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Positive Solutions of a Nonlinear Three-Point Integral Boundary Value Problem
Boundary Value Problems volume 2010, Article number: 519210 (2010)
Abstract
We study the existence of positive solutions to the three-point integral boundary value problem ,
,
,
, where
and
. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones.
1. Introduction
The study of the existence of solutions of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [1]. Then Gupta [2] studied three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, nonlinear second-order three-point boundary value problems have also been studied by several authors. We refer the reader to [3–19] and the references therein. However, all these papers are concerned with problems with three-point boundary condition restrictions on the slope of the solutions and the solutions themselves, for example,

and so forth.
In this paper, we consider the existence of positive solutions to the equation

with the three-point integral boundary condition

where . We note that the new three-point boundary conditions are related to the area under the curve of solutions
from
to
.
The aim of this paper is to give some results for existence of positive solutions to (1.2)-(1.3), assuming that and
is either superlinear or sublinear. Set

Then and
correspond to the superlinear case, and
and
correspond to the sublinear case. By the positive solution of (1.2)-(1.3) we mean that a function
is positive on
and satisfies the problem (1.2)-(1.3).
Throughout this paper, we suppose the following conditions hold:
;
and there exists
such that
.
The proof of the main theorem is based upon an application of the following Krasnoselskii's fixed point theorem in a cone.
Theorem 1.1 (see [20]).
Let be a Banach space, and let
be a cone. Assume
,
are open subsets of
with
, and let

be a completely continuous operator such that
(i),
, and
,
or
(ii),
, and
,
.
Then has a fixed point in
.
2. Preliminaries
We now state and prove several lemmas before stating our main results.
Lemma 2.1.
Let . Then for
, the problem


has a unique solution

Proof.
From (2.1), we have

For , integration from
to
, gives

For , integration from
to
yields that

that is,

So,

Integrating (2.7) from to
, where
, we have

From (2.2), we obtain that

Thus,

Therefore, (2.1)-(2.2) has a unique solution

Lemma 2.2.
Let . If
and
on
, then the unique solution
of (2.1)-(2.2) satisfies
for
.
Proof.
If , then, by the concavity of
and the fact that
, we have
for
.
Moreover, we know that the graph of is concave down on
, we get

where is the area of triangle under the curve
from
to
for
.
Assume that . From (2.2), we have

By concavity of and
, it implies that
.
Hence,

which contradicts the concavity of .
Lemma 2.3.
Let . If
and
for
, then (2.1)-(2.2) has no positive solution.
Proof.
Assume (2.1)-(2.2) has a positive solution .
If , then
, it implies that
and

which contradicts the concavity of .
If , then
, this is
for all
. If there exists
such that
, then
, which contradicts the concavity of
. Therefore, no positive solutions exist.
In the rest of the paper, we assume that . Moreover, we will work in the Banach space
, and only the sup norm is used.
Lemma 2.4.
Let . If
and
, then the unique solution
of the problem (2.1)-(2.2) satisfies

where

Proof.
Set . We divide the proof into three cases.
Case 1.
If and
, then the concavity of
implies that

Thus,

Case 2.
If and
, then (2.2), (2.13), and the concavity of
implies

Therefore,

Case 3.
If , then
. Using the concavity of
and (2.2), (2.13), we have

This implies that

This completes the proof.
3. Main Results
Now we are in the position to establish the main result.
Theorem 3.1.
Assume and
hold. Then the problem (1.2)-(1.3) has at least one positive solution in the case
(i) and
(superlinear), or
(ii) and
(sublinear).
Proof.
It is known that . From Lemma 2.1,
is a solution to the boundary value problem (1.2)-(1.3) if and only if
is a fixed point of operator
, where
is defined by

Denote that

where is defined in (2.18).
It is obvious that is a cone in
. Moreover, by Lemmas 2.2 and 2.4,
. It is also easy to check that
is completely continuous.
Superlinear Case ( and
)
Since , we may choose
so that
, for
, where
satisfies

Thus, if we let

then, for , we get

Thus ,
.
Further, since , there exists
such that
, for
, where
is chosen so that

Let and
. Then
implies that

and so

Hence, ,
. By the first past of Theorem 1.1,
has a fixed point in
such that
.
Sublinear Case ( and
)
Since , choose
such that
for
, where
satisfies

Let

then for , we get

Thus, ,
. Now, since
, there exists
so that
for
, where
satisfies

Choose . Let

then implies that

Therefore,

Thus ,
. By the second part of Theorem 1.1,
has a fixed point
in
, such that
. This completes the sublinear part of the theorem. Therefore, the problem (1.2)-(1.3) has at least one positive solution.
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Acknowledgments
The authors would like to thank the referee for their comments and suggestions on the paper. Especially, the authors would like to thank Dr. Elvin James Moore for valuable advice. This research is supported by the Centre of Excellence in Mathematics, Thailand.
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Tariboon, J., Sitthiwirattham, T. Positive Solutions of a Nonlinear Three-Point Integral Boundary Value Problem. Bound Value Probl 2010, 519210 (2010). https://doi.org/10.1155/2010/519210
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DOI: https://doi.org/10.1155/2010/519210