Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 580956, 14 pages
doi:10.1155/2010/580956
Review Article

Normal Structure and Common Fixed Point Properties for Semigroups of Nonexpansive Mappings in Banach Spaces

Anthony To-Ming Lau

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

Received 9 October 2009; Accepted 10 December 2009

Academic Editor: Mohamed A. Khamsi

Copyright © 2010 Anthony To-Ming Lau. 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

In 1965, Kirk proved that if C is a nonempty weakly compact convex subset of a Banach space with normal structure, then every nonexpansive mapping T:C→C has a fixed point. The purpose of this paper is to outline various generalizations of Kirk's fixed point theorem to semigroup of nonexpansive mappings and for Banach spaces associated to a locally compact group.