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Abstract and Applied Analysis
Volume 2003, Issue 2, Pages 83-91
http://dx.doi.org/10.1155/S1085337503205054

Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets

Instytut Matematyki, Uniwersytet M. Curie–Skłodowskiej (UMCS), Lublin 20-031, Poland

Received 30 November 2001

Copyright © 2003 Hindawi Publishing Corporation. 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

It is shown that if X is a Banach space and C is a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets {Ci:1in} of X, and each Ci has the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping of C has a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.