The Existence and Uniqueness of the Solution of Difference Equation in Neutrosophic Environment via Generalized Hukuhara Difference Ideology

Abdul  Alamin ¹ ( Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia-741249, West Bengal, India. )
Mostafijur  Rahaman ² ( Department of Mathematics and Statistics, School of Applied Sciences and Humanities, Vignans Foundation for Science, Technology and Research, Guntur, Andhra Pradesh 522213, India. )
Mohammed  Rabih ³ ( Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia. )
Kamal Hossain  Gazi ⁴ ( Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia-741249, West Bengal, India. )
Aditi  Biswas ⁵ ( Department of Basic Science and Humanities, Greater Kolkata College of Engineering & Management, Baruipur 743387, West Bengal, India. )
Sankar Prasad Mondal ⁶ ( Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia-741249, West Bengal, India. )

Submited date : 2025-04-27 Accepted date : 2025-07-16

Keywords: Neutrosophic set, Metric space, Hukuhara difference, Logistic difference equation.,

Abstract :

In real-world scenarios, the neutrosophic set or neutrosophic numbers have been widely used to deal with the uncertain difference equations of the corresponding uncertain discrete dynamical system. A situation where discrete changes occur with vague information of a neutrosophic sense can be dealt with by the neutrosophic difference equation. In this paper, a new metric is defined for the neutrosophic set, and the sense of generalized Hukuhara difference for the fuzzy numbers is extended to the neutrosophic numbers. The generalized Hukuhara difference of the type-I and type-II and their corresponding neutrosophic parametric representation are discussed. The existence and uniqueness conditions to obtain a solution of the difference equation in a neutrosophic environment are argued by some theorems. The theoretical concept has been applied to the logistic difference equation in a neutrosophic environment. We have applied both the type-I and type-II Hukuhara differences to the two different generalized Hukuhara difference forms of the logistic difference equation. The equilibrium points and their corresponding stability criteria are established to perceive the effect of the Hukuhara differences. Finally, the numerical examples and their graphical portrayal are provided to recognize the intuition of the introduced theory in this paper.

References:

[1] Hassell M. P., Comins H. N., Discrete time models for two-species competition, Theoretical Population Biology, 1976; 9(2): 202221. https://doi.org/10.1177/1687813222114212
[2] Kocic V. L., Ladas G., Global Behaviour of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic Publishers, Dordrecht, Netherlands, 1993.
[3] Popa D., Rasa I., Viorel A., Approximate solutions of the logistic equation and Ulam stability, Applied Mathematics Letters, 2018; 85: 6469. https://doi.org/10.1016/j.aml.2018.05.018
[4] Sun G. Su, T., Qin B., Eventually periodic solutions of a max-type system of difference equations of higher order, Discrete Dynamics in Nature and Society, 2018; 2018: 6. https://doi.org/10.1155/ 2018/8467682
[5] El-Dessoky M. M., Elabbasy E. M., and Asiri A., Dynamics and solutions of a fifth-order nonlinear difference equation, Discrete Dynamics in Nature and Society, 2018; 2018: 21. https://doi.org/10. 1155/2018/9129354
[6] Matsunaga H., Suzuki R., Classification of global behavior of a system of rational difference equations, Applied Mathematics Letters, 2018; 85: 5763. https://doi.org/10.1016/j.aml.2018.05.020
[7] Al-Hdaibat B., Al-Ashhab S., Sabra R., Explicit solutions and bifurcations for a system of rational difference equations, Mathematics, 2019; 69(7). https://doi.org/10.3390/math7010069
[8] Zhang Q., Yang L., Liu J., Dynamics of a system of rational third-order difference equations, Advances in Difference Equations, 2012; 2012: 136.
https://doi.org/10.1186/1687-1847-2012-136
[9] Yalcinkaya I., Caliskan V., Tollu D. T., On a nonlinear fuzzy difference equation, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 2022; 71(1), 68-78. https://doi.org/10.31801/cfsuasmas.861915
[10] Zadeh L.A., Fuzzy sets. Inf. Control, 1965; 8: 338353. https://doi.org/10.1016/S0019-9958(65) 90241-X
[11] Bede B., Mathematics of Fuzzy Sets and Fuzzy Logic, Springer, London, 2013.
[12] Diamond P., Kloeden P., Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.
[13] Lakshmikantam V., Vatsala A. S., Basic Theory of Fuzzy Difference Equations, J. Diff. Equ. Appl., 2002; 8 (11): 957968. https://doi.org/10.1080/1023619021000048850
[14] Abdullayev I., Osadchy E., Shcherbakova N., Kosorukova, I., An Innovative Approach to Financial Distress Prediction Using Relative Weighted Neutrosophic Valued Distances, International Journal of Neutrosophic Science (IJNS), 2025; 25(1). https://doi.org/10.54216/IJNS.250133
[15] Chrysafis. K. A., Papadopoulos B. K., Papaschinopoulos G., On the fuzzy difference equation of finance, Fuzzy Sets Syst., 2008; 159, 3259-3270. https://doi.org/10.1016/j.fss.2008.06.007
[16] Zhang Q., Lin F., On dynamical behavior of discrete time fuzzy logistic equation, Discrete Dynamics in Nature and Society, 2018; 2018(8). https://doi.org/10.1155/2018/8742397
[17] Deeba E. Y., Korvin A. De, Analysis by fuzzy difference equations of a model of CO2 level in the blood, Appl. Math. Lett., 12(1999): 33-40. https://doi.org/10.1016/S0893-9659(98)00168-2
[18] Dayan F., Rafiq M., Ahmed N., Raza A., Ahmad M. O., A dynamical study of a fuzzy epidemic model of Mosquito-Borne Disease, Computers in Biology and Medicine, 2022; 148, 105673. https://doi.org/10.1016/j.compbiomed.2022.105673
[19] Abu Arqub O., Singh J., Maayah B., Alhodaly, M., Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the MittagLeffler kernel differential operator, Mathematical Methods in the Applied Sciences, 2023; 46(7), 7965-7986. https://doi.org/10.1002/mma.7305
[20] Khastan A., Fuzzy logistic diference equation. Iranian Journal of Fuzzy System, 2018; 15(7): 5566.
[21] Wang, H., Rodr´ıguez-L´opez, R., Khastan, A., Existence of solutions to a class of interval-valued differential equation with impulses relative to length constraints, Fuzzy Sets and Systems, 2024; 484, 108943. https://doi.org/10.1016/j.fss.2024.108943
[22] Soroush, S., Allahviranloo, T., Azari, H., Rostamy-Malkhalifeh, M., Generalized fuzzy difference method for solving fuzzy initial value problem, Computational and Applied Mathematics, 2024; 43(3), 129. https://doi.org/10.1007/s40314-024-02645-2
[23] Shams, M., Kausar, N., Araci, S., Oros, G. I., Artificial hybrid neural network-based simultaneous scheme for solving nonlinear equations: Applications in engineering, Alexandria Engineering Journal, 2024; 108:292-305. https://doi.org/10.1016/j.aej.2024.07.078
[24] Palanikumar, M., Suguna, M., Jana, C., Generalized Neutrosophic Sets and Their Applications for Aggregated Operators Based on Diagnostic Disease Problem, In Fuzzy Optimization, Decisionmaking and Operations Research: Theory and Applications, 2023; 219-240. https://doi.org/10.1007/978-3-031-35668-1_11
[25] Atanassov K., Intuitionistic fuzzy sets. Fuzzy Sets Syst., 1986; 20: 8796. https://doi.org/10.1016/ 0165-0114(89)90215-7
[26] Liu F., Yuan X. H., Fuzzy number intuitionistic fuzzy set. Fuzzy Syst. Math, 2007; 21: 8891.
[27] Ejegwa, P. A., Kausar, N., Aydin, N., Deveci, M., A novel Pythagorean fuzzy correlation coefficient based on Spearmans technique of correlation coefficient with applications in supplier selection process, Journal of Industrial Information Integration, 2025; 44(100762). https://doi.org/10.1016/j.jii. 2024.100762
[28] Ye J., Prioritized aggregation operators of trapezoidal intuitionistic fuzzy sets and their application to multi criteria decision making. Neural Comput. Appl., 2014; 25: 14471454. https://doi.org/10.1007/ s00521-014-1635-8
[29] Li D.F., Cheng C.T., New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognit. Lett., 2002; 23(13), 221225. https://doi.org/10.1016/S0167-8655(01) 00110-6
[30] De S.K., Biswas R., Roy A.R., An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets Syst., 2001; 117(2): 209213.
https://doi.org/10.1016/S0165-0114(98)00235-8
[31] Kharal A., Homeopathic drug selection using intuitionistic fuzzy sets. Homeopathy, 2009; 98(1): 3539. https://doi.org/10.1016/j.homp.2008.10.003
[32] Ejegwa, P. A., Kausar, N., Agba, J. A., Ugwuh, F., Ozbilge, E., Ozbilge, E., Determination of medical emergency via new intuitionistic fuzzy correlation measures based on Spearmans correlation coefficient, AIMS Math, 2024; 9(6):15639-15670. https://doi.org/10.3934/math.2024755
[33] Kausar, N., Garg, H., Contra-hormonic generalized fuzzy numerical scheme for solving mechanical engineering problems, Journal of Applied Mathematics and Computing, 2024; 70(5):4629-4653. https://doi.org/10.1007/s12190-024-02148-7
[34] Revathi, A. N., Karpagam, A., Suguna, M., Salahuddin, An Effective Method of Analysing Student Performance Using Fuzzy Numbers, Advances in Fuzzy Systems, 2024; 2024(1), 1513765. https://doi. org/10.1155/2024/1513765
[35] Smarandache F., A Unifying Field in Logics Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, Delaware, 1998.
[36] T¨urkan Y. S., Alioˇgulları E., T¨uyl¨u D., Optimizing Location Selection for International Education Fairs: An Interval-Valued Neutrosophic Fuzzy Technique for Order of Preference by Similarity to Ideal Solution Approach, Sustainability, 2024; 16(23), 10227. https://doi.org/10.3390/su162310227
[37] Wang H., Smarandache F., Zhang Q., Sunderraman R., Single valued neutrosophic sets, MultiSpace and Multistructure, 2010; 4: 410413.
[38] Kumar M., Singh S. B., Khati A., Triangular neutrosophic set and its application to reliability analysis of gas turbine system, International Journal of System Assurance Engineering and Management, 2024; 1-14. https://doi.org/10.1007/s13198-024-02563-w
[39] Beigmohamadi, R., Khastan, A., Nieto, J. J., Rodr´ıguez-L´opez, R., Existence and uniqueness of nonperiodic solutions to boundary value problems for discrete fractional difference equations with uncertainty, Information Sciences, 2023; 634, 14-26. https://doi.org/10.1016/j.ins.2023.03.028
[40] Narzary G., Basumatary B., Nordo, G., A neutrosophic solution of heat equation by neutrosophic Laplace transform, Neutrosophic Sets and Systems, 2024; 73(1), 50. https://digitalrepository.unm.edu/ nss_journal/vol73/iss1/50
[41] Safikhani, L., Vahidi, A., Allahviranloo, T., Afshar Kermani, M., Multi-step gH-difference-based methods for fuzzy differential equations, Computational and Applied Mathematics, 2023; 42(1), 27. https://doi. org/10.1007/s40314-022-02167-9
[42] Bhat H. A., Iqbal, A., Generalized Hukuhara directional differentiability of interval-valued functions on Riemannian manifolds, Optimization, 2025; 1-27. https://doi.org/10.1080/02331934.2024.2447996
[43] Shams, M., Kausar, N., Kousar, S., Pamucar, D., Ozbilge, E., Tantay, B., Computationally seminumerical technique for solving system of intuitionistic fuzzy differential equations with engineering applications, Advances in Mechanical Engineering, 2022; 14(12), 16878132221142128. https://doi. org/10.1177/1687813222114212
[44] Kar A., Rai R. N., QFD-based hybrid neutrosophic MCDM approach with Six Sigma evaluation for sustainable product design in Industry 4.0, Kybernetes, 2025; 54(6), 3238-3266. https://doi.org/10. 1108/K-09-2023-1757
[45] Gonul Bilgin, N., Pamuˇcar, D., Riaz, M., Fermatean neutrosophic topological spaces and an application of neutrosophic kano method, Symmetry, 2022; 14(11), 2442. https://doi.org/10.3390/sym14112442
[46] Singh, P., Gazi, K. H., Rahaman, M., Salahshour, S., Mondal, S. P., A fuzzy fractional power series approximation and Taylor expansion for solving fuzzy fractional differential equation, Decision Analytics Journal, 2024; 10, 100402. https://doi.org/10.1016/j.dajour.2024.100402
[47] Nguyen H.T., A note on the extension principle for fuzzy sets, J. Math Anal. Appl. 1978; 64(2): 369-380.
[48] Rahaman, M., Mondal, S.P., Chatterjee, B., Alam, S., The solution techniques for linear and quadratic equations with coefficients as Cauchy neutrosphic numbers, Granular comput., 2021; 1-19. https:// doi.org/10.1007/s41066-021-00276-0
[49] Rahaman, M., Haque, R., Salahshour, S., Sobczak, A., Azizzadeh, F., Alam, S., Mondal, S.P., Solution of an uncertain EPQ model using the Neutrosophic differential equation approach, Operations Research and Decisions, 2025; 35(2):1-36. https://doi.org/10.37190/ord250203