A new investigation of impulsive fractional stochastic delayed systems in the framework of $ (\delta, \psi) $-Hilfer derivative and Lévy processes

A. M. Sayed Ahmed; Hamdy M. Ahmed; Taher A. Nofal; Soliman Alkhatib; Hisham H. Hussein; A. M. Sayed Ahmed; Hamdy M. Ahmed; Taher A. Nofal; Soliman Alkhatib; Hisham H. Hussein

doi:10.3934/math.2025885

AIMS Mathematics

2025, Volume 10, Issue 8: 19845-19866. doi: 10.3934/math.2025885

Previous Article Next Article

Research article Special Issues

A new investigation of impulsive fractional stochastic delayed systems in the framework of $ (\delta, \psi) $-Hilfer derivative and Lévy processes

1.
Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt
3.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia
4.
College of Engineering and Technology, American University in the Emirates (AUE), Dubai International Academic City, P.O. Box 503000, Dubai, UAE
5.
School of Mathematical and Computational Sciences, University of Prince Edward Island (UPEI), Cairo Campus, New Administrative Capital, Egypt

Received: 08 March 2025 Revised: 30 June 2025 Accepted: 29 July 2025 Published: 28 August 2025
MSC : 34A08, 60G22, 34A12, 60H15, 34A37

In this paper, the averaging result for impulsive $ (\delta, \psi) $-Hilfer fractional stochastic delayed differential equations (FSDDEs) caused by the Lévy process was derived. In the sense of mean square, the relationship between the equivalent solutions of the original equations and the averaged equation solutions was demonstrated. Our findings allowed us to shift our attention from the original, more complicated system to the averaged system. Additionally, to demonstrate the relevance and practicality of our findings, an example was given.
- stochastic system,
- generalized Hilfer fractional derivative,
- averaging principle,
- mild solutions
Citation: A. M. Sayed Ahmed, Hamdy M. Ahmed, Taher A. Nofal, Soliman Alkhatib, Hisham H. Hussein. A new investigation of impulsive fractional stochastic delayed systems in the framework of $ (\delta, \psi) $-Hilfer derivative and Lévy processes[J]. AIMS Mathematics, 2025, 10(8): 19845-19866. doi: 10.3934/math.2025885

Related Papers:

Abstract

In this paper, the averaging result for impulsive $ (\delta, \psi) $-Hilfer fractional stochastic delayed differential equations (FSDDEs) caused by the Lévy process was derived. In the sense of mean square, the relationship between the equivalent solutions of the original equations and the averaged equation solutions was demonstrated. Our findings allowed us to shift our attention from the original, more complicated system to the averaged system. Additionally, to demonstrate the relevance and practicality of our findings, an example was given.

References

[1]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000. https://doi.org/10.1142/3779
[2]	H. A. Fallahgoul, S. M. Focardi, F. J. Fabozzi, Fractional calculus and fractional processes with applications to financial economics: theory and application, London, UK: Academic Press, 2017.
[3]	X. Li, Z. Liu, J. Li, C. Tisdell, Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces, Acta Math. Sci., 39 (2019), 229–242. https://doi.org/10.1007/s10473-019-0118-5 doi: 10.1007/s10473-019-0118-5
[4]	R. Magin, Fractional calculus in bioengineering, Danbury, CT, USA: Begell House, 2004. http://doi.org/10.1615/CritRevBiomedEng.v32.i1.10
[5]	H. M. Ahmed, Controllability for Sobolev-type fractional integro-differential systems in a Banach space, Adv. Differ. Equ., 2012 (2012), 1–10. https://doi.org/10.1186/1687-1847-2012-167 doi: 10.1186/1687-1847-2012-167
[6]	S. Mukhtar, R. Mukhtar, S. Noor, The numerical investigation of a fractional-order multi-dimensional model of Navier–Stokes equation via novel techniques, Symmetry, 14 (2022), 1102. https://doi.org/10.3390/sym14061102 doi: 10.3390/sym14061102
[7]	J. Kongson, W. Sudsutad, C. Thaiprayoon, J. Alzabut, C. Tearnbucha, On analysis of a nonlinear fractional system for social media addiction involving Atangana–Baleanu–Caputo derivative, Adv. Differ. Equ., 2021 (2021), 356. https://doi.org/10.1186/s13662-021-03515-5 doi: 10.1186/s13662-021-03515-5
[8]	I. Podlubny, Fractional differential equations, New York, NY, USA: Academic Press, 1999.
[9]	K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, New York, NY, USA: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[10]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014. https://doi.org/10.1142/9069
[11]	J. V. C. Sousa, E. C. de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
[12]	K. D. Kucche, A. D. Mali, On the nonlinear $(k, \psi)$-Hilfer fractional differential equations, Chaos Soliton. Fract., 152 (2021), 111335. https://doi.org/10.1016/j.chaos.2021.111335 doi: 10.1016/j.chaos.2021.111335
[13]	J. T. Machado, F. Mainardi, V. Kiryakova, Fractional calculus: Quo vadimus? (Where are we going?), Fract. Calc. Appl. Anal., 18 (2015), 495–526. https://doi.org/10.1515/fca-2015-0031 doi: 10.1515/fca-2015-0031
[14]	T. M. Atanackovic, S. Pilipovic, B. Stankovic, D. Zorica, Fractional calculus with applications in mechanics: vibrations and diffusion processes, Wiley-ISTE, 2014. https://doi.org/10.1002/9781118577530
[15]	I. Nourdin, Selected aspects of fractional Brownian motion, Milano: Springer, 2012. https://doi.org/10.1007/978-88-470-2823-4
[16]	J. Liu, L. Yan, Y. Cang, On a jump-type stochastic fractional partial differential equation with fractional noises, Nonlinear Anal. Real World Appl., 75 (2012), 6060–6070. https://doi.org/10.1016/j.na.2012.06.012 doi: 10.1016/j.na.2012.06.012
[17]	K. Li, Stochastic delay fractional evolution equations driven by fractional Brownian motion, Math. Methods Appl. Sci., 38 (2015), 1582–1591. https://doi.org/10.1002/mma.3169 doi: 10.1002/mma.3169
[18]	L. Xu, Z. Li, Stochastic fractional evolution equations with fractional Brownian motion and infinite delay, Appl. Math. Comput., 336 (2018), 36–46. https://doi.org/10.1016/j.amc.2018.04.060 doi: 10.1016/j.amc.2018.04.060
[19]	A. M. Sayed Ahmed, Existence and uniqueness of mild solutions to neutral impulsive fractional stochastic delay differential equations driven by both Brownian motion and fractional Brownian motion, Differ. Equ. Appl., 14 (2022), 433–446. https://dx.doi.org/10.7153/dea-2022-14-30 doi: 10.7153/dea-2022-14-30
[20]	A. M. Sayed Ahmed, H. M. Ahmed, Non-instantaneous impulsive Hilfer–Katugampola fractional stochastic differential equations with fractional Brownian motion and Poisson jumps, J. Control Decis., 11 (2024), 317–327. https://doi.org/10.1080/23307706.2023.2171920 doi: 10.1080/23307706.2023.2171920
[21]	A. M. Sayed Ahmed, Stochastic delayed fractional-order differential equations driven by fractional Brownian motion, Malaya J. Math., 10 (2022), 187–197. https://doi.org/10.26637/mjm1003/001 doi: 10.26637/mjm1003/001
[22]	A. M. Sayed Ahmed, H. M. Ahmed, Hilfer–Katugampola fractional stochastic differential equations with nonlocal conditions, Int. J. Nonlinear Anal. Appl., 14 (2023), 1205–1214. https://doi.org/10.22075/ijnaa.2022.27337.3562 doi: 10.22075/ijnaa.2022.27337.3562
[23]	H. M. Ahmed, A. M. Sayed Ahmed, M. A. Ragusa, On some non-instantaneous impulsive differential equations with fractional Brownian motion and Poisson jumps, TWMS J. Pure Appl. Math., 14 (2023), 125–140. https://doi.org/10.30546/2219-1259.14.1.2023.125 doi: 10.30546/2219-1259.14.1.2023.125
[24]	L. Lu, Z. Liu, Existence and controllability results for stochastic fractional evolution hemivariational inequalities, Appl. Math. Comput., 268 (2015), 1164–1176. https://doi.org/10.1016/j.amc.2015.07.023 doi: 10.1016/j.amc.2015.07.023
[25]	R. Kasinathan, R. Kasinathan, D. Chalishajar, D. Kasinathan, Various controllability results for Fredholm–Volterra type stochastic elastic damped integro-differential systems with applications, Int. J. Dynam. Control, 13 (2025), 117. https://doi.org/10.1007/s40435-025-01618-5 doi: 10.1007/s40435-025-01618-5
[26]	D. Chalishajar, D. Kasinathan, R. Kasinathan, R. Kasinathan, Viscoelastic Kelvin–Voigt model on Ulam–Hyers stability and T-controllability for a coupled integro-fractional stochastic systems, Chaos Soliton. Fract., 191 (2025), 115785. https://doi.org/10.1016/j.chaos.2024.115785 doi: 10.1016/j.chaos.2024.115785
[27]	D. Kasinathan, R. Kasinathan, R. Kasinathan, D. Chalishajar, D. Baleanu, Dynamical behaviors of fractional neutral stochastic integro-differential delay systems with impulses, Bound. Value Probl., 2025 (2025), 49. https://doi.org/10.1186/s13661-025-02037-3 doi: 10.1186/s13661-025-02037-3
[28]	M. L. Deng, Q. F. Lü, W. Q. Zhu, Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index $H \in (1/2, 1)$, Int. J. Nonlinear Mech., 98 (2018), 43–50. https://doi.org/10.1016/j.ijnonlinmec.2017.10.004 doi: 10.1016/j.ijnonlinmec.2017.10.004
[29]	M. A. Ragusa, On weak solutions of ultraparabolic equations, Nonlinear Anal. Real World Appl., 47 (2001), 503–511. https://doi.org/10.1016/S0362-546X(01)00195-X doi: 10.1016/S0362-546X(01)00195-X
[30]	P. Balasubramaniam, Hilfer fractional stochastic system driven by mixed Brownian motion and Lévy noise suffered by non-instantaneous impulses, Stoch. Anal. Appl., 41 (2023), 60–79. https://doi.org/10.1080/07362994.2021.1990082 doi: 10.1080/07362994.2021.1990082
[31]	Y. Xu, B. Pei, G. Guo, Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise, Appl. Math. Comput., 263 (2015), 398–409. https://doi.org/10.1016/j.amc.2015.04.070 doi: 10.1016/j.amc.2015.04.070
[32]	D. Chalishajar, D. Kasinathan, R. Kasinathan, R. Kasinathan, Ulam–Hyers–Rassias stability of Hilfer fractional stochastic impulsive differential equations with non-local condition via time-changed Brownian motion, Chaos Soliton. Fract., 197 (2025), 116468. https://doi.org/10.1016/j.chaos.2025.116468 doi: 10.1016/j.chaos.2025.116468
[33]	H. M. Ahmed, Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Appl. Math. Lett., 112 (2021), 106755. https://doi.org/10.1016/j.aml.2020.106755 doi: 10.1016/j.aml.2020.106755
[34]	D. F. Luo, Q. X. Zhu, Z. G. Luo, A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients, Appl. Math. Lett., 122 (2021), 107549. https://doi.org/10.1016/j.aml.2021.107549 doi: 10.1016/j.aml.2021.107549
[35]	G. Shen, J. L. Wu, R. Xiao, X. Yin, An averaging principle for neutral stochastic fractional-order differential equations with variable delays driven by Lévy noise, Stoch. Dyn., 22 (2021), 2250009. https://doi.org/10.1142/S0219493722500095 doi: 10.1142/S0219493722500095
[36]	Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Lévy processes, IEEE Trans. Autom. Control, 70 (2025), 1176–1183. https://doi.org/10.1109/TAC.2024.3448128 doi: 10.1109/TAC.2024.3448128
[37]	D. Applebaum, Lévy processes and stochastic calculus, Cambridge, UK: Cambridge University Press, 2009. https://doi.org/10.1017/CBO9780511809781

Reader Comments

Your name:*

Email:*
© 2025 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)