Nonlinear ergodic theorems for a semitopological semigroup of non-Lipschitzian mappings without convexity

Abstract

Let $G$ be a semitopological semigroup, $C$ a nonempty subset of a real Hilbert space $H$, and $\Im =\left\{T_t:t\in G\right\}$ a representation of $G$ as asymptotically nonexpansive type mappings of $C$ into itself. Let $L\left(x\right)=\left\{z\in H:{inf}_{s\in G}{sup}_{t\in G}\Vert T_{ts}x-z\Vert ={inf}_{t\in G}\Vert T_{t}x-z\Vert \right\}$ for each $x\in C$ and $L\left(\Im \right)={\cap }_{x\in C}L\left(x\right)$. In this paper, we prove that ${\cap }_{s\in G}\overline{\text{conv}}\left\{T_{ts}x:t\in G\right\}\cap L\left(\Im \right)$ is nonempty for each $x\in C$ if and only if there exists a unique nonexpansive retraction $P$ of $C$ into $L\left(\Im \right)$ such that $P{T}_s=P$ for all $s\in G$ and $P\left(x\right)\in \overline{\text{conv}}\left\{T_{s}x:s\in G\right\}$ for every $x\in C$. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.