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 of a smooth, strictly convex and reflexive Banach pace into itself has a fixed point if and only if 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.