Nonexpansive actions of topological semigroups on strictly convex Banach spaces and fixed points

Abstract:

Let $C$ be a closed convex subset of a strictly convex Banach space $X$ and $\left \{ {{T_s}:s \in S} \right \}$ be a continuous representation of a semitopological semigroup $S$ as nonexpansive mappings of $C$ into itself. The main result establishes the fact that if for some $x \in C$ the trajectory $\left \{ {{T_s}x:s \in S} \right \}$ is relatively compact and $AP(S)$ has a left invariant mean then $K = \overline {\operatorname {conv} \{ {T_s}x:s \in S\} }$ contains a common fixed point for ${\left \{ {{T_s}} \right \}_{s \in S}}$.