An abstract fixed point theorem for nonexpansive mappings

Abstract

A class S \mathcal {S} of subsets of a bounded metric space is said to be normal if each member of S \mathcal {S} contains a nondiametral point. An induction proof is given for the following. Suppose M M is a nonempty bounded metric space which contains a class S \mathcal {S} of subsets which is countably compact, normal, stable under arbitrary intersections, and which contains the closed balls in M M . Then every nonexpansive self-mapping of M M has a fixed point.