Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the set ∩^N_i=1 Fix (S _i) of fixed points of a finite family of nonexpansive mappings {S_i}^N_{i = 1} in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element of ∩^N_i=1 Fix (Sⁱ) ∩ GMEP, which is the unique solution of a variational inequality.