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International Journal of Differential Equations
Volume 2012 (2012), Article ID 471975, 14 pages
http://dx.doi.org/10.1155/2012/471975
Research Article

Existence and Positivity of Solutions for a Second-Order Boundary Value Problem with Integral Condition

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 14 January 2012; Accepted 23 February 2012

Academic Editor: Yuji Liu

Copyright © 2012 Assia 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

This work is devoted to the study of uniqueness and existence of positive solutions for a second-order boundary value problem with integral condition. The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in cone. Two examples are also given to illustrate the main results.

1. Introduction

Boundary value problem with integral boundary conditions is a mathematical model for of various phenomena of physics, ecology, biology, chemistry, and so forth. Integral conditions come up when values of the function on the boundary are connected to values inside the domain or when direct measurements on the boundary are not possible. The presence of an integral term in the boundary condition leads to great difficulties. Our aim, in this work, is the study of existence, uniqueness, and positivity of solution for the following second-order boundary value problem: with boundary conditions of type where is a given function. Using the nonlinear alternative of Leray Schauder, we establish the existence of nontrivial solution of the BVP (1.1)-(1.2), under the condition where , to prove the uniqueness of solution, we apply Banach contraction principle, by using Guo-Krasnosel'skii fixed point theorem in cone we study the existence of positive solution. As applications, some examples to illustrate our results are given.

Various types of boundary value problems with integral boundary conditions were studied by many authors using different methods see [19]. In [2] Benchohra et al. have studied (1.1) with the integral condition , the authors assumed that the function depends only on and and the condition (1.3) holds for , so our work is new and more general than [2]. Similar boundary value problems for third-order differential equations with one of the following conditions , , , or , , , were investigated by Zhao et al. in [6], they established the existence and nonexistence and the multiplicity of positive solutions in ordered Banach spaces basing on fixed point theory in cone. For more knowledge about the nonlocal boundary value problem, we refer to the references [1017].

This paper is organized as follows. In Section 2, we give some notations, recall some concepts and preparation results. In the third Section, we give two main results, the first result based on Banach contraction principle and the second based nonlinear alternative of Leray-Schauder type. In Section 4, we treat the positivity of solutions with the help of Guo-Krasnosel'skii fixed point theorem in cone. Some examples are given to demonstrate the application of our main results, ending this paper.

2. Preliminaries Lemmas and Materials

In this section, we introduce notations, definitions, and preliminary facts that will be used in the sequel.

Definition 2.1. A mapping defined on a Banach space is completely continuous if it is continuous and maps bounded sets into relatively compact sets.

Theorem 2.2 (Arzela-Ascoli Theorem 1). Let be a compact set. A subset is relatively compact if and only if it is pointwise bounded and equicontinuous, where denotes the space of all continuous functions on .

Theorem 2.3 (Arzela-Ascoli Theorem 2). If a sequence in is bounded and equicontinuous then it has a uniformly convergent subsequence.

Now we state the nonlinear alternative of Leray-Schauder.

Lemma 2.4 (see [18]). 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 of .

Definition 2.5. A function is called -Carathéodory if the map is measurable for all ; the map is continuous on for almost each ; for each , there exists an such that for almost each and .

We recall the definition of positive solution.

Definition 2.6. A function is called positive solution of (1.1) if .

We expose the well-known Guo-Krasnosel'skii fixed point Theorem on cone [19].

Theorem 2.7. Let be a Banach space, and let , be a cone. Assume and are open subsets of with and let be a completely continuous operator such that, and ; or, and .
Then has a fixed point in .

Throughout this paper, let , with the norm , where denotes the norm in defined by . One can obtain the following result.

Lemma 2.8. Let , then the solution of the following boundary value problem: is given by where

Proof. Integrating two times the equation from 0 to , one can obtain The condition gives . The second condition implies Substituting and by their values in (2.5) we obtain Multiplying (2.7) by then integrating the resultant equality over we get Substituting the second term in the right-hand side of (2.7) by (2.8) it yields it is easy to get that is where is given by (2.4).

We have the following result which is useful in what follows.

Remark 2.9. The function is continuous, nonnegative and satisfies for any , .

3. Existence and Uniqueness Theorems

This section deals with the existence and uniqueness of solutions for the problem (1.1)-(1.2).

Theorem 3.1. Suppose that the following hypotheses hold. is an -Carathéodory function. There exist two nonnegative functions ,   such that , , , , one has .
Then the problem (1.1)-(1.2) has a unique solution in .

Proof. Transform the problem (1.1)-(1.2) into a fixed point problem. Consider the operator defined by From Lemma 2.8, the problem (1.1)-(1.2) has a solution if and only if the operator has a fixed point in . Let , then for each we have Hypothesis (H2) and Remark 2.9 imply applying hypothesis (H3) to the right-hand side of the above inequality, we obtain On the other hand we have for any :
Then for one can write
Applying hypothesis H3 again gives Combining inequalities (3.5) and (3.8) we obtain thus, is a contraction mapping on . By applying the well-known Banach's contraction mapping principle we know that the operator has a unique fixed point on . Therefore, the problem (1.1)-(1.2) has an unique solution.

Theorem 3.2. Suppose that the following hypotheses hold: is an -Carathéodory function, the map is continuous and , for any ;  There exist three nonnegative functions and , such that one has
Then the BVP (1.1)-(1.2) has at least one nontrivial solution.

Proof. First we show that is a completely continuous mapping that we will prove in some steps:
(1) is continuous. In fact, let be a convergent sequence in such that , then and for each we have On the other hand we have From the above discussion one can write
Due to (P1) is Cathéodory, then as .
(2) maps bounded sets into bounded sets in , to establish this step we use Arzela-Ascoli Theorem. Let , then from (P2), we have for any and consequently that implies maps bounded sets into bounded sets.
(3) maps bounded sets into equicontinuous sets of . Let and , and , then using (P2) it yields In addition, we have
Since is continuous, then tend to 0 when , and we have immediately that , this yields that is equicontinuous. Then is completely continuous.
Secondly, we apply the nonlinear alternative of Leray-Schauder to prove the existence of solution. Let us make the following notations: Since , then there exists an interval such that hence . From hypothesis (P3), we know that . Putting . Setting and let , such that . Using the same argument that to get (3.16), it yields as then . First, if then , hence , this contradicts the fact that . By Lemma 2.4 we conclude that has a fixed point and then problem (1.1)-(1.2) has a nontrivial solution .
Second, if then . By arguing as above we complete the proof.

4. Existence of Positive Solutions

In this section the existence results for positive solutions for problem (1.1)-(1.2) are presented. We make the following hypotheses:

(Q1) where and ;

(Q2) There exists such that .

The following result gives a priori estimates for solutions of problem (1.1)-(1.2).

Lemma 4.1. Assume that hypotheses (Q1)-(Q2) hold then for , one has for , one has

Proof. (i) Let , it is easy to check that inequalities in (4.1) hold.
(ii) Let , such that , then
by similar ideas, it yields
The proof is complete.

Lemma 4.2. Assume that hypotheses (Q1)-(Q2) hold, then the solution of the problem (1.1)-(1.2) is positive and satisfies where .

Proof. It follows from Lemma 4.1 that In view of (4.2) we obtain so, Similarly, one finds
Further, according to inequalities (4.9) and (4.10) we get
ending the proof Lemma 4.2.

Define the quantities and by The case and is called superlinear case and the case and is called sublinear case. The main result of this section is the following.

Theorem 4.3. Assume that hypotheses (Q1)-(Q2) hold, then problem (1.1)-(1.2) has at least one positive solution in the both cases superlinear as well as sublinear.

To prove this theorem we apply the well-known Guo-Krasnosel'skii fixed point Theorem in cone.

Proof. Denote and define the cone by where is given in Lemma 4.2.
It is easy to check that is a nonempty closed and convex subset of .
Using Lemma 4.2 we see that . Applying Arzela-Ascoli Theorem we know that is completely continuous for . On the basis of hypothesis (Q1), one can write
Let us consider the superlinear case. Since , then for any , there exists , such that if then . Let , then for any , we have Moreover, we obtain By virtue of (4.15) and (4.16) we deduce
Taking hypothesis (Q2) into account, one can choose such The inequalities (4.17) and (4.18) imply that , for any .
Second, since , then for any , there exists , such that for .
Let , where and denote by . If then Using similar techniques as in the proof of the second statement of Lemma 4.1 we obtain thus moreover, It follows from (4.21) and (4.22) that
Let us choose such that that implies . Hence, The first statement of Theorem 2.7 implies that has a fixed point in such that . Applying similar techniques as above, we prove the sublinear case. The proof of Theorem 4.3 is complete.

To illustrate the main results, we consider the following examples.

Example 4.4. Consider the following boundary value problem (P1): Choosing , , it is easy to see that and hypotheses (H1)–(H3) of Theorem 3.1 are satisfied, then, problem (P1) has a unique solution in .

Example 4.5. The following boundary value problem (P2): has at least one solution in . In fact, we have , where , , by calculation we obtain and . From Theorem 3.2, we deduce the existence of at least one solution in .

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