Convergence theorems of the sequence of iterates for a finite family asymptotically nonexpansive mappings

Abstract

Let E be a uniformly convex Banach space, C a nonempty closed convex subset of E. In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by {Tj:C→C}j=1r as follows: Un(j)=an(j)I+bn(j)TjnUn(j−1)+cn(j)un(j), j=1,2,…,r, x1∈C, xn+1=an(r)xn+bn(r)TrnUn(r−1)xn+cn(r)un(r), n≥1, where Un(0):=I, I the identity map; and {un(j)} are bounded sequences in C; and {an(j)}, {bn(j)}, and {cn(j)} are suitable sequences in [0,1]. We first consider the behaviour of iteration scheme above for a finite family of asymptotically nonexpansive mappings. Then we generalize theorems of Schu and Rhoades.