Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 414232, 14 pages
doi:10.1155/2010/414232
On the fixed-point set of a family of relatively nonexpansive and generalized nonexpansive mappings
Weerayuth Nilsrakoo1
and Satit Saejung2
1Department of Mathematics, Statistics and Computer, Ubon Rajathanee University, Ubon Ratchathani 34190, Thailand
2Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
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.