A common fixed-point theorem for compact convex semigroups of nonexpansive mappings

Abstract:

Let $C$ be a bounded closed convex subset of a strictly convex Banach space and let $S$ be a semigroup of nonexpansive self-mappings of $C$ which is convex and compact in the topology of weak point-wise convergence. If $S$ has the property that $\overline {\operatorname {co} } \mathcal {R}({s_1}) \cap \overline {\operatorname {co} } \mathcal {R}({s_2}\;) \ne \emptyset$ whenever ${s_1},\;{s_2}\epsilon S$, then $S$ has a common fixed point and $F(S)$ is a nonexpansive retract of $C$.