Fixed point iteration processes for asymptotically nonexpansive mappings

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 $T:C \to C$ an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by ${x_{n + 1}} = {t_n}{T^n}{x_n} + (1 - {t_n}){x_n}$ and ${x_{n + 1}} = {t_n}{T^n}({s_n}{T^n}{x_n} + (1 - {s_n}){x_n}) + (1 - {t_n}){x_n}$, respectively, converge weakly to a fixed point of T.