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Abstract and Applied Analysis
Volume 2014, Article ID 978257, 5 pages
Research Article

On Stability of Fixed Points for Multi-Valued Mappings with an Application

College of Science, Guilin University of Technology, Guilin 541004, China

Received 12 November 2013; Accepted 22 January 2014; Published 27 February 2014

Academic Editor: Salvador Romaguera

Copyright © 2014 Qi-Qing Song. 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.


This paper studies the stability of fixed points for multi-valued mappings in relation to selections. For multi-valued mappings admitting Michael selections, some examples are given to show that the fixed point mapping of these mappings are neither upper semi-continuous nor almost lower semi-continuous. Though the set of fixed points may be not compact for multi-valued mappings admitting Lipschitz selections, by finding sub-mappings of such mappings, the existence of minimal essential sets of fixed points is proved, and we show that there exists at least an essentially stable fixed point for almost all these mappings. As an application, we deduce an essentially stable result for differential inclusion problems.