Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 713-720
713
SOLUTION MATCHING FOR BOUNDARY VALUE
PROBLEMS FOR LINEAR EQUATIONS
DeparUeent
o
JOHNNY HENDERSON
Algebra, Comblnatorics and Analysis
Auburn University
Auburn, Alabama
36849
U.S.A.
(Received March 17, 1988)
ABSTRACT.
where
n
>
3, solutions
o
n
y(n)
For the linear differential equation,
at(x)Y(i-1)’
(1.1),
multipont boundary value problems on an Interval
[a,c]
are obtained, via the use of Liapunov-like functions, by matching solutions of
certain boundary value problems for
boundary value problems for (1.1) on
KEY WORDS AND PHRASES.
(1.1) on
[b,c].
[a,b]
wlth solutions of other
Multipoint boundary value problem, solution matching,
linear equation.
1980 MATHEMATICS SUBJECT CLASSIFICATION.
I.
34B05, 34BI0.
INTRODUCTION.
We will be concerned with the existence of solutions of k-point boundary value
<
problems, 2
k
<
a, on an interval [a,c] for the nth order linear differential
equation
yCn)
where the
at(x)
The point
b
C[a,c].
(a,c)
y.
Yl(X) of
solution Y2(X)
[a,b] with a
[b,c] such that, y(x)
_
n_>
(.)
3,
will fixed throughout, and we will employ techniques
which match a solution
on
a Cx)y (t-t)
a
(k-l)-point boundary value problem for (1.1)
of a
2-point
defined by
y(x)
IYl(X)’
a
Y2(X)’
b
<_ x
boundary value problem for (1.1) on
b,
x_ c,
is a solution of a k-point boundary value problem for
(1.1) on
[a,c].
714
J. HENDERSON
Solution matching techniques were first applied by Bailey, Shampine, and
Waltman [I] where they dealt with solutions of 2-polnt boundary value problems for
y"
the second order equation
f(x, y, y’)
by matching solutlons of initial value
Since then, a number of papers have appeared in which solutions of
problems.
[a,c] for the nth order ordinary differential
(n-i)
y’,
were obtained by matching solutions of
f(x, y,
y
2-point problems on [a,b] with 2-point problems on [b,c]; see, for example, Barr
and Sherman [2], Das and Lalli [3|, Henderson [4], Moorti and Garner [5], and Kao,
Murthy and Rao [6]. In those papers [2
6], a monotoniclty condition imposed on f
3-point boundary value problems on
y
equation
(n)
plays an important role in the solution matching techniques in obtaining solutions
of the 3-point problems.
For example, in obtaining solutions of certain 3-point
conjugate problems, 8arr and Sherman
[2] assumed that f
(-l)n-J(uj
Un_ 1 < Vn_l, and
Un_l, w) < f(x, vl,
Ul,
f(x,
vj)
vn_l,
0, j
satisfies the conditions:
1, 2,
n-2, imply
w), for all x
(a,b]
and all
w
and all
w e
R,
and
Un_ < Vn_l, and uj < vj, j I, 2, ..., u-2,
...,
ul,
Un_l, w) < f(x, Vl,
Vn_l, w), for all x
(ii)
f(x,
[5],
Moorti and Garner
Imply
e
[b,c)
by assuming (t) and (1t) with respect to third order
equations, obtained solutions of certain 3-point focal problems by matching solutions.
Das and Lalli [3] also assumed (t) and (Ii) with respect to third order
equations, (however in relaxing other assumptions which were made in
[2],
the proof
of Theorem 2.1 In [3] does not appear to be valid).
ao,
Murthy
and
ao [6],
In the paper by
conditions (t) and (tt) were modified some and solution matching was
applied to obtain solutions of 3-point conjugate problems for third order ordinary
differential equations.
Then Henderson
[4] generalized the onotontctty
conditions
of [6] and used solution matching to obtain the existence of solutions of a large
class of 3-point boundary value problems on
[a,c]
y
for
(n)
f(x, y, y’,
Ca-i)
[a,c], for 3 < k < n, by
Y2(x) on [b,c] under
Yl(X) on is,b]
assumptions (t) and (ti) or the more general monotontcIty conditions in [4], one
(n-2)
(u-2)
(b)
cannot necessarily conclude tt
(b), hence cannot conclude a
Y2
Yl
solution on [a,c]. For the linear equation (1.1), usl Llapunov-ltke functions,
Barr and Miletta [Tlmatched solutions of (n-l)-potnt boundary value problems with
However, to
obtain solutions of k-point problems on
matching a solution
with a solution
solutions of 2-point boundary value problems
thus obtaining solutions of n-point
In this paper, using techutques similar to those of Barr
and Mtletta [7], we obtain solutions of k-point boundary value problems for (1.1) on
[a,c]. In particular, given 2 < k n, positive integers ml,...,m such that
boundary value problems.
k
k
m
i
n and m
k
=I,
]
for (I.I) satisfying
points a
x
<
< Xk_
b
<
x
k
c, and Y
IJ
R
SOLUTION MATCHING FOR BOUNDARY VALUE PROBLEMS FOR LINEAR EQUATIONS
y
(i)
(xj)
and for
y
(i)
e
YiJ’ 0 <_. i <_. mj -I,
{O,l,...,n-2} fixed,
(xj
Ylj
0
<
i
< mj
I
_
<j <
k-l, y(x k)
<
k-l,
j
<
715
(1.2)
Yo,k’
yC)CXk)= Y0,k"
C1.3)
The existence of unique solutions of certain (k-l)-point and 2-point boundary
value problems for (I.I) used in the matching to obtain solutlons
(I.I),(1.3)
of(l.l), (1.2) and
will be established through the use of Liapunov-lie or control
func tlons.
DEFiNiTiONS.
Given H
V
M
0 and
[a,ll
(x,Yl,...,yn):
[a,cJ,
a control function
n
[a,8]
R
is a function which is continuous, locally Lipschitz with respect to
(yl,...,yn)
and satisfles
M
(x,Yl,...,y n)
M
(x,Yl,.
i) V
it) V
Corresponding to
VM(x,y I,
’Yu)
,y
0, if
n) >
O, if
and a solution
Yn-I
M,
Yn-I >
M.
y(x) of
the differential equation
1. ), def
V (x,y(x),...,y(n-l)(x))
llm inf
h/O+
h
iV (x+h,y(x+O),...,y(n-l)(x+h))
-v (x,y(x),...,y(n-1)(x)) ].
Its proof is a simple
Extensive use will be made of the following lemma.
extension of the one given for n=l in Yoshizawa
LEMMA I.I.
and
[3].
Suppose that y(x) is a solution of (1.1), and that for some M
[a,8]( [a,c], VM(X,Yl,...,y n)
is a control function.
>
0
Then
(n-l)
(x)) is nondecreaslng, (nonlncreaslng), if
VM(X,y(x),...,y
(n-l)
(n-l)
v’(x,y(x) ,...,y
(x)) _> o, (v(x,y(x),...,y
(x)) _< 0).
and only if
In section 2, we carry out the construction for matching solutions of
(k-l)-point boundary value problems with 2-point boundary value problems in obtaining a solution of
(I.I), (1.2).
Then in section 3, results completely analogous to
those obtained in section 2 are stated for solutions of
2.
(I.I), (1.3).
EXISTENCE OF SOLUTIONS OF (I.I), (1.2).
In this section, for 2
<
k
<
x
n, let
ml,...,mk
be positive integers such that
k
I mi=
n aad m =I.
k
Let a
I<...<
Xk_
b
< xk_<
c and Y
iJ
e
R
J. HENDERSON
716
<j<
< i < mj
0
y
-1,
Ci)
Ylj’ 0 <_ i < mj -I,
z(x) of (I.I) satisfying
(1)
!J
(xj)
with solutlons
z
We match solutions y(x) of (i.l) satisfying
k be given.
y(i) (Xk_l)
(Xk_ I)
where m e R
0
<i<
n- 3, z
<- k-L, yCn-X) (Xk_l)=m,
(n--)
(2.2)
(Xk-l)=m’ z(xa)= Yo,k’
([.I), (1.2).
to obtain a solution of
(2.1)
’the use made of a family of
control functions In establishing existence of unique solutions of (k-l)-point and
2-polnt problems and in establishing des[tel monotone properties Is seen in the next
four theorems.
Assume that there exists a control function
THEOREM 2.1.
[a,b] such that V0 (x,y(x),...,y
(xj)
y
>
(x))
on
O, for all solutions y(x) of
the boundary value problem for (I.I) satisfying
R
Then for each m e
V0(x,y[...,y n)
Yij’
0
<_ i < mj
-I, I
<_ j <_
k-I,
y{n-2) (Xk_
(2 3)
)=m,
has a unique solution.
It suffices to show that the boundary value problem for (I.I) satisfy-
PROOF.
tag
y(t)(xj)
y(n-2) (xk_ I)
0, has only me
< j < k-l,
< i < mj_
Assume on the contrary that this boundary value problem has a
O, 0
trivial solution.
y(x).
nontrivial solutlon
Xl < I < 2 < 3 < xk
(n-2)
(n-2)
(l)=
y
(n-2)
(3)= yCn-l) (2)=
O, and
(n-X)
(x) has a positive local maximum at x
2"
(n-2)
(x) has a positive local maximum
Assume without loss of generality that y
Ca-t)
at x
(ri))
2" Now from our hypotheses, Vo(t, y(l),...,y
(n-I)
(n-I)
(2)) > O. However,
(3)) 0 and V0(2, y(2),...,y
V0(3, y(3), ,y
(n-t)
(n-l)
(x)) is nondecreastng;
(x)) > O, Vo(x,y(x),...,y
,y
since
(x,w(x)
(n-t)
the
y
(x) or -y
It follows that there exist polnts
such that y
V
consequently,
(3)) >
Vo(Z 3, y(3),...,y
0 whic is a contradiction.
Thus,
assertion of the theorem is true.
Assume that for each m
THEOREM 2.2.
R
there exists a solution
Yl(X,m)
(I.I), (2.[). If for each M > 0, there exists a control
(n-)
(x)) >_ 0 for all
V(x,Yl,...,y n) on [a,bl such that V M (x,y(x),..., y
boundary value problem
function
solutions y(x) of
of
(l.1), then
(n-2)
Yl
(_l,m)
is a strictly increasing function
m.
I<
m and assume
Let m
2
Yl
consider the nontrivlal solution w(x)
PROOF.
that w
of
(n-2)
(n-2)
(n-2)
(Xk-l’ml)"
(Xk-l’m2) < Yl
Yl(X’ml) Yl(X’m2)
of (I.I).
Then
It follows
(Xk_ I) > O.
Since w
(i)
(xj)
O, 0
I_< mj -I,
<J<
k-l, it follows by successive
717
SOLUTION MATCHING FOR BOUNDARY VALUE PROBLEMS FOR LINEAR EQUATIONS
applications of Rolle’s Tneore.a that there exists
a
(n-2)
Yet, from w
(i)=0.
(n-2)
the fact that by construction w
(T I)
(n-t)
O, the
assumption that w
(Xk_ 1)
m
0 and
r2 (rl, xk_ I) such
at x
T2" Now, let i < < r2< 8 < Xk_ De
(n-2)
M and w
(2) > M. Since there exists a
VM(a,w(a),...,w
VM(2, w(2),...,
w
(x,w(x),...,w(n-l)(x)) is
VM(B,w(),... w(n-l)(B)) >
Thus, w
<
(n-2)
yl(n-2)(Xk_l
REMARK.
(xk_ I) <
m2).
(8))
such that
(n-2)
(xk_ I) >
O, and
have that there exists
has a positive local maximum
(n-2)
(a)
control function
VMC,w(),...,w
(2)) > O. However,
V
O, we
m2<
such that w
(n-l)
Ca))=
(n-l)
1-
w(n-2)(x)
that w(n-1) (2)
(n-l)
l g (Xl’Xk-I)
w
VM,
(n-2)
we have
0 and
since
[a,bl,
nondeereasing on
it follows rat
O; again a contradlctlon.
0, which In turn Implies that
Yl (n-2)(XK_l,m 1)
The proof is complete.
We remark that, under the hypotheses of Theorem 2.2, it can be argued
yl(n-J)(Xk_l,m),
by finite Induction that
increasing functions of
2
< j < n-_l
are all strictly
m.
The next two theorems follow from arguments very similar to those used in
Theorems 2.1 and 2.2.
THEOREM 2.3.
function
solutions
,...,r n-3
Let r
mk_
Wo(x,Yi,...,y n)
on
y(x) of (I.I).
[b,c]
be given.
such that W
0
Then, for each m R
Assume that there exists a control
(x,y(x),...,y
(n-l)
(x))
0 for all
the boundary value problem for
(I.I) satisfying
Yi,k-l’
(i)
(x_ I
r
0 i t
<_ mk_:l
mk_ti _<
i
i
l
’Y
(=-2 Xk-1 )=
=,
y()
Y0,k’
(2.4)
n-3
has a unique solution.
THEOREM 2.4.
Let
Yl(X,m) be as in Theorem 2.2 and assume that for each m
Y2(x,m) of boundary value problem (I.I), (2.2). If for
on [b,c] such that
control function WM(X,Yl,...y
n
there exists a solution
M
>
0, there exists a
(n-l)
(x))
(x,y(x),...,y
W (n-2)
Y2
(Xk_l,
PROOF.
m)
Let m
Theorem 2.2 that w
R
each
0 for all solutions y(x) of (i.l), then
is a strictly decreasing function of m.
I < 2
(n-2)
m
and then set w(x)
(Xk_ l) <
=- Y2(x,m2) Y2 (x’ml)"
Then argue as In
0.
We are now prepared to match solutions and otaln a solution of (I.I), (1.2).
J. HENDERSON
718
THEOREM 2.5.
Assume that, for each m e R
there exists a unique solution of
(I.I), (2.1) on [a,b], and that, for each m R there exists a solution of
(l.l), (2.2) on [b,c]. Assume, moreover, that the boundary value problem for (I.I)
on [b,c] satisfying
(i)
(2.5)
0,
y
n-3, n-l, y(x k)
(Xk_ I) 0, I 0,
has only the trlvlal solutlon. If for each M > O, there exlst control functlons
V (x,y ,...,y
and W (x,y ,...,y
on [a,b] and [b,c] respectively, such that
(n-l)
(x)) _> 0 for all solutions y(x)
(x,y(x),...,y
(x)) _> 0 and
V(x,y(x),...,y
of (1.I), then the k-polnt boundary value problem (1.1), (1.2) has a solution on
[a,c].
PROOF. If Yl(X’m) is a solution of te boundary value proolem (I.1), (2.1),
W
(n-2)
then by Theorem 2.2,
(Xk_l, m) is
(n-2)
(Xk_l,
Yl
Yl
m) is a continuous function of m
contend, furthermore, that
To see this, it suffices to show the latter; that is,
range all of R
(Xk_ l,m)
y
m e
g
u
0_<
(xj)-ytj,
(n-2)(xk_ 1)
Consider now the solution
In summation, Yl
with range all of
w(x)
u(x)
(xj)-0,
0_<
u
(n-l)
(Xk_l))
-yl(x,u
of equation (1.1).
, t_<
(xk_ 1)
u
{
J_<k- t,
(n-l)
(Xk_ 1
0
R}
m
Y2(x,m)
(I.I), (2.4) and (I.I), (2.5),
such that
yl
is a solution of the boundary value
then it will follow that, from the existence of unique
Ing, continuous function of
R
m)I
R
problem (I.I), (2.2),
m
0
(n-2)(
is a strictly increasing, continuous function of
(Xk-I ,m)
w(x)
(2.1),
i_<mj-
Similarly, if as in Theorem 2.4,
solutions of
of
,
j_<g-
0, and hence
By the hypotheses of the theorem, #(x)
(n-l)
(n-2)
(n-l)
u(x)
(Xk_l))
(Xk_l, u
(Xk_l)). Consequently, Yl
Yl(X,U
(n-2)
m
u
(Xk_ I) r, and it follows that r
Yl
xk-I
(n-2)
u(x)
r.
w(n-l)(xk_l)
and
, t_<
<aj
satisfies the boundary conditions of type
(1)
with
R
From Theorem 2.1, there is a unique solution
R
Thus, let r
(I.I), (2.3) satisfying
u
We
a strictly increasing function of m.
m
Y2
(n-2)
with range all of
(-I’ m)
R
(n-2)(xk_1, m0)" Y2 (n-2) (Xk_ I,
is a strictly decreas-
Thus, there is a unique
m
0).
Then
SOLUTION MATCHING FOR BOUNDARY VALUE PROBLEMS FOR LINEAR EQUATIONS
y(x)
I Yt(x’’O)
a
_<
(Y2(X,m0)
b
<_
is a solution of the boundary value problem
3.
(l.l), (l.2)
x
_<
x
_<
719
D,
c,
[a,c].
on
EKISCENCE OF SOLUTIONS OF (l.l), (1.3).
In this section, let k,
ml,... ink, a Xl...< Xk_
{0,1,...,u-2} be
be as in the previous section. Let p e
b
< x k_<
given.
c and
c R
YiJ
We state theorems
analogous to those in section 2 in which solutions y(x) of (l.l) satisfying
(i)
y(n-t) (Xk_ I)
_<
0
i
< j < k-l,
mj -I
are matched with solutions v(x) of (I.I) satisfying
y
(xj) YlJ
, (1) (Xk_ ,)
R
where m e
, v(n-t)(xk_)- ,, v(.) (Xk)
y(1) (Xk_), 0 _< i _< n
yielding a solution of (l.l), (I.3).
Moreover, Theorems 2.1
these theorems.
THEOREM 3.1.
(2.1)
m,
(3.i)
Y0.k’
We will omit the proofs of
and 2.2 are applicable in this section.
Assume the hypotheses of Theorem 2.3.
Then, for each m
R
the boundary value problem for (1.1) satisfying
Y
(I)
0
Yi,k-I
(Xk_
<_ i <_ mk_l
)=
r
ink_ <
I
i
I
,y
(n-2)
(Xk_
)= m,y
(p)
< n-3)
(Xk)
(3.2)
Y0,k’
has a unique solution
Theorem 3.2.
Let
Yl(X,m)
there exists a solution
M
>
O, there
be as in Theorem 2.2 and assume that for each
R
m
v(x,m) of boundary value problem (I.I), (3.1).If for
WM(X,Yl,...,yn) on [b,c] such that
each
exists a control function
M (x,y(x),...,y
W
(n-l)
(x))
>
0 for all solutions y(x) of
(l.l), then v
(n-2)
(Xk_2,
is a strictly decreaslug function of m.
THEO&EM 3.3.
Assume that, for each m c
R
there exists a unique solution of
(I.I), (2.1) on [a,b], and that, for each m c R there exists a solution of
(I.I), (3.1) on [b,c]. Assume, moreover,, that the boundary value problem for
(I.I) on [b,c] satlsfylng
y
(1)
(Xk_ I)
O, I
has only the trivial solution.
functions
V’ (x,y(x)
VH(X, Yl,...,yn)
y
(n-l)
(x))
on
>
0,..,n-3, n-l,
If for each H
[a,b]
0 and W
and
M
>
y() (x k)
0, there exist control
WM(X,Yl,...,yn)
(x,y(x)
0,
,y
(n-l)
on
[b,c],
(x))
>
such that
0 for all solutions
y(x) of (l.l), then the boundary value problem (I.I), (1.3) has a solution on [a,c].
720
J. HENDERSON
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