Weak convergence and non-linear ergodic theorems for reversible semigroups of non-Lipschitzian mappings

Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banach space E with a Frechet differentiable norm and I = {T-t:t is an element of G} be a continuous representation of G as asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F(I) of (I) in C is nonempty. We prove in this paper that if G is right reversible, then for every almost-orbit u(.) of I, boolean AND(s is an element of G) <(co)over bar{u(t):t greater than or equal to s}> boolean AND F(I) consists of at most one point. Further, boolean AND(s is an element of G) <(co)over bar{T(t)x:t greater than or equal to s}> boolean AND F(I) is nonempty for each x is an element of C if and only if there exists a nonexpansive retraction P of C onto F(I) such that PTs = TsP = P for all s is an element of G and P(x) is in the closed convex hull of {T(s)x: s is an element of G}, x is an element of C. This result is applied to study the problem of weak convergence of the net {u(t): t is an element of G} to a common fixed point of I. (C) 1997 Academic Press.