STRONG CONVERGENCE THEOREM OF AN M-STEP HALPERN-TYPE ITERATION PROCESS FOR FINITE FAMILIES OF TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPS

Abstract

In this paper we introduce an m-step Halpern-type iteration process and prove strong convergence of the scheme in a real Hilbert space $H$ to the common  fixed point of a finite family of Total asymptotically nonexpansive mappings.   Our results improve previously known ones obtained for the class of nonexpansive and asymptotically nonexpansive mappings. As application iterative methods for approximation of: solution of variational inequality problems, fixed point of finite family of continuous pseudocontractive mappings, solutions of classical equilibrium problems and solutions of convex minimization problems are proposed. Our theorems unify and complement many recently announced results.