Strong Convergence of an Iterative Method for Solving Generalized Mixed Equilibrium Problems and Split Feasibility Problems

In this paper, we investigate iterative methods for solving generalized mixed equilibrium problems, split feasibility problems, and fixed point problems in Banach spaces. We introduce a new extragradient algorithm using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions to the split feasibility problem, the set of fixed points of a resolvent operator, and the set of solutions of the generalized mixed equilibrium problem. The algorithm is analyzed in a real 2-uniformly convex and uniformly smooth Banach space, taking into account computational errors. A numerical example is provided to illustrate the applicability and performance of the proposed method.