Fixed Point Theory: Existence, Uniqueness, and Applications

Abstract

Fixed point theory is a foundational area of mathematical analysis concerned with conditions under which a mapping admits a point that remains invariant under the action of the mapping. Such points—called fixed points—play a central role across pure and applied mathematics, including nonlinear analysis, differential and integral equations, optimization, game theory, economics, and computer science. This paper presents an elaborated study of fixed point theory with emphasis on existence and uniqueness results, core theorems and methodologies, and a wide spectrum of applications. Beginning with basic definitions and metric-space techniques, the paper advances through contraction principles, topological fixed point theorems, and order-theoretic approaches, and concludes with contemporary applications and extensions. The discussion highlights how fixed point results provide unifying principles for proving solvability, stability, and convergence in mathematical models.