Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 414232, 14 pages
doi:10.1155/2010/414232
Research Article

On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings

Weerayuth Nilsrakoo1 and Satit Saejung2,3

1Department of Mathematics, Statistics and Computer, Ubon Rajathanee University, Ubon Ratchathani 34190, Thailand
2Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
3The Centre of Excellence in Mathematics, Commission on Higher Education (CHE), Sri Ayudthaya Road, Bangkok 10400, Thailand

Received 13 November 2009; Accepted 18 January 2010

Academic Editor: Tomonari Suzuki

Copyright © 2010 Weerayuth Nilsrakoo and Satit Saejung. 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

We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.