Fractional stochastic functional differential equations with non-Lipschitz condition

Rahman Ullah; Muhammad Farooq; Faiz Faizullah; Maryam A Alghafli; Nabil Mlaiki; Rahman Ullah; Muhammad Farooq; Faiz Faizullah; Maryam A Alghafli; Nabil Mlaiki

doi:10.3934/math.2025325

AIMS Mathematics

2025, Volume 10, Issue 3: 7127-7143. doi: 10.3934/math.2025325

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Research article

Fractional stochastic functional differential equations with non-Lipschitz condition

1.
School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China
2.
Department of BS&H, College of E&ME, National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Received: 11 January 2025 Revised: 24 February 2025 Accepted: 07 March 2025 Published: 27 March 2025
MSC : 60H35, 60H20, 60H10, 62L20

This article investigates fractional stochastic functional differential equations (FSFDEs) with a non-Lipschitz condition. The analysis explores the boundedness of solutions. Within this framework, results on the existence and uniqueness of solutions are presented. Furthermore, we derive error estimates between the Picard approximate solutions $ y^n(t), \, \, n\geq 1 $, and the exact solution $ y(t) $. Finally, it is demonstrated that the solutions exhibit mean square stability. To illustrate the applicability of the proposed theory, a detailed example is presented.
- stability,
- existence-uniqueness,
- boundedness,
- error estimates,
- fractional SFDEs
Citation: Rahman Ullah, Muhammad Farooq, Faiz Faizullah, Maryam A Alghafli, Nabil Mlaiki. Fractional stochastic functional differential equations with non-Lipschitz condition[J]. AIMS Mathematics, 2025, 10(3): 7127-7143. doi: 10.3934/math.2025325

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Abstract

This article investigates fractional stochastic functional differential equations (FSFDEs) with a non-Lipschitz condition. The analysis explores the boundedness of solutions. Within this framework, results on the existence and uniqueness of solutions are presented. Furthermore, we derive error estimates between the Picard approximate solutions $ y^n(t), \, \, n\geq 1 $, and the exact solution $ y(t) $. Finally, it is demonstrated that the solutions exhibit mean square stability. To illustrate the applicability of the proposed theory, a detailed example is presented.

References

[1]	M. Kijima, Stochastic processes with applications to finance, Chapman and Hall/CRC, 2002. https://doi.org/10.1201/b14785
[2]	H. Tahir, A. Din, K. Shah, B. Abdalla, T. Abdeljawad, Advances in stochastic epidemic modeling: Tackling worm transmission in wireless sensor networks, Math. Comp. Model. Dyn. Syst., 30 (2024), 658–682. https://doi.org/10.1080/13873954.2024.2396480 doi: 10.1080/13873954.2024.2396480
[3]	M. Sarwar, S. Hussain, K. Abodayeh, S. Moonsuwan, T. Sitthiwirattham, Controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via Atangana-Baleanu Caputo fractional derivative, Alex. Eng. J., 94 (2024), 149–158. https://doi.org/10.1016/j.aej.2024.03.022 doi: 10.1016/j.aej.2024.03.022
[4]	K. Diethelm, N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 2 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
[5]	J. Liouville, Sur le calcul des differentielles a indices quelconques, J. Ecole Polytechnique, 1832.
[6]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Ann. Geophys., 19 (1966), 383–393.
[7]	K. B. Oldham, J. Spanier, The fractional calculus: theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
[8]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[9]	B. $\emptyset$ksendal, Stochastic differential equations: An introduction with applications, Springer Science & Business Media, 2013.
[10]	F. Biagini, Y. Hu, B. $\emptyset$ksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications, Springer Science & Business Media, 2008.
[11]	J. Z. Zhang, G. A. Chechkin, On fractional stochastic differential equations, Chaos Soliton. Fract., 102 (2017), 29–38.
[12]	A. Ali, K. Hayat, A. Zahir, K. Shah, T. Abdeljawad, Qualitative analysis of fractional stochastic differential equations with variable order fractional derivative, Qual. Theory Dyn. Syst., 23 (2024), 120. https://doi.org/10.1007/s12346-024-00982-5 doi: 10.1007/s12346-024-00982-5
[13]	X. M. Zhang, P. Agarwal, Z. H. Liu, H. Peng, F. You, Y. J. Zhu, Existence and uniqueness of solutions for stochastic differential equations of fractional-order $q > 1$ with finite delays, Adv. Differ. Equ., 2017 (2017). https://doi.org/10.1186/s13662-017-1169-3 doi: 10.1186/s13662-017-1169-3
[14]	A. Saci, R. Amei, B. Hacene, K. Omar, Fractional stochastic differential equations driven by G-Brownian motion with delay, Probab. Math. Stat., 43 (2023), 1–21, https://doi.org/10.37190/0208-4147.00092 doi: 10.37190/0208-4147.00092
[15]	J. Z. Huang, D. F. Luo, Relatively exact controllability of fractional stochastic delay system driven by Levy noise, Math. Meth. Appl. Sci., 46 (2023), 11188–11211. https://doi.org/10.1002/mma.9175 doi: 10.1002/mma.9175
[16]	J. Zou, D. F. Luo, A new result on averaging principle for Caputo-type fractional delay stochastic differential equations with Brownian motion, Appl. Anal., 103 (2024), 1397–1417. https://doi.org/10.1080/00036811.2023.2245845 doi: 10.1080/00036811.2023.2245845
[17]	W. W. Mohammed, C. Cesarano, F. M. Al-Askar, Solutions to the (4+1)-dimensional time-fractional Fokas equation with M-Truncated derivative, Mathematics, 11 (2023), 194. https://doi.org/10.3390/math11010194 doi: 10.3390/math11010194
[18]	F. Z. Wang, I. Ahmad, H. Ahmad, M. D. Alsulami, K. S. Alimgeer, C. Cesarano, et al., Meshless method based on RBFs for solving three-dimensional multi-term time fractional PDEs arising in engineering phenomenons, J. King Saud Univ. Sci., 33 (2021). https://doi.org/10.1016/j.jksus.2021.101604 doi: 10.1016/j.jksus.2021.101604
[19]	H. P. Ye, J. M. Gao, Y. S. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075–1081. https://doi.org/10.1016/j.jmaa.2006.05.061 doi: 10.1016/j.jmaa.2006.05.061
[20]	X. Mao, Stochastic differential equations and their Applications. 2nd edition, Elsevier, 2007.
[21]	A. Columbu, R. D. Fuentes, S. Frassu, Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction-repulsion chemotaxis models with logistics, Nonlinear Anal., 79 (2024), 104135. https://doi.org/10.1016/j.nonrwa.2024.104135 doi: 10.1016/j.nonrwa.2024.104135
[22]	T. X. Li, S. Frassu, G. Viglialoro, Combining effects ensuring boundedness in an attraction-repulsion chemotaxis model with production and consumption, Z. Angew. Math. Phys., 74 (2023), 109. https://doi.org/10.1007/s00033-023-01976-0 doi: 10.1007/s00033-023-01976-0
[23]	R. Ullah, F. Faizullah, N. U. Islam, The Caratheodory approximation scheme for stochastic differential equations with G-Levy process, Math. Meth. Appl. Sci., 46 (2023), 14120–14130. https://doi.org/10.1002/mma.9308 doi: 10.1002/mma.9308
[24]	R. Ullah, F. Faizullah, On existence and approximate solutions for stochastic differential equations in the framework of G-Brownian motion, Eur. Phys. J. Plus, 132 (2017), 435–443. https://doi.org/10.1140/epjp/i2017-11700-9 doi: 10.1140/epjp/i2017-11700-9
[25]	U. Rahmam, F. Faizullah, I. Ali, M. Farooq, M. A. Rana, F. A. Awad, On the existence-uniqueness and exponential estimate for solutions to stochastic functional differential equations driven by G-Lévy process, Adv. Cont. Discr. Mod., 2025 (2025). https://doi.org/10.1186/s13662-024-03856-x doi: 10.1186/s13662-024-03856-x
[26]	F. Faizullah, On boundedness and convergence of solutions for neutral stochastic functional differential equations driven by G-Brownian motion, Adv. Differ. Equ., 2019 (2019). https://doi.org/10.1186/s13662-019-2218-x doi: 10.1186/s13662-019-2218-x
[27]	F. Faizullah, I. Khan, M. M. Salah, Z. A. Alhussain, Estimates for the difference between approximate and exact solutions to stochastic differential equations in the G-framework, J. Taibah Univ. Sci., 13 (2018), 20–26. https://doi.org/10.1080/16583655.2018.1519884 doi: 10.1080/16583655.2018.1519884
[28]	X. Long, S. H. Gong, New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays, Appl. Math. Lett., 100 (2020), 106027. https://doi.org/10.1016/j.aml.2019.106027 doi: 10.1016/j.aml.2019.106027

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