A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space

Let $C$ be a nonempty closed convex subset of a Hilbert space, $S$ a right reversible semitopological semigroup, $\mathcal{S} = \{ {T_t}:t \in S\}$ a continuous representation of $S$ as nonexpansive mappings on a closed convex subset $C$ into $C$, and $F(\mathcal{S})$ the set of common fixed points of mappings ${T_t},\;t \in S$. Then we deal with the existence of a nonexpansive retraction $P$ of $C$ onto $F(\mathcal{S})$ such that $P{T_t} = {T_t}P = P$ for each $t \in S$ and ${P_x}$ is contained in the closure of the convex hull of $\{ {T_t}x:t \in S\}$ for each $x \in C$.