CONVERGENCE THEOREMS OF IMPLICIT ITERATION PROCESS WITH ERRORS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE IN BANACH SPACES

The aim of this article is to study an implicit iteration processwith errors for a nite family of non-Lipschitzian asymptotically non-expansive mappings in the intermediate sense in Banach spaces. Alsowe establish some strong convergence theorems and a weak convergencetheorem for said scheme to converge to a common xed point for non-Lipschitzian asymptotically nonexpansive mappings in the intermediatesense. The results presented in this paper extend and improve the corre-sponding results of [1], [3]-[8], [10]-[11], [13]-[14], [16] and many others. 1. Introduction and preliminariesLet K be a nonempty subset of a real Banach space E. Let T: K !Kbe a mapping. We use F(T) to denote the set of xed points of T, that is,F(T) = fx2K: Tx= xg. Recall the following concepts.(1) Tis nonexpansive ifkTx Tyk kx yk; (1)for all x;y2K.(2) Tis asymptotically nonexpansive if there exists a sequence fa n gin [1;1)with a n !1 as n!1such thatkT n x T n yk a n kx yk; (2)for all x;y2Kand n 1.(3) Tis uniformly L-Lipschitzian if there exists a constant L>0 such thatkT