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

Abstract

This paper delves into the theoretical foundations and practical applications ofNeutrosophic MR-Metric Spaces (NMR-MS), a novel structure that synergizes the geometric properties of MR-metrics with the nuanced uncertainty modeling of neutrosophic logic.We establish two pivotal theorems within this framework. First, we prove a fixed-pointtheorem for contraction mappings in complete NMR-MS, guaranteeing the existence anduniqueness of a fixed point and the convergence of iterative sequences. Second, we introduce a measure-compression theorem, which describes how certain mappings can reduce theinternal uncertainty of sets as quantified by a neutrosophic measure. The power of thisunified framework is demonstrated through a series of examples and diverse applications, including the stabilization of uncertain control systems, image segmentation under ambiguity, the analysis of neutrosophic iterative function systems, and robust consensus protocolsin complex networks. Our results extend and generalize previous work in fixed-point theory and uncertainty analysis, providing a robust tool for modeling systems where truth, falsity,and indeterminacy coexist.