A Comprehensive Review of Fixed Point Theory: Foundations, Applications, and Emerging Trends in Mathematical Spaces

Abstract

Fixed point theory is a crucial branch of mathematical analysis that investigates the conditions under which a function returns a point to itself, symbolizing stability and equilibrium. This paper delves into various fixed point theorems within the contexts of metric spaces, Banach spaces, and Hilbert spaces, emphasizing their foundational importance and wide-ranging applications. The Banach Fixed Point Theorem guarantees the existence and uniqueness of fixed points for contraction mappings in complete metric spaces, while Brouwer's Fixed Point Theorem states that any continuous function mapping a compact convex set to it will have at least one fixed point. Recent developments have expanded these classical results to encompass new types of contraction mappings and generalized distance functions, enhancing their relevance in dynamic systems, control theory, and optimization challenges. Additionally, the paper discusses the Schauder Fixed Point Theorem in Banach spaces, highlighting its significance in analysing nonlinear operators. In Hilbert spaces, fixed point results are examined in relation to nonlinear integral equations and optimization methods, showcasing their practical implications in engineering and variation techniques. Emerging trends include the study of fixed point results in fuzzy and probabilistic environments, as well as the integration of computational approaches with traditional fixed point methods. This paper illustrates the continuous evolution of fixed point theory, connecting abstract mathematical principles with practical problem-solving across various fields. Finally, it proposes future research directions to further explore the potential of fixed point theory in modern mathematics.