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Implicit hybrid algorithm for problem and a countable family of relatively nonexpansive mappings in banach spaces

In this paper, we introduce a new implicit shrinking algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of relatively nonexpansive mappings in the framework of Banach spaces. Our results are refinement as well as generalization of several well-known results in the current literature. As a consequence, we give some applications for solving variational inequality problems and convex minimization problems in Banach spaces. | Implicit hybrid algorithm for problem and a countable family of relatively nonexpansive mappings in Banach spaces Nguyen Duc Lang University of Science, Thainguyen University, Vietnam Abstract : In this paper, we introduce a new implicit shrinking algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of relatively nonexpansive mappings in the framework of Banach spaces. Our results are refinement as well as generalization of several well-known results in the current literature. As a consequence, we give some applications for solving variational inequality problems and convex minimization problems in Banach spaces. Keywords: Relatively nonexpansive mapping, Implicit hybrid algorithm, Asymptotic fixed point, Equilibrium problems, Shrinking projection method. 2010 Mathematics Subject Classification : 47H05; 47J25. 1 Introduction and Preliminaries Over the past few decades, iterative algorithms play a key role in solving nonlinear equation in various fields of investigation. Therefore, algorithmic construction for the approximation of fixed points of various mappings is a problem of interest in various setting of spaces. Numerous implicit and explicit algorithms have been developed for the approximation of fixed point results. Most of the problems in applied sciences such as monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems as well as certain fixed point problems reduce in terms of finding solution of an equilibrium problem which is defined as follows: Let C be a nonempty closed and convex subset of a real Banach space E and let f : C × C → R (the set of reals) be a bifunction. The equilibrium problem for f is to find its equilibrium points, i.e. the set EP (f ) = {x ∈ C : f (x, y) ≥ 0, for all y ∈ C} . For solving the equilibrium problem, let us .