THE FIXED POINT PROPERTY OF STRICTLY CONVEX REFLEXIVE BANACH SPACES FOR NON-EXPANSIVE SELF-MAPPINGS

Abstract

We prove that if X is a strictly convex reflexive Banach space, C is a bounded, closed, convex subset of X with finite extreme points, then all the points in C except extreme points can be non-diametral points, hence C has normal structure, every non-expansive selfmapping T on C has a fixed point. Also, if C has countable extreme points, then C is compact, every non-expansive self- mapping T on C has a fixed point. Furthermore, if T is also surjective, we show which points are fixed points of T.