Second-Order Boundary Value Problem with Integral Boundary Conditions
© Mouffak Benchohra et al. 2011
Received: 28 May 2010
Accepted: 1 October 2010
Published: 13 October 2010
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.
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 .
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.
A map is said to be -Carathéodory if
(i) is measurable for each
(ii) is continuous for almost each
(iii)for every there exists such that
3. Existence and Uniqueness Results
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.
Our first result reads
Assume that is an -Carathéodory function and the following hypothesis
then the BVP (1.1) has a unique solution.
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).
Suppose that hypotheses
(H1) The function is an -Carathéodory,
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 1 ( is continuous).
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 , we deduce that has a fixed point in which is a solution of the problem (1.1).
Thus is compact.
We present some examples to illustrate the applicability of our results.
Hence, by Theorem 3.3, the BVP (4.1) has a unique solution on .
Hence, by Theorem 3.4, the BVP (4.4) has at least one solution on . Moreover, its solutions set is compact.
The authors are grateful to the referees for their remarks.
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