International Journal of Nonlinear Analysis and Applications

Fuzzy fractional pantograph stochastic differential equations: Existence, uniqueness and averaging principle

Document Type : Research Paper

Authors

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, PO Box 532,23000 Beni Mellal, Morocco

10.22075/ijnaa.2023.28270.3850

Abstract

Fuzzy fractional pantograph stochastic differential equations $($FFPSDEs$)$ is investigated here. The initial objective is to show the existence and uniqueness of solutions using Banach fixed point theorem. The second objective is discussing averaging principle of FFPSDEs, precisely, we will prove that the solutions of FFPSDEs can be approximated in the sense of mean square by the solutions of averaged fuzzy fractional stochastic system.

Keywords

[1] F. Acharya, V. Kushawaha, J. Panchal, and D. Chalishajar, Controllability of fuzzy solutions for neutral impulsive functional differential equations with nonlocal conditions, Axioms 10 (2021), 84.
[2] H. Afshari, H.R. Marasi, and J. Alzabut, Applications of new contraction mappings on existence and uniqueness results for implicit ϕ-Hilfer fractional pantograph differential equations, J. Inequal. Appl. 2021 (2021), no. 1, 1–14.
[3] K. Agilan and V. Parthiban, Existence of solutions of fuzzy fractional pantograph equation, Comput. Sci. 15 (2020), no. 4, 1117–1122.
[4] H.M. Ahmed and Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Appl. Math. Lett. 112 (2021), Article ID 106755.
[5] E. Arhrrabi, M. Elomari, and S. Melliani, Averaging principle for fuzzy stochastic differential equations, Contrib. Math. 5 (2022), 25–29.
[6] E. Arhrrabi, M. Elomari, S. Melliani, and L.S. Chadli, Existence and stability of solutions of fuzzy fractional stochastic differential equations with fractional Brownian motions, Adv. Fuzzy Syst. 2021 (2021), 1–9.
[7] E. Arhrrabi, M. Elomari, S. Melliani, and L.S. Chadli, Existence and uniqueness results of fuzzy fractional stochastic differential equations with impulsive, Int. Conf. Partial Differ. Equ. Appl. Model. Simul., Springer, Cham, 2023, pp. 147–163.
[8] E. Arhrrabi, M. Elomari, S. Melliani, and L.S. Chadli, Fuzzy fractional boundary value problem, 7th Int. Conf. Optim. Appl. (ICOA), IEEE, 2021, May, pp. 1–6.
[9] E. Arhrrabi, M. Elomari, S. Melliani, and L.S. Chadli, Existence and stability of solutions for a coupled system of fuzzy fractional Pantograph stochastic differential equations, Asia Pac. J. Math. 9 (2022), 20.
[10] E. Arhrrabi, M. Elomari, S. Melliani, and L.S. Chadli, Existence and uniqueness results of fuzzy fractional stochastic differential equations with impulsive, Int. Conf. Partial Differ. Equ. Appl. Model. Simul., Springer, Cham, 2023, pp. 147–163.
[11] E. Arhrrabi, M. Elomari, S. Melliani, and L.S. Chadli, Existence and controllability results for fuzzy neutral stochastic differential equations with impulses, Bol. Soc. Paranaense Mat. 41 (2023), 1–14.
[12] E. Arhrrabi, M. Elomari, S. Melliani, and L.S. Chadli, Fuzzy fractional boundary value problems with Hilfer fractional derivatives, Asia Pacific J. Math. 10 (2023), 4.
[13] I. Ahmed, P. Kumam, J. Abubakar, P. Borisut, and K. Sitthithakerngkiet, Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition, Adv. Differ. Equ. 2020 (2020), 1–15.
[14] K. Balachandran, S. Kiruthika, and J.J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci. 33 (2013), no. 3, 712–720.
[15] E.M. Elsayed, S. Harikrishnan, and K. Kanagarajan, Analysis of nonlinear neutral pantograph differential equations with Hilfer fractional derivative, MathLAB 1 (2018), no. 2, 231–240.
[16] D. Gao, J. Li, Z. Luo, and D. Luo, The averaging principle for stochastic pantograph equations with non-Lipschitz conditions, Math. Probl. Eng. Art. 2021 (2021), ID 5578936, 1–7.
[17] S. Harikrishnan, K. Kanagarajan, and D. Vivek, Solutions of nonlocal initial value problems for fractional pantograph equation, J. Nonlinear Anal. Appl. 2 (2018), 136–144.
[18] S. Harikrishnan, R. Ibrahim, and K. Kanagarajan, Establishing the existence of Hilfer fractional pantograph equations with impulses, Fund. J. Math. Appl. 1 (2018), no. 1, 36–42.
[19] M. Houas, Existence and stability of fractional pantograph differential equations with Caputo-Hadamard type derivative, Turk. J. Ineq. 4 (2020), no. 1, 1–10.
[20] R. Hosseinzadeh and M. Zarebnia, Application and comparison of the two efficient algorithms to solve the pantograph Volterra fuzzy integro-differential equation, Soft Comput. 25 (2021), no. 10, 6851–6863.
[21] W. Hu, Q. Zhu, and H.R. Karimi, Some improved Razumikhin stability criteria for impulsive stochastic delay differential systems, IEEE Trans. Automatic Control 64 (2019), no. 12, 5207–5213.
[22] D. Luo, Q. Zhu, and Z. Luo, An averaging principle for stochastic fractional differential equations with time-delays, Appl. Math. Lett. 105 (2020), Article ID 106290.
[23] D. Luo, Q. Zhu, and Z. Luo, A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients, Appl. Math. Lett. 122 (2021), 107549.
[24] W. Mao, L. Hu, and X. Xu, The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Levy jumps, Appl. Math. Comput. 268 (2015), 883–896.
[25] S. Ma and Y. Kang, Periodic averaging method for impulsive stochastic differential equations with Levy noise, Appl. Math. Lett. 93 (2019), 91–97.
[26] M.T Malinowski and M. Michta, Stochastic fuzzy differential equations with an application, Kybernetika 47 (2011), 123–143.
[27] X. Meng, S. Hu, and P. Wu, Pathwise estimation of stochastic differential equations with unbounded delay and its application to stochastic pantograph equations, Acta Appl. Math. 113 (2011), 231–246.
[28] J.R. Ockendon and A.B. Taylor, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. Ser. A 322 (1971), 447–468.
[29] B. Pei, Y. Xu, and J.L. Wu, Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Appl. Math. Lett. 100 (2020), Article ID 106006.
[30] J. Priyadharsini and P. Balasubramaniam, Solvability of fuzzy fractional stochastic Pantograph differential system, Iran. J. Fuzzy Syst. 19 (2022), no. 1, 47-60.
[31] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23–130.
[32] D. Vivek, K. Kanagarajan, and S. Sivasundaram, Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23 (2016), no. 4, 685–698.
[33] D. Vivek, K. Kanagarajan, and S. Sivasundaram, On the behavior of solutions of Hilfer-Hadamard type fractional neutral pantograph equations with boundary conditions, Commun. Appl. Anal. 22 (2018), no. 2, 211–232.
[34] W. Xu, W. Xu, and S. Zhang, The averaging principle for stochastic differential equations with Caputo fractional derivative, Appl. Math. Lett. 93 (2019), 79–84.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 01 December 2023
  • Receive Date: 01 September 2022
  • Accept Date: 04 April 2023