Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 615-618
615
RESEARCH NOTES
A NOTE ON GLOBAL EXISTENCE FOR BOUNDARY VALUE PROBLEMS
CHUAN J. CHYAN and JOHNNY HENDERSON
Department of Algebra, Combinatorlcs and Analysis
Auburn University
Auburn, Alabama
36849
U.S.A.
(Received May 21, 1987)
ABSTRACT.
Upper and lower solutions are used in establlsning global existence
results for certain two-polnt boundary value problems for
y(n)
and
_-f(x, y, y’,
KEY WORDS AND PHRASES.
f(x, y, y’, y’’)
Boundary value problem, global existence.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
y’’’
..., yn-l))."
34B15.
INTRODUCTION.
In this paper, we will be concerned primarily
with the global existence of
solutions of boundary value problems for the third order ordinary differential
equation
f(x, y, y’, y’’),
y’’’
(1.1)
satisfying boundary conditions of the form
The result we obtain
y(a)
Y3, a < b.
Y2, y’(b)
Yl, y’(a)
an
(1.2)
for (I.I),
is
extension, in some sense, of
(1.2)
those for
boundary value problems for second order equations which appeared in a recent paper
by Umamaheswaram and Sunaslni [I].
[I] made use of, or were compared
The results in
to, results dealing with upper and lower solutions for second order equations
obtained 0y Jackson and ScOrader
I],
[2], Lees [3],
and Schrader [4-b].
In [I, Theorem
the following is proved.
THEOREM I.I.
Assume that with respect to the second order equation,
y" -g(x, y, y’), the following are satisfied:
g2
R is continuous.
(A.I) g: [u, 8]
(B.I) Solutions of initial value problems exist on [a, B] or become unbounded.
+ =, such that
(C.I) There exists a sequence [Mj} of real numbers
f(x, Mj, 0) > 0, for every j > 1 and all a < x < 8.
-=, such that
(D.I) There exists a sequence [Nj} of real numbers
a
x
all
and
8.
for
every
j >
< <_
f(x, Nj, 0) < 0,
Then the boundary value problela
y"
Y(Xl)
where
a
<
x
<
x2
<
8, and
g(x, y, y’),
Yl,
Yl, Y2 e R
Y(X2)
Y2,
has a solution.
In Section 2, we extend Theorem I.I to boundary value problems (I.I), (1.2).
C.J. CHYAN AND J. HENDERSON
616
For this extension, we generalize (C.I) and (D.I) so that the conditions set forth by
[7] for (I.I), (1.2) are satisfied for any
Klaasen
Yl
e R
i=l, 2, 3.
In Section 3, the results we obtained for (I.I), (1.2) are generalized somewhat
to boundary value problems for the nth order equation
yCn)
y(n-1)),
fCx, y, y’,
(1.3)
satisfying
Y(i-l)(a)
1
Yi’
<_ i <_
y(n-21b) Yn’
n-l,
<
a
(1.4)
b.
We conclude Section 3 with an example.
GLOBAL ExISrENCE FOR (I.I), (1.2).
2.
In this section, a theorem is proved concerning the global existence of
(I.I), 41.2).
solutions of
We assume in this section that with respect to (I.I), the
following are satisfied.
(A.2)
(B.2)
f(x,
R3
u3): [a, b]
Ul, u2,
R
is continuous.
Solutions of initial value problems for 41.1) extend to
[a, b]
or become
unbounded.
(C.2)
+ ,
M.
and
a
<x<
(D.2)
There eist sequences
-,
I.
a<x<
x +
Mj
(Kj}
f(x,
such that
Li, Mj,
of real numbers with both
O)
>
O, for all
i, j
>
I
L3
and
+
all
Nj
x +
of real numbers, with both
{Nj}
and
Ki, Nj,
O)
_<
0, for all
i, j
_>
K.j
and all
b.
x, u2, u3.
for each fixed
u
is nonincreaslng in
has a solution for any choice of
(I.I), (1.2)
Let
PROOF.
I, J
(D.2) are satisfied and that
Assume that (A.2)
THEOREM 2.1.
exist
f(x,
such that
{M}
and
b.
N
and
{L}
There exist sequences
N
g
e R
YI’ Y2’ Y3
be given.
f(x, Ul, u2, u3)
Then the boundary value problem
YI’ Y2’ Y3 e R
By hypotheses (C.2) and (D.2), there
such that
-<
and
(x) =Njx + K I
[a, b], for i 0, I.
Defining
Nja + K I < Yl Mja + LI,
Nj < mln {Y2’ Y3 < max {Y2’ Y3
and (x)
Mjx + L1, it follows
-< Mj.
(1)(x) < (1)(x)
(D.2), y(x)
on
41.1)on
[a, hi.
of
lower and upper solutions, respectively,
Furthermore, by (C.2) and
and
(x)
on
are
It follows from results due to Klaasen [7] that there exists a solution y(x)
of (I.I), (1.2), for this choice of YI’ Y2’ Y3’ and furthermore y(x) < y(x) _< (x)
on [’a, 6f. ["de proof s comZeCe.
yfx)" S
and Nj
<
S
GLOBAL EXISTENCE FOR (1.3), (1.4).
In this section, we will be concerned with the existence of solutions of (1.3),
(1.4). For this consideration, results due to Kelley [8] will De used. We assume
3.
here that with respect to
(A.3)
(B.3)
(1.3),
the following are satisfied.
Rn
R
is continuous.
[a, b]
f(x, u I, u2,
Solutions of initial value problems for (1.3) extend to
Un):
[a, b] or
become
unbounded.
(C.3)
that
There exist sequences
Mk, j
+
,
{MI,j}
{M
2,j},
[Mn_l,j}
of real numbers, such
n-I
I
<
k
<
n-l, and such that if
PJlJ2
Jn-I
()=
I k ’Jk
k=l
x
GLOBAL EXISTENCE FOR BOUNDARY VALUE PROBLEMS
f(x, pj
then
>--
J n-I
Jl’
(D.3)
(x)
Jn-I
PJla
all
and
<x<
{N
There exist sequences
PJl-(n-2)""’J-n(),
(x),
Jn-I
617
>
0)
O, for
b.
of real numbers, such
{Nn_l, j}
}, {N 2 ,j
,j
n-I
<
that
Nk,
then
f(x,
qj
>
n-l, and such that if
(x)
Jn-I
Jn-I
Jl
<
k
(x),
qJl
all
and
qJlJ2
Jn-I
(n-2)
qJl Jn-I (x),
Jn-I
_< x _<
a
(D.3) that, if
b] suco that
ounded on I. Further,more, assume
Assume in addition to conditions (A.3)
THEOREM 3.1.
that for
has
I, then y(n-l)(x) is
n-2, f(x, Ul, u2, ..., u n)
is bounded on
<i<
earn
...,
Ui_l, Ui+l, ..., Un.
a solution for any choice of
Yl R
PROOF. Let
Yi e R 1 _< i _< n, be
fixed
x, Ul,
that there exist
Jn-I
Jl’ J2’
(n-2)
(n-2)
(a)
Jn-I
e N
for each
(1.3), (1.4)
n.
such that
<
(a)
Jn-I
<
min
{Yn_l, Yn
<
max
{Yn-l’ Yn
7(x)
Jn-I
<
n-Z,
(n-2)
(n-2)!
pj
>
i
[a, hi.
(x),
0, for all
It
follows
(a).
dn_l,Jn_l PJl
a
Jn-I
it follows from
<
x
(i-l)(a) > y(i-l)(a), < i < n-2, that y(i-l)(x) < (i-l)(x)
< i < n-7. Furthermore? fro--m (C.3) and (D.3), (x and (x)
solutions, respectively, of (1.3) on
ul,
It follows from (C.3) and (0.3)
given.
(x) and (x) -=
qJl
Jn-I
(n-2)(x) (n-2)(x) (n-2)(a) y(n-2)(a)
Defining
is noancreaslng in
< I<
< Yl <
--PJl
Nn_l,Jn_l
C [a,
I
Then the boundary value problem
(i-1)
qi-1) Jn-I(a)
qj
Jk
K=I
O)< 0, for all
b.
a solution of (1.3) wlttl f,axlmal interval of existence
y(n-2)(x)
k-I
<
b, and from
[a, hi, for
on
are lower and upper
from the other hypotheses
of the Theorem and from a result due to Kelley [8] that there exists a solution
of (1.3), (1.4), for this choice of
y(i-l)(x) < (i-l)(x)
EXAMPLE.
R
g(u)--
<
n-l.
Tnls completes the proof.
(ru/e"’)
Moreover,
[0, ]
+ 2e
n+
<
u
-2u
-2e
..., Un):
f(x, u
i
< _<
be defined by
Sin
and let
<
[a, b], for
on
g: R
Let
Yl
n.
i
y(x)
(i-l)(x) <
I
Sin
R
0,
0iuie
(u/e), u > e
be defined by
...,
+
f(x, ul,
Un_ 1, o n
The conditions of Theorem 3.1 are satisfied with respect to the differential equation
y(n)
In particular,
’i
f(x, y, y’
1
<
i
<
<_ k < n-2,
{j}, for
wereas,
the sequences
satisfy condition (D.3).
.-,
y(n-1))
g
2Un.
(y(n-2)
+
2y(n-l)
n, are plecewlse continuous and bounded on
hence initial value problems of
{Mk, j}
g(Un_ 1)
{Nk, j}
(3.1) exist on
and
{Mn_l, j}
{-j}, for
1
[0, ].
R
n,
Also, the sequences
(l+4j)e
Z-(n_Z)!
< k <_
(3.1)
J
[0,
n-2, and
satisfy condition (C.3),
[Nn_l,
-j e
Hence, by Theorem 3.1, boundary value problems for (3.1)
618
C.J. CHYAN AND J. HENDERSON
satisfying
Y(i-I)(O)
In fact,
/
c
k
[eXslnx
+2
C
(-4
21-2
x
41-3
]],
[eX(slnx-
n=
_<
n-l,
[
+221-I
+
(i-z
11,
x
41-2
2,
x 41
(-I) I
[221 ((i):
i=1
41-2
x
y(n-2)(.) ’n
4i-I
+
x
4+ 2, = o, :,
k
cosx) +
i
[22i-I
(_l)i
+
<
1
Yi’
are solvable.
41-I
+ x
<i-):
4k + 3, k-- O, 1, 2,
n
y(x)
C
[- x
+221-I
C
where
0
y(i-l)(o
<
C
<
+
41-I
x
x
(_:) [2
+ (t)
(i
[eX(sl
+22i-2
-:
k
o
],
4i+I
Irnvr.
n=
k
+ cosx) +
4i
+
raw.)
4(+t),
= 0,
[ (-l)i [2 21-1
t, 2,
41-2
4i-3
x
x
+ )
((41"Z)l
4i-4
]],
I, 2,
I, are infinitely ny solations of (3.1) tisfying
O, I < i < I.
n
4k+l, k
y[n-2)()
REFERENCES
I.
UMAMESWARAM, S.
and SUflASINI, M.V.S. A global existence theorem for the
boundary value problems of y"
f(x, y, y’), Nonlinear Anal. 10 (1986),
679-681.
2.
JACKSON, L. and SCHRADER, K.
equations, J. Diff.
3.
4.
Eqns.
3
LEES, M.
A boundary value problem for nonlinear ordinary differential
equations, J. Math. Mech. I0 (1961), 423-430.
SCHRADER, K.
Boundary value problelas for second order ordinary differential
equations, J. Diff. Eqns. 3
5.
6.
7.
8.
Comparison theorems for nonlinear differential
(1967), 248-255.
(1967), 403-413.
SCHRADER, K. A note on second order boundary value problems, Am. Math. Mort. 75
(1986), 867-869.
SCHRADER, K. Existence theorems for second order boundary value problems, J.
Diff. Eqns. 5 (1969), 572-584.
KLAASEN, G. Differential inequalities and existence theorems for second and
third order boundary value problems, J. Diff. Eqns. I0 (1971), 529-537.
KELLEY W. Some existence theorelas for nth-order boundary value problems, J.
Diff. Eqns. 18 (1975), 158-169.