Fixed point iteration processes for asymptotically nonexpansive mappings

Abstract

Let X be a uniformly convex Banach space which satisfies Opial’s condition or has a Fréchet differentiable norm, C a bounded closed convex subset of X , and T : C → C T:C \to C an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by x n + 1 = t n T n x n + ( 1 − t n ) x n {x_{n + 1}} = {t_n}{T^n}{x_n} + (1 - {t_n}){x_n} and x n + 1 = t n T n ( s n T n x n + ( 1 − s n ) x n ) + ( 1 − t n ) x n {x_{n + 1}} = {t_n}{T^n}({s_n}{T^n}{x_n} + (1 - {s_n}){x_n}) + (1 - {t_n}){x_n} , respectively, converge weakly to a fixed point of T .