STRONG CONVERGENCE THEOREMS BY A RELAXED EXTRAGRADIENT-LIKE

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 a general system of generalized equilibria with inverse-strongly monotone mappings in a real Hilbert space. First, this system of generalized equilibria is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient-like method converges strongly to a solution of this system of generalized equilibria. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.