On the Convergence of an Implicit Iteration Scheme for a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

In this paper we define a new implicit iterative process with errors in a real Banach space and establish a necessary and sufficient condition for the strong convergence of the iteration to a common fixed point for the case of a finite family of asymptotically nonexpansive mappings in a arbitrary real Banach space. Also we study the weak and strong convergence results of this implicit iterative scheme for the same family of mappings in the setting of a uniformly convex Banach space. Our results extend and generalize a number of existing results.