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Abstract and Applied Analysis
Volume 2014, Article ID 301675, 6 pages
http://dx.doi.org/10.1155/2014/301675
Research Article

Implicit Vector Integral Equations Associated with Discontinuous Operators

1Department of Mathematics and Computer Science, University of Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina, Italy
2Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 19 February 2014; Accepted 25 March 2014; Published 14 April 2014

Academic Editor: Chong Li

Copyright © 2014 Paolo Cubiotti and Jen-Chih Yao. 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

Let . We consider the vector integral equation for a.e. where and are given functions and are suitable subsets of . We prove an existence result for solutions , where the continuity of with respect to the second variable is not assumed. More precisely, is assumed to be a.e. equal (with respect to second variable) to a function which is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a function can be discontinuous at each point . Our result, based on a very recent selection theorem, extends a previous result, valid for scalar case .