume that W : [a, b] -> Rn is such that
the inverse (W(r))-' exists for every r E [a, b] and
inax{IlW(r)II, (W(r})-' } < M
(7.51)
where M > 0 is a constant. Let V : [a, b] x 3 -* L(R') be such
that for every 9 > 0 there exists a gauge b on [a, b] such that
k
(IV(T,i),[ai->,ai])-W{aJ)(W(«i-i))- <9
(7.52)
i='
provided
D = {ao, rl ,ale ... ak-2, Tk-1, ak-1, Tk, ak }
is a 6-fine partition of [a, b].
Then the Perron product integral fa V(t, dt) exists and is
equal to W(b)(W(a))-'.
Proof. Let 0 < 9 <
.
1
For a b-fine partition D ( b is the
gauge corresponding to 9) define
1'i = (W(ai))-'V(ri, (ai-,,c ])W(ai-1) -I
VII. Product integration
238
for i = 1, ... , k.
Then
Will = II
(W(ai))-' V(ri, [ai
ai])W(ai-1) - III =
= II(W(ai))-' [V(Ti, tai-1, ai])-
-W(ai)(W(ai-i))-']W(ai-1)II <
< M2IIV(Ti, [ai-1, ai]) - W(ai)(W(ai-i))-' II
for i = 1, . . . , k and therefore by (7.52)
k
k
< M2
IIV(ri,[ai-i,ail) - W(ai)(W(ai-1))-' II
<-
<M20<1.
Take
X =(I+Yk)...(I+Yi)-I =
_
(W(ak))-'V(rk,[ak-1,ak])W(ak-t)...
(W(al ))-'V(Ti, [ao, al])W(ao) - I.
Then Lemma 7.13 yields
IIXII = II
(W(b))-'P(V, D)W(a) - 111 :5
2M26.
Consequently
II P(V, D) - W(b)(W(a))-l II =
= IIW(b)[(W(b))-'P(V,D)W(a) - I](W(a))-'II <
< IIW(b)IIII(W(b))-'P(V,D)W(a) - IIIII(W(a))-' II < 2M49
and this proves the theorem.
0
239
VII. Product integration
7.17 Lemma. Assume that L > 1 is a constant such that for
every matrix Z E L(Rn), Z = (z;k)i,k=1,...,n the inequality
1
max
L i,k=1,...,n
IZikI < IIZII : L:,k=1,...,n
max IZikI
(7.53)
holds. Let
0<0<
91
(7.54)
,
... , Zr E L(R") and assume that for every p-tuple
{j1ij2i...,jp} C {1,2,...,r} the inequality
Z1, Z2,
II (I + Zip)(I + Zip-,) ... (I + Zi,) - III < e
holds. Then
(7.55)
r
IIZiII : Me,
(7.56)
i=1
where M = 4n2L2.
Proof. Let us denote Zj = (zik))i,k=1,...,n, i.e. zT is the element belonging to the i-th row and k-th column of the matrix
Zi.
Denote further by J(1, m) the set of such values j E { 1, ... , r}
for which
(j) j
kmax IzikI
= Iz2 I,
i
,n
i.e. J(1, m) is the set of values j E {1,... , r} for which the absolute value of the elements of the matrix Zi reaches its maximum
in the 1-th row and m-th column. It is clear that for a given pair
(1, m) the set J(1, m) can be empty.
Assume that the estimate (7.56) is not valid, i.e. that
r
n
n
4n2L20<EWill =EE
j=1
1=1 m=1 jEJ(t,m)
Pill
VII. Product integration
240
holds. Then evidently there is a pair (1, m) such that
JIZj JI < > L i kmax f Iz,k)) =
4L20 <
jEJ(t,m)
jEJ(t,m)
= L E izInt I,
jEJ(l,m)
where (7.53) was used. Denote now
E J(1, m), zIm > 0}
J+
and
J = J(l, m) \ J+
E J(l, m), z(I,'n) < 0}.
Using the last inequality given above we can conclude that at
least one of the inequalities
-
zI,',t) > 2L8,
jEJ+
zIn,)
> 2L0
jEJ-
holds. Assume that the first of these inequalities is satisfied. The
case when the second one holds can be treated similarly.
By the assumption (7.55) we have
Il Zj 11 < 0,
j = 1, 21 ... , r,
and by (7.53) also
zingg < LO for j E J+
and therefore we have a set J+ C J+ such that
2L9 <
zIm) < 3L9
jEJ+
(7.57)
VII. Product integration
241
and consequently by (7.53) and (7.54) we have
zIM < 3L29 < 1.
L
IIZj11
jEJ+
jEJ+
From this inequality we can see that the matrices Zj with
j E J+ satisfy the assumptions of Lemma 7.13 and by this lemma
we obtain
11 (I+Z,)-I= > Z' +x
jEJ+
jEJ+
and
IIxIi <
IIZjII)2 < 9L
402.
(7.58)
jEJ+
Hence by (7.55) we obtain
Zjll=11 11 (I+Zj)-I-xll <
II
jEJ+
jEJ+
< ll fl (I+zj)-III+IIxII <
jEJ+
< B + 9L4B2.
On the other hand, by (7.57) we have
20 <
1
L
z2 <
max
L i,k=1,...,n
zit) I < ii
I
jEJ+
jEJ+
Zj Il
jEJ+
and this together with the previous inequality leads to
20 < 0 + 9L4B2,
that is
1
9L4
<0
which contradicts (7.54) and therefore the estimate (7.56) holds.
0
VII. Product integration
242
7.18 Theorem. Let V : [a, b] x 3 -- L(IR") be Perron product
integrable over [a, b] with fa V(t, dt) = Q where Q E L(R") is
invertible and assume that the condition C is satisfied.
Let us define. c : [a, b] --> L(R') by the relations
4)(s) _ fj V(t, dt) for s E (a, b].
a
II- where L is the constant given in
Let e E (0, 9L
(7.51) and let S : [, b] --4 (0, -boo) be a gauge on [a, b] such that
IIP(V, D) -,t(b)II < e
for every S-fine partition D of [a, b].
If
a <01
is a S-fine system
C1 57'1 :592 502:572 :5 ...
[,j, 7j]), j = 1, ... , na}, i.e.
C; E [j;,7;] C [C; - b(e,),C; +5(e;)],
7 = 1,...,m
then
m
-III
II
j=1
MII (1(b))-' IIe (7.59)
where M is the constant from Lemma 7.16 and
In
Ili
=1
6;
I II
V (t, dt)II <- K2MII(4(b))-' Ii--
(7.60)
VII. Product integration
243
where K is the constant from Theorem 7.8.
Proof. Let us set
[Q;,'Yj])I(A;) - I
Z3 =
for j = 1, 2, ... , m. Since all assumptions of Lemma 7.9 are satisfied, we obtain by (7.28) the inequalities
II(I + Z,.)(I + Z;p-,) ... (I + Z;,) - Ill < ll(,k(b))-' IIe
for every p-tuple {jl,j2i...lip) C {1,2,...,m}, J1 < j2 <
jp, and by the choice of e > 0 we also have
Hence Lemma 7.17 yields
m
1112;11 <- MII(4P(b))-' IIe
;-i
and (7.59) holds.
Since
-r;
II V (t, dt)
a;
for j = 1, ... , m and therefore also
7i
V(e;, [Q;,'Yi])
- J1 V(t,dt) _
ai
II((D(b))-' lie <
<
9L
VII. Product integration
244
for j = I.... , in, we obtain by Theorem 7.8 the inequality
- ft V(t,dt)II < K 211Zj11
Ii
f o r j = 1, ... , m, which together with (7.59) implies (7.60).
Remark. Lemma 7.17 and its proof is strictly based on the
structure of n x n-matrices which represent linear operators from
L(Rn). All statements given before Lemma 7.17 are independent
of the structure of n x n-matrices and they can be formulated,
including the definition of the Perron product integral, in terms
of general bounded linear operators from the space L(X) of all
bounded linear operators on a Banach space X.
It can be shown that if X is a Banach space which is not
finite-dimensional then a statement analogous to Lemma 7.17 is
not true. A counterexample demonstrating this fact is given in
[148].
7.19 Definition. Two functions V1, V2 : [a, b] x 3 -> L(Rn)
are called equivalent if for every 9 > 0 there is a gauge b on [a, b]
such that
k
IIV1(r ,(a-l,ail) - V2(r,[a9-1,a1])II < 9
j=1
for every 6-fine partition
D = {(r,,
of the interval [a, b].
k)
VII. Product integration
245
7.20 Theorem. Assume that the functions V1, V2 : [a, b] x
3 - L(R") are equivalent.
If the function V1 satisfies condition C and [a V, (t, dt) = Q
exists and Q E L(R) is invertible then also fa V2 (t, dt) exists and
b
ftV2(t,dt) _ ft V, (t, dt).
a
a
Proof. Assume that 8 > 0 is given. By (7.60) from Theorem
7.18 there is a gauge 6 on [a, b] such that for every 6-fine partition
D we have
k
VI (7-j, [aj-1, aj])
-'k1(aj)(Iti(aj_1))-III < 0
j=1
where it 1(s) = fl V1(t,dt) for s E (a,b] and 11(a) = I.
By the equivalence of V1 and V2 the gauge 6 can be given in
such a way that
k
E IIV1(rj, [aj-1, aj]) - V2(rj, [aj-1, aj])II < d
j=1
for every 6-fine partition D.
Hence
k
E IIV2(7-j, [aj-1, aj])
(aj-1 ))-1 II C
j=1
k
E II V2(r , [aj-1, aj]) - V1(Tj, [aj-1, aj])II +
j=1
VII. Product integration
246
k
II Vi (ri, [ai-i , a.i]) - V2(ri, [ai-i, a,i])II < 20
+
j=1
and by Theorem 7.16 the Perron product integral fl' V2(t, dt)
exists and
a
b
f V2 (t, dt) _ bI (b)(,b (a))-1 =
a
b
'1(b) _ [JV1(t,dt).
a
Let us now introduce another condition for functions V : [a, b] x
- L(Rn).
7.21 Definition. A function V : [a, b] x J -> L(JR") satisfies the condition C+ if there exists a nondecreasing function
g : [a, b] -- R such that for every r E [a, b] there is a p = p(r) > 0
such that
II V(r, [a, a]) - III <- g(/) - g(a)
(7.61)
for alla,#3,rE [a,b] withr -p<a<r < Q<r+p.
7.22 Lemma. Assume that a function V : [a, b] x 3 -i L(R")
is Perron product integrable over [a, b] with an invertible value
of the product integral and that the conditions C and C+ are
satisfied.
Then the function -tP : [a, b] -, L(]8") given by
4(a) = I,
-!D(s) _ J V(t, dt), s E (a, b]
(7.62)
a
is of bounded variation on [a, b], i.e. 4D E BV([a, b]; L(R")) and
also-' E BV([a,b];L(R")).
247
VII. Product integration
Proof. Assume that a, p E [a, b], a < 0. Then if r E [a,,0], we
have
I].i(a) _
`gy(p) - (a) _
Q
V(t,dt) - I]4(a) _
=
Ck
a
[V(r, [a, /3]) - I]4(a).
V(t, dt) - V(r,
Cr
By Theorem 7.8 and by the condition C+ we have
II,P(p) - -,k(a)II <
10
< K [II 11 V(t, dt) - V(r, [a, /31)11 + g(p) - g(a)]
(7.63)
Q
provided r - p(r) < a < r < /3 < r + p(r) and p is the function
from Definition 7.21.
Assume further that e > 0 is given and that b
:
[a, b] -+
(0, +oo) is such a gauge on [a, b] that
II P(V, D) -,,D(b)II < e
for every b-fine partition D of [a, b] and that b(r) < p(r) for
r E [a, b] where p(r) > 0 is given in condition C+.
Letnowa=so<sl
given and let
P
P
P
P
Dp= f aPo,rl,a1,...,rkp,ak,J
be an arbitrary b-fine partition of [Sp_1, Sp], p= 1,. .. , m. Then
by (7.63) we have
kp
II,(sP) - Csp-1)II < >
j=1
-
:5-
VII. Product integration
248
kP
a7
< K>[II fl V(t,dt)-V(rjP,[aJ-1,aP])II+
j=1
a'
kP
= K>[II fj V(t,dt) -V(Tj,[ai-1+a;])II +h[g(sP)-g(sp-1)]
for every p = 1, ... , m and henceforth
m
E
P(SP-1 )II <
P=1
m kP
ai
P=1 j=1
a;_1
< K E E[II fi V (t, dt) - V(-r,", [d-1, a;])II + K[g(b) - g(a)].
(7.64)
Using Theorem 7.18 we obtain the estimate
in
cal
kP
II [[
P=1.7=1
c
K2MII(4'(b))-'IIE
_1
because evidently D = DI o D2 o . o D' is a 6-fine partition of
[a, b]. By (7.64) we therefore have
m
P=1
< Ii 3MII (,D(b))-1 Ile + K[g(b) - g(a)]
249
VII. Product integration
for an arbitrary choice of points a = so < s1 <
consequently
vary
i.e.
< K3MII (,b(b))-' Ile + K[g(b)
< s,,, = b, and
- g(a)] < oo,
(7.65)
E BV([a, b]; L(R"))
It is easy to observe that (7.65) yields the inequality
var'' < K[g(b) - g(a)]
because e > 0 in (7.65) can be taken arbitrarily small.
Since we have (ob(s))_' = (4,(b))-1 f, V(t, dt), the boundedness of vary-1 can be shown similarly.
7.23 Lemma. Assume that a function V : [a, b] x 3 --' L(R")
is Perron product integrable over [a, b] with an invertible value
of the product integral and that the conditions C and C+ are
satisfied.
Then for every t E [a, b] the Perron-Stieltjes integral
I
t
A(t) E L(R")
(7.66)
exists.
For A : [a, b] - L(R") we have A_E BV([a, b]; L(R")) and
[I + A+A(t)]-1 exist for t E (a, b],
the inverses [I t E [a, b), respectively.
Proof. By Lemma 7.22 the n x n-matrix valued functions
and 40 are of bounded variation on [a, b]. Therefore the PerronStieltjes integral in (7.66) exists by Corollary 1.34. The fact that
A from (7.66) is of bounded variation easily follows from Corollary
1.36 (see also Remark 1.37).
VII. Product integration
250
By the definition of A we have for any i > 0 the equality
A(t) - A(t -
=
j
d[(r)]({r))t
and therefore by Theorem 1.16 we have
_
0-A(t)
t
lien
JO-1
i.e.
By (7.37) in Lemma 7.12 we have 4b(t-) =
I - 0-A(t) =
=
_ (V (t))-1lk (t)(c(t))-1 =
(V-(t))-1
for t E (a, b] and V-(t) is invertible by (7.4-).
In a completely analogous way we obtain also
I + 0+A(t) = V+(t)
for t E [a, b), where V+ (t) is invertible by (7.4+).
0
7.24 Theorem. Assume that a function V : [a, b] x 3 --.
L(Rn) is Perron product integrable over [a, b] with an invertible
value of the product integral and that the conditions C and C+
are satisfied.
Then for every s E [a, b] the relation
s
_
4k(s) = ik(a) + f d[A(r)]k(r)
a
(7.67)
251
VII. Product integration
holds, where -t : [a, b] -> L(R") is given by (7.62) and A :
L(IR") is defined by (7.66) in Lemma 7.23.
[a, b]
Proof. Let us consider the integral
=
I ae d[A(r)]4 (r)
j d[
f
a
To the right hand side the substitution theorem for Perron-Stieltjes
integrals can be applied ( see e.g. [155 ,I.4.25]) to obtain
I.
d[A(r)]4 (r) =
f
e
fi(r) =
a
= f d['(r)] = f(s) a
for every s E [a, b], i.e. (7.67) holds.
0
Looking at the concepts of Chap. 6 we can reformulate the
results of Theorem 7.24 and Lemma 7.23 as follows.
7.25 Theorem. Assume that a function V : [a, b] x 3 L(R") is Perron product integrable over [a, b] with an invertible
value and that the conditions C and C+ are satisfied.
Then the function -t : [a, b) --i L(R") given by
fJ V(t, dt), s E (a, b]
(D(a) = I,
a
is a fundamental matrix of the generalized linear differential equation
dx = D[A(t)x],
aT
VII. Product integration
252
where A : [a, b] -+ L(R")is defined by (7.66) in Lemma 7.23. The
matrix valued function A satisfies the condition (6.13).
This result shows that for a V : [a, b] x 3 -+ L(R") the "indefinite" Perron product integral fa V (t, dt) is a fundamental matrix
corresponding to a well-behaved generalized linear ordinary differential equation. The condition (6.13) described and used in
Chap. 6 is a necessary and sufficient condition for the existence
of a unique solution of the corresponding nonhomogeneous equation for every right hand side and every initial condition.
Theorem 7.25 naturally suggests the following problem.
Given A : [a, b] --> L(R") such that
A is of bounded variation on [a, b],
(7.68)
[I + 0+A(t)]
exists for every t E [a, b),
(7.69-)
[I - A-A(t)]
exists for every t E (a, b]
(7.69+)
holds.
Construct a function V : [a, b] x 3 - L(R") which is Perron
product integrable over [a, b] with an invertible value of its Perron
product integral for which the conditions C and C+ are satisfied
and such that for the function -t : [a, b] - L(R") given by
(a) = I, (s) _
V(t, dt), s E (a, b)
a
the equality
(s) _
holds.
9
1.a
d[A(r)]t(r)
VII. Product integration
253
In other words, we are looking for the representation of the
fundamental matrix of the generalized linear ordinary differential
equation
dx = DA(t)x
dr
for the case when this linear equation satisfies the necessary and
sufficient conditions for the existence of a unique solution on [a, b]
for all possible initial conditions (see e.g. Theorem 6.14 and the
results given in Chap. 6).
Assume that (7.68) and (7.69) hold for A : [a, b] -4 L(R"). For
a, /j, r E [a, b], a < r < A define
W(r, [a, Q]) = (I + A(/3) - A(r)][I + A(a) - A(r}]-'.
(7.70)
Since A satisfies (7.68) we have IIA-A(t)II < 2 except a finite
set of points tl,t2i...,tt E [a,b]. For t
{t1,t2,...,t1} we then
have
00
[I - 0-A(t)]-' = T(0-A(t))k
k=0
and
II [I - o-A(t)]-' II < E00 IIo-A(t)Ilk < 2.
k=0
Taking K_ = max{2, II[I- o A(t, )]-l II,
,
II[I- o A(ti)]-' II}
we have
II [I - A-A(t)]-l 11 5 K_
for every t E (a, b]. Similarly it can be also shown that
11 [1 + 0+A(t)]-111 < K+
for every t E [a, b) where K+ is a constant.
VI!. Product integration
254
Since the onesided limits of A exist in [a, b] we can easily state
that there is a constant L > 0 such that for every r E [a, b] there
is a S1 (r) > 0 such that
[I + A(a) - A(r)]-', [I + A(3) - A(T)]-'
exist and
11 [1 + A(a) - A(r)]-' II : L, II [I + A(/3)
- A(r)]-1II <- L (7.71)
whenever a, /3 E [a, b], r - b, (r) < a < r < 0 < r + b, (r).
For W : [a, b] x - L(R") given by (7.70) the following identities hold:
W(r, [r, r]) = I,
r E [a, b],
W(T, [a, r]) = [I + A(a) - A(T)]-',
W(r, [r, /3]) = [I + A(/3) - A(-r)]
and therefore
W (T, [a,,31) = W (T, [T, 0)) W(r, [a, T ] )
provided a,,3 E [a, b], T - S1 (T) < a < T < /3 < T + bl (T).
Finally, we have
,6
lim W(r, [r, 13]) = lim [I + A(/3) - A(r)] = I + A+A(T)
r+
fl-r+
for r E [a, b) and by Lemma 7.11 also
lim W(r, [a, r]) = cflimr_[I + A(a) - A(r)]-' _ [I - A-A(T)]
airfor r E (a, b]. In this way we have verified that W given by (7.70)
satisfies the condition C from Definition 7.4.
255
VII. Product integration
Moreover, by (7.71) we have
[I+A(/3)-A(r)][I+A(a)-A(-r)]-'- III
IIW(r, [a,#])-III = II
[I+A(/3)-A(r)-(I+A(a)-A(r))][I+A(a)-A(r)]-'II <
< IIA(Q) - A(a)IIL < L(var' A - var4 A)
whenever a, Q E [a, b], r - b1 (r) < a < r _< Q < r + bi (,r), and
this yields that W satisfies the condition C+ given in Definition
7.21 with the nondecreasing function g : [a, b] -> R given by
g(s) = L varQ A for s E [a, b].
Now let ' : [a, b] - L(R") be the fundamental matrix of the
generalized linear ordinary differential equation
dx
dr
= DA(t)x
(7.72)
with 11(a) = I (see Theorem 6.11). Since the Perron-Stieltjes in-
tegral fa d[A(r)]W (r) exists, the Saks-Henstock lemma 1.13 yields
for every e > 0 there is a gauge 62 on [a, b], 62(r) _< b1(r),
r E [a, b] such that if
a <Q1 <e !571 <Q2 «2 <72 <...
<Nvn <
, <1'm
<b
is a b -fine system {(Cj, [fj, -yj] ),
j E [#j, ^tj] C [6j - b(ci ), 6j + b(6i )],
1 = 1, ... , m
then we have
yJ
in
II[A('Yj) j=1
J
d[A(r)]1(r)II < e.
,
(7.73)
VII. Product integration
256
7.26 Lemma. Assume that A : [a, bl -+ L(R") satisfies (7.68)
and (7.69).
Let' : [a, b) -+ L(R') be the fundamental matrix of the generalized linear ordinary differential equation (7.72) with %F(a) = I.
Then for every 9 > 0 there is a gauge b on [a, b] such that
k
Ell W (ri, [ai-1, ai]) -
<e
(7.74)
j=1
for every 6-fine partition D = {ao, r1, al, ... , ak-1 , rk, ak} of
[a, b] where W [a, b] x 3 -- L(R") is given by (7.70).
Proof. Let e > 0 be arbitrary and let b be a gauge on [a, b]
such that b(r) < b2(T) for r E [a, b] where b2 is given in (7.73).
If D is a b-fine partition of [a, b] then W(ri, [ai-1i a;]) is well
defined (cf. (7.71)) for j = 1, ... , k. By definition and (7.71) we
have
II W(ri, [a,...
,ai]) -
'(ai)(
I'(ai-1))-1II
= II [I + A(ai) - A(ri)][I + A(ai-1) (1P(a.i-1
= II
))-1
A(ri)]-1
=
- `I'(a.i)
II =
[I+A(a;)-A(ri)-T(ai)(W(ri))-1
][I+A(a.i-1)-A(r;)J-1 +
+1P(a;)(1I'(r; ))-1J([I + A(a;-1) -
A(ri)J-1
- T(ri)
('I'(ai-1))-1)l1 <
< LIII + A(ai) - A(T,) +I) `I'(a.i)(`I'(Tr))-1 IIII[I + A(a;-1)
I['I'(ai-1 )(W (r1))-1- [I +A(a.i-1)
-
A(ri)]-1
II
Iyy
))-1)II
VII. Product integration
257
By Theorem 6.16 there is a constant M > 0 such that
IIT(t)(NS))-' II < M
for all s, t E [a, b] and therefore we get from the inequality derived
above the following one (let us mention that II`f'(s)II < M and
II(`F(s))-'
II < M, because we have %F (a) = I):
IIW (Tj, [aj_1, aj]) -''(aj)('I'(aj-1 ))-' II <
< LIIW(TJ) + (A(aj)
-
A(Tj)),'(Tj) - W(a,)fII(W(T,))-' II+
))-''(T,)
+LMII[
)II <
< LMII(A(aj) - A(Tj))W(Tj) -
f
aj
d[A(r)]W(r)II+
r,
I. Ti
MZLII (A(Tj)
- A(aj-1))I'(Tj) -
j
d[A(r)]`F(r')II
for every j = 1, ... , k.
Using (7.73) in this situation we finally obtain
k
IIW (T), [aj-1, aj])
j=1
- I'(al)(I'(aj-1 ))-' II <
k
< LM >
aj
II(A(aj) - A(Tj))`F(Tj)
- f d[A(r)]41(r)II +
j=1
k
+M2L
ri
II(A(T1) - A(aj-1))`1'(Tj)
j=1
- 4-1 d[A(r)]T(r)II <
258
VI!. Product integration
< eLM(M + 1).
Taking 0 < E < LM(M -}- 1) + 1 for an arbitrary 0 > 0 we obtain
(7.74) for partitions D which are 6-fine with S corresponding by
0
(7.73) to this choice of e.
By Lemma 7.26 and by Theorem 7.16 we now immediately
obtain
7.27 Theorem. Assume that A : (a, b] -- L(R") satisfies
(7.68) and (7.69).
Let IF : [a, b] - L(R") be the uniquely determined fundamental matrix of the generalized linear ordinary differential equation
(7.72) with 1i(a) = I.
Then the function W[a, b] x 3 - L(R") given by (7.70) is
Perron product integrable over [a, b] with the value %F(b) and for
every s E [a, b] we have
W(t, dt) = 41(s).
(7.75)
Remark. Taking into account the result of Theorem 7.27 together with the result of Theorem 7.25 we can see that there
is a one-to-one correspondence between the "indefinite" Perron
product integrals fa V (t, dt) of functions V (a, b] x 3 - L(1R")
which satisfy conditions C and C+ and the fundamental matrices of generalized linear ordinary differential equations (7.72) with
A : [a, b] - L(R") satisfying (7.68) an (7.69). Let us now turn
our attention to the case when A : [a, b) -, L(R") is such that
A is of bounded variation on [a, b],
(7.68)
A is continuous at every point in [a, b].
(7.76)
and
VII. Product integration
259
For A : [a, b] - L(R") satisfying (7.68) and (7.76) define V1, V2
[a, b] x 3 -> L(R") by the relations
Vi (r, [a, /3]) = I + A(/3) - A(a)
(7.77)
for r E [a,,31 C [a, b] and
A(R)-A(a)
V2(r,
=
E
k=0
A(a)]k
[A(#)
k!
(7.78)
for r EE [a,#] C [a, b].
By (7.76) for every E E (0, 1) and r E [a, b] there is a b(r) > 0
such that
(7.79)
II A(Q) - A(a)JI < --
for every [a, /3] C [a, b], r - b(r) < a < r < 0 < r + b(r).
If the pair (r, [a,,3]) is b-fine then
Vi (r, [a, Q])
- W(r, [a, 8]) =
I+A(/3)-A(a)-[I+A(/3)-A(r)][I+A(a)-A(r)]-1
=
_ {[I + A(/3) - A(a)] [I + A(a) - A(r)] - [I + A(/3) - A(r)]}.
.[I+ A(a) - A(r)]-1 =
_ [A(0) - A(a)][A(a) - A(7-)) [I + A(a) -
A(r)]-1
and
V2(r,
= A(#)-A(c,)
Vi (r, [a,#]) =
- [I + A(/3) - A(a)] _
= E [A(/3) - A(a)]k
k=O
- [I + A(Q) - A(a)] _
(7.80)
Vll. Product integration
260
_
A(a)]k
[A(Q)
(7.81)
k
k=2
It is not difficult to see that 6(r) > 0 can be choosen so small
that if (r, [a, Q]) is b--fine then II [I + A(a) - A(T)]-1 II < L where
L is a constant (see also (7.71)).
By (7.80) we have
IIVV(T, [a,Q]) - W(T, [a, Q])II < ELIIA(/3) - A(a)II
(7.82)
while (7.82) leads to
Vi (7-,
V2
II A(A) - A(a)Ilk
k=2
k!
=
= ellA(,6)-A(a)II -1- IIA(a) - A(a)II <
< IIA(g) - A(a)112 < EII A(fl) - A(a)II
(7.83)
whenever the the pair (T, [a, /3]) is b-fine.
Assume now that D =
= 1,... , k) is an
arbitrary b-fine partition of [a, b]. Then by (7.82) we obtain
k
i=1
k
< EL
II A(ai) - A(ai-1)Il < eL var A
and (7.83) yields
k
E 11V1 (Ti, [ai-1, ail)
j=1
- V2(Ti, [a.i-1, ai])II <
VII. Product integration
261
k
A(c j) - A(aj-1)II < evarb A.
e
j_1
Using this two inequalities we can conclude now that
the functions W, V1, V2 : [a, b] x 3 -> L(R") given by the relations (7.70), (7.77), (7.78), respectively, are equivalent in the
sense of Definition 7.19.
Using Theorems 7.20 and 7.27 we arrive at the following statement.
7.28 Proposition. Assume that A : [a, b] - L(R") satisfies
(7.68) and (7.76). Then the functions W, V1, V2 [a, b] x 3 --+
L(Rn) given by (7.70), (7.77), (7.78), respectively, are Perron
product integrable over [a, b] to the value T(b) and for every s E
:
[a, b] we have
9
11 W (t, dt)
a
9
_ 11 V, (t, dt) _
a
9
11 V2 (t, dt) = W(s).
a
(7.84)
CHAPTER VIII
CONTINUOUS DEPENDENCE ON
PARAMETERS FOR GENERALIZED
ORDINARY DIFFERENTIAL EQUATIONS
Let us assume that G = B x (a, b) where B = B, = {x E
R"; I]xil < c}, c > 0, (a, b) C R, -oo < a < b < +oo and
h : [a, b] -> R is a nondecreasing function defined on [a, b], w :
[0, +oo) - R is a continuous, increasing function with w(0) = 0.
8.1 Lemma. Assume that Fk : G --> R" belongs to the class
F(G, h, w) for k = 0,1, ... and that
k m Fk(x,t) = Fo(x,t)
(8.1)
for (x, t) E G.
If x : [c, d] -- B is a function of bounded variation on [c, d] C
(a, b) then
lim J dDFk(x(r), t) = I DFo(x(r), t).
koo c
Jc
(8.2)
Proof. Let e > 0 be given. Assume that rJ > 0 is such that
w(77)
E
Since x : [c, d] --i Bc is a function of
2(h(d) - h(c) + 1}.
bounded variation on [c, d], for every i > 0 there is a finite step
function V : [c, d] - Bc such that
II x(r) - (p(T)]I
r)
262
for T E [c, d].
(8.3)
VIII. Continuous dependence for generalized ODE'S
263
Therefore
IIFk(x(T),t2)-Fk(x(T),t,)-Fk('P(T),t2)+Fk(p(7),tl)II <
< w(II x(T) - W(T )II )I h(t2) - h(tl )I :5
< w(i7)I h(t2) - h(ti)I
for r E [c, d], tI, t2 E [c, d] and k = 0,1, ... because Fk E
.F(G, h, w) (see Definition 3.8). The integrals f d DFk(x(T), t)
f d DFk(V(rr), t) evidently exist by Corollary 3.16, and Corollary
1.35 yields the estimate
d
II
D[Fk(x(T), t) - Fk(V(T), t)]II
J
Id
<
w(,q) dh(s) = w(rj)(h(d) - h(c))
(8.4)
for every k = 0,1,....
Since Fk E .F(G, h, w) we have
II Fk(x, t2) - Fk(x, tJ )II < Ih(t2) - h(ti )I
for every x E B, and tl , t2 E (a, b) and this leads to the conclusion
that limp.o+ Fk(x, t + p) = Fk(x, t+) and limp.o+ Fk(x, t - p) =
Fk(x,t-) for every (x, t) E G uniformly with respect to k =
0,1, .... Hence by (8.1) we obtain
lim Fk(x, t+) = lim lim Fk(x, t + p) _
k-»oo p-+0+
= lim lim Fk(x, t + p) _
=pll
Fo(x,t+p)=Fo(x,t+)
264
VIII. Continuous dependence for generalized ODE'S
and similarly also
lim Fk(x, t-) = Fo(x, t-)
k--.oo
provided (x, t) E G. Using these inequalities and assuming that
V(r) = cj for r E (sj_1,sj), (sj_1,sj) C [c, d] we obtain (see the
proof of Corollary 3.15)
sj
DFk(W(T), t) _
= Fk(c.j, sj-) - Fk(cj, oo) - Fk(4o(s.i), sj-) + Fk(co(s.i), Sj)+
+Fk(cj,vo) - Fk(cj,sj-1+)+
+Fk(p(sj-1), sj-1 +) - Fk(co(sj-1 ), sj-1) =
Fk(cj, sj-) - Fk(cj, sj-1 /+) +
Fk(c'(Sj-1 ), sj-1+)-
-Fk(W(S.i-1 ), sj-1) - Fk((p(Sj), sj-) + Fk((p(Sj), Sj).
and therefore
lim
k-.oo
D[Fk(cp(r), t) - Fo(cp(T), t)] = 0.
is"
(8.5)
_1
Since cp is a finite step function we obtain from (8.5) using the
additivity of the integral the relation
km
f
d
D[Fk(4p(r), t) - Fo(cP(r), t)] = 0.
By (8.4) we get
f d DFk(x(r), t) - f d DFo(x(r), t)II :5
(8.6)
VIII. Continuous dependence for generalized ODE's
:511
f
d
DFk(x(r), t) -
f
d
+11 f DFo(x(r), t)
c
-f
265
d
DFk(v(r), t)II+
d
t)iJ+
c
d
d
+111 DFk(c2(r),t)- f DFo('P(T),t)ii <
d
< 2w(,)(h(d) - h(c)) + 11
Jc
DFk(sp(T), t)
d
+11 f DFk(SP(T),t)-f
-
f
d
DFo((p(r), t)II
c
d
c
by the choice of i. Taking k -> 00 on both sides of this inequality
we obtain
d
DFk(x(r), t)
lim
k-.oo
c
-f
d
DFo(x(r), t)11<
c
d
6 + k-.oo
lim fl f DFk(V(T),t) c
f
dDFo(V(T),t)iI,
c
and since e can be taken arbitrarily small we obtain the result
f DFk(x(r), t) = f
d
km
d
DFo(x(r), t).
0
8.2 Theorem. Assume that Fk : G -> Rn belongs to the class
.F(G, h, w) fork = 0,1, ... and that
lim
= Fo(x,t)
(8.7)
VIII. Continuous dependence for generalized ODE's
266
for (x, t) E G.
Let xk : [a, l3] -. Rn, k = 1, 2, ... be solutions of the generalized differential equation
dx = DFk(x,t)
dr
on [a,#] C (a, b) such that
urn
xk(s) = x(s),
and (x(s), s) E G for s E [a,#]
s E [a,#],
(8.9)
-
Then x : [a, /3] - Rn is of bounded variation on [a, /3] and it
is a solution of the generalized differential equation
dx
dT
= DFo(x, t)
(8.10)
on
Proof. By Lemma 3.10 we have
Ilxk(s2) - xk(s, )II < Ih(s2) - h(sl )I
f o r every k = 1, 2, ... and s2, sl E [a,#] .
Hence
IIxk(s)II < Ilxk(a)II + h(s) - h(a) < Ilxk(a)II +h(/3) - h(a)
and
vara Xk < h(/3) - h(a).
(8.11)
By (8.9) we have xk(a) ---* x(a) for k -- oo and therefore
the sequence (xk) of functions on [a,#] is equibounded and by
(8.11) of uniformly bounded variation on [a, /3]. By Helly's Choice
VIII. Continuous dependence for generalized ODE's
267
Theorem (see [104] ) there exists a subsequence of (xk) which
converges pointwise to a function of bounded variation on [a, 0].
Hence we conclude by (8.9) that the function x : [a, 0] - Rn is
of bounded variation on [a, /3], and Corollary 3.16 leads to the
conclusion that the integral f DFo(x(r), t) exists.
By definition of a solution of the generalized differential equation (8.8) we have
82
Xk(S2) - xk(sl) = J DFk(xk(r),t)
(8.12)
for every sl, s2 E [a, /3] and k = 1, 2, .... Our aim is to show that
82
lim f DFk(xk(r) , t)
koo
d
DFo (x(r) , t)
(8 . 13)
dl
.,
for any $1, s2 E [a, /3] because passing to the limit k -+ oo in
(8.12) we obtain
82
x(s2) - x(sy) =
DFo(x(r),t)
J9,
for every 81, s2 E [a, /3] provided (8.13) is true, and this means
that x : [a, /3] --+ R" is a solution of (8.10) on the interval [a,#].
To prove (8.13) let us consider the difference
82
DFk(xk(r),t)
I
-
f32
DFo(x(r),t) _
sd2
D[Fk(xk(r), t) - Fk(x(r), t)]+
,
1,12
D[Fk( x(r), t) - Fo(x(r), t)]
+
,
268
VIII. Continuous dependence for generalized ODE'S
for a<sl <s2<8.
By Lemma 8.1 we have
f2
D[Fk(x(T), t) - Fo(x(-r), t)] = 0.
Jim
k-.oo s,
J
(8.14)
Since Fk : G -- Rn belongs to the class.F(G, h, w) for k = 0, 1, .. .
we have
II Fk(xk(T), t2) - Fk(xk(T), tJ) - Fk(x(T), t2) + Fk(x(T ), tI )II <
(8.15)
< w(Il xk(T) - x(T)II)I h(t2) - h(tl)I
for T,tl,t2 E [a,,Q].
The functions xk - x, k = 1, 2, ... are of bounded variation on
[a, Q] and therefore the functions w(II xk (T) - x(r)II) are regulated
nonnegative real functions because they have onesided limits at
every point [a, fl]. Hence by Corollary 1.34 the integrals
/32
Js,
w(Ilxk(S) - x(S)II)dh(S)
exist f o r every k = 1, 2, .... Using (8.15) it can be shown that
the assumptions of Corollary 3.15 hold and by this corollary we
obtain the inequality
92
II
J
D[Fk(xk(T),t) -- Fk(x(r),t)]II <
l
f
<_ J '
w(Ilxk(s)- x(s)I!)ah(s)
9,
for every sl, 82 E [a,,0] and k = 1, 2, ... .
Moreover, (8.9) implies
kli w(Ilxk(S) - x(S)II) = 0,
s E (a,#]
(8.16)
VIII. Continuous dependence for generalized ODE's
269
and we evidently have also
0 C w(Ilxk(S) - x(s)II) C K = const.
for s E [ca, J3]. Hence by Corollary 1.32 of the convergence theorem
we obtain
klim
-oo
w(Ilxk(s) - x(s)II)dh(s) = 0
and by (8.16) also
lm
D[Fk(xk(r), t) - Fk(x(r), t)] = 0.
k-oo f'2
i
This relation together with (8.14) yields (8.13) and concludes the
proof.
D
8.3 Examples.
1. For k = 1,2.... let us define functions hk : [-1,1] --+ IR as
follows:
hk(t) = t
hk(t)=t+
fort E [-1,01,
for t E (0, 11
and set ho(t) = t for t E [-1, 1]. Now define
Fk(x, t) = xhk(t),
IxI < 1, t E [-1, 1], k = 0, 1'...
.
For the functions Fk : (-1, 1) x [-1, 1] -+ R we have
IFk(x,t2) - Fk(x,tI)I <- (hk(t2) - hk(tl))IxI : hk(t2) - hk(tl)
270
V111.
Continuous dependence for generalized ODE'S
and
IFk(x,t2) - Fk(x,tl) - Fk(y,t2)+Fk(y,tl)I
Ix - yl(hk(t2) - hk(tl))
provided x, y E (-1, 1) and -1 < t1 < t2 < 1. Let us set
h(t) = t
fort E [-1,0],
h(t) = t + 1 fort E (0,1].
It is easy to see that we have Fk E .F(G, h, w) with w(r) = r for
r > 0, G = (-1,1) x [-1,1] and also
Jim Fk(x, t) = x Jim hk(t) = xho(t) = Fo(x, t).
k
k
Let us consider the generalized differential equation
dx
TIT
= DFk(x, t) = D[xhk(t)]
(8.17)
with the initial condition x(-1) = xo with Ixol <
. A solution
2e2
xk : [-1,1] --+ (-1,1) C R of (8.17) exists on [-1,1] and satisfies
by definition the integral equation
x(S) = x0 +
r D[x(r)hk(t)] ' x0 + J
J
x(r) dhk(T)
1
1
It is easy to check that the function xk
by
Xk(t) = et+lxo
fort E [-1, 0],
xk(t) = e'+1(1 + k)xo
for t E (0,1]
1) given
VIII. Continuous dependence for generalized ODE's
271
is a solution of (8.17) with xk(-l) = xo and
lira xk(t) = et+lxo for t E [-1,1].
The limit function xo(t) = et+l xo, t E [-1, 1] is evidently a
solution of the classical initial value problem
x=x, x(-1)=xo
and by Theorem 8.2 also of the corresponding generalized form
dx
d = DFo(x, t) = D[xt],
x(-1) = xo.
(8.18)
2. For k = 1, 2, ... let us define functions Ilk [-1,1] - R as
:
follows:
hk(t) = t
for t E [-1, k],
hk(t) = t +
for t E (k, l]
k
and set ho(t) = t for t E [-1, 1]. Define as in the first example
Fk(x, t) = xhk(t),
IxI < 1, t E [-1, 1], k = 0, 1, ...
.
For the functions Fk : (-1,1) x [-1, 1] -* R we have
I Fk(x, t2)
- Fk(x, tl )I < (hk(t2) - hk(tl ))IxI < hk(t2)
- hk(tl )
and
IFk(x,t2)-Fk(x,tl)-Fk(y,t2)+Fk(y,tl)I <
< Ix - yI (hk(t2) - hk(tl ))
V111. Continuous dependence for generalized ODE'S
272
provided x, y E (-1,1) and -1 < t1 < t2 < 1.
Moreover, we again have
Jim Fk(x, t) = x
k
lin hk(t) = xho(t) = Fo(x, t)
k
oo
provided x E (-1, 1) and -1 < t < 1.
It is easy to check that the function xk : [-1, 11 -+ (-1, 1)
given by
xk(t) = et+lxo
for t E [-1, k],
xk(t)=et+'(1+)xo
for
tE(k,1]
is a solution of (8.17) with xk(-1) = xo and
Jim xk(t) = et+lxo for t E [-1,1].
The limit function xo(t) = et+lxo, t E [-1, 1] is again a solution of the initial value problem
x = x,
x(-1) = xo
and also of the corresponding generalized form
dx =
DFo(x,t) = D[xt],
x(-1) = xo.
(8.18)
Nevertheless, this evident convergence result cannot be explained by Theorem 8.2 because it is not possible to have a nondecreasing function h : [-1,1] --* R such that Fk E .F(G, h,w) for all
k = 0,1,
....
Indeed, for the functions Fk : (-1, 1) x [-1,1] -+ R
we have
Fk(x, t2) - Fk(x, t1)j = (hk(t2) - hk(ti ))I XI
VIII. Continuous dependence for generalized ODE'S
273
for x E (-1, 1) and -1 < t 1 _< t2 < 1. If Fk E .F(G, h, w)were
valid for all k = 0,1, ... , then we should have
IFk(x,t2)
- Fk(x,tl)I = (hk(t2) - hk(tl))Ixl
h(t2) - h(ti)
for xE(-1, 1) and -1<t1 <t2<1.
Hence e.g. for x = 2 the above inequality implies
1
1
2k - 2
2k-1
1hk(2k(k
- 1))
2k-1
<h(2k-1
-
2k+1
hk(2k(k + 1) )
1))-h(
2k+1
2k(k+ 1)
for every k > 1 and this means that for an arbitrary positive
constant C > 0 we obtain the inequality
h(l) - h(-1) > C
which contradicts the fact that h is a finite function. Therefore
we cannot have a nondecreasing h : [-1,1) -- R such that Fk E
.F(G, h, w) for all k = 0,11 ....
Looking at the second example we can see that in this case we
have
lim sup(hk(t2) - hk(tl )) < ho(t2) - ho(ti )
k-+oo
for -1 < ti < t2 < 1, and we notice that the function ho is
continuous in [-1, 1]. This example motivates our subsequent
considerations.
Let us introduce some notation. Let w : [0, +oo) --+ [0, +oo)
be continuous, nondecreasing and w(0) = 0. Assume that ho :
[a, b] -+ R is a nondecreasing function which is continuous from
VIII. Continuous dependence for generalized ODE'S
274
the left on [a, b] and that an arbitrary 77 > 0 is given. If [a, /3] C
[a, b] then denote
D = D(i, ho, [a,,8]) = It E [a, Q]; w(ho(t+) - ho(t)) > i}.
It is clear that for every q > 0 the set D is finite. The existence of
the onesided limits of the function ho at every point in [a, /3] and
the continuity of w yields the existence of a function 81 : [a,,3]
(0, +oo) with the following properties:
If t D then
w(ho(t + 61(t)) - ho(t - b1(t))) < ij
(8.19)
provided t E (a, /3) and
(ho (a + 61(a)) - ho(a)) < 17,
w(ho(/3)
< 17
- ho(/3 - bl(/3))) (8.20)
if a D, /3 D, respectively.
If t E D then
w(ho(t + b1 (t)) - ho(t+)) < 77,
w(ho(t) - ho(t - b1(t))) < r7,
t E [a, 0),
t E (a,/3].
(8.21)
If t E D is one of the endpoints of the interval [a, /3] then only
the respective one of these inequalities is required.
Further, define b2 : [a, 3] - (0, +oo) by the relations
b2 (r) = dist(r, D) for r 0 D,
b2(r)
for r E D
(8.22)
and take
b+i,ho (r) = min(b1(r), b2(r)) for r E [a,#] .
(8.23)
VIII. Continuous dependence for generalized ODE'S
275
The function bn,ho : [a, /3] -4 (0, +oo) is evidently a gauge on
[a,#] and by Cousin's Lemma 1.4 the set of bq,ho--fine partitions
of [a, /3] is nonempty. Assume that
A = {ao, Ti, ai, ... , ak-i , Tk, ak }
is a b,,,ho-fine partition of [a, /3]. Then
D C {Ti, T2, ... , Tk}
(8.24)
because 69,ho (T) < bi (r) for r E [a, 0] by (8.23) and (8.22) holds.
This can be shown in the same way as in the proof of Theorem
1.11. Moreover, by virtue of the properties of the gauge b,,,ho the
partition A has the following properties:
if Tm
D then w(ho(am) - ho(am_i)) < t]
(8.25)
and
if
E D then w(ho(am) - ho(Tm+)) < i and
w(ho(Tm)
- ho(am-t )) <,q.
(8.26)
The property (8.25) easily follows from (8.19) and (8.20) while
(8.26) is a consequence of (8.21).
Since the right hand side limits of ho exist at every point of
[a,,8) we can see that the bv,ho-fine partition A of [a, /31 can be
chosen such that
am > Tm if Tm E D (1 [a, /3).
(8.27)
Concluding these preparatory considerations we state that for
every nondecreasing function ho : [a, /3] -4 1 which is continuous
from the left and for every q > 0 there exists a partition of the
interval [a, /3] wich satisfies (8.24), (8.25), (8.26) and (8.27).
Since the set of points of continuity of ho is dense in [a,)3] and
all the inequalities in (8.25) - (8.27) are sharp, the partition can
be chosen in such a way that the points a1,. .. ak_i are points
of continuity of ho.
VIII. Continuous dependence for generalized ODE'S
276
8.4 Lemma. Assume that the functions h, ho : [a, b] - R are
nondecreasing and continuous from the left. Let F E .F(G, h,w)
and let x : [a, /3] -- Rn, [a,,31 C [a, b] be such that (x(r), r) E G
for r E [a,#] and
IIx(t2) - x(t1)II < ho(t2) - ho(ts),
a < t1 < t2 < Q.
(8.28)
Then the integral fa DF(x(r), t) exists and if ri > 0 is arbitrary and A = {ao, rs, a,,. , ak-1 i rk, ak} is a partition of [a, /3]
- -
which satisfies (8.24), (8.25), (8.26) and (8.27) then
(
f DF(x(r), t) -
S. II : n(h(Q) - h(a) + ho(N) - ho(a))
M=1
(8.29)
where
Sn, = F(x(rm), ant) - F(x(Tm), am-1 )
if
S. =F(x(rm+), am) - F(x(Tm+), r,n+)+
+ F(x(rm ), rm+) - F(x(rm ), am-1 )
provided r E D.
Proof. The existence of the integral fa DF(x(r), t) is clear by
Corollary 3.16 because by (8.28) the function x is of bounded
variation on [a, /3].
Let q > 0 be given and assume that the partition A satisfies the
conditions given in the statement. The integral f m-1 DF(x(r), t)
exists for every in = 1,... , k (see Theorem 1.10 ). Let m E
{ 1, ... , k } be fixed and assume r E D. Then for every e > 0
there exist gauges b-, b+ on [a,n-s, rm], [7-,,,, an], respectively,
such that
DF(x(r),t) - S(A )II <
II
f
am_l
2
VIII. Continuous dependence for generalized ODE'S
277
for every 6--fine partition A- of [CYin-1 , T,n] where S(A-) is the
integral sum corresponding to A- and
am
II
IM DF(x(T), t) - S(A+)II < 2
for every b+-fine partition A- of [T,,,, am] where S(A+) is the
integral sum corresponding to A+. From these inequalities we
immediately get
am
II f m _1 DF(x(T), t) - S(A-) - S(A+)II < e.
(8.30)
Without loss of generality we assume that the value 8+(T,,,) >
0 is such that
h(a) - h(T,n+) < ?7
for every a E (Tm, Tm +
ho(T,n+) - ho(T,n)
w(ho(Tm+) - ho(Tn,))
b+(Tr=))
Assume further that
is a 6--fine partition of [am-1, T,nJ with
am-1 =ao <Tl <ixl <... <at-1 <tt=at = Tm
and that
A+ = {at, Tt+1, at+1, ... 16,1+3-1) rt+s, al+,}
is a b+-fine partition of [Tm, am] with
Tm =at < T1+1 < a1+1 < T1+2 <
< ... < a1+s-1 5 Tl+s = al+a = am.
278
VIII. Continuous dependence for generalized ODE'S
We give an estimate for II S(A-) + S(A+) - Sm II using the
properties of the class F(G, h, w) . We have
IIS(A ) + S(A+) - S. II
IF(x(Tj),aj) - F(x(Tj),aj-1)]-
II
j=1
-(F(x(Tm ), T,n+) - F(x(T, ), am-1))+
J+s
+
IF(x(Tj ), aj)
- F(x(Tj ), aj-1 )]-
j=J+1
-(F(x(Tm+),a,,,) - F(x(T.n+),T,n+))II =
J
= II EIF(x(Tj), aj) - F(x(Tj), aj_1)j=1
-F(x(Tm),,&j) + F(x(Tni), aj_1)]-F(x(Tm),Tm+) - F(x(Tm),&J)+
+F(x(TJ+1), &,+1)
- F(x(i1+1), &J)-
-F(x(Tm+), &J+1) + F(x(Tm+),,r., +)+
J+s
+ E [F(x(T`j ), aj) - F(x(Tj ), aj-1 )j=1+2
-F(x(Tm+), aj) + F(x(Tm+), &j-1 )] II C
II F(x(Tj ), aj)
- F(x(Tj ), aj-1)-
j=1
-F(x(Tn, ), aj) + F(x(Tm), &j-1)II+
VIII. Continuous dependence for generalized ODE'S
279
+II F(x(,rm), a1+1) - F(x(T,n),Tn,+)-F(x(7-,1, +), &1+1) +
F(x(r11 +), T,,1+)II+
1+s
+
II F(x(Ti ), ai) - F(x(Ti ),
j=1+2
-F(x(Tm+), a.i) + F(x(Tm+), ai-1 )II
1
:5 E w(IIx(TJ)
- x(T,,,)II)(h(al) - h(aJ-1 ))+
l=1
+w(II x(Tne+) - x(Tm )II)(h(a1+1) - h(T,n+))+
1+s
+ E w(II x(TJ) - x(Tm+)II)(h(a7)
j=1+2
< t1(h(Tm) - h(am-1 ))+
+w(ho(Tm+) - ho(Tm))y
ho(Tm+) - ho(Tm)
+
w(ho(Tm+) - ho(Tm))
+i (h(a,,,) - h(&,+,)) C
:5 71(h(am) - h(an,-1) + ho(T,,,+) - ho(Tn,)) <
ii(h(a,n) - h(am-1) + ho(an,) - ho(am-1 ))
This estimate together with (8.30) leads to the inequality
jarn
II
m_1
DF( x(T),t)-S,,, I
+(IS(A)+S(A)-SmI)
< e+7)(h(anl) - h(a,,,-1)+ho(a..) - ho(am_1)).
<
VIII. Continuous dependence for generalized ODE'S
280
Since s > 0 can be chosen arbitrarily small this inequality yields
Om
11Z
<
m_I
DF(x(r),t)
- S,n11 <
i1(h(a) - h(an-, ) + ho(a:) - ho(a,n-i )).
(8.31)
For the case when r D the same inequality can be proved with
less effort. Using (8.31) and the inequality
Q
II
J
k
DF(x(r), t) - E s,,, II
n=1
k
<-
cl' m
DF(x(r), t) - s=II
II
j
m
n=1
1
we can see immediately that (8.29) holds.
Let us now derive another theorem on continuous dependence
on a parameter which is based on the above Lemma 8.3. The
structure of the statement is close to that of Theorem 8.2.
8.5 Theorem. Assume that Fk : G -> Rn belongs to the class
,17(G, hk, w) for k = 0, 1,... where hk : [a, b] --+ R are nondecreasing functions which are continuous from the left when k = 1, 2, .. .
and the function ho : [a, b] --> R is nondecreasing and continuous
on [a, b].
Assume further that
limsup(hk(t2) - hk(ti)) < ho(t2) - ho(ti)
k-oo
for every a < ti < t2 < b.
Suppose that
lim Fo(x,t) = Fo(x,t)
k-oo
for(x,t)EG.
(8.32)
VIII. Continuous dependence for generalized ODE's
281
Let Xk : [a,#] -+ R", k = 1, 2,... be a sequence of solutions
of the generalized differential equation
dx = DFk(x,t)
dT
(8.8)
on [a, /9] C (a, b) such that
lira xk(s) = x(s),
s E [a, (3],
k-»oo
(8.9)
and (x(s), s) E G for s E [a, 13].
Then x : [a, /3] -4R" is a continuous function of bounded variation on [a, Q], and it is a solution of the generalized differential
equation
dx
d
_
= DFo(x,t)
(8.10)
on the interval [a,,3].
Proof. Since Xk is a solution of (8.8) we have by Lemma 3.10
Il xk(t2) - xk(tl )11 < hk(t2) - hk(tl) for k = 1, 2.... and t1, t2 E
t1 < t2. Therefore by (8.9) and (8.32) we have also
IIx(t2) - x(tl)11 < ho(t2) - ho(ti)
for t1i t2 E [a, /3], tl _< t2. Hence the function x is continuous on
[a, /9] and the integral fa DFo(x(r), t) exists by Corollary 3.18.
We will show that x : [a, ,Q] -> R" is a solution of (8.10). To do
this, it is sufficient to prove that for every s1 , s2 E [a,#], s1 < s2
we have
82
82
lim
koo I'l
DFk(xk(r), t) =
DFo(x(r), t).
(8.33)
JJJ,
Suppose that s1i $2 E [a,#], s1 < s2 and choose an arbitrary
rj > 0. Determine a partition
A = {ao,r1,a11...,al-1, r,, at}
VIII. Continuous dependence for generalized ODE'S
282
of the interval [Si, s21 for which
w(ho(a,,,) - ho(a,.-1)) < TI,
in = 1, ... ,1.
Since the function ho is continuous, the set
D = D(t, ho, [a,,3]) = {t E [a,#]; w(ho(t+) - ho(t)) >
17}
is empty and (8.25), (8.26) is automatically satisfied. The assumption (8.32) implies that there is a ko E N such that for
k > ko we have
w(hk(am) - hk(am-1)) < ,j,
m = 1, ... ,1
and we have also w(hk(T,,,+) - hk(Tm )) < 71 for in = 1, ... 71
because hk(T,,,+) - hk(am) _< hk(a,,,) - hk(am_t) and this means
that T,,, D(i7, hk, [a, Q]) for in = 1, . . . , 1 and k > ko. Hence for
all sufficiently large k E N the partition A satisfies (8.25), (8.26)
and (8.27) also with the functions hk instead of ho.
Due to (8.9) it can be also assumed that
w(Il xk(Tf,) -
)II) < ii
for all in =1,...,land k>ko.
By Lemma 8.4 we obtain for k > ko the inequality
I
DFk(xk(T),t) - E Sm,kll < 211(hk(s2) - hk(31))
M=1
where S,, ,k = Fk(xk(r, ), am) - Fk(xk(T.,),am-1), and also
S2
Il
f DFo(x(T),t) S1
I
S,,,,011 < 27(ho(s2)
m=1
- ho(st))
VIII. Continuous dependence for generalized ODE'S
where Sm,o = Fo(x(r,n),a,,,)
Hence
82
32
J
II
283
DFk(xk(r), t) 91
JS
DFo(x(T), t)II <
1
C II DSnt,k - Sm,O)II + 277(hk(S2) - hk(SI) + ho(s2) - ho(s1 ))
m=I
(8.34)
Let us consider the first term on the right hand side of (8.34).
We have
(Sm,k - Sm,o)II
II
m=1
1
E IIFk(xk(Tm), am) - Fk(xk(Tm), am-1 )m=]
-Fo(x(rm ), am) + Fo(x(rm ), am-I )II <
IIFk(xk(T., ),am)
Fk(xk(Tm),an _1 )-
m=1
-Fk(x(Tm ), am) + Fk(x(Tm), am-1)II+
I
+ ? IIFk(x(Tm),am) - Fk(x(Tm),am-1)`
m=1
-Fo(x(Tn, ), am) + Fo(x(Tm ), am-1 )II <
t
E w(II xk(Tm) - x(Tm)II)(hk(ant) - hk(a,n-I ))+
m=I
1
+ E (II
m=1
Fk(x(rm ), am) - Fo (x(Tm ), am) II +
VIII. Continuous dependence for generalized ODE'S
284
+IIFk(x(T., ), an, - ) - Fo(x(T,,),am-1)I[] <
I
<
7i(hk(an,) - hk(a,,,-1))+
m=I
+ 1: [II Fk(x(T, ), a,,)
-
am )II +
M=1
Fo(x(T,n),
By (8.7) the suns on the right hand side of this inequality converges to zero for k -* oo. Using the fact that rt > 0 can be taken
arbitrarily small we obtain by (8.34) that for every e > 0 there is
a k1 E N such that for k > k1 we have
fs
82
II
DFk(xk(T), t) -
82
Js,
DFo(x(7-), t)II < e,
i.e. (8.33) holds and the theorem is proved.
0
Theorems 8.2 and 8.5 are in a certain sense weak forms of continuous dependence results for generalized ordinary differential
equations. The most important assumption is the relation (8.7)
which ensures that if a sequence X k : [a, /3] - Rn of solutions
of (8.8), k = 1, 2, ... converges pointwise to a certain function
x : [a, /3] -, Rn then this limit is a solution of the equation (8.10).
There are different additional conditions on the right hand sides
Fk of (8.8) and Fo of (8.10) in Theorems 8.2 and 8.5.
Now we present results with an additional uniqueness condition for the "limit" equation (8.10).
8.6 Theorem. Assume that Fk : G -- Rn belongs to the class
.F(G, h, w) for k = 0, 1, ... where h : [a, b] -, R is nondecreasing,
continuous from the left, and that
klim Fk(x, t) = Fo(x, t)
-oo
(8.7)
VIII. Continuous dependence for generalized ODE'S
285
for (x, t) E G.
Let x : [a, /3] -, IR' , [a, /3] C [a, b] be a solution of the generalized differential equation
dx
r= DFo(x,t)
(8.10)
d
on [a, l3] which has the following uniqueness property:
if x : [a, -y] - R", [a, y] C [a, /3] is a solution of (8.10) such
that y(a) = x(a) then y(t) = x(t) for every t E [a, y].
Assume further that there is a o > 0 such that ifs E [a, /3] and
IIy - x(s)II < o then (y, s) E G = Bc x [a,#], and let Yk E IR",
k = 1, 2,
...
satisfy
lira Yk = x(a).
k-.oo
Then for sufficiently large k E N there exists a solution xk of
the generalized differential equation
dx
= DFk(x,t)
on [a,,81 with xk(a) = Yk and
kym xk(s) = x(s),
s E [a,,3].
Proof. By assumption we have (y, a) E G provided
fly - x(a)II < 2
or
fly - x(a+)II = fly - x(a) - Fo(x(a), a+) + Fo(x(a), a)]I < 2
VIII. Continuous dependence for generalized ODE'S
286
Since Yk -> x(a) for k - oo, we have by (8.7) also
Yk + Fk(yk, a+) - Fk(yk, a) -> x(a) + Fo(x(a), a+) - FO (X (a), a)
for k - oo because
IIFk(yk, a+) - Fk(x(a), a+) - Fk(yk, a) + Fk(x(a), a)II
< w(Ilyk
- x(a)II)(h(a+) - h(a))
and
Fk(x(a), a+) - Fk(x(a), a) - Fo(x(a), a+) + Fo(x(a), a) - 0
for k --> oo. Hence we can conclude that there is a k, E N such
that for k > kl we have (yk, a) E G as well as
(Yk +Fk(yk,a+) - Fk(yk,a),a) E G.
Since the set B, is open there exists d > a such that if t E [a, d]
and
Iix - (yk + Fk(yk, a+) - Fk(yk, a))II
h(t) - h(a+)
then (x, t) E G for k > kl. Using Theorem 4.2 on local existence
for generalized ordinary differential equations we obtain that for
k > kl there exists a solution xk : [a, d] -+ R" of the generalized
differential equation (8.8) on [a, d] such that xk(a) = yk, k > k1.
We claim that limk-oo xk(t) = x(t) for t E [a, d].
Let us mention that the solutions xk of (8.8) exist on the interval [a, d] and that this interval is the same for all k > kl . Indeed,
looking at the proof of Theorem 4.2 it is easy to check that the
value d > a depends only on the function h which is common for
all right hand sides Fk of (8.8).
VIII. Continuous dependence for generalized ODE's
287
By Theorem 8.2, if the sequence (xk) contains a pointwise convergent subsequence on [a, d] then the limit of this subsequence
is necessarily x(t) for t E [a, d] by the uniqueness assumption on
the solution x of (8.10).
By Lemma 3.10 the sequence (xk), k > k, of functions on
[a, d] is equibounded and of uniformly bounded variation on [a, d].
Therefore by Helly's Choice Theorem (see [104] ) the sequence
(xk) contains a pointwise convergent subsequence and x(t) is
therefore the only accumulation point of the sequence xk(t) for
every t E [a, d], i.e.
xk(t) = x(t) for t E [a, d].
In this way we have shown that the theorem holds on
[a, d], d > a. Let us assume that the convergence result does not
hold on the whole interval [a, Q]. Then there exists d* E (a,,6)
such that for every d < d* there is a solution Xk of (8.8) with
xk(a) = Yk on [a, d] provided k E N is sufficiently large and
xk(t) = x(t) fort E [a, d] but this does not hold on [a, d]
for d > d*. By Lemma 3.10 we have
Ilxk(t2) - xk(t, )II < Ih(t2) - h(t1)I,
t2, t1 E [a, d*)
for k E N sufficiently large. Therefore the limits xk(d*-) exist
and we also have
lim xk(d*-) = x(d*-) = x(d*)
k-oo
since the solution x is continuous from the left. Defining xk(d*) =
xk(d*-) we obtain limk.,,. xk(d*) = x(d*) and this means that
Theorem 8.6 holds on the closed interval [a, d*], too. Using now
d* < /3 as the starting point we can show in the same way as
above that the theorem holds also on the interval [d*, d* + A]
with some A > 0 and this contradicts our assumption. Therefore
the theorem holds on the whole interval [a, Q].
0
8.7 Remark. Let us mention that in Theorem 8.6 the continuity from the left of the function h is assumed. This assumption
288
VIII. Continuous dependence for generalized ODE'S
is made here to avoid technical problems of the same kind as in
Chap. IV. In this case we have to take care for onesided (right)
discontinuities of a solution only. In the general case the idea is
the same but some additional reasoning has to be done for the
possible discontinuities from the left.
Theorem 8.6 is derived from the result given in Theorem 8.2.
Now we give a similar result based on Theorem 8.5. Assuming
again the uniqueness of a solution of the "limit" equation we
obtain a result which is stronger than the result of the previous
Theorem 8.6.
8.8 Theorem. Assume that Fk : G --> Rn belongs to the class
F(G, hk, w) for k = 0,1, ... where hk : [a, b]
lR are nondecreasing functions which are continuous from the left when k = 1, 2, .. .
and the function ho : [a, b] --> R is nondecreasing and continuous
on [a, b].
Assume further that
limsup(hk(t2) - hk(ti )) < ho(t2) - ho(ts)
k-woo
(8.32)
forevery a_<tl <t2 <b.
Suppose that
km Fk(x,t) = Fo(x,t)
00
8.7)
for (x, t) E G.
Let x : [a, a] -+ 1Rn, [a, a] C [a, b] be a solution of the generalized differential equation
dx
dr
=
DFo(x,t)
(8.10)
on [a, Q] which has the following uniqueness property:
if y : [a, -y] - ]Rn, [0, -fl C [a,,0] is a solution of (8.10) such
that y(a) = x(a) then y(t) = x(t) for every t E [a, -y].
VIII. Continuous dependence for generalized ODE's
289
Assume further that there is a e > 0 such that ifs E [a, p] and
and let yk E R z,
Ily - x(s)II < P then (y, s) E G = B, x
k = 112.... satisfy
lain Yk = x(a)
Then for every p > 0 there exists a k,, E N such that for k E
N, k > k* there exists a solution Xk of the generalized differential
equation
dx = DFk(x, t)
(8.8)
dr
on [a f3] with xk(a) = Yk and
iiXk(s) - x(s)II <
11,
s E [a,#]
(8.35)
holds.
Proof. The existence of the solutions xk of the equation (8.8)
for sufficiently large k E N and the pointwise convergence
k
im xk(s) = x(s),
sE
[a,,19]
can be shown in the same way as in Theorem 8.6 with minor
changes arising from the assumption (8.32).
For showing (8.35) let us consider the difference xk(s) - x(s)
for sufficiently large k E N and for s E [a, p]. By the definition
of a solution we have
xk(s) - x(s) =
9
= yk - x(a) +J D[Fk(xk(r), t-Fo(x(r), t)]
a
for every s E [a,,0].
(8.36)
290
VIII. Continuous dependence for generalized ODE'S
Let an arbitrary q > 0 be given. Choose a K > 0 such that
is < 77, w(rc) < ij and determine a partition
A = {ao,rl,al,...,al_l,ri,al )
of [a, /3] such that
ho(am) - ho(am-1) < K, am-1 = Tm < am,
m = 1,2,...,1.
(8.37)
It is evident that such a partition A exists because the function
ho is uniformly continuous on [a, /3]. The choice of rc gives also
w(ho(am) - ho(am-i )) < w(K) < 1J,
rn = 1,2,...,1.
(8.38)
By the continuity of ho we have D = D(11, ho, [a, /3]) = 0 and
the partition A satisfies (8.25), (8.26) and (8.27) . Since (8.32) is
assumed, there exists a ko E N such that for k > ko, k E N we
have
hk(am) -
K, m
and consequently also
hk(am_1)) < ij, m = 1, 2, ... ,1.
(8.39)
Further, rm = a,,,_17 m = 1,2,... ,1 and the inequality
hk(a'm-1 +) - hk(a.,-1) < hk(am) - hk(am-1) < rc
yields
w(hk(rm+) - hk(Tm)) < w(rc) < 77
D(T1, hk, [a, /3]) for m = 1, 2, ... ,1.
Hence for k > ko the partition A satisfies (8.25), (8.26) and (8.27)
and therefore T., = a,,,_1
with hk instead of ho.
VIII. Continuous dependence for generalized ODE's
291
Due to the pointwise convergence of solutions xk to x there is
a k1 > ko, kl E N such that f o r every k > kl and m = 1,2,...,1
we have
Il xk(Tm) - x(r,,,)II < h;
and therefore also
w(Ilxk(rm_) - x(rm)II) < w(K) < ,J.
(8.40)
Assume now that s E (a, ,Q] is given. Then there is a
p E { 1, 2, ... ,1} such that s E (ap_1, ap]. Denoting
As = {ao,rl,al,...,ap-1,rp,ap = s}
we obtain a partition of [a, s] and this partition satisfies (8.25),
(8.26) and (8.27) and also the same relations with Ilk instead of
ho provided k > k1.
We will consider the difference
8
f D[Fk(xk(r), t) - Fo(x(r), t)]
Since by the properties of a solution we have
llxk(t2) - xk(tl )II
hk(t2) - hk(tl )
and
IIx(t2) - x(t1)II < ho(t2) - ho(t1)
for a < tl < t2 < s and k > k1, Lemma 8.4 yields the inequality
s
II
Ja
DFk(xk(r), t)
- L Sm,k II < 277(hk(s) - hk(a))
m=1
for k > k1 where
S.,k = Fk(xk(rm),a,>e) - Fk{xk(r,,),am-1) =
VIII. Continuous dependence for generalized ODE'S
292
= Fk(xk (am-1 ), am) - Fk(xk(am-1), am-1 )
for m = 1,2,...,p- l and
Sp,k = Fk(xk(ap-1), S) - Fk(xk(ap-1), ap-1),
and also
s
II
f DF0(x(T),t) - L Sm,oII < 2rq(ho(s) - ho(a))
m=1
Q
where
Sm,o = Fo (x(Tm ), am) - Fo(x(Tm ), am-1) =
= Fo(x(am-1 ), am) - Fo(x(am-1 ), am-1)
form=1,2,...,p-land
Sp,o = Fo(x(ap-1), s) - Fo(x(aP-1), ap-1).
For k > k1 this yields
II
f sDFk(xk(r),t) - f DFo(x(T),t)II
a
a
P
E II Sm,k - Sm,oll + 277(hk(s) - hk(a) + ho(s) - ho(a)) _
M=I
P-1
_
E
IIFk(xk(am-1), am) - Fk(xk(am-1), am-1 )-
m=1
-Fo(x(am-1), am) + Fo(x(am-1 ), am-1)II+
+II
Fk(xk(ap-1), S)
- Fk(xk (ap-1), ap-1)-
VIII. Continuous dependence for generalized ODE'S
293
-Fo(xk(ap-1 ), S) + Fk(xk(ap-1 ), ap-1 )II+
+2r/(hk(s) - hk(a) + ho(s) - ho(a)).
(8.41)
By the properties of the class F(G, hk, w) and by (8.40) we
obtain fork > kl and m = 1, 2, ... , p - 1
II Fk(xk(am-1 ), am) - Fk(xk(am-i ), am-1 )-Fo(x(am-1 ), ani) + Fo(x(am-1 ), am-1 )II C
II Fk(xk(am-l ), am) - Fk(xk(arrt-1 ), am-1 )-
-Fk(x(ant-l ), arn) + Fk(x(ant-1 ), ant-i )II+
+IIFk(x(am-1 ),ant) - Fo(x(a,n_1 ), an1)II+
+IIFk(x(am-1 ), ant-1) + Fo(x(am-1 ), am-1 )II
C w(Ilxk(ant-l) - x(am-1 )II)(hk(am) - hk(am-1))+
+IIFk(x(am-1 ), am) - Fo(x(am-1 ), a,, )II+
+II Fk(x(am-1 ), ant-1) + Fo(x(an1-l ), ant-1 )II
j(hk(am)
hk(am-1 ))+
+IIFk(x(am-i ), ant) - Fo(x(am-_1 ), ant)II+
+II Fk(x(ant-1 ), ant-1) + Fo(x(am-1 ), am-1)II
and similarly also
IIFk(xk(ap-1), S) - Fk(xk(ap-1 ), ap-1 )-
-Fo(x(ap_1), s/) + Fo(x(ap-1), ap-1)II <
:5 w(II
xk(ap-1) - x(ap-1)II)(hk(ap) - hk(ap-1 ))+
+IIFk(x(ap-1 ), ap) - Fo(x(ap-1 ), ap)II+
(8.42)
VIII. Continuous dependence for generalized ODE'S
294
+II Fk(x(ap-1), ap-1) + Fo(x(ap-1), ap-1)II+
+hk(ap) - hk(s) + h0(ap) - ho(s)
:5 ?7(hk(ap)
- hk(ap-1))+
+IIFk(x(ap-1 ), aP) - Fo(x(ap-1), ap)II+
Fo(x(ap__1),
+IIFk(x(ap-1), ap-1) +
ap-1)II+
+hk(ap) - hk(S) + ho(ap) - ho(s).
(8.43)
Hence (8.38), (8.39), (8.41), (8.42) and (8.42) yield
II S DFk(xk(r), t) - jS DF0(x(r), t)II :5
a
a
< 2q(hk(s) - hk(a) + ho(s) - ho(a))+
P
+77 L (hk(am) - hk(am-1))+
m=1
P
+ L [IIFk(x(am-1), am) - Fo(x(am-1), am)II+
m=1
+IIFk(x(am-1), am-1) + Fo(x(am-1), am-1)III+
+hk(ap) - hk(ap-1) + ho(ap) - ho(ap-1) :5
< 277[1 + ho(Q)
- ho(a)] +7l[hk($) - hk(a))+
1
+
[IIFk(x(am-1), am) - Fo(x(am-1), am)II+
m=1
+IIFk(x(am-1), am-1) + Fo(x(am_1), am-1)II].
(8.44)
VIII. Continuous dependence for generalized ODE'S
295
By the assumption (8.7), for every it > 0 there is a k2 E N,
k2 > k1 such that
t
i [IIFk(x(am-1 ), arn) - Fo(x(am-1 ), a.) 11+
M=1
+II Fk(x(am-1 ), am-1) + Fo(x(am-1 ), am-1)II] <
4
for k > k2 and by (8.32) there is a k3 E N, k3 > k2 such that for
kEN,k>k3wehave
hk(Q) - hk(a) < ho(/3) - ho(a) + 1.
Since q > 0 can be arbitrary, we choose it so that
µ
77 <
20(1- ho(A) - ho(a))
By the inequality (8.44) we then obtain for every s E [a,#]
II
Ja
9
DFk(xk(r), t) - f 9 DFo(x(r), t)II <
a
< µ + 5p(1 + ho(Q) - ho(a))
- 4 20(1 + ho(Q) - ho(a))
-µ
-2
Hence by (8.36) we have
Ilxk(s) - x(s)II < IIYk - x(a)II + 2
for every s E [a,,3] and, finally, we take k,, E N such that k* > k3
and II yk - x(a)II < 2 for k > k*. Then we obtain
Ilxk(q) -x(s)II <µ
for every s E [a, /3], k > k* and the theorem is proved.
0
296
VI!!. Continuous dependence for generalized ODE'S
Applications to classical ODE's
Now we will use the previous results for deriving continuous dependence results for classical ordinary differential equations. The
link between generalized differential equations and the classical
Caratheodory concept is given by Theorem 5.14.
Let P be a metric space, po E P na accumulation point of P.
Let B be the closure of the set B.
Assume that a function f : B x [a, b] x P --* R" satisfies the
following conditions:
f (x, , p) is Lebesgue measurable on [a, b] for (x, p) E B X P,
there exists a Lebesgue measurable function m : [a, b] -'
R such that ' m(s) ds < +oo and
Ilf(S, xII < m(s)
for (x, SIP) E B x [a, b] x P,
there exists a Lebesgue measurable function 1 : [a, b] -> R
such that fQ 1(s) ds < +oo and
IIf(S,x,p) - f(S,y,p)II < 1(s)w(IIx -VII)
for (x, s, p), (y, s, p) E B x [a, b] x P.
Let us define
t
F(x, t, p) = f R XI SIP) ds
for (x, t, p) E B x [a, b] x P.
(8.45)
297
Applications to classical ODE'S
8.9 Theorem. Assume that for some c E [a, b] we have
/t
lim
J
t
f(x,s,p) ds = J f(x,s,po) ds
c
(8.46)
c
for (x, t) E B x [a, b].
Let x(t, p) : (a,) x P
IN
it,
p
po be solutions of
i = f (x, t, p)
(8.47)
on [a, 0] C [a, b] such that
lim x(t,p) = y(t)
P-'PO
for t E [a Q],
(8.48)
andy(t)EBfortE[a,/3].
Then y : [a,,3] -> R" is a solution of
(8.49)
x = f(x,t,po)
on [a, 0].
Proof. By the assumptions made above, Proposition 5.5 yields
that the function F given by (8.45) belongs to F(B x [a, b], h, w)
for all p E P where
t
h (t)
= f m(s) ds +
c
j
i(s) ds,
t E [a, b].
The relation (8.46) can be written in the form
lim F(x, t, p) = F(x, t, po )
P-Po
when (8.45) is taken into account. By Theorem 5.14 the equation
x = f (x, t, p) has the same set of solutions as the generalized
ordinary differential equation
dx
= DF(x, t, p)
(8.50)
298
VIII. Continuous dependence for generalized ODE's
for all p E P. Consequently, using (8.48) and Theorem 8.2 we
obtain that the function y : [a, 131 -- R" is a solution of the
generalized differential equation
dx
dT
= DF(x, t, po)
Therefore again by Theorem 5.14, y is a solution of (8.49) on
[a, /3] and this proves the theorem.
8.10 Theorem. Assume that for some c E [a, b] we have
It
lim
PPo Jc
f (x, s, p) ds =
I
t
f (x, s, po) ds
(8.46)
c
for (x, t) E B x [a, b].
Let x : [a, /j] --+ ]R",
c [a, b] be a solution of the differential equation (8.49) on [a, /3] which has the following uniqueness
property:
if y : [a, -y] -+ ]R", [a, ry] C [a,#] is a solution of (8.10) such
that y(a) = x(a) then y(t) = x(t) for every t E [a, -y].
Assume further that there is a p > 0 such that ifs E [a, /3] and
Ily - x(s)II < P then (y, s) E G = B x [a,,61, and let y, E R",
p E P satisfy
lim yP = x(a).
P-'Po
Then for every ,u > 0 there exists an q, > 0 such that for
dist(p, po) < q* (by dist the metric in P is denoted) there exists
a solution x(t,p) of the differential equation (8.47) on [a, /3] with
x(a,p) = yP and
Ilx(s, p) - x(s)II < it,
s E [a,#]
holds.
Proof. It is easy to check that the function F given by (8.45)
satisfies the assumptions of Theorem 8.8 with the parameter p E
Averaging for generalized ODE'S
299
P instead of the sequential setting with k E N in Theorem 8.8.
Theorem 8.8 together with Theorem 5.14 yields the statement of
our theorem. We are not going into details here. Looking at the
proof of Theorem 8.8 it can be easily seen that it can be repeated
for the case of the parameter p E P with with minor technical
0
changes.
8.11 Remark. Both the previous Theorems 8.9 and 8.10 are
corollaries of continuous dependence results for generalized ordinary differential equations. They represent continuous dependence theorems for classical ordinary differential equations under
the relatively weak "integral continuity" assumption represented
by (8.46). In this context let us mention the results of M.A. Krasnoselskij and S.G. Krejn [66], J. Kurzweil and Z. Vorel [22] as well
as of Z. Artstein [1], [2], [3], [4].
Averaging for generalized
ordinary differential equations
Now we derive a special result for generalized ordinary differential equations with a small parameter which is analogous to the
known results substantiating the method of averaging for classical
ordinary differential equations.
8.12 Theorem. Assume that G = B x [0, +oo), B = {x E
R"; jjxii < c}, c > 0 and that F E .P(G, h, w) where
h : [0, +oo) -> IR is continuous from the left, nondecreasing, and
w [0, +oo) --> [0, +oo) is continuous, increasing and w(0) = 0.
Assume that
h(r + a) - h(a)
<
r
- C, C = const
r-oo
lien sup
(8.51)
for every a > 0 and
lim
F(x, r) = Fo(x),
r
x E B.
(8.52)
VIII. Continuous dependence for generalized ODE'S
300
Let y : [0, +oo) -+ Rn be a uniquely determined solution of
the autonomous ordinary differential equation
y = Fo(y)
(8.53)
which belongs to B together with its p-neighbourhood with p >
0, i.e. there is a p > 0 such that {x E R"; lIx - y(t)II < p} C B
for every t E [0, +oo).
Then for every y > 0 and L > 0 there is an co > 0 such that
for E E (0, co) the inequality
II x,(t) - £(t)I < µ
holds for t E [0, L ] where x, is a solution of the generalized
E
ordinary differential equation
dx
= D[EF(x, t)]
aT
(8.54)
such that x,(0) = y(0), and Ct is a solution of the
autonomous ordinary differential equation
x = EFo(x)
(8.55)
on [0, L] such that C,(0) = y(0).
E
Proof. For y E B, t E [0, +oo) and s > 0 define
Gf(y, t) = EF(y,
t
and take h,(t) = Eh(t) for t >r 0. The function h, is evidently
E
nondecreasing and continuous from the left on [0, -boo).
301
Averaging for generalized ODE'S
Since F E F(G, h, w) we obtain by definition
IIG,(y, t2) - G,,(y, ti )II = EIIF(y,
EIh(t
!) - F(y,
E-' )II <_
h(E')I = IhE(t2)-h,(tl)I,
and similarly also
IIGE(y, t2) - GE(y, ti) - G,(x, t2) + GE(x, ti)II <
< w(II y
- xII)Ihe(t2) - he(ti )I
whenever x, y E B, tl, t2 E [0, +oo). These inequalities mean
that we have Ge E F(G, hS, w) for e > 0.
If Y E B, then (8.48) yields
lim F(y, r) - F(y, 0) = lim F(y, r) = Fo(y)
r-'oo
r
r
r- oo
and therefore (8.51) and (8.52) imply that for every y > 0 there
is an R > 0 such that for r > R we have
IIFo(y)II < IIFo(y) -
F(y, r)
F(y, 0)
r
+ IIF(y, r) - F(y, 0)II < + h(r) - h(0)
-
r
77
II+
< 217 + C
r
because F E .F(G, h, w) implies IIF(y, r) - F(y, 0)II < h(r) - h(0).
Since i > 0 can be chosen arbitrarily small, we have
IIFo(y)II < C,
Y E B.
(8.56)
Analogously, if x, y E B then for every 77 > 0 there is an R > 0
such that for r > R we have
IIFo(x) - FO(y)11 < n +
II F(y, r) - F(y, 0) - F(x, r) + F(x, 0) 11 <
r
302
VIII. Continuous dependence for generalized ODE'S
<-y+w(IIy-x11)h(r)
h(O)
r
<y(1+w(IIY-xil))+Cw(IIY-xII),
and again since n > 0 can be arbitrarily small, we obtain
IIFo(x) - Fo(y)II <- Cw(II y - xll )
(8.57)
provided x, y E B.
ForyEB, t>0weobtain
lim G, (y, t) = lim eF(y,
a-o+
tl-o+ t t fly,
e
t) = tFo(y)
and also
lim G, (y, 0) = lim eF(0,
e o+
t) =
e-*o+
t) = 0.
e
Denote Go(y,t) = tFo(y) for y E B, t _> 0. Then the relations
given above imply
lim Go(y,t) = Go(y,t).
e--.0+
(8.58)
By (8.56) and (8.57) we have Go E .F(G, h, w) where ho(t)
Ct, t>0.
_
Further, for 0 < tl < t2 < +oo we obtain by definition
h5(t2) - h,(tl) = e(h(tE)
(t2 - tl
t2
tl
(h(t2
tl
- h(e' )) _
+ 1) - h(S
and the assumption (8.51) yields
lim sup(he(t2) - h1(tl )) < C(t2 - ti) = ho(t2) - ho(ts) (8.59)
CO+
Averaging for generalized ODE's
because we have
303
t2 - tl
= +00.
e-0+ 6
It is easy to se that (8.59) is satisfied in the case tl = t2 as well.
lira
Using the fact that y : [0, +oo) --> B is a solution of (8.53) we
obtain by the properties of the generalized Perron integral the
equality
92
eai
y(s2) - y(si) = f Fo(y(r)) dr =
9
32
2
=
D[Fo(y(r))t] = J
DGo(y(r), t)
Si
for sl, 32 E [0, +00), i.e. y is a solution of the generalized ordinary
differential equation
Jr
(8.60)
Go(y, t)
on [0, +oo), and by the assumption this solution is uniquely determined. In this way we have shown that all assumptions of
Theorem 8.8 are satisfied for the case of the continuous parameter 6 --> 0+. Using the result of Theorem 8.8 we obtain that
for every p > 0 and L > 0 there exists a value co > 0 such that
for E E (0, co) there is a solution y, of the generalized differential
equation
dry = DGe(y,t)
(8.61)
on the interval [0, L] such that y,(0) = y(O) and
tIye(s)
- y(s)1I : µ
(8.62)
for all s E [0, L].
For the solution ye : [0, L] --+ B of (8.61) we have
at
a2
113
ye(S2) - ye(S1) =
DG.(ye(r),t) = e f
t
DF(ye(r), E))
It
VIII. Continuous dependence for generalized ODE'S
304
whenever s1, s2 E [0, L]. For t E [0, L] denote xe(t) = ye(Et).
E
Then
xa(t2)
-
xe(ti) = ye(Et2) - yE(Etl) _
Etz
= E et2 DF(ye(a), s) = e
et,
I
fats
DF(x,(- ), s
E
for ti, t2 E to, L]. Applying the Substitution Theorem 1.18 with
e
o
the continuous monotone substitution so(a) = we obtain
et2
f
et,
Q S
DF(xe(-),
-) =
E
r
IP(et2)
t2
DF(xa(7),t) _ f DF(xa(T),t)
t,
E
for any tl,t2 E to, L]. This together with the previous equality
yields
xe(t2) - xa(t,) = of DF(xa(r),t)
tl
it
for tl, t2 E [0,
tion xe
:
] and xe(0) = y,(0) = y(0). Therefore the func-
to, L]E -+ B is a solution of the generalized differential
e
equation (8.54) on [0, L
Analogously it can be shown that the function la : [0, L] -* B
E
given by ee(t) = y(et) is a solution of the autonomous ordinary
differential equation (8.55) on [0, L].
E
Finally, by (8.62) we obtain
I(xe(t) - G(01I = IIya(et) - y(et)II < i
Averaging for generalized ODE'S
305
for every t E [0, L] and the theorem is proved.
0
E
8.13 Remark. Theorem 8.12 forms an analogue of the known
result of N.N. Bogoljubov on the method of averaging. The classical autonomous differential equation (8.55) is the averaged equation for (8.54), where the process of averaging is described by the
relation (8.52).
As an application of Theorem 8.12 we give a result for averaging in the case of differential equations with impulses. We use the
equivalence of such type of equations with generalized ordinary
differential equations as they are described in Chap. V. Results
on averaging of differential equations with impulses were given for
the first time by A.M. Samojlenko (see especially the monograph
[1201).
8.14 Theorem. Assume that G = B x [0, +oo), B = tx E
R";IIxII < c}, c>0andK>0. Let f :G->R" be such a
function that
II.f(x,s)II < K,
and
II f(x, s) - .f(y, s)II < KIIx - yII
for x, y E B, S E [0, +oo). Further, assume that a sequence of
points 0 < t1 < t2 <
is given such that
lim sup
l E1<d
r-oo r a<t; <a+r
for every a > 0 and that I; : B -4R" i = 1, 2,
of functions such that
III=(x)II < K,
and
III;(x) - I=(y)II -< KII x -- yII
is a sequence
VIII. Continuous dependence for generalized ODE'S
306
for x, y C B, i = 1, 2, -. Suppose that
lim
1
r-oo r
J
0
f (x, s) ds = fo(x), x E B
and
Ii(x) = Io(x), x E B.
lim 1
r-oo r
O<ti <r
Let y : [0, +oo) - - R" be a uniquely determined solution of
the autonomous differential equation
y = fo(y) + Io(y)
which belongs to B together with its p -neighbourhood with p >
0.
Then for every i > 0 and L > 0 there is an co > 0 such that
for e E (O,eo) the inequality
II xe(t) - ee(t)II < i
holds for t E [0, L ] where xE is a solution of the differential equae
tion with impulses
i = e f (x, t), t # ti
Ax l t;
(8.63)
= x(ti+) - x(ti) = eIi(x(ti )), i = 1, 2, .. .
on [0, LI such that x,(0) = y(0), and t is a solution of the "ave
eraged" system of the autonomous ordinary differential equation
i = e[fo(x) + IOW)
(8.64)
Averaging for generalized ODE'S
307
on [0, L] such that &r(0) = y(0).
Proof. By Theorem 5.20 the system (8.63) is equivalent to
dx
dT
where
= DEF(x, t)
t
F(x,t)=jf()d+EHt(t)
i-1
and
0 for t E [0, v], H,(t) = 0 for t > v.By the assumptions it follows that the function F belongs to ,F(G, h, w)
where
00
h(t) = Kt+K>2Ht;(t)
i=1
for t > 0 is evidently continuous from the left and nondecreasing
by the properties of the function H and w(r) = r for r > 0.
The assumptions further yield
lim 1 F(x, r) = fo(x) + Io(x) = Fo(x)
r-+oo r
and
h(r + a) - h(a)
< K(1 + d)
r- m
r
for x E B and a E [0, +oo). All assumptions of Theorem 8.12
being satisfied we obtain the desired result immediately when uslim sup
ing the above mentioned equivalence of the system with impulses
(8.63) and the corresponding generalized differential equation.
CHAPTER IX
EMPHATIC CONVERGENCE FOR GENERALIZED
ORDINARY DIFFERENTIAL EQUATIONS
Let us consider in this chapter the convergence effect which
occurs in the theory of ordinary differential equations when the
right hand sides of the equations converge to the Dirac delta
function.
We consider these phenomena in the framework of generalized
ordinary differential equations and the results presented below are
in fact a continuation of the previous Chap. VIII on continuous
dependence on a parameter.
Let us start with a simple example which shows the convergence effect mentioned above.
9.1 Example. Let bk
:
[-1, 1] -- R, k = 1, 2,... be the b-
sequence given by
bk(t)=k
bk(t) = 0
fortE(0,1],
fort E 1-1,1]\(O,
J.
Assume that A, B E L(R") are given constant n x n- matrices. Let us consider the sequence of linear ordinary differential
equations
x = [A + bk(t)B]x
(9.1)
308
309
IX. Emphatic convergence for generalized ODE'S
where x E R", t E [-1, 1] and the initial condition x(-1) = x E
R" is given. It is easy to check that if for a given k E N we define
the function xk : [-1, 1] ---* R" by the relations
xk(t) = eA(t+1)x for t E [-1,0],
Xk(t) = e
(A+kB)t
xk(0) = e (A+kB)t e Ax ^- for t E (0,
k
1
x t
eA(t k)x
1
eAte BeAx for t E
1
1]
(9.2)
IR", k = 1, 2,... is a solution of (9.1) with
then xk
the given initial condition x(-1) = x E R. Passing to the limit
k - oo we easily get
lira xk(t) = eA(t+l)x = eAte Ax for t E [-1, 01
and
lim xk(t) = eAteBeAx for t E (0, 1].
k-oo
Denoting
x(t) = eAte Ax for t E [-1,0],
x(t) = eAteBeAx for t E (0, 1]
(9.3)
we have simply
lim xk(t) = x(t) fort E [-1,1]
k-»oo
and we can ask wether the limit x is a solution of some differential
equation.
The function x : [-1,1] -+ R" given by (9.3) is continuous on
the intervals [-1, 0] and (0, 1] and can exhibit a discontinuity at
the point t = 0 for which
t0+ eAte BeAx - eAx =
x(0+) - x(0) = lim
310
IX. Emphatic convergence for generalized ODE'S
_ (e11 - I)eAx = (eB - I)x(0),
I denotes the n x n identity matrix.
From this relation we can immediately see that x is continuous at t = 0 if either x(0) = 0 (equivalently Y = 0) or the matrix
eB - I is singular and x(0) = eAx belongs to its null space. Otherwise the function x has a discontinuity at t = 0 and therefore it
cannot be a solution of a classical ordinary differential equation
on the interval [-1, 1]. Nonetheless, the function x : [-1, 1] -- ]Etn
given by (9.3) is a solution of a generalized ordinary differential
equation which can be easily constructed using the facts given in
Chap. V.
It can be easily observed that the function x : [-1, 1] -+ Rn
given by (9.3) is a solution of the linear ordinary differential equation
i=Ax
on the intervals [-1, 0] and (0, 1] with
x(0+) = x(0) + (eB - I)x(0).
This can be treated as a differential system with an impulse act-
ing at t = 0 as in Chap. V. By the results given in Chap. V
this system is equivalent to the generalized ordinary differential
equation
dx
dr
= D[At + (eB - I)H(t)]x
(9.4)
where H(t) = 0 for t < 0 and H(t) = 1 for t > 0. Using the result
stated in Theorem 5.20 we obtain that the limit x of the sequence
(xk) of solutions of the linear ordinary differential equations (9.1)
is a solution of the generalized ordinary differential equation (9.4).
The ordinary differential equation (9.1) is for every k E N
equivalent to a generalized ordinary differential equation by virtue
IX. Emphatic convergence for generalized ODE's
311
of the results given in Theorem 5.14. For the construction of this
generalized ordinary differential equation let us define
Fk(x, t) =
j
[A + bk(s)B]x ds = [At + Hk(t)B]x
where Hk(t) = f 1 bk(s) ds and by definition we have
fort E [-1,0],
Hk(t) = 0
fe
Hk(t)=J bk(s)ds=J kds=kt fortE(0,11,
0
0
Hk(t)=1
for t
k01.
By Theorem 5.14 the solutions xk : [-1, 1] - > R", k = 1, 2, ... of
(9.1) on the interval [-1, 1] are also solutions of the generalized
ordinary differential equations
dx
dT= D[At + Hk(t)B]x
(9.5)
on the interval [-1, 1] for every k E N.
In (9.5) a sequance of generalized ordinary differential equations is given and the righjt hand sides of this sequence of equations obviously satisfy
lim [At + Hk(t)B]x = [At + H(t)B]x.
(9.6)
In spite of this fact, having in mind the convergence results for
generalized ordinary differential equations desribed in Chap. VIII
one would expect that the solutions Xk
R" of the
initial value problem
dx
dT
= D[At + Hk(t)B]x,
x(-1) = i
(9.7)
312
IX. Emphatic convergence for generalized ODE's
should converge to a solution of the "limit" problem
dx
dT
= D[At + H(t)BJx,
x(-I) _
(9.8)
on the interval [-1,1].
It is easy to check that the solution of the initial value problem
(9.8) is the function x : [-1, 1] -a R" given by
x(t) = eA(t+1)x for t E [-1,0],
x(t) = eAt(I + B)eAx for t E (0, 1].
Comparing this with the actual limit of the sequence of solutions
of (9.7) which is given by the relations (9.3), we can see that they
are different unless the exceptional case I + B = eB occurs.
The discrepancy is caused by the discontinuity of the limit
function [At + H(t)B]x of the right hand sides [At + Hk(t)B]x of
the generalized ordinary differential equations (9.5) at the point
t = 0. Nevertheless, the structure of the equations (9.4) and (9.8)
is similar.
The true and effective limit equation to the sequence of equations (9.5) is the generalized ordinary differential equation (9.4)
and the convergence theorems 8.2 and 8.5 given in Chap. VIII
cannot be applied to the case described in this example.
The discontinuity of the solution of the limit equation (9.4) is
in the example given by the limit
lyrn
[xk(I) - xk(0)]
Since we are treating linear equations in the example, this limit
can be calculated exactly. In the general nonlinear case such a
calculation cannot be carried out, nevertheless the idea works in
this case, too.
We will use the concept of emphatic convergence which was
introduced by J. Kurzweil in [69].
IX. Emphatic convergence for generalized ODE's
313
9.2 Definition. Assume that G = Bc x (a, b), B, _ {x E
R"; Ilxil < c}, c > 0 and Fk : G - Rn, k = 1, 2,.... The
sequence of functions Fk, k = 1, 2, ... converges emphatically to
Fo for k -> oo if the following conditions are satisfied:
(i) there exist an increasing continuous function
w : [0, +oo) -> [0, +oo), w(0) = 0 and functions hk
[a, b] -* R, k = 0, 1, 2.... which are nondecreasing and
continuous from the left, such that
F k E .F(G, lzk, w),
k = 0,1, 2, ... ,
limsup(hk(t2) - hk(tl )) < ho(t2) - ho(t1)
provided ho is continuous at the points t1 and t2, a <
ti < t2 < b,
(iii)
lim Fk(x,t) = Fo(x,t) + F*(x,t)
k-.oo
if (x, t) E G, t is a point of continuity of the function ho
and F. : G - R' is such that
JIF*(x, t2) - F*(x, ti )II
Ih*(t2) - h*(tz )I
for t1, t2 E (a, b) where h* : (a, b) -> R' is the break
function corresponding to ho,
(iv) x + Fo(x, t+) - Fo(x, t) E B, for every x E B, t E (a, b),
(v) if ho(to+) > ho(to), (xo,to) E G then for every e > 0
there is a S > 0 such that for each 6' E (0, b) there is a
ko E N with the following property:
if y : [to - b', to + S'] - R" is a solution of the generalized
ordinary differential equation
dx
= DFk(x, t)
aT
IX. Emphatic convergence for generalized ODE's
314
on [to - b', to + b'], k > ko and IIy(to - b') - xo Il < b then
Il y(to + b') - y(to - b') - [Fo(xo,to+) - Fo(xo,to)]ll < 6-
9.3 Theorem. Assume that G = Bc x (a, b). Let hk : (a, b) -i
R, k = 0, 1, 2,... be nondecreasing functions continuous from
the left. Let d E (a, b) be such that ho(t+) = h(t) for t # d.
Assume further that Fk E .F(G, hk, w), k = 0, 1, 2.... and that
the sequence (Fk) converges emphatically to Fo for k -- oo.
Let X k : [a, ,3] ---> R" be solutions of
dx
d= DFk(x,t)
on an interval [a, /3] C (a, b), k = 1, 2, ... such that
lira xk(t) = z(t)
k-+oo
for t E
(9.10)
t # d and d E (a, 0). Then the function x : [a, fl) -
R" defined by x(t) = z(t) for t E [a,,81, t # d and x(d) =
x(s) is a function of bounded variation on [a,#], and
it is a solution of the generalized ordinary differential equation
dx = DFo(x,t)
aT
(9.11)
on the interval [a,#].
Proof. By Lemma 3.10 we have
Ilxk(s2) - xk(sl)II C Ihk(s2) - hk(sl)I
for s1i s2 E [a, Q] and (ii) from Definition 9.2 gives for k
inequality
ll z(s2) - z(sl )II < Iho(s2) - ho(sl )I
oo the
IX. Emphatic convergence for generalized ODE's
315
for s1, s2 E [a,#], s1i s2 # d. This yields the existence of the
onesided limits lim,-d_ z(s) = x(d) and
z(s) = x(d+).
Therefore x : [a, #] -> R" is of bounded variation on [a, /3].
Assume that a _< sl < d < s2 _< Q. By (iii) in Definition
9.2 and by Theorem 8.5 we obtain that for any A > 0 the limit
function x is a solution of the generalized ordinary differential
equation
dx
aT
= D[Fo(x, t) + F. (x, t)]
on the intervals [a, d - 0] and [d+ 0, ,0]. Therefore for any 0 > 0
with 0 <min(s2 - d,d - sl) we have
x(d - 0) - x(sl) =
D[Fo(x(r), t) + F,(x(r), t)] =
d-,&
_I
DFo(x(-r),t)
(9.12)
1
and
x(s2) - x(d + 0) = f
32
+
D[Fo(x(r), t) + F(x(-r), t)] _
s21+0
I
DFo(x(r),t)
(9.13)
because evidently
d -A
r
81
s21
D[F*(x(-r), t)
= Id+0 D[F*(x(r), t) = 0
by the properties of the function F*(x, t) given in Definition 9.2.
For a given e > 0 there is a 61 > 0 such that
sha(d) - ho(d - p)I < 2 , Iho(d + p) - ho(d+)j < 2
(9.14)
316
IX. Emphatic convergence for generalized ODE'S
for every p E (0, S1). Assume that S E (0, 61) corresponds to e by
the requirement (v) from Definition 9.2. From the existence of
the limit limy-d_ z(s) = x(d) we obtain that there is a 0 E (0, S)
such that
IIx(d- 0) - x(d)II < 2
and by (9.10) there is a k1 E N, kl > ko such that
II xk(d - 0) - x(d - ')II <
2
for k > k1. Hence for k > k1 we have
Ilxk(d - 0) - x(d)II < S.
Using (v) from Definition 9.2 we obtain
II xk(d + A) - x(d - 0) -
f
d+0
DFo(x(T), t)II -<
-o
< II xk(d + 0) - x(d - 0) - (Fo(x(d), d+) - Fo(x(d), d))II +
+IIFo(x(d), d+) - Fo(x(d), d)
-J
d
d+A
-O
DFo(x(r), t)II <_
d
< e + II
J_o
DFo(x(T ), t)II+
d+A
+II Fo(x(d), d+) - Fo(x(d), d) -
DFo(x(r), t)II
J
d
Id
+ Iho(d) - ho(d - o)I+
d+A
+IIFo(x(d), d+) - Fo(x(d), d) - f
d
DFo(x(r), t)II <
IX. Emphatic convergence for generalized ODE'S
317
+A
e + + II Fo(x(d), d+) - Fo(x(d), d) -
DFo(x(r), t)II.
LA
By Theorem 1.16 we obtain
d+A
lim
sd+
J
DFo(x(r), t) _
S
d+ All
= Fo(x(d), d+) - Fo(x(d), d) - f
DFo(x(r), t)
d
and because
II
f
d+0
DFo(x(r),t)II
Iho(d+ 0) - ho(s)I
9
we obtain by (9.14) the inequality
d+O
II Fo(x(d), d+) - Fo(x(d), d) -
dA
DFo(x(r), t)II <
< ho (d + O) - ho (d+) < -'
because we also have 0 < 61.
Hence for every k > k1, k E N we get
d+0
II xk(d + A) - xk(d - A) -
d-0
DFo(x(r), t)II < 2e.
By (9.10) there exists k2 E N, k2 > k1 such that
Ilxk(d+L1)-x(d+o)II < 2'
I1xk(d-0)-x(d-A)II
<2
IX. Emphatic convergence for generalized ODE's
318
whenever k > k1 and therefore
d+ a
II x(d + A) - x(d - A) - Id-A DFo(x(r), t) 11 < 3e,
d+0
x(d + A) - x(d -A)=
d
-0
DFo(x(r), t)
(9.15)
since e > 0 was given arbitrarily.
Using (9.12), (9.13) and (9.15) we finally obtain
rs2
DFo(x(r),t).
X(S2) - x(sl) =
(9.16)
Js,
The case when a < sl = d < s2 < 3 can be treated similarly.
The remaining cases of possible positions of si, s2 in the interval
[a,,8] are covered directly by Theorem 8.5 and we obtain that
(9.16) holds for every s 1, s2 E [a, ,l], which proves the theorem.
0
9.4 Remark. Theorem 9.3 represents a continuous dependence
result for generalized ordinary differential equations provided the
right hand sides converge emphatically to a function
Fo E .F(G, ho, w), in the case when the function ho exhibits a
discontinuity at a single point d E (a, b). The idea of the proof of
the general case of a nondecreasing function ho with the possibly
infinite number of discontinuities is essentially the same but the
proof is technically rather complicated. The general result reads
as follows:
Assume that G = B, x (a, b). Let hk : (a, b) --> R, k =
0,1,2.... be nondecreasing functions continuous from the left.
Assume further that Fk E F(G, hk, w), k = 0, 1, 2, ... and that
the sequence (Fk) converges emphatically to Fo for k -> oo.
IX. Emphatic convergence for generalized ODE's
319
Let xk : [a, l31 - Rn be solutions of
dx = DFk(x,t)
dT
on an interval [a, /9] C (a, b), k = 1, 2.... such that
lira xk(t) = z(t)
k-oo
for t E [a, /3] whenever ho(t+) = ho(t). Then x :
[a, /3]
Rn
defined by x(t) = z(t) for t E [a, /3] with ho(t+) = ho(t) and
continuous from the left in [a, Q] is a function of bounded variation on [a,,0], and it is a solution of the generalized ordinary
differential equation
dx
= DFo(x,t)
dT
on the interval [a, /3].
The main tool for the proof of this result is Lemma 8.4 and
the fact that the set of points at which the jump ho(t+) - ho(t) is
greater than a given positive constant is finite. See also Theorem
4.1 in [69] where a detailed proof is given.
Looking at the special case described in Theorem 9.3 and at
Definition 9.2 we can see that we have to determine the value
of the difference Fo(y, d+) - Fo(y, d) which determines the jump
x(d+)-x(d) of the function x at a point d provided x(d) = y (see
Lemma 3.12). This value can be determined via the requirement
(v) in Definition 9.2, i.e. we have to know the limit (for k --' oo)
of increments of the solutions of (9.9) on a short interval containig
the point d. In the general case, to obtain such an information
can be complicated.
IX. Emphatic convergence for generalized ODE'S
320
Looking at Example 9.1 we can see that the jump of the limit
equation at the point d = 0 was evaluated by taking the limit of
the difference xk( ) - xk(0) for k --+ oo. In Example 9.1 we have
Fk(x, t) = [At + Hk(t)B]x,
Fo(x,t) + Fk(x,t) = [At + H(t)BJx,
urn Fk(x,t) = Fo(x,t) + Fk(x,t)
k-.oo
and
Fo(x, t) = [At + H(t)(eB - I)]x
because according to Theorem 9.3 and (v) from Definition 9.2
obviously
Fo(y, d+) - Fo(y, d) =
xk(0)) = (eB - I )y
for a solution Xk of (9.1) for which xk(0) = Y-
9.5 Example. Let G = ]R" x [-1, 1]. Assume that
F E F(G, h, w) where h : [-1,11 --+ R is nondecreasing and continuous in [-1, 1].
For k = 1, 2,... let us have ak E [-1,0), Qk E [0,1] where ak <
Qk and limk-,w ak = limk,m /3k = 0. Let "k : [-1, 1] -+ [0, 1] be
continuous on [-1,1], increasing on [ak, /3k] and such that
Ik(t) = 0 for t E [-1,ak],
file(t) = 1 for t E [,3k,1].
For the restriction fik : [ak, Qk] --> [0, 1] let us denote by file
[0, 1] -+ [ak, 3k] the inverse function to 44. The function 1k 1 is
1
continuous and increasing on [0, 1].
321
IX. Emphatic convergence for generalized ODE's
Define
H(t) =0 fort <0, H(t)=1 fort>O.
Then
inn 4k(t) = H(t)
for every t E [-1, 11, t # 0.
Let a function g : IR" --+ R" be given such that
11g(x)II : K,
11g(x)
- 9(y)II : LII x - yII
(9.17)
for x, y E R.
For (x, t) E G define
Fk(x, t) = F(x, t) + 9(x)$k(t),
k = 1, 2, ...
.
Then by definition we have Fk E F(G, hk,1k), where hk(t) =
h(t) + K'Pk(t) and 11(p) = w(p) + Lp. By definition we further
have
(9.18)
iyi n Fk(x, t) = F(x, t) + g(x)H(t)
k
for every (x, t) E G with t # 0, and the function F(x, t)+g(x)H(t)
clerly belongs to .F(G, h + KH,1 ). Moreover, we have
limsup(hk(t2) - hk(ti)) < ho(t2) - ho(ts)
k-soo
for ti, t2 E [-1, 0) U (0,1] where ho(t) = h(t) + KH(t), t E [-1,1].
We consider the problem whether there is a function Fu(x, t) to
which the sequence Fk(x, t) converges emphatically for k -- oo.
Let us consider the autonomous ordinary differential equation
y = g(y)
(9.19)
322
IX. Emphatic convergence for generalized ODE'S
and assume that for x E R n the function v(s, x), s E [0,1] is the
uniquely determined solution of (9.19) on [0, 1] for which we have
v(0, x) = X.
Since the function h is continuous at 0, for every q > 0 there
exists b > 0 such that
h(b) - h(-b) < i,
(9.20)
and of course also h(Q) - h(a) < ij for every interval [a,,8] C
[-b, b]. Let 6' E (0, b) be given and let ko E N be such that for
k > ko we have
[ak, Nk] C [-b', b'].
Assume that xo E R" is given and that y : [-6', b'] -p R' is a
solution of the generalized ordinary differential equation
DFk(y, t) = D[F(y, t) +
dy =
(9.21)
such that Ily(-6') - xo 11 < 6. Then by the definition of a solution
we have
y(r) = y(-6') + ja DF(y(T ), t) +
y(-6') +
f
f
+J DF(y(r), t) f
r
_
r
a'
DF(y(r), t) +
9(y(T )) d'bk(T) =
a'
r
= y(-6')
D[9(y(7-))'1k(t)]
J-6,
6'
r
+
9(y(T))dtk(r)
ak
for every r E
since $k(t) = 0 for t < ak. We use the
notation of the Stieltjes integral in the second integral.
If now s E [0,1] then 4Pk1(s) E [ak,,Qk] C [-6',6'] and we have
y(IDk 1(s)) =
IX. Emphatic convergence for generalized ODE's
y(-b') +
J
DF(y(T ), t) + f
The
g(y(T )) d bk (T) _
ak
k`(J)
4k`(8)
y(-b') + J
323
DF(y(r),t) +
af
g(y(T))dck(r).
4'_1(0)
14k
Applying the Substitution Theorem 1.18 to the last integral we
obtain
y(Ikk'(s)) = y(-8')+
DF(y(T),t)+J
J a'
0
(9.22)
Since v(s, xo) is a solution of (9.19) on [0, 11, we have
v(s, xo) = x0 + jg(v(c,xo))dc
for every s E [0, 1]. (9.22) and (9.23) yield further
y(.tk' (S)) - v(S, xO) =
*k `(8)
xo +
+
f
DF(y(T ), t)+
J 6'
8
g(y(tk 1(cr))) dcr -
0
J0
a
g(v(cr, xo )) da
and therefore
Ily(Nk 1(s)) - v(S, xo )II
IM-8')
+ I"
J0
l
- xoII + II J
1(a)))
DF(y(r),t)II+
b'
- g(v(o,, xo))II do,
(9.23)
IX. Emphatic convergence for generalized ODE's
324
for all s E [0, 11. Consequently, taking into account (9.17) and
(9.20) we obtain
Ily(Ic- ' (s)) - V(s, xo)II <
b + h(,Dk' (s)) - h(-b)+
+L
f
(c)) - v(a, xo)II dv <
0
9
b + tj + L f
(or)) - v(a, xo )II da.
0
Using the Gronwall lemma (see also Corollary 1.43) we obtain
from the last inequality the estimate
IIy('Dk' (s)) - v(s, xo )II 5 (b + y)e1
,
s E [0,1]
and for s = 1 also
IIy(Ok) -v(1,xo)11= IIy(''k'(1)) -v(1,xo)II < (b+r7)e`
Further we have
6'
Y(6')
- y(Qk) = J
D[F(y(T), t) + 9(y(7-))4kt]
ak
=
f'
DF(y(T), t)
k
because 4kk(t) = 1 for t > 13k, and consequently, by (9.20),
6'
IIy(b') - y(ak)11 <-1I f DF(y(T),t)II <
h(b')
- h(/3k) <
.
IX. Emphatic convergence for generalized ODE'S
325
Hence
Il y(S') - v(1, xo)II
+lIy(Qk) - v(1, xo)ii
and
11Y(b1)
IIy(S') - Y(Qk)ll+
y)eL
y + (S +
- y(-S') - (v(1, xo) - v(0, xo))ll <
IIy(S') - v(1, xo)11 + 11
y(-S')
- X0 11
<77 +S+(S+?l)eL =(E+77)(1 +eL).
For a given e > 0 the values of q > 0 and b > 0 can be taken so
small that (S + ii)(1 + eL) < e and we can easily conclude that
for every e > 0 there is a S > 0 such that if b' E (0, S) and k > ko
then for every solution y : [-S', S'] -- Rn of (9.21) on the interval
[-S',S'] such that Ily(-S') - roil < S the inequality
Ily(S')
- y(-S') - (v(1, xo) - xo)II < e
(9.24)
holds.
For (x,t) E G define
Fo(x, t) = F(x, t) + (v(1, x) - x)H(t)
x) is the uniquely determined solution of (9.19) with
v(0, x) = x. Then
where
Fo(x, 0+) - Fo(x, 0) = v(1, x) - x.
It is easy to see that (iv) from Definition 9.2 holds and using the
definition of FO we can write (9.24) in the form
IIy(S') - y(-S') - (Fo(xo, 0+) - Fo(xo, 0))ii < e
IX. Emphatic convergence for generalized ODE'S
326
and the results presented above show that (v) from Definition
9.2 is fulfilled. The remaining parts of Definition 9.2 are easy to
check with
F. (x, t) = [g(x) - (v(1, x) - x)]H(t)
for (x, t) E G and finally it can be concluded that the functions
Fk converge emphatically to F0 for k -> oo. Therefore the continuous dependence result given in Theorem 9.3 can be used in
this situation.
The case described by this example is applicable for a sequence
of ordinary differential equations
x = f(x, t) + 9(x)tpk(t), k = 1, 2, ...
(9.25)
where Vk : [-1, 1] --> R", k = 1, 2, ... is a sequence of Lebesgue
integrable functions which tends positively to the Dirac function,
i.e.
cpk(t) > 0,
t E [-1,1],
and the sequence of functions (bk : [-1, 1] - R given by
1k(t)
cpk(s) ds, k = 1, 2, ... , t E [-1,1]
satisfies the assumptions given at the beginning of this example.
For g : R" - R" we assume (9.17).
If f : G -, R" satisfies the Caratheodory conditions and
II f (x, t) 11 < r(t),
t E [-1,1]
where r is a Lebesgue integrable function in [-1, 1] then we can
define
F(x, t)
f(x, s) ds
IX. Emphatic convergence for generalized ODE's
327
and by Theorem 5.14 the ordinary differential equation (9.25) is
equivalent to the generalized ordinary differential equation
dx
d7
= D[F(x, t) + g(x)4k(t)1,
k = 1, 2,...
.
The right hand sides of this sequence of generalized ordinary differential equations emphatically converge to the function
F(x, t) + (v(1, x) - x)H(t)
where v(t, x) is the solution of (9.19) defined on [0, 1] and such
that v(0, x) = X.
Let us define a function x : [-1, 1] -- R" as follows:
Let u : [-1, 0] -> R" be a unique (for increasing values of t)
solution of the ordinary differential equation
x = f(x,t)
(9.26)
on [-1, 0]. Let v(t, u(0)) be the unique (for increasing values of
t) solution of (9.19) defined on [0, 1] such that v(0, u(0)) = u(0).
Let further w : [0, 1] --+ R' be a unique (for increasing values of t)
solution of (9.26) on the interval [0, 1] for which w(0) = v(1, u(0)).
Let us set
x(t) = u(t) for t E [-1, 0],
x(t) = w(t) for t E (0, 1].
1] -+ R' is a solution of the generalized ordinary
Then x
differential equation
dx
dT
= D[F(x, t) + (v(1, x) - x)H(t)]
(see Theorem 5.20). It can be further shown that if yk -+ x(-1)
for k -- oo then for sufficiently large k E N there exists a solution
R" of (9.25) on [-1, 11 and
Xk
krn xk(t) = x(t)
328
IX. Emphatic convergence for generalized ODE'S
for t E [-1,1], t :AO.
This convergence phenomenon expresses the fact that the dynamics of the system (9.25) in a small neighbourhood of 0 is
emphatically forced by the large term g(x)epk(t) which influences
the system in a short time in the same way as the term g(x) does
in a time interval the length of which is close to the integral of
cPk, i.e. close to 1.
Remark. At the end of Chapter V the work of D. Frankova on
generalized ordinary differential equations with a substitution was
mentioned. Her approach is a good tool for explaining emphatic
convergence effects of the type described by Definition 9.2. The
corresponding convergence results are given in the paper [26].
CHAPTER X
VARIATIONAL STABILITY FOR GENERALIZED
ORDINARY DIFFERENTIAL EQUATIONS
Concepts of stability
Let us assume that G = B, x (a, b) where B, = {x E R'n; lixII <
c}, c > 0. Let F : B, x [0, +oo) -+ R' be given.
We assume further that h : [0, +oo) -- R is a nondecreasing function defined on [0, +oo), w : [0, +oo) -> R is a continuous,increasing function with w(0) = 0 and F E .F(G, h, w) where
G = B, x [0, +oo).
In addition to these usual conditions we assume that
F(0, t2) - F(0, tl) = 0
for every tu, t2 > 0.
(10.1)
This assumption evidently implies that
r$2
Js,
DF(0, t)
F(0, s2) - F(0, sl) = 0,
sl, s2 E [0, +oo)
and therefore the function x given by x(s) = 0 for s > 0 is a
solution of the generalized ordinary differential equation
dx
= DF(x,t)
dT
(10.2)
on the whole half-axis [0, +oo).
Let us introduce some concepts of stability of the trivial solution x(s) = 0, s E [0, +oo) of the equation (10.2).
329
X. Variational stability for generalized ODE'S
330
10.1 Definition. The solution x - 0 of (10.2) is called variationally stable if for every E > 0 there exists 6 = 6(E) > 0 such
that if y
:
[to, tl ]
---+ B, 0 < to < tl < +oo is a function of
bounded variation on [to,tl], continuous from the left on (to,tl]
with
1'y(to)1I < b
and
var(y(s) -
j
DF(y(r),t)) < S
o
then we have
for t E [to,t1].
Ily(t)II < e
10.2 Definition. The solution x - 0 of (10.2) is called variationally attracting if there exists ao > 0 and for every e > 0 there
isaT=T(E)>0and-y=-y(e)>0such that
0 _< to < tl < +oo is a function of bounded variation on [to, tl],
continuous from the left on (to, tI ] with
IIy(to)II < ao
and
vaio(y(s)
-J
a
:o
DF(y(r),t)) <'y
then
IIy(t)II < e
for all t E [to, tl ] n [to + T(e), +oo) and to > 0.
10.3 Definition. The solution x = 0 of (10.2) is called variationally-asymptotically stable if it is variationally stable and variationally attracting.
Following the same lines we now give definitions, which introduce the concept of stability of the trivial solution x =_ 0 of (10.2)
Concepts of stability
331
with respect to perturbations of the equation (10.2) which are in
some sense small.
Together with (10.2) we consider the perturbed equation
dx
dT
= D[F(x, t) + P(t)]
(10.3)
where P : [0, +oo) - R" is a function which is locally of bounded
variation and continuous from the left on [0, +oo).
Clearly we have
II F(x, t2) + P(t2) - F(x, ti) - P(t1)II <
< I h(t2) + varo2 P - h(t1) - varo' PI
for x E B, and t1,t2 E [0,+oo) and
II F(x, t2) + P(t2) - F(x, t1) - P(t1)-(F(y, t2) + P(t2) - F(y, t1) - P(t1))II 5
< w(II x - yII )I h(t2) - h(tl )I <_
- yII)Ih(t2) + var12 P - h(t1) -
varoPI
and therefore the right hand side F(x, t) + P(t) of the gener: w(IIx
alized ordinary differential equation (10.3) belongs to the class
.F(G, h, w) where h(t) = h(t) + va4 P, and all fundamental results (e.g. the local existence of a solution) hold for the equation
(10.3).
10.4 Definition. The solution x - 0 of (10.2) is called stable
with respect to perturbations if for every e > 0 there exists b =
b(e) > 0 such that if I I yo 11 < b, yo E R" and P E BV ([to, t j ]) is a
function of bounded variation on [to, tI], continuous from the left
on (to, t1 ] and such that
vartaP<b
then we have
IIy(t,to,yo)II < e fort E [to,t1]
where y(t, to, yo) is a solution of (10.3) with y(to, to, yo) = yo
X. Variational stability for generalized ODE'S
332
10.5 Definition. The solution x - 0 of (10.2) is called attractive with respect to perturbations if there exists bo > 0 and for
every e > 0 there is a T = T(e) > 0 and -y =,y(e) > 0 such that
if
Ilyolt<8o,YoER"
and P E BV ([to, t i ]) is a function of bounded variation on [to, tl ],
continuous from the left on (to,tl] and such that
vart, P < 7
then
IIy(t, to, yo)II < e
for all t E [to, ti ] 11 (to + T(e), +oo) and to > 0 where y(t, to, yo) is
a solution of (10.3) with y(to,to,yo) = yo.
10.6 Definition. The solution x
0 of (10.2) is called asymp-
totically stable with respect to perturbations if it is stable and attractive with respect to perturbations.
10.7 Remark. The concept of variational stability comes from
the following intuitive idea: if a certain function given on some
[to, ti] C [0, +oo) is such that the initial value y(to) is close to zero
and on the interval [to, ti ] the function y is "almost" a solution
of (10.2), i.e. the variation of the function
y(s)
- y(to) - J DF(y(r), t)
to
on [to, tl ] is small enough, then y is close to zero on the interval
[to, tl].
On the other hand, the stability with respect to perturbations
is motivated by the desire that the solutions of the perturbed
equation (10.3) be close to zero on a certain [to,tl] whenever the
Concepts of stability
333
value y(to) is close to zero and the perturbing term P in (10.3)
is small in the sense that vario P is small.
The following result shows that both these natural concepts of
stability are equivalent.
10.8 Theorem. The trivial solution x - 0 of (10.2) is variationally stable if and only if it is stable with respect to perturbations.
The trivial solution x - 0 of (10.2) is variationally attracting
if and only if it is attracting with respect to perturbations.
Proof. Let us prove the first statement.
1) Assume that x - 0 is variationally stable. For a given
e > 0 let b > 0 be given by Definition 10.1. Assume that yo E
J", Ilyo II < 6 and vario P < b and that y(t) = y(t, to, yo) is
a solution of (10.3) on [to,tl]. Then IIy(to)II = IIy(to,to,yo)II =
Iyo II < 6 and by the definition of a solution we have for any
S102 E [to, ti]
j82
y(s2) - y(sl) =
l
DF(y(r), t) + P(s2) - P(sl ).
Hence
92
y(s2) - J
DF(y(r), t) - y(sI) +
f
11
to
to
DF(y(r), t) _
= P(s2) - P(si)
for sl, s2 E [to, tl] and consequently
vario (y(s)
- j DF(y(r), t)) = vario P < b.
o
The variational stability yields
IIy(t)II = II y(t, to, yo)II < e, t E [to, ti],
X. Varialional stability for generalized ODE'S
334
and this means that the solution x =_ 0 of (10.2) is stable with
respect to perturbations.
2) Assume that the solution x - 0 of (10.2) is stable with respect to perturbations. For e > 0 let b > 0 be given by Definition
10.4. Let y : [to, t1] --- R" be of bounded variation on [to, t1 ],
continuous from the left and such that
rs
IIy(to)II < 6 and varto(y(s) -
Jo
DF(y(r),t)) < b-
For 81,32 E [to,t1] we have
32
y(s2) - y(s1) =
f DF(y(r), t) _
J1
a,
,2
DF(y(r),t) + y(s2)
J
- J DF(y(r),t) - y(s1)+
p
$1
+
ral
Jto
92
DF(y(r), t) =
Jst
DF(y(r), t) + P(S2) - P(S1) (10.4)
where P(s) = y(s) -- fro DF(y(r), t) for s E [to, t1]. By Lemma
3.9 this function P is of bounded variation on [to, t1], continuous
from the left, and (10.4) shows that the function y is a solution
of (10.3) on [to,t1] with this P and IIy(to)II < 6. Moreover,
varP = var(y(s) -
j
DF(y(r), t)) < b.
o
Hence by the assumption of stability with respect to perturbations we get IIy(t)II < e for t E [to,t1] and x = 0 is variationally
stable.
Now we prove the second part of the statement.
1) Assume that x = 0 is variationally attracting. Then there
exists bo > 0 and for a given e > 0 also T > 0 and ^y > 0 by
Concepts of stability
335
Definition 10.2. If now yo E R" is such that Ilyoll < S, P is of
bounded variation on [to, t1] and continuous from the left on this
interval where vary P < b and y(t) = y(t, to, yo) is a solution of
(10.3) on [to,tl] then lly(to)Il = llyoll < bo and
varto(y(s)
-J
s
DF(y(r),t)) = varto P < y.
to
Hence by Definition 10.2 we have lly(t,to,yo)ll = lly(t)ll < e for
allt>to+T and to>0.
2) If x - 0 is attractive with respect to perturbations then
we can set P(s) = y(s) - fto DF(y(r), t) for s E [to, tl] to show
that x - 0 is variationally attractive directly from the properties
described in Definition 10.5.
Theorem 10.8 and Definitions 10.3 and 10.6 simply yield the
following result.
10.9 Theorem. The trivial solution x - 0 of (10.2) is variational-asymptotically stable if and only if it is asymptotically
stable with respect to perturbations.
10.10 Remark. The concept of variational stability was introduced by H. Okamura. I. Vrkoc considered Caratheodory equations in [172] and pointed out that Okamura's variational stability
is equivalent to his concept of integral stability. There is an improvement of Vrkoc's results given by S.-N. Chow and J.A. Yorke
[17].
In the case of classical ordinary differential equations variational stability is a somewhat exotic concept but in the case of
generalized ordinary differential equations it seems to be very
natural because in our setting the solutions of such equations are
functions of bounded variation and in the case of variational stability we are measuring the distance of two solutions using the
norm in the space By. Now we turn our attention to the method
X. Variational stability for generalized ODE'S
336
of Ljapunov functions for the stability -concepts described above.
First we derive some auxiliary results.
10.11 Proposition. Assume that -oo < a < b < +oo and
that f , g : [a, b] -- R are functions which are continuous from the
left in (a, b].
If for every a E [a, b] there exists b(a) > 0 such that for every
q E (0, b(a)) the inequality
f(a+q)-f(a) <g(a+q)-g(a)
holds, then
f(s) - f(a) : g(s) - g(a)
for all s E [a, b].
Proof. Let us denote
M = {s E [a, b]; f (Q) - f (a) < g(o) - g(a), a E [a, s])
and set S = sup M. Since
f(a + q) - f(a) < g(a + q) - g(a)
for q E (0, 6(a)) and 6(a) > 0, the set M is nonempty, S > a and
f (s) - f (a) < g(s) - g(a) for every s < S. Using the continuity
from the left of the functions f and g we have also
f(S) - f(a) < g(S) - g(a).
If S < b then by assumption we have
As + q) - f(S) < g(S + q) - g(S)
for every q E (0, b(S)), b(S) > 0 and therefore also
f(S+q)-f(a) =f(S+q)-f(S)+f(S)-f(a) <
< g(S + q) - g(S) + g(S) - g(a) = g(S + q) - g(a).
This implies that S + q E M for q E (0, b(S)), i.e. S < sup M
0
and this contradiction yields S = b and M = [a, b].
For deriving Ljapunov type theorems we need another technical lemma.
Concepts of stability
337
10.12 Lemma. Suppose that V : [0, +oo) x R" --> R is such
x) : [0, +oo) - R is
that for every x E R" the function
continuous from the left in (0, +oo). Assume that
IV(t, x) - V(t, y)f <
.
jIx - yll
(10.5)
forx,yER",tE[0,+oo) with a constant K >0.
Further assume that there is a real function : R" -> R such
that for every solution x : (a, 0) -> R" of the generalized ordinary
differential equation (10.2) on (a, /3) C [0, +oo) we have
lim sup
V(t +9, x(t + 77))
n-o+
- V(t, x(t)) -< '1'(40)
(10.6)
fort E (c,/3).
If y : [to, t1 ] --+ R", 0 _< to < tl < +oo is continuous from
the left on (to,t1] and of bounded variation on [to,t1], then the
inequality
V(tj)x(t1))
< V(to, x(to)) + Kvari[y(s)
o
-
r
8
DF(y(r), t)] + M(tl - to)
o
(10.7)
holds, where M = suptE[to,t,l
t(y(t)).
Proof. Let y : [to,t1] -+ R" be given and let a E [to, t I] be an
arbitrary point. It is clear that the function V(t, y(t)) : [to, t1]
R is continuous from the left on (to, t1].
Assume that x : [a, a + 171 (a)] --+ R" is a solution of (10.2) on
the interval [a, a + ql (a)], ql (a) > 0 with the initial condition
x(a) = y(o). The existence of such a solution is guaranteed by
the local existence theorem 4.2. By the assumption (10.5) we
then have
V (O' + q, y(a + q)) - V(v + q, x(u + q)) <
X. Variational stability for generalized ODE'S
338
< KIl y(a + il)
- x(a + i)II =
r +Tj
= k IIy(a + il) - y(a) -
J0
DF(x(T),t)II
for every q E [0,
By this inequality and by (10.6) we obtain
771(a)].
V (a + 9, y(a + ii)) - V (a, x(a)) =
= V (a +,q, y(a + ii)) - V (a +'I, x(a + 7J))+
+V(a +il,x(a+,l)) - V(a,x(a)) <
< KIy(a + ?) - y(a) -
j
+n
DF(x(T), t)II +
<
rv+n
DF(x(r), t)II + qM + ijE
J0
where e > 0 is arbitrary and n E (0,772(0')) with 172(0') < rll(a),
712(0) > 0 is sufficiently small. Denote
< Klly(a + il) - y(a) -
P(s) = y(s) -
J
DF(y(r),t) for s E [to, tl].
o
The function P : [to, t1] --+ R' is continuous from the left on
(to,tl], of bounded variation on [to,t1], and the last inequality
can be used to derive
V(a + i, y(a + 77)) - V(a, x(a)) <
o+q
< KII y(a + 77) - y(a) - f
0
DF(y(r), t)II +
r +n
+KII
J0
D[F(y(r),t) -F(x(T),t)]II
Concepts of stability
339
<KIIP(c+77)-P(cr)II +i7M+ie+
r+n
+KII
D[F(y(T), t) - F(x(T), t)1II s
J
< K(vario n P - vario P) + t7M + 77E+
a+
+KII
D[F(y(T),t)
Js
- F(x(T),t)]II
(10.8)
for every i E (0, 772(x)).
Let us consider the last term in (10.8). Since F E .F(G,h,w)
we obtain by Theorems 1.16 and 1.35 the estimate
II
f
D[F(y(T), t) - F(x(T), t)JII <
s
ro+n
<J
w(II y(T) - x(r)II) dh(T) _
0
v+a
lim [f
w(Ily(r) - x(T)II)dh(T)+
s
f
a+n
+ u+a
w{ IIy(T)
- x(T)II) dh(T )] _
= w(II y(a) - x(o)II)(h(cr+) - h(o))+
+ a0+
lim
+n
Jo+a w(IIy(T) - x(T)II)dh(T)
o+A
= lim
a-+ 0+f
<
sup
PE(o,o+'J
w(II y(r) - x(T)II) dh(T) <
+a
w(II y(P) - x(P)II) 0--00+
li(h(a + y) - h(a + a)) _
X. Variational stability for generalized ODE'S
340
sup
PE(a,a+n]
w(II y(P) - x(P)II)(h(a + il) - h(a+)),
(10.9)
because y(a) = x(a) and w(II y(a) - x(a)II) = 0.
For P E [a, a + 712(a)] we have
P
Y(P) - x(P) = y(P) - y(a) - f DF(x(r), t)
0
and therefore
lira (y(P) - X(P)) _
y(a+) - y(o) - Plim (F(x(a), p)
= y(a+) - y(a) - (F(x(a), a+) -
- F(x(a), a)) _
a)) = P(a+) - P(o)
and also
lim IIy(P) - x(P)II = II P(a+) - P(a)II
P- ff+
(10.10)
For every e > 0 we define
a=
e
K(h(t,) - h(to) + 1) >
0
( 0 . 11 )
and assume that r = r(a) > 0 is such that w(r) < a. Further,
we choose ry E (0, r2 )
Since (10.10) holds, there is an 93(0r) E (0,'72(a)) such that
IMP) - x(P)II < II P(a+) - P(a)II + y
(10.12)
for p E (a, a + 773 (a)) and also
w(IIy(P) - x(P)II) < w(IIP(a+) - P(a)II +7)
for p E (a, Or + 773(x))
(10.13)
Concepts of stability
341
Let us denote
N(a) = {a E [to, tj]; IIP(a+) - P(a)II ?
2
}.
Since P is of bounded variation on [to, t1 ] the set N(a) is finite
and we denote by l(a) the number of elements of N(a).
If a E [to, tl ] \ N(a) and p E (a, a + 7)3(a)) then by (10.13) we
have
w(II y(P) - x(P)II) < w(r + y) < (2 + 2) = w(r) < a
and by (10.9) also
II
f
0
o+
D[F(y(T),t) - F(x(r),t)]1I S a(h(a +7l) - h(cr+))
(10.14)
whenever 77 E (0, 773(a)).
If a E [to, ti] fl N(a) then there exists 774 (or) E (003 (a)) such
that for r E (0, i4(a)) we have
h(o + ?I)
- h(a+) = I h(a + 71) - h(a+)I <
a
(1(a) + 1)w(IIP(a+) - P(a)II + -Y)
and (a, a + 774(a)) fl N(a) = 0. Hence (10.9) and (10.13) yield
II
f
a
D[F(y(-r), t) - F(x(T), t)] II
< w(JIP(a+) - P(a)11 +'Y)
a
O(a) + 1)w(1IP(a+) - P(a)II + -r)
a
1(a) + 1
-
(10.15)
342
X. Variational stability for generalized ODE'S
for every i E (a, a + 774 (a))
Define now
-
a
ha(t) = 1(a) + 1
Ha(t)
vEN(a)
for tE [to,tI]where H,,, (t) = 0 for t < a and H, (t) = 1 for t > a.
The function ha : [to, tI] - R is nondecreasing, continuous from
the left and
varto ha = ha(te) - ha(to) =
1(a) +
11(a) < a.
(10.16)
The points of discontinuity of the function ha are clearly only the
points belonging to N(a), and for t E N(a) we have
ha (t) - h (t)
or
a
1(a)
+1
Using the function ha we can set
ha(t) = ah,(t) + ha(t)
for t E [to, ti ] where by h, the continuous part of the function h
is denoted. By definition the function ha is nondecreasing and
continuous from the left on [to, ii] and by (10.16) and (10.11) we
obtain
ha(ts) - ha(to) = a[hr (t1} -
ha(ts)
< a[h(ti) - h(to) + 1] = h,.
- ha(to) <
(10.17)
If a E [to, t1] \ N(a) then set 6(a) _ 713(0') > 0 and if
or E
[to, ti] n N(a) then set b(a) =q4 (v) > 0. By this choice of b(a) >
Concepts of stability
343
0 for a E [to,t1], by (10.14), (10.15) and by the definition of ha
we obtain the inequality
o+n
D[F(y(T), t) - F(x(T ), t)] Ii
ha(O + ?))
0
- ha(a)
for r) E [0, b(c)], and (10.8) gives
V(O+1),y(o+ii))-V(O,x(O)) <
<Ii(var o+''P-varoP)+qM+
+i + K(ha(a + 77) - ha(a)) = g(a + q) - g(u)
(10.18)
for all O E [to, tI] and 71 E [0, b(O)] where
g(t) = K varro P + Mt + Et + Ii ha(t).
The function g is of bounded variation on [to, tr ] and continuous
from the left on (to, ti ). From Proposition 10.11 and from (10.17)
we obtain by (10.18) the inequality
V(tr, y(tr )) - V(to, y(to)) < g(ti) - g(to) =
= K varro P + M(ti - to) + e(ti - to) + K(ha(ti) - ha(to)) <
< K varro P + M(ti - to) + e(ti - to) + E.
Since e > 0 can be arbitrary small, we obtain from this inequality
the result given in (10.7).
0
X. Variational stability for generalized ODE's
344
Ljapunov type theorems
10.13 Theorem. Assume that V [0, -boo) x Ba -. R, 0 <
a < c is such that for every x E Ba = {y ER'; IIyII <a} the
function V(., x) is continuous from the left.
Assume that the function V(t, x) is positive definite, i.e. there
exists a continuous increasing real function b : [0, +oo) --4R such
that b(p) = 0 if and only if p = 0 and
V(t,x) > b(IIxII
(10.19)
for all (t, x) E [0, +oo) x Ba,
V (t' 0) = 0
(10.20)
IIV(t,x)-V(t,y)II <KIIx - yII
(10.21)
and
for x, y E Ba, K > 0 being a constant.
If the function V(t, x(t)) is nonincreasing along every solution
x(t) of the equation (10.2) then the trivial solution x - 0 of (10.2)
is variationally stable.
Proof. Since we assume that the function V(t, x(t)) is nonincreasing whenever x : [a, $] -.' R' is a solution of (10.2) we have
V (t + rt, x(t - )) - V (t, x(t)) < 0
lin
e sup
(10.22)
o
fort E [a, 01.
Let us check that under these assumptions the properties required in Definition 10.1 are satisfied.
Let s > 0 be given and let y : [to, t1 ] - R" be of bounded
variation on [to, tl ] and continuous from the left in (to, t,1. Since
Ljapunov type theorems
345
the function V satisfies the assumptions of Lemma 10.12 with fi
0 in the relation (10.6) (see (10.22)) we obtain by (10.7),(10.20)
and (10.21) the inequality
J DF(y(r), t)) <
V(r, y(r)) < V (to, y(to )) + K
fo
KIIy(to)II + Kvarto(y(s) -
j
DF(y(r),t))
(10.23)
o
which holds for every r E [to, ti I.
Let us define a(e) = infr<e b(r). Then a(e) > 0 for e > 0 and
lim,-o+ a(e) = 0. Further, choose 6(e) > 0 such that 2Kb(e) <
a(e).
If in this situation the function y is such that IIy(to)II < 6(e)
and
varto(y(s)
- J DF(y(r),t)) < b(e)
eo
then by (10.23) we obtain the inequality
V(r, y(r)) < 2K6(e)
(10.24)
provided r E [to, t,].
If there exists a t E [to,tl] such that Iiy(iII ? e then by (10.19)
we get the inequality
V(t, y(t)) > b(IIy(i)JI) ? iuf b(r) = a(e)
r<e
which contradicts (10.24). Hence IIy(t)II < e for all t E [to, tl] and
by Definition 10.1 the solution x = 0 is variationally stable. 0
X. Variational stability for generalized ODE'S
346
10.14 Theorem. Let V : [0, -f-oo) x BQ - R, 0 < a < c be a
function with the properties given in Theorem 10.13.
If for every solution x : [to, tI ] --> Ba of the equation (10.2) the
inequality
limsup
V(t + rt,x(t +7/)) - V(t,x(t)) < -lt(x(t))
11-o+
(10.25)
7)
holds for every t E [to, tl ], where 4 : R' i IR is continuous,
c(0) = 0, 4 (x) > 0 for x j4 0, then the solution x - 0 of (10.2) is
variationally-asymptotically stable.
Proof. From (10.25) it is clear that the function V(t, x(t)) is
nonincreasing along every solution x(t) of (10.2) and therefore by
Theorem 10.13 the trivial solution x - 0 of (10.2) is variationally
stable. By Definition 10.3 it remains to show that the solution
x = 0 of (10.2) is variationally attracting in the sense of Definition
10.2.
From the variational stability of the solution x - 0 of (10.2)
there is a bo E (0, a) such that if y : [to, tI ] - Rn is of bounded
variation on [to, t1 ] where 0 _< to < tl < +oo, y is continuous
from the left on (to, ti] and such that IIy(to)II < bo,
vario(y(s)
-
DF(y(r),t)) < bo,
eo
then IIy(t)II < a fort E [to,tI], i.e. Y(t) E Ba for every t E [to,ti].
Let e > 0 be arbitrary. From the variational stability of the
trivial solution we obtain that there is a 6(s) > 0 such that for
every y : [t2i t3] --> R" of bounded variation on [t2, t3] where
0 < t2 < t3 < +oo, y continuous from the left on (t2, t3] and such
that
II y(to )II < b(s)
(10.26)
Ljapunov type theorems
and
347
s
varto(y(s)
DF(y(r),t)) < S(e),
- J
(10.27)
o
we have
IIy(t)II <e
(10.28)
for t E [t2, t3]. Define y(E) = niin(bo, b(e)) and
T(E)=-Kao+-(e) >0 where
M = sup{-fi(x); ry(e) < IIxII < e} _
inf{fi(x); y(E) < IIxII < e} < 0
R" is of bounded variation on
and assume that y [to, ti]
[to,tl] where 0 < to < tj < +oo, y is continuous from the left on
:
(to,t1) and such that Ily(to)II < bo,
var'o(y(s)
rs
- J DF(y(r),t)) < y(e).
(10.29)
co
Assume that T(e) < tl - to, i.e. to + T(e) < tl . We show that
there exists a t* E [to, to+T(e)] such that Ily(t*)II < ry(e). Assume
the contrary, i.e. Ily(s)II > y(e) for every s E [to,to + T(e)].
Lemma 10.12 yields
V(to + T(e), y(to + T(e))) - V(to,y(to)) <
<
varto+T(e)
s
:o
< K7(--) + M
-h
(bM
y(e)) _ -Kbo.
Hence
V(to +T(e),y(to + T(e))) < V(to, y(to)) - Kbo <
X. Variational stability for generalized ODE'S
348
< KII y(to)II - l bo < Iibo - Ki o = 0
and this contradicts the inequality
V(to + T(e), y(to + T(E))) ? b(II y(to + T(E))II) ? b(y(E)) > 0.
Hence necessarily there is a t* E [to, to + T(e)] such that
IIy(t*)II < y(E)
and by (10.29) we have Ily(t)II < E fort E [t*,t1], because (10.26),
(10.27) hold in view of the choice of y(e) and (10.28) is satisfied
for the case tj = t*, t3 = tl. Consequently, also Ily(t)II < e for
t > to +T(e), because t* E [to, to +T(e)] and therefore the trivial
solution x - 0 is a variationally attracting solution of (10.2). 0
Converse Ljapunov theorems
This part is devoted to the conversion of the Ljapunov type
stability results, namely Theorems 10.13 and 10.14. Our goal is
to show that the variational stability and the asymptotic variational stability imply the existence of Ljapunov functions with
the properties described in Theorems 10.13 and 10.14.
First we show some auxiliary results.
Let us introduce a modified notion of the variation of a function.
10.15 Definition. Assume that -oc < a < b < +oo and that
G : [a, b] -> 1R' is given.
For a given decomposition
of the interval [a, b] and for every ) > 0 define
k
E e-\(b-nj-,)IG(n;)
.i=I
- G(aj-I )I = vA(G, D)
Converse Ljapunov theorems
349
and set
ea varQ G = sup va(G, D)
D
where the suprelnuin is taken over all finite decompositions D of
the interval [a, b]. The number e,\ varQ G is called the ex-variation
of the function G over the interval [a, b].
10.16 Lemma. If -oo < a < b < -boo and G :
[a, b] -* ][8"
then for every A > 0 we have
e-A(b-a) varQ G < ex varQ G < varQ G.
(10.30)
If a < c < b then for A > 0 the identity
eavarQG = e-\(b-c)eavarQG+eAvarbG
(10.31)
holds.
Proof. For every A > 0 and every decomposition D of [a, b] we
have
e-A(b-arj-j) < eO
e-J\(b-a) <
for j = 1, 2,... k.
Therefore
e-A(b-a)vo(G, D) < v,\ (G, D) <
k
< vo(G, D) = E IG(aj)
- G(aj-1 )I
j=1
and passing to the supremum over all finite decompositions D of
[a, b] we obtain the inequality (10.30).
It is easy to see that for proving the second statement we can
restrict ourselves to the case of decompositions D which contain
the point c as a node, i.e.
D:a=ao<a, <...<a!-I <at=c<at+1 <...<ak=b.
350
X. Variational stability for generalized ODE's
Then
e-JG(aj) - G(aj-1)) _
E
j=1
k
D)
va(G,
I
_
Ee-A(b-`xj-')IG(aj) -G(aj-1)I+
j=1
k
+ E eA(b-aj-I)IG(aj) - G(aj-i )I =
j=t+1
I
= e-A(b-c) I` e-A(C-a;_I IG(aj) - G(aj-1 )I+
jj=1
k
+E
CA(b-°j-I)IG(nj)
_
G(aj-1 )I =
j=1+1
= e-A(b-c)v'\(G, D1) + va(G, D2)
(10.32)
where
D1 :a=ao <ca,
<at=c
and
D2:c=at<a1+1 <...<ak=b
are decompositions of [a, c] and [c, b], respectively. On the other
hand, any two such decompositions D1 and D2 form a decomposition D of the interval [a, b]. The equality (10.31) now easily
follows from (10.32) when we pass to the corresponding suprema.
0
Converse Ljapunov theorems
351
10.17 Corollary. If a < c < b and A > 0 then
(10.33)
ea vara G < e,\ varb G.
For a > 0, t > 0, x E Ba denote by Aa (t, x) the set of all functions cp : [0, +oo) -i Rn which are locally of bounded variation
on [0, +oo), cp(O) = 0, v(t) = x, cp is continuous from the left and
sup,E[o,t]
a.
Moreover, for A > 0. s > 0 and x E Ba define
/'a
{eA varo(cp(o) - J DF(cp(r), t))}
inf
V.(s,x)=
SPEA,(s,x)
ifs>0
lixil
o
ifs = 0.
(10.34)
Note that this definition of VQ(s, x) snakes sense because for
cp E Aa(s, x) the integral fo DF(cp(r), t) is a function of bounded
variation in the variable o and therefore the function p(a) fo DF(cp(r), t) is of bounded variation on [0, s] as well and the
eA-variation of this function is bounded.
The function cp = 0 evidently belongs to A,, (s, 0) and therefore
we have
VA(s,0) = 0
(10.35)
for every s > 0 and A > 0 because cp(o)
- fQ DF(cp(r), t) = 0 for
Since ex vara(cp(o) - fo DF(cp(r), t)) > 0 for every
cp E Aa(s, x) we have by the definition (10.34) also the inequality
V\ (s, x) > 0
forevery s>0andxER'.
(10.36)
X. Variational stability for generalized ODE's
352
10.18 Lemma. For x, y E Ba = {x E l[8"; Ilx[I < a}, s E
[0, +oo) and A > 0 the inequality
INS, X) - VA(s, y)I 5 Ilx
- yll
(10.37)
holds.
Proof. Assume that s > 0 and 0 < q < s. Let cp E Aa(s,x) be
arbitrary. Define
cp,i(a) = V(a) for a E [0, s - rl],
and
1(y-cP(a-t7))(a-s+77) for a E [s-77,s].
n(a) = (a-7))+
77
The function cp,, coincides with cp on [0, s - ij and is linear with
S ,,(s) = y on Is - 77, .s]. By definition clearly cpj E Aa(s, y) and by
(10.31) from Lemma 10.15 we obtain
-f
VA(s, y) <_ ex varo(cpn(a)
a
DF(co,, (r), t))
0
.o.
=
DF(p(r), t))+
J
0
+ea var;_,1((pn(a) - f
DF(4'n(r), t)) C
0
< e-A'lcAvaro-'1(W(a)
-f
a
DF(cp(r),t))+
0
+ var;_, cp, +
ro
0
DF(cp(r), t)) <
_
353
Converse Gjapunov theorems
< e-A''ea
-
varo-'I
/0
(cp(a) -
J
DF(y (r), t))+
in
+l y
- V(s - ri)1 + h(s) - h(s - y).
Since for every ?1 > 0 we have
a
DF(ip(r), t)) _
10
0
jDF((r)t))-
= eA
-eAvar:_(7)
DF(p(r),t)) <
0
< eavaro(cp(o) - J DF(cp(r),t))
0
by (10.35), we obtain for every 77 > 0 the inequality
ro
VA,(s, y) < e.\ varo(cp(o) -
+Iy-co(s-r!)I
J0
DF(V(r), t))+
+h(s)-h(s-17).
The functions V and h are assumed to be continuous from the
left and therefore liinr.8_ cp(r) = tp(s) = x; moreover the last
inequality is valid for every y > 0 and consequently we can pass
to the limit q -> 0+ in order to obtain
VA(s, y) < ea varo(W(v) - J DF(cp(r), t)) + I y - xI
0
for every cp E A.(s, x). Taking the infimum for all V E A. (s, x)
on the right hand side of the last inequality we arrive at
VA(s, Y) < VA(s, X) + Ix - yi.
(10.38)
X. Variational stability for generalized ODE'S
354
Since this reasoning is fully symmetric with respect to x and y
we similarly obtain also
VA(s, X) < V\(s, Y) + Ix - A
and this together with (10.38) yields (10.37) for s > 0.
If s = 0, then we have by definition
1VA(0, Y)
- VA(0, x)i = ii l - lxII C Ix - Y1.
0
This proves the statement.
10.19 Corollary. Since VA(s, 0) = 0 for every s > 0, we have
by (10.35) and (10.37)
0 < VA(s, x) <
(10.39)
114.
10.20 Lemma. For Y E Ba, s, r. E [0, +oo) and A > 0 the
inequality
IVA(r, y) - VA(s, y)l < (1
- e-air-'1)a + Jh(r) - h(s)l
(10.40)
holds.
Proof. Suppose that 0 < s < r and let cp E Aa(r, y) be given.
Then by Lemma 10.16 we have
e.\ varo(cp(Q) -J DF(p(r), t)) _
= e-a(r-')ea varo(v(o,) -
J
0
a
+ea var;(cp(a) - fo DF(y2(r), t)) >
t)) +
Converse Ljapunov theorems
>
355
GP(s)) + eA vars(,p (a) - f DF((p(r), t)) >
0
> e- \(r-s) [VA(s, cp(s)) + ears W
/
or
- ears J DF((p(r), t)] >
0
> e-A(r-s)[VA(s, cp(s)) + Jy - sp(s)I + (h(r) - h(s))] >
> e-A(r-s)[VA(s, y) + (h(r) - h(s))].
(10.41)
The inequality (10.36) from Lemma 10.18 leads to
VA(S, sO(s)) + ly - sO(S) I ? VA(s, Y).
Taking the infimum over cp E Aa(r, y) on the left hand side of
(10.41) we obtain
V\ (r, y) ?
[V,\ (s, y) + (h(r) - h(s))]
>
VA(s, y) + (h(r)
- h(s)).
(10.42)
Now let cp E Aa(S, y) be arbitrary. Let us define
(0)
p(a) for a E [0, s],
y for or E (s,r].
We have evidently cp*(s) = V(s) = y, cp* E Aa(r, y) and by
(10.30), (10.35) we obtain
/'o
VA(r, y)
J DF(v*(r), t)) _
eA
0
=e
or
eavar
a
in
DF
r t))+
356
X. Variational stability for generalized ODE's
+eA vaxs(cp*(a) -
f
t)) <
0
< e-A(r-s)eA varoMa)
0
-f
DF(ep(r), t))+
0
r OF
+ vary V* + var;
t) <
J
0
< e-A(r-9)eA varo(v(a) - J DF((p(r), t)) + h(r) - h(s).
0
Taking the infinmum over all cp E Aa (s, y) on the right hand side
of this inequality we obtain
VA(r, y) <
e-Mr-9)
VA(s, y) + (h(r) - h(s)).
Together with (10.42) we have
IVA(r, y) -
e-A(r-s)VA(s,
y)I < h(r) - h(s).
Hence by (10.39) we get the inequality
IVA(r,y)
< IVa(r, y)
-
e-A(r-')VA(s,
- VA(S,y)I <
y)I + 11
e-A(r-s)
II VA(s, y)I -<
< h(r) - h(s) + (1 - e-A(r-'))I yI <
< h(r) - h(s) + (1 - e-A(r-°))a
because IyI < a. In this way we have obtained (10.40).
Assume that s = 0 and r > 0. Then by (10.39) and by the
definition given in (10.34) we get
VA(r, y) - VA(s, y) = VA(r, y) - VA(0, y) _
Converse Ljapunov theorems
= VA(r, y) - IyI < 0.
357
(10.43)
We derive an estimate from below. Assume that cp E Aa(r, y).
We have
eA varo(cp(a) -
J
DF(w(r), t)) >
0
17
eA varo V - e,\ varo(J
DF((p(r), t)) >
0
ra
> e-Ar varo co - vara(J
DF(p(r), t)) >
0
>
c'(0)I
- (h(r) - h(0)) _
= e-Arlyl - (h(r) - h(0))
by (10.30), Lemma 10.16 and Lemma 3.9. Passing again to the
infimum for cp E Aa(r, y) on the left hand side of this inequality
we get
VA(r, y) >- a-\rlyl - (h(r) - h(0))
and
VA(r, y)
- V\(0, y) = VA(r, y) - I yI
1)IyI
_ -(1 -
- (h(r) - h(0)) =
e-,\r)lyl
- (h(r) - h(0)).
This together with (10.43) yields
I VA(r, Y)
- V,\(0, y)I < (1 - e-Ar)a - (h(r) - h(0)),
and this means that the inequality (10.40) holds in this case,
too. The remaining case of r = s = 0 is evident. Finally, let us
mention that the case when r < s can be dealt with in the same
way because the situation is symmetric in s and r.
By the previous Lemmas 10.18 and 10.20 we immediately conclude that the following holds.
X. Variational stability for generalized ODE'S
358
10.21 Corollary. For x, y E Ba = {x E Rn; {Ixil < a}, r, s E
[0, +oo) and A > 0 the inequality
IVA(s, x) - VA(r, y)l C jjx - yuf + (1 -
e-air-'1)a
+ )h(r) - h(s)l
(10.44)
holds.
x) deNow we will discuss the behaviour of the function
fined by (10.34) along the solutions of the generalized ordinary
differential equation
dx
= DF(x, t).
(10.2)
We still assume that the assumptions given at the beginning
of this chapter are satisfied for the right hand side F(x, t).
The next statement will be of interest for the forthcoming considerations.
10.22 Lemma. If ' : Is, s + r7(s)] --4R" is a solution of the
generalized ordinary differential equation (10.2), s > 0, r7(s) > 0,
then for every A the inequality
lim sup
n--o+
Va(s + 77, O(s + z1))
- VA(s, O(S))
< -AVA(s, O(s))
q
(10.45)
holds.
Proof. Let s E [0, +oo) and x E R" be given. Let us choose
a> 0 such that a> jxi+h(s+1)-h(s). Assume that ca E Aa(s, x)
is given and let 0 : Is, s + r7(s)] -> R" be a solution of (10.2) on
Is, s + r7(s)] with O(s) = x where 0 < q(s) < 1. The existence of
such a solution is guaranteed by the local existence theorem 4.2.
For 0 < r1 < it(s) define
V(a) for a E [0, $],
Pn(a) = 1 0(Q) for
a E Is, s + t ];
Converse Ljapunov theorems
359
we have yo(s) _ sb(s) = cp,7(s) = x. Evidently cp,, E Aa(s+rj, c(s+
ri)) since 0 is continuous from the left and by the definition of a
solution we have
a
II0(a)II = II x + f DF(b(r), t)II <
e
< Ijxjj + h(a) - h(s) < IIxI) + h(s + 1) - h(s) < a
for a E Is, s + r)] and
VA(s + 77, O(S + rl )) <
a
f DF(cp,r(r), t)) _
< ex
0
a
= e-A'?eA varo((p(a) - fo DF((p(T ), t))+
+ea var;+n(O(a)
- f DF(cP(T ), t) - j
o
=
varo(cp(a) -
r
DF(O(r), t)) _
DF(cp(r), t))+
J
0
r9
+ea var;+'I(x -
J
DF(cp(r), t)) _
0
= e-''Ilea
/a
J DF(V(r), t))
0
Taking the infimum for all W E Aa(s, x) on the right hand side of
this inequality we obtain
VA(s + ii,'+b(s + ii)) <
x) = e-,\,,VA(s, tI(s)).
X. Variational stability for generalized ODE'S
360
This inequality yields
VA(s + i), (s + q)) - VA(s, %(s)) <
1)VA(s, 0(s))
and also
V\ (s + rl, V) (s + q))
- V\ (s, V (s))
< e-an - 1
q
rl
V\ (s, 0(s))
for every 0 < r) < r)(s).
Since lim,1-o
e-an-1
= -A we immediately obtain (10.45).
77
0
Now we are in position when the converse theorems to Theoreins 10.13 and 10.14 can be proved.
10.23 Theorem. If the trivial solution x = 0 of the generalized ordinary differential equation (10.2) is variationally stable
then for every 0 < a < c there exists a function V : [0, +oo) x
Ba --> R satisfying the following conditions:
1) for every x E B. the function V(., x) is continuous from
the left and V(., x) is locally of bounded variation on
[0, +oo),
2) V(t,0) = 0 and
IV(t,x) - V(t,y)I < IIx - yll
for x, y E Ba, t E [0, +oo),
3) the function V is nonincreasing along the solutions of the
equation (10.2),
4) the function V(t,x) is positive definite, i.e. there is a
continuous non decreasing real-valued function
[0, +oo) -> R such that b(p) = 0 if and only if p = 0
and
b:
b(IIxII) < V(t, x)
for every x E Ba, t E [0, +oo).
Converse Ljapunov theorems
361
Proof. The candidate for the function V is the function
Vo(s, x) defined by (10.34) for A = 0, i.e. we take V(s,x) _
Vo(s, x). The properties stated in 1) are easy consequences of
Corollary 10.21. The properties given in 2) follow from (10.35)
and from Lemma 10.18. By Lemma 10.22 for every solution 0 :
Is, .s + b] -- R" of the generalized ordinary differential equation
(10.2) we have
lim sup VA (s + 77, 0(s + r!)) - VA(s, 0(s)) < 0
n-o+
rl
and therefore 3) is also satisfied. It remains to show that the
function V(t, x) given in this way is positive definite. This is the
only point where the variational stability of the solution x - 0 of
the equation (10.2) is used. Assume that there is an e, 0 < e < a,
and a sequence (tk, xk), k = 1, 2, ... , e < IIxkdI < a, tk --9 oo for
k --; oo such that V(tk, xk) --- 0 for k - oo.
Let b(e) > 0 correspond to e by Definition 10.4 of stability
with respect to perturbations (the variational stability of x - 0
is equivalent to the stability with respect to perturbations of this
solution by Theorem 10.8).
Assume that ko E N is such that for k > ko we have V(tk, xk) <
b(e). Then there exists Wk E AQ(tk, xk) such that
o
DF(4pk(r),t)) < b(e).
varak(cpk(o)
fo
Let us set
P(a) = Wk(a) -
J0
DF(cpk(r), t) for a E [0, tk],
tk
P(v) = xk - fo D F(cpk(r), t) for a E [tk, +oo).
X. Variational stability for generalized ODE'S
362
We evidently have
varo P = varok((Pk(a)
-f
DF(Vk(r),t)) < li(e).
0
and the function P is continuous from the left. For o E [0, t] we
have
a
47
Vk(cr) = f DF(ok(T), t+ Pk(a) J
t) _
0
t) + P(a) - P(0) _
J=
rc
Pk(0) +
J0
t) + P(t)]
because Wk(O) = 0. Hence cPk is a solution of the equation
dT
D[F(y, t) + P(t)]
and therefore, by the variational stability we have
E
for every s E [0,tk]. Hence we also have IlVk(tk)II = IIxk1I < e and
this contradicts our assumption. In this way we obtain that the
0
function V(t, x) is positive definite and 4) is also satisfied.
The next statement is the converse for Theorem 10.14 on variational-asymptotic stability.
10.24 Theorem. If the trivial solution x = 0 of the generalized ordinary differential equation (10.2) is variationally-asymptotically stable then for every 0 < a < c there exists a function
U : [0, +oo) x B. -4 R satisfying the following conditions:
1) for every x E Ba the function U(., x) is continuous from
the left and U(., x) is locally of bounded variation on
[0, +oo),
Converse Ljapunov theorems
363
2) U(t, 0) = 0 and
IU(t, X)
- U(t, y)I <_ Ilx - yll
forx,yEBa, tE[0,+o0),
3) for every solution di(rt) of the equation (10.2) defined for
a > t, where fi(t) = x E Bo the relation
lim sup
U(t + rj, O(t + q)) - U(t, x) < _U(t, x)
holds,
4) the function U(t, x) is positive definite.
Proof. For x E Ba, s > 0 let us set
U(s, X) = Vj (s, x)
where Vo(s, x) is the function defined by (10.34) for A = 1. In the
same way as in the proof of Theorem 10.23 we can see that the
function U satisfies 1), 2) and 3). ( The item 3) is exactly the
statement given in Lemma 10.22.)
It remains to show that 4) is satisfied for this choice of the
function U.
Since the solution x - 0 of (10.2) is assumed to be variationally
attracting, it is by Theorem 10.8 also attracting with respect to
perturbations and therefore there exists bo > 0 and for every
e > 0 there is a T = T(e) > 0 and -y = ry(e) > 0 such that if
ilyo li < bo, yo E R" and P E BV([to, tl]) is a function of bounded
variation on [to, t1 ], continuous from the left on (to, ti ] and such
that varia P < y(e) then
lly(t,to,yo)II < --
X. Variational stability for generalized ODE'S
364
for all t E [to, t1] fl [to +T(e), +oo) and to > 0 where y(t,to, yo) is
a solution of
dx
T
= D[F(x, t) + P(t)]
(10.3)
with y(to,to,yo) = yo.
Assume that U is not positive definite. Then there exists e,
0<e<a=boandasequence
(tk,xk),k=1,2,...,E<IIxk11 <
a,tk->oofor k-'oosuch that U(tk,xk)-+Ofor k-goo.
Let us choose ko E N such that for k E N, k > ko we have
tk > T(E) + 1 and
U(tk,xk) <,),(e)e-(T(f)+I)
According to the definition of U let us choose ep C Aa(tk, xk) such
that
el varo(tp(a)
-f
7(E)e-y(
0
Define to = tk - (T(e) + 1). Then to > 0 because tk > T(e) + 1
and also tk = to + T(e) + 1 > to + T(e). Further, evidently
el varia
f
DF(So(T), t)) <
(T(a)+1)7
0
and by (10.30) (see Lemnia 10.16) also
varto M09
- J o DF(cp(T ), t)) _
0
= e-(tk-to) varto MO)
-
OF
J
t)) < y(e)e-(T(E)+t)
0
and therefore
vario (V(a)
- J DF(co(T), t)) < y(e).
0
(10.46)
365
Converse Ljapunov theorems
For a E [to, tk] we define
f
P(a)
DF(W(T ), t))
0
The function P : [to, tk] -> R" is evidently continuous from
the left and by the inequality (10.46) we have
varto P < ry(e).
Moreover,
(a) =
f
0
0
t) + '(a) - fo
o
0
=
t) _
j DF((T), t) + P()
and also
DF(V(r), t) + P(s) - P(to) _
p(s) - P(to) _
eo
re
=
J
D[F((p(r), t) + P(t)),
eo
and this means that the function w : [to, tk] -+ R' is a solution of the equation (10.3) with II'(to)II _< a = 6o because
(p E A. (t, k, xk ). By the definition of the attractivity the inequality IIW(to)II < e holds for every t > to+T(e). This is of course valid
also for the value t = tk > to + T(e), i.e. Ilsv(tk)II = Ilxk II < e
and this contradicts the assumption IIxkII > e. This yields the
positive definiteness of U.
0