MEASURE-THEORETIC AND FIXED POINT ANALYSIS IN NEUTROSOPHIC MR-METRIC SPACES

Abstract

This research delves into the intricate structure of Neutrosophic MR-MetricSpaces (NMR-MS), a novel framework that synergizes the ternary relational logic of MR-metrics with the uncertainty modeling capabilities of neutrosophic set theory. We establishfundamental measure-theoretic foundations for these spaces by constructing a unique, regular, and Borel-regular measure derived from the neutrosophic topology. This is achievedthrough a specialized application of Carathéodory's extension theorem, tailored to the neu-trosophic context. Furthermore, we prove a powerful fixed point theorem for neutrosophiccontraction mappings within complete NMR-MS. This theorem not only guarantees theexistence and uniqueness of fixed points but also establishes their convergence μ-almost ev-erywhere, provides measure-theoretic stability, and demonstrates an exponential convergencerate in measure. The theoretical developments are substantiated with illustrative examplesand applied to solve fractional differential equations and analyze optimization algorithms, showcasing the versatility and applicability of the proposed framework in analysis and com-putational mathematics.