Fixed point theorems for nonexpansive mappings in a locally convex space

Abstract

Several fixed point theorems for nonexpansive self mappings in metric spaces and in uniform spaces are known. In this context the concept of orbital diameters in a metric space was introduced by Belluce and Kirk. The concept of normal structure was utilized earlier by Brodskiĭ and Mil'man. In the present paper, both these concepts have been extended to obtain definitions of β-orbital diameter and β-normal structure in a uniform space having β as base for the uniformity. The closed symmetric neighbourhoods of zero in a locally convex space determine a base β of a compatible uniformity. For 3-nonexpansive self mappings of a locally convex space, fixed point theorems have been obtained using the concepts of β-orbital diameter and β-normal structure. These theorems generalise certain theorems of Belluce and Kirk.