A Hybrid Method for Generalized Equilibrium, Variational Inequality and Fixed Point Problems of Finite Family of Nonexpansive Mappings

Sommay Peathanom, Withun Phuengrattana

Abstract

In this paper, we introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality problem for an inverse-strongly monotone mapping in real Hilbert spaces. Furthermore, we prove that the proposed iterative method converges strongly to a common element of the above three sets. Our result improve and extend the corresponding results of Kangtunyakarn and Suantai [A. Kangtunyakarn, S. Suantai, A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Analysis: Theory, Methods & Applications 71 (2009) 4448–4460], Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515] and many others.