STRONG CONVERGENCE THEOREM FOR A GENERALIZED EQUILIBRIUM PROBLEM AND A PSEUDOCONTRACTIVE MAPPING IN A HILBERT SPACE

Abstract

Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis 69 (2008) 1025-1033] suggested and analyzed an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In this paper, we introduce an implicit viscosity approximation method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a pseudocontractive mapping in a Hilbert space. Then, we prove a strong convergence theorem which is the improvements and development of Takahashi and Takahashi’s (2008) corresponding result. Using this theorem, we prove three new strong convergence theorems in fixed point problems, variational inequalities and equilibrium problems.