Existence And Uniqueness of Fixed Points on Non-Expansive Mapping In Quasi-Normed Spaces

Abstract

This study examines the existence and uniqueness of fixed points of non-expansive mappings in quasi-normed spaces, to establish the existence of a solution to a non-expansive function in a quasi-normed space. The research method employed is a literature review, which provides some theorems with proofs and formal examples. The research began by outlining fundamental notions, such as convergence, Cauchy sequences, boundedness, and completeness, in the context of quasi-norms. Furthermore, the properties of compactness and their implications were elaborated as part of a theoretical framework. In the section on mappings, the characteristics of operators in quasi-normed spaces were first explained, including continuous and bounded mappings, along with their equivalence. Non-expansive and contraction mappings were then formally defined, serving as the basis for demonstrating the existence and uniqueness of fixed points. By applying a sequence approach and the completeness property, it was proven that every non-expansive mapping on a quasi-Banach space possesses a unique fixed point. Finally, it was shown that a quasi-normed space that is both compact and convex guarantees the existence of fixed points for non-expansive mappings defined on such spaces.