Convergence of Picard Iterates of Nonexpansive Mappings

Let X be a Banach space, C a closed subset of X , and T : C -+ C a nonexpansive mapping. Conditions are given which assure that 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 ) . If T is asymptotically regular, it suffices to assume that the closed subsets of X are densely proximinal and that nested spheres in X have compact interfaces. Such spaces include, among others, those which have Rolewicz's property ( P ) . If X has strictly convex norm the asymptotic regularity assum tion can be dropped and the nested sphere property holds trivially. Consequently the re ult holds for all reflexive locally uniformly convex spaces. 9