Study of Proximity Points and Fixed Points - Sumitra Jena (Recent Trends in Mathematics, 2025)
This paper explores the concepts of proximity points and fixed points, which are fundamental in mathematical analysis and nonlinear functional analysis. Fixed-point theorems play a crucial role in optimization, game theory, differential equations, and dynamic systems. Proximity points, an extension of fixed points, provide a more generalized approach, allowing near-coincidence rather than exact identity. The study discusses classical fixed-point theorems, such as Banach’s contraction principle, Brouwer’s fixed-point theorem, and Schauder’s theorem, along with their generalizations. Applications in computational mathematics, metric and non-metric spaces, and fuzzy logic are explored. The significance of these theorems in optimization, stability analysis, and numerical approximations is examined. Differences between fixed and proximity points are highlighted, with emphasis on their impact in applied mathematics. Additionally, proofs and derivations of key results are provided to ensure mathematical rigor. The paper concludes with a discussion on recent developments and future research directions in proximity and fixed-point theory.Beyond these applications, the study emphasizes how fixed-point and proximity point theories serve as bridges between pure and applied mathematics, linking abstract analysis to practical problem-solving. Their role in iterative algorithms is especially important, as many computational methods rely on fixed-point iterations for convergence guarantees. In economics and game theory, fixed points often represent equilibria, while proximity points allow models to account for approximate strategies or bounded rationality. Furthermore, extensions of these concepts into fuzzy metric spaces and probabilistic settings have opened new avenues for handling uncertainty and imprecision in real-world systems. Current research increasingly focuses on hybrid models, combining fixed-point results with modern computational tools such as machine learning and variational analysis, highlighting the growing relevance of these mathematical frameworks in contemporary science and engineering.
Keywords: Banach theorem; Fixed Point; contraction mapping; metric space; proximity point