373
Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 373-380
K-COMPONENT DISCONJUGACY FOR SYSTEMS
OF ORDINARY DIFFERENTIAL EQUATIONS
JOHNNY HENDERSON
Department of Mathematics
Auburn Unlverslty
Auburn, Alabama 36849
(Received October 3, 1985)
BSTKACT:
DisconJugacy of the
dlfferenttal equations
Yn )’
YI’
f(x,
a solution
z(n)
o(X)
(n)
component of the
kth
lj(I.I), YI’
Yn
(x’
Rmn
): (a,b)
order system of nth order
ruth
y(n-l)),
’
f(x, Y
(I.I)
Rm
is defined, where
are continuous.
Given a
k-component dlsconJugacy of the variational equation
of
Y0(n-l)(x))
?n..i=ifyi( x’ 0(x),
Z
(i-l),
(1.2),
is also studied.
Conditions
are given for continuous dependence and dlfferentlabillty of solutions of (I.I) with
respect to boundary conditions, and then intervals on which (I.I) is k-component
dlsconJugate are characterized in terms of intervals on which (1.2) is k-component
dlsconJugate.
System of differential equations, variational equation,
k-component disconJugacy (right dlsfocallty), continuity and dlfferentlabillty with
KEY WORDS AND PHRASES.
respect to boundary conditions.
1980 MS SUBJECt CLASSIFICATION:
1.
34BI0, 34B15.
INTRODUCTION.
In the past several years, a number of results have been proven concerning the
dlsconJugacy of an
order scalar ordinary differential equation when certain
nth
dlsconJugacy assumptions are made for the corresponding linear variational equation.
in this paper we investigate similar concepts for systems of ordinary differential
equations.
In particular, we shall be concerned with solutions of boundary value
problems for the
ruth
order system of
y(n)
order differential equations
nth
y(n-1))
f(x, ,Y’
where we assume throughout:
(A)
f(x,
(B)
__f (x,
Yt
I’
Y ): (a,b)
n
YI’’’’’
Yn ): (a,b)
Rmn+
R
Rmn/
m
R
is continuous;
m,
< J <
m,
<
i
<
n, are continuous,
JOHNNY HENDERSON
374
(C)
(D)
x
(a,b)
<
and
<
k
m
such that
k
[c,d], for
(i-l) (x)}
(Yrjk
I
all
Y0(x)
solutions of the linear
of
z(n)
,Rather
(I.I),
0(x)
n
i=l fYi (x, Y(x) ’z0’
<
i
<
n, denotes the
(a,b), for I
nth
-0
Z
(i-l)
Jacobian matrix
m
m
(I.I),
k
<
Yk (xj)
Yk(X)
Given a solution
k,
<
m.
we will be concerned with the
Motivation for our
and Sukup [4].
We say that (I.I) is k-component
n, points a < x <
< x < b,
<
mq
partitioning
n, and solutions
k
k
(1)
It follows that
<
be given.
m
dlsconJugate on (atb), if given 2 < q
(a,b), positive integers ml,
(x) and Z (x) of (I.I) satisfying
and
(1.2)
’Y]
[I-2], Spencer [3],
considerations here are papers by Peterson
<
n.
order equations called
v(n-1) (x))
(x),
than with the dlsconJugacy of
Let
<i<
and given by
dlsconJugacy of one of the components of the system (I.I).
DEFINITION.
such that
we will also be concerned with
order system of
ruth
the variational equation along
fYi
(x)}
Y
then there is a subsequence
converges uniformly on each compact subinterval of
Given a solution
Y0(x)
will
k
X (x)} of (I.I), a point
r
[c,d] C (a,b), M > 0,
a compact subinterval
^
then
(a,b);
If there exist a sequence of solutions
u
where
Yk-l’ Yk+l’
Solutions of (I) extend to
and
Ym Rm’
Ym )’)
(YI’
If
(YI’
(m-l)-tuple
will denote the
on
(Note:
Ylm );
Y! (Yil’
where
0
z
Zk(X).
0(x) of
<
i
< mj-I
<
<
q
(I.I), k-component dlsconJugacy of (1,2) along
is defined similarly.
In this paper, we first sh that If the syst (I.I) Is k-component
dlsconJugate, for some
<
k
<
m, then solutions of certain boundary value problems
for (I.I) can be differentiated with respect to boundary conditions.
partial derivatives as functions of
problems for the system (1.2).
x
The resulting
are solutions of related boundary value
The main results of this paper appear in section 3,
where we show that intervals on which
(1.1)
is k-component dlsconJugate can be
characterized in terms of Intervals on which (1.2) is k-component dlsconJugate.
Then
in our last section, we state without proof some analogues of the results in section
3 in terms of k-component right dlsfocallty for the system (I.I).
2.
CONTINUITY AND DIFFERENTIABILITY WITH RESPECT TO BOUNDARY CONDITIONS.
Our first result is concerned wih the continuous dependence of
solutions of (I.I) on boundary conditions. Its proof is a standard application
of the Brouwer Invarlance of Domain Theorem.
THEOREM I.
<
a
< Xq <
<
x
v i),<, 1_< i_<
,
_<
i_< n,
Z!i)(x)"
lira
m, the system (I.I) is k-component
be a solution of (I.I).
Y (x)
Let
<
k
b, a e (a,b), positive integers
()(x)- clJ
n,
Z(x)
(1)(tj)
c
Zk
Y(i)(x)
<
Yk
there exists a unique solution
(Vi)
<
Assume that for some
dlsconJugate on (a,b).
375
SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
DISC()NJUGACY FOR
iJ
,
<
.....
<
< mj
i
<
I,
<
Zd)
of (1.1) satisfying
0
partitioning
ml,...,mq
0<i<.
<
_< q _<
2
Given
n, points
n, and
q, imply
I-I) (a)
and
q
uniformly on each compact subinterval of (a b)
for O<i<n-1
::,.0+
In addition to the continuous dependence in Theorem I, connectedness properties
have played an important role in establishing dlsconJugacy or dlsfocallty results in
[5], Peterson [2], and Sukup [4l.
the papers of Henderson
(a,b).
dlsconJugate on
xI,
’4,
a, and
Xq,
the set
{v
Sp
(mp-1)
k
v
i)(xj)
0
<
show that if
z
sup
0
{ol[o,y
be as in Theorem I.
(rap-l)
<
i
(Xp)] C. Sp}
q’
P’
i-l).a.(
V
2}
is an open interval.
(xp),
]
S
is open.
$
P
z’
> z0’
then o’
Sp.
and
P
< o0,
o’
and
that
, ,k(t) (x )_(t)
(_,k ),
Vk(1)(Xp) Yk(1)(Xp) 0 < < m-2,
p
i
z0’
P
We will make the
’ > z0
We suppose that there exists
_< _<
It suffices to
then z’
V (x) of (I.I) satisfying
Now, from the definition of
Let
{z)
Sp
such that
> 0,
0
{wkP
(x))
Yk
(mp-I)
_< _< %-
0
and
(mp-I) (x)
v
k
condition (D)
i
{Wk(X)}
(Xp)
p
<
’.
and
0
w
i
and
],
is uniformly bounded on
there exists a subsequence
uniformly on compact subintervals of
N
(a,b),for
<
<
q,
if for some
[Xp, Xp +
is uniformly bounded on
boundary conditions that
it ollows from the
[Xp, Xp
(x)} such that
I
_< i _< n.
+ ].
{w(i-l)(x)}’"
ujk
By
converges
In particular, this
convergence is uniform on any compact subinterval containing a, and consequently,
of (I.I)
converges uniformly to a solution
the subsequence
No(X)
N(x)}
on compact subintervals of
(a,b).
P
of solutions
i
uk
S
there exists a strictly monotone
be the corresponding
P,
z’
and
z(1.1)< z0satisfyingz
N
sequence
n,
<
(gu)i-l)(a) i-l)(a), <_
(1)k (xj)- yk(1)(xj), < < mj-l,
w(i)(Xp), yk(i)(Xp) _< _< mp 2, Wk(rap-l) (Xp) -. Now
(m -)
creaslng sequence
q,
p’
mp
{zl[yk
(}.
v-) () t-}
_< p _<
Then, for
(i.I),
is a solution of
(m p-I)
Then there is a solution
-<
m, the system (I.I) is k-component
Yk
inf
0
k_
the argument for the first case.
!
<
It follows immediately from Theorem
PROOF.
O
k
be a solution of (I.I) and let
mq
V (x)
(Xp)
(Xp) Yk(1)(Xp),
v
(x)
Let
ml,
_< _< n,
and
<
Assume that for some
THEOREM 2.
Thus, it follows that
0
S
P
which is
376
JOHNNY HENDERSON
contradictory to the fact that
is not
on
uniformly bounded on
(m -1)
P
(x +6
k
(x
P
or (ii)
v
k
is open.
P
Hence,
[Xp, Xp + ].
(mp-1) (x) < (m P -1
Ince, for each
e N
Yk
p
Wuk
there exists a sequence
that
(i)
S
(rap-l)
Yk
P
+
Xp
with
(Xp+j)
and
6j+0
ymp
wj k
on
(1)
llm w
/
(mp-I)
(Xp+dj)
=v
(x
+ 6 ), for all
x
P
k
P
Yk(1)(x)
p
ujk(Xp + J)
and hence by Theorem
(Xp+dj)
-1)
(x)
0
<
<
i
{wuj k
compact subinterval of
v
>
0, {w
(m -I)
P
(x)}
uk
(m-I)
< (rap-l)
wj k (x) <
(mp-I) (x)
Yk
<
v
it
follows
(m p -1)
(x)
k
(mp-I) (x) <
wj k
We then have by continuity that
J
(i) (x)}
I,
and
<
(x)
p
e
(x)
k
p
such that, either
for all
(m -1
P
(mp-1) (x),
{6j}
given
m -2
p
converges to
< <
(a,b), for 0
i
Yk(1)(x)
uniformly on each
n-l; a contradiction.
This
completes the proof.
Our next result deals with
differentiation of solutions of (1.1) with
to boundary conditions in the presence
of k-component dlsconJugacy.
The proof follows along the lines of
those given in Henderson [5-6] and
respect
Peterson [I]
and we will omit it here.
THEOREM 3. Let
_< k < m be given and assume that (I.I) and the variational
equation (1.2) along all solutions
Y (x) of (I.I) is k-component
dlsconJugate on
(a,b). Let Y (x) be a solution of (I.I),
and let 2 < q _< n, points
a < x <
< xq < b, a (a,b), and positive integers
partitlonlng
ml, ...,
n be given. For
< p < q, let S be as in Theorem 2, and for
mq
each
p
s
(x,s) denote the corresponding solutlon of (1.1)
where v (mp-1)
k
(Xp,S)
V
’rhen -------x,s) exists and Z (x,s)
V_(x,s
let
S
p
7
s
is the solution of
(1.2) along V (x,s)
condl tlous
z
(xj
(x p
zk
0
O, 0
(m -i)
P
(x)
p
0
<
<
i
i
<
s.
s
and satisfies the boundary
< mj
,
<
j
<
q,
p,
m-2,
p
=I
3.
INTERVALS OF K-COMPONENT DISCONJUGACY.
In this section, we determine subintervals of (a,b) on which (I.I) is
k-component disconJugate in terms of subintervals on which (1.2) is
k-component disconJugate.
like those in Peterson
Our arguments for this characterization are much
[2] and Spencer [3].
For notational purposes, given
a
(a,b),
let
Y (x; a,
solution of the initial value problem for (I.I) satisfying
(vii,
Vim)
<_ <_
i
n.
Then, under our assumptions (A)
Vl,
Y(i-l)(a)
V
denote the
V
i
(D), for each
DISCONJUGACY FOR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
<
<
<
<
and
n
(x; =,
nd is a solution of (1.2) along Y
<
O,
<
,,
n,
DEFINITIONS.
(i>
<
Let
n k(t
Deflne
U(-l)(a)
k
<
o(X) o
(ll) Given a solutlon
o
be esbllshed In
i() <
that
this pape= Is
o parts.
It,b),
o(X)
proo
The
of
b.
o(X))
on
(t)
ma
n(t)
we set
((t;
In
o(X>
0())}
Similar to the argument In Spen=er
0(x))}"
(t;
Inf
me
in, i
[3],
whi=h wlii
we first prove
eorem o me
how ha trit Inequali Is not possible, hence the equaii will
oe4.
?ROOF.
i!k!
Let
inf
0(x)
Ineal
It, +),
[t,+),
solution
Z (x;
Y0(x))
i
i
2 m-i,
2
and let
2q 2,
>
E
0
be given.
tExl <
xq <
<
,,
p, rti=ionlng
0(x)).
0,
2 i 2 n,
and
ose In
[9], It follows at ere Is
order zero at
x
vn=iCIk UIk(x;a,
+
x"
Now for suitable consents
in"
V
V
n),
where
i-l)(a)
[7], Muldowney [8],
Y0(x))
W (x;
su=h at
of (1.2)
Wli-l)(=;
2 I2i, ,n
Clk _<
i i n. For
Vi,
ti’
consider n the difference quotient
Y (x;a,(
!
Vl),Vlk+hC1k,...,( Vn),Vnk+hCnk
Y1(X;’( Vl)’Vlk+hC1k’
Ym(X;a,( Vl),Vlk+hClk
,(
By adding and subtracting, to
quotient, tes of
Vn),
Vnk
YO(X)
solution
zi)(x; 0()) -0,
Henderson
a solution
< tn <
Yo(X) and points t_< t <
2 i in, wk(x; 0(x)) s a slmpie ze=o t
h
+,
and a non-trlvlal
i22q.
along
!
esbllshed.
Then on the
variational equation (1.2) along a
By dlseonugay arguments similar
and Peterson
points
mq
ml,
e
of
i-i)(=;
of (i.i), su=h mat
0
Y0(x))}
,etlo,
0 (> >i"
0(x>
there exist
positive integers
(t>i. ((;
The.
bgi..
{(t;
-O,
(a,b).
t
(t;
(i.i), deine
(1.2) Is not k-component dls=onJugate along
The In =esuit
exists
(I I) Is not k-component dlseonugste on
If (I.I) Is k-component dls=onug, te on
[t, ti.
satisfying
n)
O(i-l)()
V
..., mv)
Iv
be given and let
m
(t,b)
t
Inf
Yn
VI,
(
e
, V I,
(x;
(x; a, V l’***’ Vn)
m, U
377
+
e
hCnk),
form
Vn)’v+hCnk)-Yl (x;a’ VI"’’’
Vn
Vn),v+hCnk)-Ym(X;a
Vn)
e Jth
V I,
component,
yj(x;a, VI,...,( Vs), vs k
we obin
n
m
+
of
e
hCsk,...
YO(X))
O,
,
an od
h
O,
JOHNNY HENDERSON
378
i_{
(x;a,(
V1),Vlk+hC1k,...,( Vn),Vnk+hCnk )-
h
+
Vl)’Vlk
Ulkl(X;a
Y (x;a
Ikl
V2)’V2k+hC2k
+ Ikg’
V2)’V2k+hC2k’
V
V )}
n
V
Vnk
+
hCnk)]
Clk
Ulkm(X;a’( Vl)’Vlk
Vl’’’’’ Vn-l’( Vn)’ Vnk
+
Vnk
+
Unkl(X;a’..
Unkm(X;a’ VI"’’’ Vn-l’( Vn)’
Cnk
<
<
v
m,
0
is between
ukv
and
nkl,
nkg
Hence,
hCuk.
Vn )’vnk + hCnk)
,(
where for each
0, the d.fference
h
as
quotient (3.1) converges uniformly on compact subintervals to
Ulk m(x;a,
V 1,..., V
vnt=lCik Uik(x;a Vl,
I (x)
difference
Vn).
Y (x;a,(
VI,... Vn
Y (x;a,
and
-=
<
O,
Pk(i)
(t)
<
Vl),Vlk
PROOF.
if
ns+
>
n, for some
.i
(t)
< +e,
<
Let
-
Let
inf
Zo(X)
9q)},
{(ml,
_<
k
m
+
),Vnk
P(-l)(a)
hCnk
_< i _< n,
-O,
and since
e
>
0
was arbitrary, we have
{nlk(t;
where
o(X))}
Then
{n (t;
inf
o(X)
and assume that
9q
91,
n l(t)
ik(t)
<
10(x)}"
o.
On
are positive integers partitioning
ordering by
q-l}
(nl, ..., Uq) > (91,
9p),
n
9i, _< i <_ s, and
i
such that
ms+ I.
_<
t
points
y ml
V
t
be given.
<
(Xq)
(m
q.
)(x I)
.=oo o
x1
1,...,gq)
is the last tuple for such distinct solutions,
(Xl)"
w
heo=,
y
< Xq <
<
0
9q),
W (x) of
hence
That being the case, it follows from the argument used in the
, m. m ,
(Xq),
n(t)
< o, there exist a last tuple (91
o, a c It,o), and distinct solutions Y (x) and
Since we are assuglng that
<
sufficiently small, the
hClk,...,(
_< q < n, we define a lextcographical
n > 91, or if there exists s
{1,
n,
h
n
l(t; Y0(x)) }.
Inf
THEOREM 5.
v
+
V 1,..., V
{n k
o(X)
the set
Thus, for
satisfies the conditions
It follows that
k
Unkg(X;,
n)
<
i
< 9q
S
k’)()
V )
,
,ouo
o .),
DISCONJUGACY FOR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
If for each
s
S
S
s’
and
If we set
(x,)
V
v
(Xq,S)
k
V
8s
(x,)
Z
v
above that
z
[t,).
(mq-1)
<
.
(s-s,) s
(x,)
Z
(1.2) along
is the solution of
_< i _<
O,
i
vmq-l" (Xq,).
(Xq,S’)
then
<
0,
(Xq,S--)
k
k
(a,)
i)(xq,)
q-l,
(mq-l)
(x,),
and satisfies
<
o
q
such that
such that, for the kth component,
(m -I
0
_
denote the corresponding solution of (I.I),
Y (x)
V (x,s) and W (x)
V (x,s’).
and Theorem 3, we conclude that there exists an
s,s’
From the connectedness of
which is between
V (x,s)
S, we let
s
then there are distinct
379
(xj,)
n,
0,
m-2, and
0_<i<_mj-l,
But we also have
O, which contradicts the dlsconJugacy of (1.2)
Thu, ou= .s.upio.
()
f.e .d
i-
Yo(X)
((t; 0 ()))"
4.
RIGHT DISFOCALITY AND INTERVALS OF RIGHT DISFOCALITY.
In this section we present analogues of the results of section 3 in
terms of what we call k-component right disfocality. Much of our notation is
that used by Muldowney [10].
DEFINITIONS.
Let z
We say that a function
<
_<
i
n,
y(J)(t i)
and
A partition
(Zl;
i; ...;T) Is
nd if
+ m
!
,
(tl,
y(x)
...;
an
l, s
where
in)
has
)
Tj
i
<
t
t
in
W (x)
(a,b),
some
(vl;
mj
...;
TI,
)
-=
Y (x)
W(kJ-l)(x)
and
has
k(x)
{nj}j.
then it follows that
w
For a sequence of integers
let
m
+
(mj) J=l
+
m
For a sequence
mj
-
>
n
n
<
or
t
Wi-1)(a)
zeros at
n
< s
zl*
t
_< tn,
<_
2
s
and
...; m) dlsfocal on
Z (x) of (I.I) such that
Z (x) satisfies
is an increasing partition of
n
O,
(ml;
We say the system (I.I) is k.-component right
(a,b), 0 <_ mj _< n-+l, if given solutions Y (x),
their difference
t
provided
J, then either
is the multipllclty of
m
(a,b).
if t
occurs m times in z.
i
is obtained by inserting -1 semicolons.
z
Increasing partition of
with
y(t i)
provided
_< _< m-l,
0, 0
of
be an n-tuple of points from
zeros at
n
j,
points in
J
0,
_<
,
(a,b)
_< i < n,
where
with
O.
satisfying
>, >_,
>
(4.1)
be a sequence of nonnegative integers satisfying
n, m 2
{nj}=
k-component right
satisfying (4.2)}.
+
/
m < n2,
m_
/
m < n_l, m < n.
(4.2)
(4.1), d.efine 8k(t)- sup{t > t (I.I) is
dlsfocal on [t,t I] for all sequences
{mj}
satisfying
(m I; ...;
m)
Given a solution
Y (x) of (I.I),
8k(t;
defined similarly for the variational equation (1.2) along
Y (x))
is
Y (x).
In much the same manner as Peterson [11] has proven for scalar equation
can be characterized In tes of
8k(t;
Y (x))
as sted In
e
following
8k(t)
eorem.
JOHNNY HENDERSON
380
<
Let
THEOREM 6.
k
<
(a,b).
m be given and let c
Then, either
(x) of (1.1) such that the variational equation (I.2)
has a noutrivial solution Z (x) satisfying the conditions
(1) there is a solution
(x)
along
i-)()
k(i-t) (c)
z k(i-1) (d)
where
here
c; Y (x)), a
d
<_
n-
J+l
<_ , (n+
Bk(c)
(ll)
o
in-.i,
0
n-J+1
[c,d], and
i
_< n+l,
satisfies
nn_j+ 2 < J _< nn_J+l,
0), or
inf
{Bk(c;
(x))}.
Y (x)
REFERENCES
I.
PETERSON, A., Comparison theorems and existence theorems for ordinary
differential equations, J. Math. Anal. Appl. 55 (1976), 773-784.
PETERSON, A., An expression for the first conjugate point for an nth order
nonlinear differential equation, Proc. Amer. Math. Soc. 61 (1976), 300-304.
SPENCER, J., Relations between boundary value functions for a nonlinear
d. lfferential
equation and its variational equations, Canad. Math. Bull.
18(1975), 269-276.
4.
SUKUP, D., On the existence of solutions to multipoint boundary value problems,
Rocky Mtn. J. Math. 6(1976), 357-375.
5.
HENDERSON, J., Right focal point boundary value problems for ordinary
differential equations and variational equations, J. Math. Anal. Appl- 98
(1984), 363-377.
6.
HENDERSON, J., DisconJugacy, disfocallty, and differentiation with respect to
boundary conditions, J. Math. Anal. Appl., to appear.
7.
HENDERSON, J., K-polnt dlsconJugacy and disconJugacy for linear differential
equations, J. Differential Eqs_. 54(1984), 87-96.
8.
MIILDOWNEY, J., On disconJugacy and k-point disconJugacy for linear ordinary
differential operators, Proc. Amer. Math. Soc. 92 (1984), 27-30.
9.
On a relation between a theorem of Hartman and a theorem of
PETERSON,
Sherman, Canad. Math. Bull. 16 (1973), 275-281.
.,
I0.
MULDOWNEY, J., On invertlbillty of linear ordinary differential boundary value
problems, SLAM. J. Math. Anal. 12(1981), 368-384.
II.
PETERSON, A., A dlsfocallty function for a nonlinear ordinary differential
equation, Rocky Mtn. J. Math. 12(1982), 741-752.