Fixed points of reversible semigroups of nonexpansive mappings

Takahashi [7, p. 384] proved that if K is a compact convex subset of a Banach space, and S is a left amenable semigroup of nonexpansive self-maps of K, then K contains a common fixed point of S. This theorem generalizes a result of DeMarr [2, p. 1139], who obtained the above implication for the case where S is commutative. In this note, we observe that Takahashi's theorem can be further extended, and the proof slightly simplified, by considering a purely algebraic property that every left amenable semigroup must possess, that of left reversibility. The proof employs suitable modifications of the methods of [2] and [7].