Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 806837, 7 pages
Research Article

Ray's Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces

Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Received 3 July 2010; Accepted 29 September 2010

Academic Editor: A. T. M. Lau

Copyright © 2010 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.


We prove that every firmly nonexpansive-like mapping from a closed convex subset C of a smooth, strictly convex and reflexive Banach pace into itself has a fixed point if and only if C is bounded. We obtain a necessary and sufficient condition for the existence of solutions of an equilibrium problem and of a variational inequality problem defined in a Banach space.