We study the existence and the algorithmic aspect of a System of Generalized Mixed Equilibrium Problems involving variational-like inequalities (SGMEPs) in the setting of Banach spaces. The approach adopted is based on the auxiliary principle technique and arguments from generalized convexity. A new existence theorem for the auxiliary problem is established; this leads us to generate an algorithm which converges strongly to a solution of (SGMEP) under weaker assumptions. When the study is reduced to the setting of reflexive Banach spaces, then it can be more relaxed by dropping the coercivity condition. The results obtained in this paper are new and improve some recent studies in this field. 1. Introduction and the Problem Statement Let be a nonempty closed convex subset of a Banach space , and let be a real-valued bifunction. By equilibrium problem, in short (EP), we mean the following problem: Equilibrium problems are suitable and common format for investigation of various applied problems arising in economics, mathematical physics, transportation, communication systems, engineering, and other fields. Moreover, equilibrium problems are closely related with other general problems in nonlinear analysis, such as fixed points, game theory, variational inequality, and optimization problems, see [1–4]. Various kinds of iterative algorithms to solve equilibrium problems and variational inequalities have been developed by many authors. There exists a vast literature on the approximation solvability of equilibrium problems and nonlinear variational inequalities using projection type methods, proximal-type methods, or resolvent operator type methods, see [5–11]. We observe that the projection method and its variant forms cannot be applied for constructing iterative algorithms of mixed variational-like inequalities or mixed equilibrium problems involving variational-like inequalities. This fact motivated many authors to develop the auxiliary principle technique to study the existence and algorithm of solutions for variational-like inequalities and its extensions to mixed equilibrium problems, see [12–15]. Kazmi and Khan [16] studied a system of generalized variational-like inequality problems in Hilbert spaces by using the auxiliary principle technique. Recently, Ding and Wang [17] and Ding [18] introduced new iterative algorithms for solving some class of system of generalized mixed variational-like inequalities and system of mixed equilibrium problems involving variational-like inequalities. In this paper, we study the existence and the algorithmic aspect of a
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