Nonlinear ergodic theorems of nonexpansive type mappings

Let S be a semitopological semigroup. Let C be a closed convex Subset of a uniformly convex Banach space E whose norm is Frechet differentiable and J = {T-t: t is an element of S} be a continuous representation of S as almost asymptotically nonexpansive type mapping of C into C such that the common fixed point set F(J) of J in C is nonempty. In this paper, we prove that if S is right reversible then for each chi is an element of C, the closed convex set boolean AND(s is an element of S) (co) over bar {T-t chi: >= s) boolean AND F(J) consists of at most one point. We also prove that if S is reversible, then the intersection boolean AND(s is an element of S) (co) over bar {T-t chi: >= s) boolean AND F(J) is nonempty for each chi is an element of C if and only if there exists a nonexpansive retraction P of C onto F(J) such that PTt = TtP = P for all t is an element of S and Px is in the closed convex hull of {T(t)x: t is an element of S} for each chi is an element of C. (C) 2009 Elsevier Inc. All rights reserved.