MR-Metric Spaces: Theory and Applications in Fractional Calculus and Fixed-Point Theorems

This paper investigates the interplay between MR-metric spaces and fractional calculus, estab lishing new theoretical results with applications in analysis and mathematical physics. We introduce a novel connection between fractional derivatives and MR-metric structures, proving three main theorems: (1) a bound on the MR-metric expression involving fractional derivatives, (2) a fixed-point theorem for mappings with fractional differentiability in complete MR-metric spaces, and (3) a characterization of continuity for fractional derivatives in the MR-metric framework. The theoretical developments are complemented by concrete examples and applications to fractional differential equations, anomalous diffusion models, and viscoelastic material analysis. Our results extend the classical theory of metric spaces and provide new tools for analyzing nonlinear problems in fractional calculus.