Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi- ϕ -Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

The purpose of this paper is first to introduce the concept of total quasi-$\varphi$-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme forfinding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi-$\varphi$-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.