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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 513189, 7 pages
http://dx.doi.org/10.1155/2012/513189
Research Article

On a Third-Order Three-Point Boundary Value Problem

1Laboratory of Advanced Materials, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria
2Laboratory LASEA, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria

Received 28 March 2012; Accepted 3 June 2012

Academic Editor: Rodica Costin

Copyright © 2012 A. Guezane-Lakoud et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a third-order three-point boundary value problem. We introduce a generalized polynomial growth condition to obtain the existence of a nontrivial solution by using Leray-Schauder nonlinear alternative, then we give an example to illustrate our results.

1. Introduction

In this work, we study the existence of nontrivial solution for the following third-order three point boundary value problem (BVP): where .

The parameters and are arbitrary in such that . Our aim is to give new conditions on the nonlinearity of , then using Leray-Schauder nonlinear alternative, we establish the existence of nontrivial solution. We only assume that and a generalized polynomial growth condition, that is, there exist two nonnegative functions such that where , so our conditions are new and more general than [1].

Such problems arise in the study of the equilibrium states of a heated bar. Very recently, there have been several papers on third-order boundary value problems. Graef and Yang [2, 3], Guo et al. [4], Hopkins and Kosmatov [5], and Sun [6] have all considered third-order problems. Anderson [7] considered the three-point boundary value problem for (1.1) in the case and the three-point conditions ; using the Krasnoselskii and Leggett-Williams fixed-point theorems, the existence of solutions to the nonlinear problem is proved. Excellent surveys of theoretical results can be found in Agarwal [8], Agarwal et al. [9], and Ma [10]. More results can be found in [1113].

This paper is organized as follows. First, we list some preliminary materials to be used later. Then in Section 3, we present and prove our main results which consist in existence theorems. We end our work with some illustrating examples.

2. Preliminary Lemmas

Let , with supremum norm , for all . Now, we state two preliminary results.

Lemma 2.1. Let . If , then the three-point BVP has a unique solution

Proof. Integrating over the interval , we see that
The constants , , and are given by the three-point boundary conditions (1.2).

We define the integral operator by

By Lemma 2.1, the BVP (1.1)-(1.2) has a solution if and only if the operator has a fixed point in . By Ascoli-Arzela theorem, we prove that is a completely continuous operator. Now we cite the Leray-Schauder as nonlinear alternative.

Lemma 2.2 (see [14]). Let be a Banach space and a bounded open subset of , . Let be a completely continuous operator. Then, either there exists ,   such that , or there exists a fixed point .

3. Main Results

In this section, we present and prove our main results.

Theorem 3.1. It is assumed that , and there exist two nonnegative functions such that Then the BVP (1.1)-(1.2) has at least one nontrivial solution .

Proof. Setting by hypothesis (3.2), we have . Since , then there exists an interval such that , then . Let ,  . Assume that such , then as , then
First, if , then ; hence, , and this contradicts the fact that . By Lemma 2.2, we deduce that has a fixed point , and then the BVP (1.1)-(1.2) has a nontrivial solution .
Second, if , then . By arguing as above, we complete the proof.

Let us define the following notation:

Theorem 3.2. Under the conditions of Theorem 3.1    and if one of the following conditions is satisfied
there exist and   such that
there exist two constants such that
the functions and satisfy
the function satisfies Then the BVP (1.1)-(1.2) has at least one nontrivial solution .

Proof. Let and be defined as in the proof of Theorem 3.1. To prove Theorem 3.2, we only need to prove that and .
By using Hölder inequality, we get Integrating, using (3.8), and remarking that , we arrive at .
Using Hölder inequality a second time, we get
Integrating and then using (3.9), we arrive at .
Taking into account (3.10), it yields On the other hand, using (3.11), we obtain
Using the same reasoning as in the proof of the second statement, we prove the third statement.
From the condition , we deduce that there exists such that for , we get , for  all  . Choosing , then . Now from the condition , we deduce that there exists such that for , we have choosing , then , for  all  ; consequently, where , setting , then for all , we get , where and . Using (3.16), we obtain then from (3.13), we get . Using (3.14), we arrive at then . Now applying the third statement, we achieve the proof of Theorem 3.2.

Example 3.3. Consider the three-point BVP, We have So, Applying the first statement of Theorem 3.2 for , to get Then the BVP (3.26) has at least one nontrivial solution in .

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