Fixed Point Results for Implicit Relations in Neutrosophic Metric Spaces and their Application

Introduction: Fixed point theory is a fundamental area in mathematical analysis with wide-ranging applications. This research aims to extend fixed point results in the framework of neutrosophic metric spaces, which incorporate degrees of membership, non-membership, and indeterminacy. The study focuses on the development of compatible maps to generalize fixed point theorems under more realistic conditions. Methods: We introduce two novel classes of compatible maps—types (α) and (β)—defined via implicit relations in neutrosophic metric spaces. New fixed point theorems are established by using these map types and extending existing theories from fuzzy and intuitionistic fuzzy metric spaces. The work includes formal definitions, interrelations among mappings, and proofs of key results. Results: The newly established theorems offer broader conditions for ensuring the existence of fixed points in neutrosophic metric spaces. These generalizations effectively account for uncertainty, expanding the applicability of fixed point theory to more complex and indeterminate settings. Discussion: The inclusion of implicit relations and compatible maps enhances the theoretical flexibility of fixed point results. To demonstrate the applicability, the proposed theorems are used to solve the Fredholm integral equation under a neutrosophic framework, validating their relevance in handling real-world problems characterized by imprecise information. Conclusion: This study significantly advances fixed point theory by developing and analyzing compatible maps of types (α) and (β) in neutrosophic metric spaces. The findings provide a robust foundation for future mathematical research and practical applications involving uncertainty and incomplete data.