Invariant means and semigroups of nonexpansive mappings on uniformly convex Banach spaces (Q1173887)

Let \(S\) be a semitopological semigroup. Let \({RUC}(S)\) denote the space of all bounded rightly continuous real valued function on \(S\). For this space, the author proves the following theorem. Theorem: Let \(S\) be a semitopological semigroup. Let \({\mathcal S}=\{T_ s;\;s\in S\}\) be a continuous representation of \(S\) as nonexpansive mappings on a closed convex subset \(C\) of a uniformly convex and uniformly smooth Banach space \(E\) into \(C\). If \({RUC}(S)\) has a left invariant mean, and \(C\) contains an element \(x\) with bounded orbit, then there exists \(u\in C\) such that \(T_ s u=u\) for all \(s\in S\).