Let $G$ be a semitopological semigroup, $C$ a nonempty subset of a
real Hilbert space $H$ , and $\Im =\{T_{t}:t\in G\}$ a representation of $G$ as asymptotically nonexpansive type mappings of $C$ into itself. Let $L(x)=\{z\in H:\inf_{s\in G}\sup_{t\in G}\|T_{ts}x-z\|=\inf_{t\in G}\|T_{t}x-z\|\}$ for each $x\in C$ and $L(\Im)=\bigcap_{x\in C}L(x)$ . In this paper, we prove that $\bigcap_{s\in G}\overline\mathrm{conv}\{T_{ts}x:t\in G\}\bigcap L(\Im)$ is nonempty for each $x\in C$ if and only if there exists a unique nonexpansive retraction $P$ of $C$ into $L(\Im)$ such that $PT_{s}=P$ for all $s\in G$ and $P(x)\in\overline\mathrm{conv}\{T_sx:s\in G\}$ for every $x\in C$ . Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.