ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUT CONVEXITY

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ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUT CONVEXITY

Let $G$ be a semitopological semigroup. Let $C$ be a nonempty subset of a Hilbert space and $ \Im =\{T_{t}:t\in G \} $ be a representation of $G$ as asymptotically nonexpansive type mappings of $C$ into itself such that the common fixed point set $ F(\Im)$ of $\Im$ in $C$ is nonempty. It is proved that $ \bigcap \limits_{s\in G} \overline{\text{co}} \{T_{ts}x:t\in G\}\bigcap F(\Im)$ is nonempty for each $ x \in C $ if and only if there exists a nonexpansive retraction $P$ of $C$ onto $F(\Im)$ such that $ PT_{s}=T_{s}P =P $ for all $ s\in G $ and $P(x)$ is in the closed convex hull of $ \{T_{s}x: s\in G \} $, $ x\in C $. This result shows that many key conditions in 1--4, 9, 12--15 ] are not necessary.