Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems
Lu-Chuan Ceng, Sy‐Ming Guu, Jen‐Chih Yao · Fixed Point Theory and Applications · 2012
Abstract 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 ⋂ i = 1 N Fix ( S i ) of fixed points of a finite family of nonexpansive mappings { S i } i = 1 N in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element of ⋂ i = 1 N Fix ( S i ) ∩ G M E P , which is the unique solution of a variational inequality. AMS subject classifications: 49J40; 47J20; 47H09.