A nonlinear ergodic theorem for asymptotically nonexpansive mappings

Abstract

Let X be a real uniformly convex Banach space satisfying the Opial's condition, C a bounded closed convex subset of X, and T: C → C an asymptotically non-expansive mapping. Then we show that for each x in C, the sequence {Tⁿx} almost converges weakly to a fixed point y of T, that is, This implies that {Tⁿx} converges weakly to y if and only if T is weakly asymptotically regular at x, that is, weak- . We also present a weak convergence theorem for asymptotically nonexpansive semigroups.