Applied Mathematics
Vol.3 No.9(2012), Article ID:23002,6 pages DOI:10.4236/am.2012.39149
Existence of Positive Solutions for a Third-Order Multi-Point Boundary Value Problem
1Laboratory of Advanced Materials, Faculty of Sciences, University Badji Mokhtar-Annaba, Annaba, Algerie
2Université 8 mai 1945 Guelma, Guelma, Algerie
Email: a_guezane@yahoo.fr, zenkoufi@yahoo.fr
Received June 25, 2012; revised July 25, 2012; accepted August 3, 2012
Keywords: Guo’s Fixed Point Theorem; Three Point Boundary Value Problem; Positive Solution; Leray Schauder Non-Linear Alternative; Contraction Principle
ABSTRACT
By using Leray-Schauder nonlinear alternative, Banach contraction theorem and Guo-Krasnosel’skii theorem, we discuss the existence, uniqueness and positivity of solution to the third-order multi-point nonhomogeneous boundary value problem (BVP1):
where
for
The interesting point lies in the fact that the nonlinear term is allowed to depend on the first order derivative
.
1. Introduction
It shows that problems related to nonlocal conditions have many applications in many problems such as in the theory of heat conduction, thermoelasticity, plasma physics, control theory, etc. The current analysis of these problems has a great interest and many methods are used to solve such problems. Recently certain three point boundary value problems for nonlinear ordinary differential equations have been studied by many authors [1-9]. The literature concerning these problems is extensive and application of theorems of functional analysis has attracted more interest. Recently, the study of existence of positive solution to third-order boundary value problems has gained much attention and is a rapidly growing field see [1,2,6,8-11]. However the approaches used in the literature are usually topological degree theory and fixed-point theorems in cone. We are interested in the existence, uniqueness and positivity of solution to the third-order multi-point nonhomogeneous boundary value problem (BVP1):
where
for
The organization of this paper is as follows. In Section 2, we present some preliminaries that will be used to prove our results. In Section 3, we discuss the existence and uniqueness of solution for the BVP1 by using Leray-Schauder nonlinear alternative and Banach contraction theorem. Finally, in Section 4 we study the positivity of solution by applying the Guo-Krasnosel’skii fixed point theorem.
2. Preliminary Lemmas
We first introduce some useful spaces. we will use the classical Banach spaces,. We also use the Banach space
, equipped with the norm
where
.
Firstly we state some preliminary results.
Lemma 1 Let and
then the problem
(2.1)
(2.2)
has a unique solution
(2.3)
where
(2.4)
(2.5)
Proof Integrating the Equation (2.1), it yields
From the boundary condition we deduce that
and
.
And the boundary condition implies
Therefore we have
Now it is easy to have
which achieves the proof of Lemma 1.
We need some properties of functions.
Lemma 2 For all t, s, such that we have
Proof It is easy to see that, if
If
then
Lemma 3 For all t, s such that,
, we have
Proof For all if
it follows from (2.4) that
and
If it follows from (2.4) that
Therefore
Lemma 4 (See [6]) We define an operator by
Lemma 5 (See [5]) The function is a solution of the (BVP1) if and only if T has a fixed point in X, i.e.
.
3. Existence Results
Now, we give some existence results for the BVP1 Theorem 6 Assume that and there exist nonnegative functions
such that
we have
and
then, the (BVP1) has a unique solution in
Proof We shall prove that T is a contraction. Let then
So we can obtain
Similarly, we have
From this we deduce
Then T is a contraction. From Banach contraction principe we deduce that T has a unique fixed point which is the unique solution of (BVP1).
We will employ the following Leray-Schauder nonlinear alternative [12].
Lemma 7 Let Fbe Banach space and be a bounded open subset of F,
.
be a completely continuous operator. Then, either there exists
,
such that
, or there exists a fixed point
Theorem 8 We assume that
and there exist nonnegative functions
such that
Then the (BVP1) has at least one nontrivial solution.
Proof Setting
Remarking that
and
then there exists an interval
such that
and
a.e.
Le
With the help of Ascoli-Arzela Theorem we show that
is a completely continuous mapping. We assume that
such that
then
we have
and
This shows that From this we get
this contradicts By applying Lemma 7, T has a fixed point
and then the BVP1 has a nontrivial solution
4. Positive Results
In this section, we discuss the existence of positive solutions for (BVP1). We make the following additional assumptions.
(Q1) where
and
(Q2)
We need some properties of functions
Lemma 9 For all, we have
where
.
Proof It is easy to see that.
If
If
Lemma 10 Let and assume that
then the unique solution u of the (BVP1) is nonnegative and satisfies
Proof Let it is obvious that
is nonnegative. For any
by (2.3) and Lemmas 2 and 3, it follows that
(4.1)
On the other hand, (2.4) and Lemma 11 imply that, for any we have
From (4.1) it yields
(4.2)
Therefore, we have
Similarly, we get
On the other hand, for and using Lemma 10 and (4.1) we obtain
(4.3)
Therefore,
Finally, regrouping (4.2) and (4.3) we have
Definition 11 Let use introduce the following sets
K is a non-empty closed and convex subset of X.
Lemma 12 (See [5]) The operator T is completely continuous and satisfies
To establish the existence of positive solutions of (BVP1), we will use the following Guo-Krasnosel’skii fixed point theorem [13].
Theorem 13 Let E be a Banach space and let be a cone. Assume that
,
are open subsets of E with
and let
be a completely continuous operator. In addition suppose either 1) and
or 2)
and
holds. Then has a fixed point in
The main result of this section is the following Theorem 14 Let (Q1) and (Q2) hold, and assume that
Then the problem (BVP1) has at least one positive solution in the case 1) and
or 2)
and
Proof We prove the superlinear case. Since then for any
such that
for
. Let
be an open set in X defined by
then, for any it yields
Therefore
So
If we choose
then it yields
Now from we have
such that
for
. Let
Denote by
the open set
If then
then Let
then
And
Choosing
we get
By the first part of Theorem13, T has at least one fixed point in
such that
This completes the superlinear case of the theorem 14. Proceeding as above we proof the sublinear case. This achieves the proof of Theorem 14.
Example 15 Consider the following boundary value problem
(E1)
Set
and
where
and and
. One can choose
It is easy to prove that
are nonnegative functions, and
Hence, by Theorem 6, the boundary value problem (E1) has a unique solution in X.
2) Now if we estimate as
then one can choose. So
are nonnegative functions. Hence, by Theorem 8, the boundary value problem (E1) has at least one nontrivial solution,
Example 16 Consider the following boundary value problem
(E2)
where, and
Then
We put,
and
, when
and when
Then
By theorem 13 1) the BVP (E2) has at least one positive solution.
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