Ergodic Retraction Theorem and Weak Convergence Theorem 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 T = {T t : t . G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F(T) of T in C is nonempty. It is shown that if G is right reversible, then for each almost.orbit u(.) of T, .s.Gco.....{u(t):t.s}.F(T) consists of at most one point. Furthermore, .s.Gco.....{Ttx:t.s}.F(T) is nonempty for each x . C if and only if there exists a nonlinear ergodic retraction P of C onto F(T) such that PT s = T s P = P for all s . G and Px.co.....{Tsx:s.G} for each x . C. This result is applied to study the problem of weak convergence of the net {u(t) : t . G} to a common fixed point of T.