# Second-Order Boundary Value Problem with Integral Boundary Conditions

- Mouffak Benchohra
^{1}Email author, - JuanJ Nieto
^{2}and - Abdelghani Ouahab
^{1}

**2011**:260309

**DOI: **10.1155/2011/260309

© Mouffak Benchohra et al. 2011

**Received: **28 May 2010

**Accepted: **1 October 2010

**Published: **13 October 2010

## Abstract

The nonlinear alternative of the Leray Schauder type and the Banach contraction principle are used to investigate the existence of solutions for second-order differential equations with integral boundary conditions. The compactness of solutions set is also investigated.

## 1. Introduction

where is a given function and is an integrable function.

Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [1–9] and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example [10–14]. The goal of this paper is to give existence and uniqueness results for the problem (1.1). Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative [15].

## 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let be the space of differentiable functions whose first derivative, , is absolutely continuous.

Definition 2.1.

A map is said to be -Carathéodory if

## 3. Existence and Uniqueness Results

Definition 3.1.

A function is said to be a solution of (1.1) if satisfies (1.1).

In what follows one assumes that One needs the following auxiliary result.

Lemma 3.2.

Proof.

Our first result reads

Theorem 3.3.

Assume that is an -Carathéodory function and the following hypothesis

then the BVP (1.1) has a unique solution.

Proof.

showing that, is a contraction and hence it has a unique fixed point which is a solution to (1.1). The proof is completed.

We now present an existence result for problem (1.1).

Theorem 3.4.

Suppose that hypotheses

(H1) The function is an -Carathéodory,

is compact.

Proof.

Transform the BVP (1.1) into a fixed-point problem. Consider the operator as defined in Theorem 3.3. We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.

Step 2 ( maps bounded sets into bounded sets in ).

Indeed, it is enough to show that there exists a positive constant such that for each one has .

Step 3 ( maps bounded set into equicontinuous sets of ).

As the right-hand side of the above inequality tends to zero. Then is equicontinuous. As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem we can conclude that is completely continuous.

Step 4 (A priori bounds on solutions).

and consider the operator From the choice of , there is no such that for some As a consequence of the nonlinear alternative of Leray-Schauder type [15], we deduce that has a fixed point in which is a solution of the problem (1.1).

## 4. Examples

## Declarations

### Acknowledgment

The authors are grateful to the referees for their remarks.

## Authors’ Affiliations

## References

- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions.
*Boundary Value Problems*2009, 2009:-11.Google Scholar - Belarbi A, Benchohra M: Existence results for nonlinear boundary-value problems with integral boundary conditions.
*Electronic Journal of Differential Equations*2005, 2005(06):10 .MathSciNetGoogle Scholar - Belarbi A, Benchohra M, Ouahab A: Multiple positive solutions for nonlinear boundary value problems with integral boundary conditions.
*Archivum Mathematicum*2008, 44(1):1-7.MathSciNetMATHGoogle Scholar - Benchohra M, Hamani S, Nieto JJ: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions.
*The Rocky Mountain Journal of Mathematics*2010, 40(1):13-26. 10.1216/RMJ-2010-40-1-13View ArticleMathSciNetMATHGoogle Scholar - Infante G: Nonlocal boundary value problems with two nonlinear boundary conditions.
*Communications in Applied Analysis*2008, 12(3):279-288.MathSciNetMATHGoogle Scholar - Lomtatidze A, Malaguti L: On a nonlocal boundary value problem for second order nonlinear singular differential equations.
*Georgian Mathematical Journal*2000, 7(1):133-154.MathSciNetMATHGoogle Scholar - Webb JRL: Positive solutions of some higher order nonlocal boundary value problems.
*Electronic Journal of Qualitative Theory of Differential Equations*2009, (29):-15.Google Scholar - Webb JRL: A unified approach to nonlocal boundary value problems. In
*Dynamic Systems and Applications. Vol. 5*. Dynamic, Atlanta, Ga, USA; 2008:510-515.Google Scholar - Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach.
*Journal of the London Mathematical Society*2006, 74(3):673-693. 10.1112/S0024610706023179View ArticleMathSciNetMATHGoogle Scholar - Brykalov SA: A second order nonlinear problem with two-point and integral boundary conditions.
*Georgian Mathematical Journal*1994, 1: 243-249. 10.1007/BF02254673View ArticleMATHGoogle Scholar - Denche M, Marhoune AL: High order mixed-type differential equations with weighted integral boundary conditions.
*Electronic Journal of Differential Equations*2000, 2000(60):-10.MathSciNetGoogle Scholar - Kiguradze I: Boundary value problems for systems of ordinary differential equations.
*Journal of Soviet Mathematics*1988, 43(2):2259-2339. 10.1007/BF01100360View ArticleGoogle Scholar - Krall AM: The adjoint of a differential operator with integral boundary conditions.
*Proceedings of the American Mathematical Society*1965, 16: 738-742. 10.1090/S0002-9939-1965-0181794-9View ArticleMathSciNetMATHGoogle Scholar - Ma R: A survey on nonlocal boundary value problems.
*Applied Mathematics E-Notes*2007, 7: 257-279.MathSciNetMATHGoogle Scholar - Granas A, Dugundji J:
*Fixed Point Theory, Springer Monographs in Mathematics*. Springer, New York, NY, USA; 2003:xvi+690.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.