SHRINKING PROJECTION METHOD OF PROXIMAL-TYPE FOR A GENERALIZED EQUILIBRIUM PROBLEM, A MAXIMAL MONOTONE OPERATOR AND A PAIR OF RELATIVELY NONEXPANSIVE MAPPINGS

Abstract

The purpose of this paper is to introduce and consider a shrinking projection method of proximal-type for finding a common element of the set $EP$ of solutions of a generalized equilibrium problem, the set $F(S) \cap F(\widetilde S)$ of common fixed points of a pair of relatively nonexpansive mappings $S,\widetilde S$ and the set $T^{-1}0$ of zeros of a maximal monotone operator $T$ in a uniformly smooth and uniformly convex Banach space. It is proven that under appropriate conditions, the sequence generated by the shrinking projection method of proximal-type, converges strongly to some point in $EP \cap F(S) \cap F(\widetilde S) \cap T^{-1}0$. This new result represents the improvement, generalization and development of the previously known ones in the literature.