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Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 154742, 10

Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 154742, 10 pages doi:10.1155/2011/154742 Research Article On a Nonlinear Integral Equation with Contractive Perturbation Huan Zhu Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China Correspondence should be addressed to Huan Zhu, mathzhuhuan@gmail.com Received 19 December 2010; Accepted 19 February 2011 Academic Editor: Jin Liang Copyright q 2011 Huan Zhu. 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. We get an existence result for solutions. | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 154742 10 pages doi 10.1155 2011 154742 Research Article On a Nonlinear Integral Equation with Contractive Perturbation Huan Zhu Department of Mathematics University of Science and Technology of China Hefei Anhui 230026 China Correspondence should be addressed to Huan Zhu mathzhuhuan@gmail.com Received 19 December 2010 Accepted 19 February 2011 Academic Editor Jin Liang Copyright 2011 Huan Zhu. 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. We get an existence result for solutions to a nonlinear integral equation with contractive perturbation by means of Krasnoselskii s fixed point theorem and especially the theory of measure of weak noncompactness. 1. Introduction The integral equations have many applications in mechanics physics engineering biology economics and so on. It is worthwhile mentioning that some problems considered in the theory of abstract differential equations also lead us to integral equations in Banach space and some foundational work has been done in 1-8 . In this paper we want to study the following integral equation x f g t x t x l t fi t J k t s f2 s x s ds t e R 1.1 in the Banach space X. This equation has been studied when X R in 9 with g 0 and 10 with a perturbation term g. Our paper extends their work to more general spaces and some modifications are also given on an error of 10 . Our paper is organized as follows. In Section 2 some notations and auxiliary results will be given. We will introduce the main tools measure of weak noncompactness and Krasnoselskii s fixed point theorem in Section 3 and Section 4. The main theorem in our paper will be established in Section 5. 2 Advances in Difference Equations 2. Preliminaries First of all we give out some notations to appear in the following. R .