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.

Abstract

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.