MR-Metric Spaces: Theory, Applications, and Fixed-Point Theorems in Fuzzy and Measure-Theoretic Frameworks (European Journal of Pure and Applied Mathematics, 2025)
This paper explores the theoretical foundations and practical applications of MR-metric spaces, a generalization of classical metric spaces introduced by Malkawi et al. [1]. We investigate key properties such as symmetry, permutation invariance, and the modified tetrahedral inequality, which are pivotal for extending fixed-point theorems, measure theory, and fuzzy analysis. Our main results include: 1. Fuzzy-Measurable Banach Contraction Theorem: A unique fuzzy fixed-point theorem under Hausdorff MR-metric contractions [2, 3]. 2. Non- Archimedean Fuzzy Measure Concentration: A result linking compactness in MR-metric spaces to fuzzy measure concentration [4]. 3. MR-Fuzzy Radon-Nikodym Theorem: A fuzzy derivative construction for σ-finite measures [5]. Applications span medical diagnosis (fuzzy symptom analysis), sensor data fusion (epicenter detection), and financial risk modeling (fuzzy Value-at-Risk). This work synthesizes advancements in fixed-point theory [6, 7], fractional calculus [8, 9], and neutrosophic metrics [10], offering a unified framework for uncertainty quantification.
Keywords: MR-metric spaces; fuzzy fixed points; Radon-Nikodym derivative; measure concentration; Hausdorff metric; neutrosophic sets