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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 123613, 8 pages
http://dx.doi.org/10.1155/2014/123613
Research Article

Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

1Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 14 September 2013; Accepted 10 December 2013; Published 16 January 2014

Academic Editor: Calogero Vetro

Copyright © 2014 Moosa Gabeleh and Naseer Shahzad. 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 A and B be two nonempty subsets of a Banach space X. A mapping T : is said to be cyclic relatively nonexpansive if T(A) and T(B) and for all ( ) . In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.