A common fixed-point theorem in reflexive locally uniformly convex Banach spaces

Abstract

Let X be a reflexive locally uniformly convex Banach space and G an ultimately nonexpansive commutative semigroup of continuous self-maps of X. If there exists a point x in X recurrent under G such that G(x) is bounded, then G has a common fixed point in co(G(x)). If X is a Hubert space then there is exactly one such point in co(G(x)).