Fixed Points in Modified Soft Neutrosophic Metric Spaces and theirApplication to Integral Equations

Introduction:: This paper introduces the Modified Soft Neutrosophic Metric Space (MSNMS), a framework that generalizes classical metric spaces by integrating soft set and neutrosophic set theory. The study aims to establish fundamental properties and investigate the topological structure of MSNMS. Methods:: Core operations and key properties of MSNMS are defined and analyzed using mathematical derivations. Illustrative examples demonstrate the structure, while practical applications, particularly in solving integral equations, validate the theoretical results. objective: see manuscript Results:: The MSNMS framework successfully establishes essential topological properties and verifies critical theorems. Its application to integral equations demonstrates improved accuracy and robustness compared to existing methods. Discussion:: MSNMS effectively handles uncertainty and imprecision, extending traditional metric space concepts. The framework’s practical applicability highlights its potential in mathematical modeling and engineering problems. Conclusion:: MSNMS provides a versatile and powerful tool, combining rigorous theory with practical effectiveness. It generalizes existing metric spaces and offers enhanced methods for solving complex problems. AMS Mathematics Subject Classification: 47H10; 54H25.