Abstract

Fixed point theorems hold a pivotal position in mathematics and its applications, serving as fundamental tools in solving equations, analyzing dynamic systems, and modeling equilibrium states. This review delves into common fixed point theorems within the framework of metric and Menger spaces, an extension of classical metric spaces that incorporate probabilistic distance measures. By accommodating uncertainty and randomness, metric and Menger  spaces provide a robust mathematical framework for addressing real-world problems characterized by variability and stochastic behavior. Theoretical foundations of fixed point theorems, such as Banach’s Contraction Principle, Kannan’s Fixed Point Theorem, and extensions like Reich’s and Geraghty’s theorems, are explored in detail. These theorems generalize classical results by introducing conditions that are suited for probabilistic and fuzzy environments. This generalization expands their applicability, enabling their use in complex systems where deterministic approaches are insufficient. The applications of these theorems span diverse fields. In optimization and computational mathematics, they provide the foundation for iterative methods like gradient descent, ensuring convergence in uncertain environments. In data sciences, fixed point principles enhance the stability and reliability of clustering algorithms, neural networks, and decision-making models. Furthermore, in game theory and economics, these theorems are instrumental in proving equilibrium states such as Nash equilibria in stochastic systems. The relevance of fixed point theorems extends to the objectives outlined in the National Education Policy (NEP) 2020. By fostering interdisciplinary learning and research-driven education, NEP 2020 encourages integrating abstract mathematical concepts with practical applications. Exploring fixed point theorems in metric and Menger spaces aligns with these goals, promoting critical thinking, innovation, and problem-solving skills. This review underscores the transformative potential of bridging theoretical mathematics with real-world applications, emphasizing its role in modern education and technological advancements.