A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an -inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results. 1. Introduction Let be a Banach space with norm , a nonempty closed convex subset of , and let denote the dual of . Let be a bifunction, be a real-valued function, and a mapping. The generalized mixed equilibrium problem, is to find such that The set of solutions to (1.1) is denoted by , that is, If , the problem (1.1) reduces into the mixed equilibrium problem for , denoted by , which is to find such that If , the problem (1.1) reduces into the mixed variational inequality of Browder type, denoted by , which is to find such that If and the problem (1.1) reduces into the equilibrium problem for , denoted by , which is to find such that If , the problem (1.3) reduces into the minimize problem, denoted by , is to find such that The above formulation (1.4) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an . In other words, the is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of ; see, for example, [1, 2] and references therein. Some solution methods have been proposed to solve the ; see, for example, [1, 3–11] and references therein. A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of . Then a Banach space is said to be smooth if the limit exists for each It is also said to be uniformly smooth if the limit exists uniformly for . Let be a Banach space. The modulus of convexity of is the function defined by A Banach space is uniformly convex if and only if