(2302-7881) Finite-time stability results for fuzzy fractional stochastic delay system under Granular differentiability concept

Document Type : Research Paper

Authors

Department of Mathematics, Guizhou University, Guiyang 550025, China

Abstract

In present manuscript, we investigate a new type of fuzzy fractional stochastic delay system (FFSDS), in which the derivative is defined by Granular differentiability. We first transform the considered system into an equivalent integral system with the aid of fuzzy Laplace transformation and its inverse involving Mittag-Leffler function. Subsequently, existence and uniqueness results of the solutions for FFSDS are derived by applying Carath'{e}odory approximation, under non-Lipschitz conditions, and contradiction method, respectively. \textcolor{black}{In addition, we establish the finite-time stability of the system by utilizing the generalized Gr\"{o}nwall delay inequality. Finally, the obtained conclusions are expound via an example.

Keywords

References

[1] M. Abouagwa, J. Li, Approximation properties for solutions to It^o Doob stochastic fractional di fferential equations with non-Lipschitz coefficients, Stochastics and Dynamics, 19(4) (2019), 1950029.
[2] R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional di erential equations with uncertainty, Nonlinear Analysis: Theory Methods and Applications, 72(6) (2010), 2859-2862.
[3] M. S. Ali, G. Narayanan, S. Saroha, B. Priya, G. K. Thakur, Finite-time stability analysis of fractional-order memristive fuzzy cellular neural networks with time delay and leakage term, Mathematics and Computers in Simulation, 185 (2021), 468-485.
[4] T. Allahviranloo, Fuzzy fractional di erential operators and equations, Springer, Cham, 2020.
[5] T. V. An, N. D. Phu, N. Van Hoa, A survey on non-instantaneous impulsive fuzzy di erential equations involving the generalized Caputo fractional derivative in the short memory case, Fuzzy Sets and Systems, 443 (2022), 160-197.
[6] E. Arhrrabi, M. Elomari, S. Melliani, L. S. Chadli, Existence and stability of solutions of fuzzy fractional stochastic di erential equations with fractional Brownian motions, Advances in Fuzzy Systems, 2021 (2021), 3948493.
[7] E. Arhrrabi, M. Elomari, S. Melliani, L. S. Chadli, Existence and stability of solutions for a coupled system of fuzzy fractional Pantograph stochastic di erential equations, Asia Pacific Journal of Mathematics, 9 (2022), 20.
[8] S. Arshad, On existence and uniqueness of solution of fuzzy fractional di erential equations, Iranian Journal of Fuzzy Systems, 10(6) (2013), 137-151.
[9] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20(2) (2016), 763-769.
[10] S. Chakraverty, S. Tapaswini, D. Behera, Fuzzy arbitrary order system: Fuzzy fractional di erential equations and applications, John Wiley, New York, 2016.
[11] N. P. Dong, N. T. K. Son, T. Allahviranloo, H. T. T. Tam, Finite-time stability of mild solution to time-delay fuzzy fractional di erential systems under Granular computing, Granular Computing, 8 (2022), 223-239.
[12] F. F. Du, J. G. Lu, Finite-time stability of fractional-order fuzzy cellular neural networks with time delays, Fuzzy Sets and Systems, 438 (2022), 107-120.
[13] W. Y. Fei, Existence and uniqueness of solution for fuzzy random di erential equations with non-Lipschitz coefficients, Information Sciences, 177(20) (2007), 4329-4337.
[14] W. Y. Fei, Existence and uniqueness for solutions to fuzzy stochastic di erential equations driven by local martingales under the non-Lipschitzian condition, Nonlinear Analysis: Theory Methods and Applications, 76 (2013),202-214.
[15] S. Hati, K. Maity, Reliability dependent imperfect production inventory optimal control fractional order model for uncertain environment under Granular di erentiability, Fuzzy Information and Engineering, 14(4) (2022), 379-406.
[16] H. Jafari, M. T. Malinowski, M. J. Ebadi, Fuzzy stochastic di erential equations driven by fractional Brownian motion, Advances in Difference Equations, 2021(1) (2021), 16.
[17] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new de nition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.
[18] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional di erential equations, Elsevier, Amsterdam, 2006.
[19] R. Liu, J. R. Wang, D. O’Regan, On the solutions of  rst-order linear impulsive fuzzy di erential equations, Fuzzy Sets and Systems, 400 (2020), 1-33.
[20] D. F. Luo, M. Q. Tian, Q. X. Zhu, Some results on  nite-time stability of stochastic fractional-order delay di erential equations, Chaos, Solitons and Fractals, 158 (2022), 111996.
[21] V. Lupulescu, L. S. Dong, N. V. Hoa, Existence and uniqueness of solutions for random fuzzy fractional integral and di erential equations, Journal of Intelligent and Fuzzy Systems, 29(1) (2015), 27-42.
[22] M. T. Malinowski, Approximation schemes for fuzzy stochastic integral equations, Applied Mathematics and Computation, 219 (2013), 11278-11290.
[23] M. T. Malinowski, Fuzzy and set-valued stochastic di erential equations with local Lipchitz condition, IEEE Transactions on Fuzzy Systems, 23(5) (2015), 1891-1898.
[24] M. T. Malinowski, Stochastic fuzzy di erential equations of a nonincreasing type, Communications in Nonlinear Science and Numerical Simulation, 33 (2016), 99-117.
[25] M. T. Malinowski, Fuzzy stochastic di erential equations of decreasing fuzziness: Non-Lipschitz coecients, Journal of Intelligent and Fuzzy Systems, 31(1) (2016), 13-25.
[26] M. T. Malinowski, M. Michta, Fuzzy stochastic integral equations, Dynamic Systems and Applications, 19(3-4) (2010), 473-493.
[27] M. T. Malinowski, M. Michta, Stochastic fuzzy di erential equations with an application, Kybernetika, 47(1) (2011), 123-143.
[28] M. Mazandarani, N. Pariz, A. V. Kamyad, Granular di erentiability of fuzzy-number-valued functions, IEEE Transactions on Fuzzy Systems, 26(1) (2018), 310-323.
[29] M. Mazandarani, L. Xiu, A review on fuzzy di erential equations, IEEE Access, 9 (2021), 62195-62211.
[30] M. Mazandarani, Y. Zhao, Z-Di erential equations, IEEE Transactions on Fuzzy Systems, 28(3) (2020), 462-473.
[31] M. Michta, On set-valued stochastic integrals and fuzzy stochastic equations, Fuzzy Sets and Systems, 177(1) (2011), 1-19.
[32] V. H. Ngo, Fuzzy fractional functional integral and di erential equations, Fuzzy Sets and Systems, 280 (2015), 58-90.
[33] J. Priyadharsini, P. Balasubramaniam, Existence of fuzzy fractional stochastic di erential system with impulses, Computational and Applied Mathematics, 39(3) (2020), 195.
[34] J. Priyadharsini, P. Balasubramaniam, Solvability of fuzzy fractional stochastic Pantograph di erential system, Iranian Journal of Fuzzy Systems, 19(1) (2022), 47-60.
[35] S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional di erential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 1372-1381.
[36] K. Shah, I. Ahmad, J. J. Nieto, G. U. Rahman, T. Abdeljawad, Qualitative investigation of nonlinear fractional coupled Pantograph impulsive di erential equations, Qualitative Theory of Dynamical Systems, 21(4) (2022), 131.
[37] N. T. K. Son, H. V. Long, N. P. Dong, Fuzzy delay di erential equations under granular di erentiability with applications, Computational and Applied Mathematics, 38(3) (2019), 1-29.
[38] S. Tyagi, S. C. Martha, Finite-time stability for a class of fractional-order fuzzy neural networks with proportional delay, Fuzzy Sets and Systems, 381 (2020), 68-77.
[39] H. Vu, N. V. Hoa, Uncertain fractional di erential equations on a time scale under granular di erentiability concept, Computational and Applied Mathematics, 38(3) (2019), 110.
[40] H. Vu, V. H. Ngo, Hyers-Ulam stability of fuzzy fractional Volterra integral equations with the kernel ψ-function via successive approximation method, Fuzzy Sets and Systems, 419 (2021), 67-98.
[41] X. Wang, D. F. Luo, Q. X. Zhu, Ulam-Hyers stability of Caputo type fuzzy fractional di erential equations with time-delays, Chaos, Solitons and Fractals, 156 (2022), 111822.
[42] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[43] J. K. Zhang, X. Y. Chen, L. F. Li, X. J. Ma, Optimality conditions for fuzzy optimization problems under granular convexity concept, Fuzzy Sets and Systems, 447 (2022), 54-75.
[44] J. K. Zhang, Y. Wang, S. Zhang, A new homotopy transformation method for solving the fuzzy fractional Black-Scholes european option pricing equations under the concept of Granular di erentiability, Fractal and Fractional, 6(6) (2022), 286.
[45] M. W. Zheng, L. X. Li, H. P. Peng, J. H. Xiao, Y. X. Yang, Y. P. Zhang, H. Zhao, Finite-time stability and synchronization of memristor-based fractional-order fuzzy cellular neural networks, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 272-291.
[46] Y. Zhou, J. R. Wang, L. Zhang, Basic theory of fractional di erential equations, World Scientific, Singapore, 2014.
Iranian Journal of Fuzzy Systems
Volume 20, Issue 5
September and October 2023
Pages 135-150

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