Neutrosophic MR-Metric Spaces: Fixed Point Theorems and Applications to Fractional Differential Equations with Uncertainty

Abstract

This paper introduces the new concept of Neuterosophic MR-metric space (NMR-MS), a powerful mathematical framework that integrates Neuterosophic logic with recently developed MR metric spaces. We establish the basic fixed point theorem for Neuterosophic contractions in this new setting, and extend classical results to accommodate three degrees of membership: truth, falsity, and uncertainty. The main theorem guarantees the existence and uniqueness of fixed points for neuterosophic contraction mappings, with constant dependence on the geometric convergence rate and parameters. An important application of our theory is presented in the context of fractional differential equations involving generalized fractional derivatives. By constructing an appropriate NMR-MS structure in the space of continuous functions, we reformulate the fractional differential equations as fixed point problems and prove the existence and uniqueness of the solutions under Lipschitz conditions. Neurosophisticated components add a natural amount of uncertainty to the solution process. We illustrate our theoretical results with various applications, including fractional decay processes, nonlinear logistic growth models, fractional heat equations, PID controller design, financial modeling, and biological tissue growth. These examples demonstrate the versatility and practical utility of the NMR-MS framework for dealing with real-world problems involving imprecision, ambiguity, and incomplete information. The results presented here generalize and unify several existing fixed point theorems in metric spaces, fuzzy metric spaces and Neuterosophic metric spaces, and provide new perspectives for research in uncertainty theory, fractional calculus and their interdisciplinary applications.