Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings
Anthony To Ming Lau, Wataru Takahashi · Pacific Journal of Mathematics · 1987
Let S be a semitopological semigroup.Let C be a closed convex subset of a uniformly convex Banach space E with a Frechet differentiable norm and y= {T a ; a e S} be a.continuous representation of S as nonexpansive mappings of C into C such that the common fixed point set F(S?) of y in C is nonempty.We prove in this paper that if S is right reversible (i.e. S has finite intersection property for closed right ideals), then for each x e C, the closed convex set W(x) Π F(5?) consists of at most one point, where W(x) = f){K s (x); s e S}, K s (x) is the closed convex hull of {T t x\ t > s) and / > s means / = 5or/e&.This result is applied to study the problem of weak convergence of the net {T s x; s e S}, with S directed as above, to a common fixed point of Sf.We also prove that if E is uniformly convex with a uniformly Frechet differentiable norm, S is reversible and the space of bounded right uniformly continuous functions on S has a right invariant mean, then the intersection W(x) Π F(S?) is nonempty for each x e C if and only if there exists a nonexpansive retraction P of C onto F(Sf) such that PT S = T S P = P for all s e S and P(x) is in the closed convex hull of {T s (x); s<ΞS}, x<=C.