Fractional-Order Neutrosophic MR-Metric Spaces: Theory, Fixed Point Theorems, and Applications

Abstract

This paper introduces the novel concept of Fractional-Order Neutrosophic MR-Metric Spaces (FoNMR-MS), which synergistically combines three powerful mathematical frameworks: MR-metric spaces, neutrosophic logic, and fractional calculus. We begin by extending the classical MR-metric structure through the incorporation of fractional integrals, defining a comprehensive fractional-order metric Mα and corresponding neutrosophic functions Tα, Iα, Fα. Fundamental properties including non negativity, identity, symmetry, and a generalized fractional triangle inequality are rigorously established. The core theoretical contribution is a comprehensive fixed point theorem for contraction mappings in complete FoNMR-MS, accompanied by detailed convergence analysis and neutrosophic consistency conditions. We further provide extensive examples and applications demonstrating the utility of our framework in modeling anomalous diffusion processes, image denoising, and machine learning under uncertainty. This work significantly generalizes existing results in fixed point theory and offers a robust mathematical foundation for handling complex systems characterized by fractional dynamics and neutrosophic uncertainty.