Some results on Fixed Points of non-expansive mappings

Abstract

This paper investigates the crucial role played by the star-shaped subsets of the underlying domain in guaranteeing the existence of fixed points for non-expansive mappings. It presents a comprehensive survey of several fundamental methods and analytical techniques that have been developed for the study of non-expansive mappings and their fixed point properties. Furthermore, the paper highlights the close and profound relationship between the structure of Banach spaces and the applicability of fixed point theorems, demonstrating how specific characteristics of these spaces significantly influence the existence, uniqueness, and behavior of fixed points under non-expansive mappings.