An examination of topological spaces' fixed point theory and its uses

Abstract

One of the most important branches of contemporary mathematics is fixed point theory, which has wide-ranging effects on both the pure and practical sciences. The study of fixed points in different structures has yielded profound insights into analysis, topology, optimization, and nonlinear functional equations. A fixed point of a mapping is a point that stays invariant under the action of the mapping. We provide a thorough analysis of fixed point theory in the context of topological spaces in this paper. We examine traditional findings like the fixed point theorems of Brouwer and Schauder as well as more recent extensions in metric, normed, and topologically ambiguous circumstances. The importance of contractive conditions, continuity, and compactness in proving the existence and uniqueness of fixed points is emphasized. We also emphasize the use of fixed point theory in dynamical systems, game theory, mathematical economics, and differential equations. Along with discussing possible study avenues, such as fixed points in Hilbert spaces, nonlinear mappings, and their relationships to contemporary computer techniques, the work not only synthesizes previous findings.