%0 Journal Article
%T Integrable Solutions of a Nonlinear Integral Equation via Noncompactness Measure and Krasnoselskii's Fixed Point Theorem
%A Mahmoud Bousselsal
%A Sidi Hamidou Jah
%J International Journal of Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/280709
%X We study the existence of solutions of a nonlinear Volterra integral equation in the space . With the help of Krasnoselskii¡¯s fixed point theorem and the theory of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes on nonlinear integral equations. Our results extend and generalize some previous works. An example is given to support our results. 1. Introduction In this paper, we present an existence result for the functional integral equation , where , , , and are given measurable functions while is an unknown function. Equation (1) is a general form of many integral equations, such as the mixed Volterra-Fredholm integral equation which has been considered by many authors, see for example, [1¨C3] and references cited therein. Moreover, (1) contains the nonlinear Volterra and Fredholm integral equation on such as which is studied in [4, 5]. The existence of solution of Urysohn¡¯s equation, was studied in [6] where he proved that (4) has a solution in space. The problem was studied in [1] where they obtained the existence of solution by using the classical Schauder fixed point principle. The nonlinear integral equation has been considered very recently by Liang et al. [7]. The main tool used in our research is a measure of weak noncompactness given by Bana£¿ and Knab [3] to find a special subset of and also by applying the Krasnoselskii's fixed point theorem on this set. The existence results generalizing several previous works [1, 8] will be proved. Let us mention that the theory of functional integral equations has many useful applications in describing numerous events and problems of the real world. For example, integral equations are often applicable in engineering, mathematical physics, economics, and biology (cf. [3, 4, 9¨C12]). The paper is organized in five sections, including the introduction. Some preliminaries, notations, and auxiliary facts are presented in Section 2; in Section 3, we will introduce the main tools: measure of weak noncompactness and Krasnoselskii¡¯s fixed point theorem. The main theorem in our paper will be established in Section 4. In Section 5, we give an example to illustrate our results. 2. Preliminaries Throughout this paper, we let be the set of all real numbers, , and denotes the space of the Lebesgue integrable functions on a measurable subset of with the standard norm The space and will be briefly denoted by and , respectively. Let be an interval of bounded or not. Definition 1. Consider a function . We say that satisfies Carath¨¦odory conditions if
%U http://www.hindawi.com/journals/ijanal/2014/280709/