A note on Picard iterates of nonexpansive mappings

Abstract

Let X be a Banach space, C a closed subset of X, and T : C → C a nonexpansive mapping. It has recently been shown that if X is reflexive and locally uniformly convex and if the fixed point set F (T) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F (T). In this paper it is shown that if T is assumed to be asymptotically regular, this condition can be weakened much further. Finally, some observations are made about the geometric conditions imposed.