A common fixed-point theorem for compact convex semigroups of nonexpansive mappings
Ronald E. Bruck · Proceedings of the American Mathematical Society · 1975
Let C C be a bounded closed convex subset of a strictly convex Banach space and let S S be a semigroup of nonexpansive self-mappings of C C which is convex and compact in the topology of weak point-wise convergence. If S S has the property that co ¯ R ( s 1 ) ∩ co ¯ R ( s 2 ) ≠ ∅ \overline {\operatorname {co} \,} \mathcal {R}({s_1}) \cap \overline {\operatorname {co} \,} \mathcal {R}({s_2}\;) e \emptyset whenever s 1 , s 2 ϵ S {s_1},\;{s_2}\epsilon S , then S S has a common fixed point and F ( S ) F(S) is a nonexpansive retract of C C .