International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 4, Pages 251-257
doi:10.1155/S0161171202107113

Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces

Wei-Shih Du,1 Young-Ye Huang,2 and Chi-Lin Yen3

1Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan
2Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
3Department of Mathematics and Science Education, National Hsinchu Teacher's College, Hsinchu 300, Taiwan

Received 25 July 2001; Revised 10 January 2002

Copyright © 2002 Wei-Shih Du et al. 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 every asymptotically regular or λ-firmly nonexpansive mapping T:C→C has a fixed point whenever C is a finite union of nonempty weakly compact convex subsets of a Banach space X which is uniformly convex in every direction. Furthermore, if {T i}i∈I is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of T i, i ∈I, have a nonempty intersection, then T i, i∈I, have a common fixed point in C.