Study of Unique Fixed-Point Theorems in Different Mathematical Spaces

Abstract

Fixed-point theory constitutes one of the most significant branches of modern mathematical analysis and has extensive applications in pure and applied mathematics, physics, engineering, economics, computer science, and optimization theory. The concept of a fixed point refers to an element that remains invariant under a given mapping. The present research paper investigates unique fixed-point theorems in various mathematical spaces, including metric spaces, Banach spaces, normed linear spaces, complete metric spaces, cone metric spaces, and fuzzy metric spaces. The paper provides a systematic discussion of classical and modern fixed-point results with emphasis on uniqueness conditions. Special attention is given to the Banach Contraction Principle, Kannan’s theorem, Chatterjea’s theorem, Edelstein’s theorem, and generalized contraction mappings. Applications of uniquefixed-point theory in differential equations, dynamic systems, integral equations, and computational mathematics are also discussed.