Stability results for neutral fractional stochastic differential equations

Omar Kahouli; Saleh Albadran; Zied Elleuch; Yassine Bouteraa; Abdellatif Ben Makhlouf; Omar Kahouli; Saleh Albadran; Zied Elleuch; Yassine Bouteraa; Abdellatif Ben Makhlouf

doi:10.3934/math.2024158

AIMS Mathematics

2024, Volume 9, Issue 2: 3253-3263. doi: 10.3934/math.2024158

Previous Article Next Article

Research article

Stability results for neutral fractional stochastic differential equations

1.
Department of Electronics Engineering, Applied College, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Electrical Engineering, College of Engineering, University of Ha'il, Ha'il 2440, Saudi Arabia
3.
Department of Computer Science, Applied College, University of Ha'il, Ha'il 2440, Saudi Arabia
4.
Department of Computer Engineering, College of Computer Engineering and sciences, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
5.
Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia

Received: 13 September 2023 Revised: 01 November 2023 Accepted: 10 November 2023 Published: 03 January 2024
MSC : 26A33, 39B82, 60H10

Many techniques have been recently employed by researchers to address the challenges posed by fractional differential equations. In this paper, we investigate the concept of Ulam-Hyers stability for a class of neutral fractional stochastic differential equations by using the Banach fixed point theorem and the stochastic analysis techniques. An example is presented at the end of the paper to show the interest and the applicability of the results.
- fractional calculus,
- stochastic systems
Citation: Omar Kahouli, Saleh Albadran, Zied Elleuch, Yassine Bouteraa, Abdellatif Ben Makhlouf. Stability results for neutral fractional stochastic differential equations[J]. AIMS Mathematics, 2024, 9(2): 3253-3263. doi: 10.3934/math.2024158

Related Papers:

Abstract

Many techniques have been recently employed by researchers to address the challenges posed by fractional differential equations. In this paper, we investigate the concept of Ulam-Hyers stability for a class of neutral fractional stochastic differential equations by using the Banach fixed point theorem and the stochastic analysis techniques. An example is presented at the end of the paper to show the interest and the applicability of the results.

References

[1]	A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos Soliton. Fract., 114 (2018), 478–482. https://doi.org/10.1016/j.chaos.2018.07.032 doi: 10.1016/j.chaos.2018.07.032
[2]	L. Chen, Y. Chai, R. Wu, T. Ma, H. Zha, Dynamic analysis of a class of fractional-order neural networks with delay, Neurocomputing, 111 (2013), 190–194. https://doi.org/10.1016/j.neucom.2012.11.034 doi: 10.1016/j.neucom.2012.11.034
[3]	T. S. Doan, P. T. Huong, P. E. Kloeden, H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stoch. Anal. Appl., 36 (2018), 654–664. https://doi.org/10.1080/07362994.2018.1440243 doi: 10.1080/07362994.2018.1440243
[4]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Vol. 204, Elsevier, 2006.
[5]	R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, ASME J. Appl. Mech., 51 (1984), 299–307. https://doi.org/10.1115/1.3167616 doi: 10.1115/1.3167616
[6]	Y. L. Li, C. Pan, X. Meng, Y. Q. Ding, H. X. Chen, A method of approximate fractional order differentiation with noise immunity, Chemometr. Intell. Lab. Syst., 144 (2015), 31–38. https://doi.org/10.1016/j.chemolab.2015.03.009 doi: 10.1016/j.chemolab.2015.03.009
[7]	C. Li, C. Tao, On the fractional Adams method, Comput. Math. Appl., 58 (2009), 1573–1588. https://doi.org/10.1016/j.camwa.2009.07.050 doi: 10.1016/j.camwa.2009.07.050
[8]	C. Li, F. Zeng, Numerical methods for fractional calculus, 1 Ed., New York: Chapman and Hall/CRC Press, 2015. https://doi.org/10.1201/b18503
[9]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 1999.
[10]	M. Abbaszadeh, A. Khodadadian, M. Parvizi, M. Dehghan, C. Heitzinger, A direct meshless local collocation method for solving stochastic Cahn-Hilliard-Cook and stochastic Swift-Hohenberg equations, Eng. Anal. Bound. Elem., 98 (2019), 253–264. https://doi.org/10.1016/j.enganabound.2018.10.021 doi: 10.1016/j.enganabound.2018.10.021
[11]	A. Khodadadian, M. Parvizi, C. Heitzinger, An adaptive multilevel Monte Carlo algorithm for the stochastic drift-diffusion-Poisson system, Comput. Methods Appl. Mech. Eng., 368 (2020), 113163. https://doi.org/10.1016/j.cma.2020.113163 doi: 10.1016/j.cma.2020.113163
[12]	A. Khodadadian, M. Parvizi, M. Abbaszadeh, M. Dehghan, C. Heitzinger, A multilevel Monte Carlo finite element method for the stochastic Cahn-Hilliard-Cook equation, Comput. Mech., 64 (2019), 937–949. https://doi.org/10.1007/s00466-019-01688-1 doi: 10.1007/s00466-019-01688-1
[13]	M. Abu-Shady, M. K. A. Kaabar, A generalized definition of the fractional derivative with applications, Math. Probl. Eng., 2023 (2023), 9444803. https://doi.org/10.1155/2021/9444803 doi: 10.1155/2021/9444803
[14]	A. Boutiara, M. M. Matar, M. K. A. Kaabar, F. Martínez, S. Etemad, S. Rezapour, Some qualitative analyses of neutral functional delay differential equation with generalized Caputo operator, J. Funct. Spaces, 2021 (2021), 9993177. https://doi.org/10.1155/2021/9993177 doi: 10.1155/2021/9993177
[15]	C. T. Deressa, S. Etemad, M. K. A. Kaabar, S. Rezapour, Qualitative analysis of a hyperchaotic Lorenz-Stenflo mathematical model via the Caputo fractional operator, J. Funct. Spaces, 2022 (2022), 4975104. https://doi.org/10.1155/2022/4975104 doi: 10.1155/2022/4975104
[16]	S. Etemad, M. S. Souid, B. Telli, M. K. A. Kaabar, S. Rezapour, Investigation of the neutral fractional differential inclusions of Katugampola-type involving both retarded and advanced arguments via Kuratowski MNC technique, Adv. Differ. Equations, 2021 (2021), 214. https://doi.org/10.1186/s13662-021-03377-x doi: 10.1186/s13662-021-03377-x
[17]	S. M. Ulam, A collection of mathematical problem, New York: Interscience, 1960.
[18]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[19]	L. Mchiri, A. Ben Makhlouf, H. Rguigui, Ulam-Hyers stability of pantograph fractional stochastic differential equations, Math. Methods Appl. Sci., 46 (2023), 4134–4144. https://doi.org/10.1002/mma.8745 doi: 10.1002/mma.8745
[20]	A. Ahmadova, N. I. Mahmudov, Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations, Stat. Probab. Lett., 168 (2021), 108949. https://doi.org/10.1016/j.spl.2020.108949 doi: 10.1016/j.spl.2020.108949
[21]	A. Ben Makhlouf, L. Mchiri, M. Rhaima, Ulam-Hyers-Rassias stability of stochastic functional differential equations via fixed point methods, J. Funct. Spaces, 2021 (2021), 5544847. https://doi.org/10.1155/2021/5544847 doi: 10.1155/2021/5544847
[22]	N. P. N. Ngoc, Ulam-Hyers-Rassias stability of a nonlinear stochastic integral equation of Volterra type, Differ. Equations Appl., 9 (2009), 183–193. https://doi.org/10.7153/dea-09-15 doi: 10.7153/dea-09-15
[23]	A. Ben Makhlouf, L. Mchiri, M. Rhaima, J. Sallay, Hyers-Ulam stability of Hadamard fractional stochastic differential equations, Filomat, 37 (2023), 10219–10228. https://doi.org/10.2298/FIL2330219B doi: 10.2298/FIL2330219B
[24]	J. B. Diaz, B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74(1968), 305–309.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)