A Comprehensive Study on Extensions and Comparative Aspects of Fixed Point Theorems in Metric Space

Abstract –Fixed point theory is a central component of nonlinear analysis, with profound implications across numerousmathematical and applied fields, including differential equations, optimization, and computational modeling. Classical results,such as Banach’s and Kannan’s contraction principles, provide powerful tools for establishing the existence and uniqueness offixed points. However, their restrictive assumptions, particularly regarding contractiveness and continuity, often limit theirapplicability in complex or real-world systems.This paper introduces a new fixed point theorem that unifies and generalizes the Banach and Kannan contraction conditionsthrough a hybrid contraction framework. By relaxing traditional constraints, the proposed theorem significantly broadens thescope of mappings for which fixed point results can be obtained in complete metric spaces. The theoretical advancement isaccompanied by a detailed comparative analysis with classical theorems, highlighting the improved generality and flexibility ofour approach.To validate its practical relevance, the theorem is applied to a series of illustrative examples, along with discussions on itsutility in areas such as economic modeling, machine learning algorithms, control theory, and numerical simulation. The studyconcludes by outlining directions for future research, including potential extensions to multivalued and generalized metricspaces. Overall, this work contributes to the expanding landscape of fixed point theory and its applications in modernmathematics.