Li, Gang and Sims, Brailey *ERGODIC THEOREM AND STRONG CONVERGENCE OF AVERAGED APPROXIMANTS FOR NON-LIPSCHITZIAN MAPPINGS IN BANACH SPACES.* to be submitted . (Unpublished)

## Abstract

Let C be a bounded closed convex subset of a uniformly convex Banach space X and let T be an asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we first provide a ergodic retraction theorem and a mean ergodic convergence theorem. Using this result, we show that the set F(T) of fixed points of T is a sunny, nonexpansive retract of C if the norm of X is uniformly Gateaux differentiable. Moreover, we discuss the strong convergence of the sequence $\{x_n\}$ defined by $x_n = a_nx + (1 - a_n)T(\mu)x_n$ for $n = 0,1,2,\dots$, where $x \in C$, $\mu$ is a Banach limit on $l^\infty$ and $a_n$ is a real sequence in $(0, 1]$.

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