PIECEWISE LINEAR APPROXIMATIONS OF SOLUTIONS OF
GENERALIZED LINEAR DIFFERENTIAL EQUATIONS
(Dedicated to the memory of Mikhail Drakhlin)
Zdeněk Halas and Milan Tvrdý
Institute of Mathematics, Academy of Sciences of the Czech Republic, Praha, Czech
Republic
Abstract. This contribution deals with systems of linear generalized linear
differential equations of the form
t
x(t ) =    d[ A(s)]x(s)  g (t )  g (a), t [a, b],
a
where   < a < b < , A is an n n -complex matrix valued function, g is an n comp-lex vector valued function, A and g have bounded variation on [a, b]. The
integrals are understood in the Kurzweil-Stieltjes sense.
Our aim is to present some new results on the continuous dependence of solutions
to linear generalized differential equations on parameters and initial data. In
particular, we generalize in several aspects the known result by Ashordia. Our main
goal consists in the more general notion of a solution to the given system. In
particular, both g and x need not be of bounded variation on [a, b] and, in general,
they can be regulated functions.
1. INTRODUCTION
Starting with Kurzweil [8], generalized differential equations has been extensively
studied by many authors, like e.g. Hildebrandt [6], Schwabik, Tvrdý and Vejvoda
[14]-[16], [18]-[20], Hönig [7], Ashordia [2], [3]. In particular, see the monographs
[16], [14], [19] and [7] and the references therein. Moreover, during several recent
decades, the inte-rest in their special cases like equations with impulses or discrete
systems increased considerably, cf. e.g. monographs [10], [22], [4], [13] or [1].
Importance of generalized linear differential equations with regulated solutions
consists in that they enable us to treat in a unified way both continuous and discrete
systems and, in addition, also systems with fast oscillating data.
Throughout the paper we keep the following notation:
As usual, N is the set of natural numbers and C stands for the set of complex
numbers. C m n is the space of complex matrices of the type m  n, Cn = Cn1 and
C1 = C. For a matrix A  C m n , the elements of A are denoted by
ai , j , i = 1,2,, m, j = 1,2,, n, and the norm | A | of A  C m n is defined by
m
| A |= max
|a
j =1,2,, n i =1
i, j
|.
The symbols I and 0 stand respectively for the identity and the zero matrix of the
proper type. For an nn -matrix A , det[ A] denotes its determinant.
If   < a < b < , then by [a, b] and (a, b) we denote the corresponding closed
and open intervals, respectively. Furthermore, [a, b) and (a, b] are the corresponding
7-1
half-open intervals. When the intervals [a, a) and (b, b] occur, they are understood to
be empty.
For a function F : [a, b]  Cmn , we set || F || = sup{| F (t ) |: t [a, b]}. The set
D = {t0 , t1 , , t p }  [a, b], is called a subdivision of [a, b], if a = t0 < t1 <  < t p = b.
If, for each t  [a, b) and each s  (a, b], the function F : [a, b]  Cmn possesses in
C m n the limits
F (t ) := lim F ( ), F ( s ) := lim F ( ),
 t 
 s 
we say that the function F is regulated on the interval [a, b]. The set of all k  n matrix valued functions regulated on the interval [a, b] is denoted by G mn [a, b].
Furthermore, we denote  F (t ) = F (t )  F (t ) for t  [a, b),  F (b) = 0 and
 F ( s) = F ( s)  F ( s ) for s  (a, b],  F (a) = 0. It is known that, for each
F  G mn [a, b], the set of all points of its discontinuity on the interval [a, b] is at most
countable. Moreover, for each  > 0 there are at most finitely many points t  [a, b)
such that |  F (t ) |  and at most finitely many points s  [ a, b] such that
|  F ( s) |  . Clearly, each function regulated on [a, b] is bounded on [a, b], i.e.
|| F || <  for F  G mn [a, b].
For a function F : [a, b]  Cmn , we denote by varab F its variation over [a, b]. We
say that F has a bounded variation on [a, b] if varab F < . The set of k  n -complex
matrix valued functions of bounded variation on [a, b] is denoted by BV mn [a, b]. By
AC mn [a, b] we denote the set of functions F: [a, b]  Cmn such that each component
f ij , i = 1,, k , j = 1,, n, of F is absolutely continuous on the interval [ a, b].
Similarly, C mn [a, b] stands for the set of functions F: [a, b]  Cmn that are
continuous on [a, b]. Analogously to the spaces of functions of bounded variation,
AC n [a, b] = AC n1[a, b], G n [a, b] = G n1[a, b] and C n [a, b] = C n1[a, b]. Obviously,
AC mn [a, b]  BV mn [a, b]  G mn [a, b]
and C mn [a, b]  G mn [a, b]. Finally, a function f : [a, b]  C is called a finite
step function on [a, b] if there is a division { 0 , 1 ,  ,  p } of [a, b] such that f is
constant on every interval ( j 1 ,  j ), j = 1,, p. The set of all finite step functions on
[ a, b] is denoted by S [ a, b], S kn [a, b] is the set of all k  n -matrix valued functions
whose arguments are finite step functions and S n1[a, b] = S n [a, b]. It is known that
the set S kn [a, b] is dense in G mn [a, b] with respect to the supremum norm, i.e.
 for each  > 0 and each F  G mn [a, b]

k n
 there is a   S [a, b] such that || F   || <  .
(1.1)
2. KURZWEIL-STIELTJES INTEGRAL
The integrals which occur in this paper are the Perron-Stieltjes ones. For the
original definition, see A.J. Ward [21] or S. Saks [12]. We use the equivalent
summation definition due to J. Kurzweil [8] (cf. also e.g. [9] or [16]). We call this
7-2
integral the Kurzweil-Stieltjes integral, in short KS-integral. For the reader's
convenience, let us recall the properties of the KS-integral needed later.
b
 fd[g ]
It is well-known that the KS-integral
a
f  G[ a, b] and
exists if
g  BV [a, b]. Taking into account [17, Theorem 2.8] or [19, Theorem 2.3.8], we can
state the following crucial existence assertion.
Theorem 2.1. If f , g  G[a, b] and at least one of the functions f , g has a
b
 fd[g ] exists. Furthermore,
 fd[ g ]  2| f (a) | var f  || g ||
bounded variation on [a, b], then the integral



a
b
b
a
a
if

f  BV [a, b] and g  G[a, b],
(2.1)
and

b
a
fd[ g ] || f || (varab g ) if
f  G[a, b] and g  BV [a, b].
In particular, if f  BV [a, b] and lim k  || g k  g || = 0, then
t
 t

lim sup  a fd[ g k ]  a fd[ g ] : t [a, b] = 0.
k 


Further basic properties of the KS-integral with respect to scalar regulated
functions were described in [17] (see also [19]).
Given a m   - matrix valued function F and a   n - matrix valued function G
defined on [a, b] and such that all the integrals
b
f
a
i ,k
(t )d[ g k , j (t )]
(i = 1,2,, m; k = 1,2,, ; j = 1,2,, n) exist, the symbol
simply
a
(or
b
 Fd[G] ) stands for the m n -matrix A with the entries
=   f d [ g ], i = 1,2, , m, j = 1,2, , n.
a

ai , j
b
 F (t )d[G(t )]
k =1
b
a
i ,k
k, j
The extension of the results obtained in [17] or [18] for scalar real valued functions
to complex vector valued or matrix valued functions is obvious and hence for the
basic facts concerning integrals with respect to regulated functions we refer to the
corresponding assertions from [17] or [18]. The next natural assertion will be very
useful for our purposes and, to our opinion, it is not available in the literature.
Lemma 2.2. Let x, xk  G n [a, b], A, Ak  BV n n [a, b] for k  N and let
lim || xk  x || = 0,
k 
*
 := sup{varab Ak : k N} < ,
lim || Ak  A || = 0.
k 
Then
t
 t

lim sup  ad[ Ak ]xk  ad [ A]x : t [a, b] = 0.
k 


7-3
(2.2)
(2.3)
(2.4)
Proof. Let  > 0 be given. By (1.1), (2.2) and (2.4), we can find   S n [a, b] and
k0  N such that
|| x   || <  , || xk   || < 
and
|| Ak  A || < 
for k  k0 .
Furthermore, since var u < , using (2.1) we can see that, for t  [a, b] and
k  k0 , the relations
b
a
t
t
t
t
t
a
a
a
a
a
 d[ Ak ]xk   d[ A]x =  d[ Ak ]( xk   )   d[ Ak  A]u   d[ A](  x)


  *  2(varab )   * = 2  *  varab  ,
wherefrom our assertion immediately follows.
□
3. GENERALIZED LINEAR DIFFERENTIAL EQUATIONS
Let A  BV nn [a, b], g  BV n [a, b] and   Cn . Consider an integral equation
t
x(t ) =    d[ A(s)]x(s)  g (t )  g (a).
a
(3.1)
We say that a function x : [a, b]  Cn is a solution of (3.1) on the interval [a, b] if
the integral
b
 d[ A(s)]x(s) has a sense and equality (3.1) is satisfied for all
a
t  [a, b].
The equation (3.1) is usually called a generalized linear differential equation. Such
equations with solutions having values in the space Rn of real n -vectors have been
thoroughly investigated e.g. in the monographs [14] or [16]. In this section we will
describe the basics needed later. A special attention is paid to the features whose
extension to the complex case seems not to be so straightforward.
For our purposes the following property is crucial
(3.2)
det I   A(t )  0 hold for each t [a, b].

(Remind that we put  A(a) = 0.) Its importance is well illustrated by the next
assertion which is a fundamental existence result for equation (3.1).


Theorem 3.1. [18, Proposition 2.5]. Let A  BV nn [a, b] satisfy (3.2) Then, for
each   Cn and each g  G n [a, b], equation (3,1) has a unique solution x on [a, b]
and x  G n [a, b]. Moreover, x  g  BV n [a, b].
Furthermore, analogously to [16, Theorem III.1.7] where g  BV n [a, b], we have:
Theorem 3.2. Let A  BV nn [a, b] satisfy (3.2). Then the relations
c A := sup{ [ I   A(t )]1 : t  [a, b]} < 
and
(3.3)
| x(t ) | c A |  | 2 || g ||  exp (c A varat A) for t  [a, b] (3.4)
hold for each  Cn [a, b], g  G n [a, b] and each solution x of () on [a, b].
1
Proof. First, notice that for t  [a, b] such that |  A(t ) |< , we have
2


1
[ I   A(t )]1 = ( A(t )) k   |  A(t ) |k =
< 2.
1 |  A(t ) |
k =1
k =1
7-4
1
Therefore, (3.3) follows easily from the fact that the set {t  [a, b] :|  A(t ) | }
2
has at most finitely many elements. Now, let x be a solution of (3.1). Put
B (a ) = A(a ) and B(t ) = A(t ) otherwise. Then, as in the proof of [16, Theorem
III.1.7], we have
A(t )  B(t ) =  A(t ) and
t
 d[ A  B]x =  A(t )

a
for t [a, b].
Consequently,
t
x(t ) = [ I   A(t )]1    g (t )  g (a)   d [ B]x 
a


and
t
| x(t ) | K1  K 2  d[h] | x | for t [a, b],
a
where K1 = cA |  | 2 || g || , K2 = cA and h(t ) = varat B for t  [a, b]. The function
h is nondecreasing and, since B is left-continuous on (a, b], h is also leftcontinuous on (a, b]. Therefore we can use the generalized Gronwall inequality (see
e.g. [16, Lemma I.4.30] or [14, Corollary 1.43]) to get the estimate (3.4).
Corollary 3.3. Let A  BV nn [a, b] satisfy (3.2). Then for g  G n [a, b] and each
solu-tion x of (3,1) on [a, b], the estimate
varab ( x  g )  c A (varab A)(|  | 2 || g ||  ) exp (c A varab A) || g ||  (3.5)
is true, where c A is defined by (3.3).
Proof. By (3.4), we have || x ||  c A (|  | 2 | g || ) exp (c Avarab A), which inserted
into the inequality varab ( x  g ) | g (a) | varab A || x || yields estimate (3.5). □
Lemma 3.4. Let A  BV nn [a, b] satisfy (3.2) and let c A be defined by (3.3) Then
 


1
c A = inf I   A(t ) x : t  [a, b], x  Cn , | x |= 1
.
(3.6)
Proof. We have
| [ I   A(t )]1 || [ I   A(t )] x |

c A = sup 
: t [a, b], x  Cn , | x |= 1

| [ I   A(t )] x |




1
 sup 
: t  [a, b], x  Cn , | x |= 1

| [ I   A(t )] x |

 

= inf | [ I   A(t )]x |: t  [a, b], x  Cn , | x |= 1
Thus, it remains to prove that the inequality
 

1
.

c A  inf I   A(t ) x : t  [a, b], x  C n , | x |= 1
1
(3.7)
is true, as well. To this aim, first, let us notice that for each t  [a, b], there is a
z  C n such that | z |= 1 and
[ I   A(t )]1 = [ I   A(t )]1 z .
(3.8)
Indeed, let t  [a, b] and let B = [ I   A(t )]1. Let i0 {1,2, , n} be such that
| B |=  j =1 | bi
n
0, j
| and let z  Cn be such that zi = sgn(bi ,2 ) for i = 1,2,, n. Then
0
| z |= 1. Furthermore,
7-5
n
| Bz |= max
|b
i =1,2,, n j =1
i, j
sgn(bi , j ) | max
0
n
|b
i =1,2,, n j =1
i, j
|=| B |
On the other hand,
n
| B |=  | bi
j =1
0
n
, j |=
sgn(b
j =1
i0 , j
)bi
0, j
| Bz | .
Therefore, we can conclude that (3.8) is true.
Now, due to (3.2), there is w  C n such that z = [ I   A(t )]w.
instead of z into (3.8) and having in mind that | z |= 1, we get

[ I   A(t )]
1
=
Inserting this
[ I   A(t )]1[ I   A(t )]w
[ I   A(t )]w


1
 sup 
: x  C n , | x |= 1.

 w 
| [ I   A(t )] x |

[ I   A(t )]

| w|
It follows that


1
c A  sup 
: t  [a, b], x  Cn , | x |= 1

| [ I   A(t )] x |

=
1


1
= inf {| [ I   A(t )] x |: t  [a, b], x  Cn , | x |= 1 ,
i.e. (3.7) is true. This completes the proof □
4. CONTINUOUS DEPENDENCE OF A SOLUTION ON PARAMETER
The main result of this paper is provided by the next theorem. It concerns
continuous dependence of solutions of generalized linear differential equations on a
parameter and generalizes that due to M. Ashordia [2, Theorem 1]. Unlike [2] and [3],
we do not utilize
the variation-of-constants formula and therefore we need not assume that, in
addition to (3.2), also the condition det[ I   A(t )]  0 for all t  [a, b] is satisfied.
Furthermore, both the nonhomogeneous part of the equation and solutions may be
regulated functions (not necessarily of bounded variation).
Theorem 4.1. Let A, Ak  BV n n [a, b], g , g k  G n [a, b],  ,  k  C n for k N.
Assume (2.3), (2.4), (3.2),
(4.1)
lim || g k  g || = 0,
k 
lim  k =  .
(4.2)
k 
Then, equation () has a unique solution x on [a, b]. Furthermore, for each k  N
sufficiently large, there exists a unique solution xk on [a, b] to the equation
t
x(t ) =  k   d[ Ak (s)]x(s)  g k (t )  g k (a)
a
(4.3)
and (2.2) is true.
Proof. Step 1. As in the first part of the proof of [2, Theorem~1], we can show that
there is a k1  N such that det[ I   Ak (t )]  0 for all t  [a, b] nd all k  k1. In
particular, (4.3) has a unique solution xk for k  k1.
7-6
Step 2. For k  k1 , put
cA := sup{[ I   Ak (t )]1 : t  (a, b]}.
k
Then, by Lemma 3.4, we have
(cA ) 1 = inf I   Ak (t ) x : t [a, b], x  Cn , | x |= 1
k



 inf I   A(t )x : t  [a, b], x  C , | x |= 1
 sup  ( A (t )  A(t )) x : t  [a, b], x  C , | x |= 1.

n

n
k
Since, due to assumption (2.4), lim k  ||  Ak   A || = 0, we conclude that there
is a k0  k1 such that (c A ) 1  (c A ) 1  (2c A ) 1 = (2c A ) 1 for k  k0 .
k
To summarize
c A  2c A <  for k  k0 .
(4.4)
k
Step 3. Set wk = ( xk  g k )  ( x  g ). Then, for k  k0 ,
t
wk (t ) =  k   d[ Ak ]wk  hk (t )  hk (a) on [a, b],
a
where
t
t
t
hk (t ) =  d [ Ak  A]( x  g )    d [ Ak ]g k   d [ A]g  for t  [a, b],
a
a
 a

 k =  k  g k (a)    g (a).
By (4.1) and (4.2), we can see that
(4.5)
lim  k = 0.
k 
Furthermore, since x  g  BV n [a, b] and lim k  || Ak  A || = 0, by Theorem 2.1,
we have
t
t
 d[ A ]( x  g )Ã  d[ A]( x  g ) on
lim  d [ A ]g =  d [ A]g .
a
k
a
t
and, by Lemma 2.2,
k  a
[a, b]
t
k
k
a
To summarize,
lim || hk || = 0.
(4.6)
k 
On the other hand, using Theorem 3.2 and taking into account the relation (4.4),
we get
|| wk ||  2c A |  k | 2 || hk ||  exp (2c A * ) for t  [a, b] and k  k0 ,
wherefrom, with respect to (4.5) and (4.6), the relation lim k  || wk || = 0 follows.
Finally, having in mind assumptions (4.1) and (4.2), we conclude that the relation
lim || xk  x || = 0 is true, as well. This completes the proof. □
Notation 4.2. For a k  N, denote by Dk a division of [a, b] given by


Dk =  0k , 1k ,,  mk , where m = 2 k and

i(b  a)

 ik = a 
for i = 0,1,, m.

2k

For given A  BV nn [a, b], g  G n [a, b] and k  N, we define
7-7
(4.7)
A(t )
if t  D k ,


k
k
A( i )  A( i 1 )
Ak (t ) = 
A( ik1 ) 
(t   ik1 ) if t  ( ik1 ,  ik ),
k
k

 i   i 1

g (t )
if t  D k ,


k
k
g ( i )  g ( i 1 )
g k (t ) = 
g ( ik1 ) 
(t   ik1 ) if t  ( ik1 ,  ik ).
k
k

 i   i 1
Then, obviously, { Ak }  AC n n [a, b] and {g k }  AC n [a, b] and
varab Ak  varab A
(4.8)
(4.9)
for all k  N.
(4.10)
The equations (4.3) with Ak and g k given by (4.8) and (4.9) are just initial value
problems for linear ordinary differential systems
(4.11)
x = Ak (t ) x  g k (t ), x(a) =  k .
In this view, next theorem says that the solutions of (3.1) can be uniformly
approximated by solutions of linear ordinary differential equations provided the
functions A and g are continuous.
Theorem 4.3. Assume that A  BV nn [a, b]  C nn [a, b] and g  C n [a, b]. Let  and
 k  C n , k  N, be such that (4.2) holds. Furthermore, let the sequence {D k } of
[ a, b]
divisions
of
the
interval
be
given
by
(4.7)
and
let
n n
n
{ Ak }  AC [a, b],{g k }  AC [a, b] be defined by (4.8) and (4.9), respectively.Then,
equation (3.1) has a unique solution x on [a, b]. Furthermore, for each k  N,
equation (4.11) has a solution xk on [a, b] and (2.2) holds.
Proof. Step 1. Since A is uniformly continuous on [a, b], we have


for each  > 0 there is a  > 0 such that | A(t )  A( s ) |<
(4.12)

2

holds for all t , s  [a, b] such that | t  s |<  .
k
Let 1/2 0 <  and let t  [a, b] be given. Furthermore, let
 1 ,    Pk = { 0 , 1 ,,  p } and t [ 1 ,   ].
0
k0
Then |     1 |= 1/2 <  and, according to (4.7), (4.8) and (4.12), we get for
k  k0
| Ak (t )  A(t ) |=| Ak (t )  Ak ( 1 )  A( 1 )  A(t ) |
k0
 t    1 

 A(  1 )  [ A(  )  A(  1 )]
  A(  1 ) 
2
     1 
 t    1    
=| A(  )  A(  1 ) | 
    = .
     1  2 2 2
As k0 was chosen independently of t , we can conclude that (2.4) is true.
Step 2. Analogously we can show that (4.1) holds for {g k } and g .
Step 3. By (4.10), condition (2.3) is satisfied. Moreover, as A and Ak , k  N, are
continuous, by Theorem 3.1, equations (3.1) and (4.3), k  N, have unique solutions
and we can complete the proof using Theorem 4.1. □
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Remark 4.4. Piecewise linear approximations (4.8) and (4.9) of A, g can be
constructed also in a general case, when A, g need not be continuous. Some results
for homogeneous equations were in this direction obtained by M. Pelant in his
unpublished thesis [11]. If A and/or g are not continuous on [a, b] the situation is
rather more complicated and, in general, the solutions of (4.3) do not converge to the
solution of (3.1), but to solutions of a somewhat modified equation. Nevertheless,
using matrix logarithms, one can always define a sequence (4.3) of approximating
initial value problems for ODE's whose solutions tend to the solution of (3.1). Of
course, even if A and g are real valued, such approximating piecewise linear
functions will be, in general, complex valued. We suppose to expose this theme in
more details later.
ACKNOWLEDGEMENTS
This work is supported by the grant No. A100190703 of the Grant Agency of the
Academy of Sciences of the Czech Republic and by the Institutional Research Plan
No. AV0Z10190503
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