Fixed Points of Nonexpansive Mappings on Weak and Weak-Star Compact Convex Sets in Banach Spaces

Abstract

Let X be a Banach space, K a nonempty bounded closed convex subset of X and T a nonexpansive selfmapping of K. The purpose of this thesis is to investigate the following question: What further assumptions (of a geometrical nature) on K (or X) can we make to ensure that there exists a point x in K for which Tx = x? Such points in K are called fixed points of T in K. X is then said to have the fixed point property (FPP) if for every nonempty bounded-closed convex subset K of X, each nonexpansive selfmapping T of K has a fixed point. This, together with the following two properties, will be the subject of our subsequent investigation. The weak fixed point property (w-FPP): For every nonempty weak compact convex subset K of X and each nonexpansive selfmapping T: K → K, there exists x ∈ K with Tx=x; and in the case of a dual space X*. The weak-star fixed point property (w*-FPP): For every nonempty weak-star compact convex subset K of X* and each nonexpansive self-mapping T: K→ K, there exists x ∈ K with Tx=x.