Neutrosophic MR-Metric Spaces: Topological Foundations, Completion, and Applications

Abed Al-Rahman M. Malkawi; Ayat M. Rabaiah

Authors

Abed Al-Rahman M. Malkawi Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman 11953, Jordan.
Ayat M. Rabaiah Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman 11953, Jordan.

Abstract

This paper introduces and systematically studies the concept of \textbf{Neutrosophic MR-Metric Spaces (NMR-MS)}, a novel structure that generalizes MR-metric spaces by incorporating neutrosophic logic to handle truth, falsity, and indeterminacy membership degrees. We establish the fundamental topological properties of these spaces, proving that every NMR-MS induces a Hausdorff topology and characterizing convergence within it. A central result is the completion theorem, demonstrating that every NMR-MS has a unique completion. Furthermore, we construct a complete NMR-MS on the function space \(C([a,b])\) and provide detailed examples and applications spanning signal processing, machine learning, image segmentation, and decision-making under uncertainty. These applications showcase the framework's versatility in modeling and analyzing complex systems where uncertainty is inherent. Our work bridges the gap between abstract metric space theory and practical computational problems, offering a robust foundation for future research in neutrosophic analysis and its applications.

References

A. Malkawi, A. Talafhah, W. Shatanawi, Coincidence and fixed point results for (y, L)-M-weak contraction mapping on Mb-metric spaces, Italian Journal of Pure and Applied Mathematics, no. 47 (2022), 751–768.

A. Malkawi, A. Rabaiah, MR-metric spaces with applications, Journal of Interdisciplinary Mathematics, 28(8) (2025), 2837–2865.

T. Qawasmeh, (H, Ωb)-interpolative contractions in Ωb-distance mappings with applications, European Journal of Pure and Applied Mathematics, 16(3) (2023), 1717–1730.

A. A. R. M. Malkawi, A. Talafhah, W. Shatanawi, Coincidence and fixed point results for generalized weak contraction mapping on b-metric spaces, Nonlinear Functional Analysis and Applications, 26(1) (2021), 177–195.

R. Al-deiakeh, M. Alquran, M. Ali, S. Qureshi, S. Momani, A. A. R. Malkawi, Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time- fractional modified Benjamin-Bona-Mahony equation, Journal of Applied Mathematics and Computational Mechanics, 23(1) (2024), 19–31.

T. Qawasmeh, H-simulation functions and Ωb -distance mappings in the setting of Gb-metric spaces and application, Nonlinear Functional Analysis and Applications, 28(2) (2023), 557–570.

A. Bataihah, T. Qawasmeh, A new type of distance spaces and fixed point results, Journal of Mathematical Analysis, 15(4) (2024), 81–90.

W. Shatanawi, T. Qawasmeh, A. Bataihah, A. Tallafha, New contractions and some fixed point results with application based on extended quasi b-metric spaces, U.P.B. Sci. Bull., Series A, 83(2) (2021), 1223–7027.

T. Qawasmeh, W. Shatanawi, A. Bataihah, A. Tallafha, Fixed point results and (a, b)-triangular admissibility in the frame of complete extended b-metric spaces and application, U.P.B. Sci. Bull., Series A, 83(1) (2021), 113–124.

A. Bataihah, W. Shatanawi, A. Tallafha, Fixed point results with simulation functions, Nonlinear Functional Analysis and Applications, 25(1) (2020), 13–23.

A. A. R. M. Malkawi, D. Mahmoud, A. M. Rabaiah, R. Al-Deiakeh, W. Shatanawi, On fixed point theorems in MR-metric spaces, Nonlinear Functional Analysis and Applications, (2024), 1125–1136.

G. M. Gharib, A. A. R. M. Malkawi, A. M. Rabaiah, W. A. Shatanawi, M. S. Alsauodi, A common fixed point theorem in an M*-metric space and an application, Nonlinear Functional Analysis and Applications, 27(2) (2022), 289–308.

K. Abodayeh, W. Shatanawi, A. Bataihah, A. H. Ansari, Some fixed point and common fixed point results through Ω-distance under nonlinear contractions, Gazi University Journal of Science, 30(1) (2017), 293–302.

A. Bataihah, A. Tallafha, W. Shatanawi, Fixed point results with Ω-distance by utilizing simulation functions, Italian Journal of Pure and Applied Mathematics, 43 (2020), 185–196.

K. Abodayeh, A. Bataihah, W. Shatanawi, Generalized Ω-distance mappings and some fixed point theorems, U.P.B. Sci. Bull. Ser. A, 79 (2017), 223–232.

T. Qawasmeh, W. Shatanawi, A. Bataihah, A. Tallafha, Common fixed point results for rational (a, b)f-mw contractions in complete quasi metric spaces, Mathematics, 7(5) (2019), 392.

A. Rabaiah, A. Tallafha, W. Shatanawi, Common fixed point results for mappings under nonlinear contraction of cyclic form in b-metric spaces, Advances in Mathematics: Scientific Journal, 26(2) (2021), 289–301.

I. Abu-Irwaq, W. Shatanawi, A. Bataihah, I. Nuseir, Fixed point results for nonlinear contractions with generalized Ω-distance mappings, U.P.B. Sci. Bull. Ser. A, 81(1) (2019), 57–64.

A. A. R. M. Malkawi, Existence and uniqueness of fixed points in MR-metric spaces and their applications, European Journal of Pure and Applied Mathematics, 18(2) (2025), Article No. 6077.

A. A. R. M. Malkawi, Convergence and fixed points of self-mappings in MR-metric spaces: theory and applications, European Journal of Pure and Applied Mathematics, 18(2) (2025), Article No. 5952.

A. A. R. M. Malkawi, Fixed point theorem in MR-metric spaces via integral type contraction, WSEAS Transactions on Mathematics, 24 (2025), 295–299.

T. Qawasmeh, A. Malkawi, Fixed point theory in MR-metric spaces: fundamental theorems and applications to integral equations and neutron transport, European Journal of Pure and Applied Mathematics, 18(3) (2025), Article No.6440.

A. Malkawi, Enhanced uncertainty modeling through neutrosophic MR-metrics: a unified framework with fuzzy embedding and contraction principles, European Journal of Pure and Applied Mathematics, 18(3) (2025), Article No. 6475.

A. Malkawi, A. Rabaiah, MR-metric spaces: theory and applications in weighted graphs, expander graphs, and fixed point theorems, European Journal of Pure and Applied Mathematics, 18(3) (2025), Article No. 6525.

A. Malkawi, Applications of MR-metric spaces in measure theory and convergence analysis, European Journal of Pure and Applied Mathematics, 18(3) (2025), Article No. 6528.

A. Malkawi, A. Rabaiah, Compactness and separability in MR-metric spaces with applications to deep learning, European Journal of Pure and Applied Mathematics, 18(3) (2025), Article No. 6592.

I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Functional Analysis, 30 (1989), 26–37.

S. Czerwik, Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1 (1993), 5–11.

S. Al-Sharif, A. Malkawi, Modification of conformable fractional derivative with classical properties, Italian Journal of Pure and Applied Mathematics, 44 (2020), 30–39.

G. M. Gharib, M. S. Alsauodi, A. Guiatni, M. A. Al-Omari, A. A. R. M. Malkawi, Using atomic solution method to solve the fractional equations, Springer Proceedings in Mathematics and Statistics, 418 (2023), 123–129.

A. A. R. M. Malkawi, Fixed point theorems for fuzzy mappings in neutrosophic MR- metric spaces, Journal of Nonlinear Modeling and Analysis, (2025), accepted.

N. Anakira, I. H. Jebril, O. Ogilat, I. M. Batiha, T. Sasa, and A. A. R. M. Malkawi, Variable Fractional-Order Reaction Diffusion System for Edge Preservation in Biomedical Imaging, International Journal of Analysis and Applications 24 (2026), 23.

A. Almalkawi, A. Amourah, A. Alsoboh, J. Salah, A. A. R. Malkawi, and T. Sasa, Analytic Estimates for Bi-Univalent Functions Associated with a New Operator Involving the q–Rabotnov Function, International Journal of Analysis and Applications 24 (2026), 1.

A. Malkawi, Fractal dimension estimation and attractor existence in MR-metric spaces: a generalization of classical fractal geometry, Nonlinear Functional Analysis and Applications, 30(4) (2025), 1205–1222.

A. Malkawi, Fixed point theorems for fuzzy mappings in MR-metric spaces, Nonlinear Functional Analysis and Applications, 30(4) (2025), 1083–1094.

A. Malkawi, A. Rabaiah, Fuzzy metric spaces: theory and applications in medical imaging, transportation, and social networks, Nonlinear Functional Analysis and Applications, 30(4) (2025), 1255–1272.

A. Malkawi, A. Rabaiah, Neutrosophic MR-metric spaces: a topos-theoretic framework with applications, International Journal of Analysis and Applications, 23 (2025), 333.