Vol. 126, No. 2, 1987

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Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings

Anthony To-Ming Lau and Wataru Takahashi

Vol. 126 (1987), No. 2, 277–294

Let S be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banach space E with a Fréchet differentiable norm and 𝒮 = {Ta;a S} be a continuous representation of S as nonexpansive mappings of C into C such that the common fixed point set F(𝒮) of 𝒮 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 C, the closed convex set W(x) F(𝒮) consists of at most one point, where W(x) = {Ks(x);s S}, Ks(x) is the closed convex hull of {Ttx;t s} and t s means t = s or t Ss. This result is applied to study the problem of weak convergence of the net{Tsx;s S}, with S directed as above, to a common fixed point of 𝒮. We also prove that if E is uniformly convex with a uniformly Fréchet 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) F(𝒮) is nonempty for each x C if and only if there exists a nonexpansive retraction P of C onto F(𝒮) such that PTs = TsP = P for all s S and P(x) is in the closed convex hull of {Ts(x);s S}, x C.

Mathematical Subject Classification 2000
Primary: 47H20
Received: 11 May 1984
Published: 1 February 1987
Anthony To-Ming Lau
Wataru Takahashi