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.
