Convexity structures in T₀-quasi-metric spaces and fixed point theorems

Abstract

In this MSc dissertation, we generalize the concept of convexity structures in metric spaces to the framework of quasi-pseudometric spaces. In the To-quasi-metric setting we explore various interesting additional conditions that convexity structures in the sense of Takahashi can satisfy. Some interesting examples of convexity structures in both pseudometric and quasi-pseudometric spaces are discussed. Furthermore we look at fixed point theorems for maps which do not increase distance: so-called nonexpansive maps in convex metric spaces. Our results generalize and extend the well-known results in the literature. In this MSc dissertation, we generalize and extend the classical fixed point theorems obtained by F. Browder in Banach spaces. Moreover, we introduce the concept of fixed point theorems for nonexpansive mappings on convex To-quasi-metric spaces. We obtain necessary conditions for the existence of a fixed point of nonexpansive maps defined on a convex To-quasi-metric space.