Well-posedness and stability of fractional stochastic integro-differential equations with general memory effects

Elkhateeb S. Aly; Muhammad Imran Liaqat; Saleh Alshammari; Mahmoud El-Morshedy; Elkhateeb S. Aly; Muhammad Imran Liaqat; Saleh Alshammari; Mahmoud El-Morshedy

doi:10.3934/math.2025992

AIMS Mathematics

2025, Volume 10, Issue 9: 22265-22293. doi: 10.3934/math.2025992

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Well-posedness and stability of fractional stochastic integro-differential equations with general memory effects

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
2.
Nanotechnology Research Unit, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
3.
Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New MuslimTown, Lahore 54600, Pakistan
4.
Department of Mathematics College of Science University of Hail Hail 2240, Saudi Arabia
5.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
6.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 17 July 2025 Revised: 10 September 2025 Accepted: 16 September 2025 Published: 26 September 2025
MSC : 34A07, 34A08, 60G22

Most existing research on well-posedness and stability has focused on fractional stochastic differential equations, with relatively fewer studies addressing fractional stochastic integro-differential equations (FSIDEs). In this work, we address this gap by establishing theoretical results on the well-posedness of FSIDEs. In particular, we derive a generalized Grönwall inequality and present results on Ulam-Hyers stability (UHS). Moreover, we extend existing findings by incorporating both the $ \Phi $-Caputo fractional derivative and the $ \mathrm{p} $th moment, thereby unifying and generalizing current results in the literature.
- qualitative analysis,
- delay term,
- fractional calculus,
- stochastic models,
- existence and uniqueness,
- stability
Citation: Elkhateeb S. Aly, Muhammad Imran Liaqat, Saleh Alshammari, Mahmoud El-Morshedy. Well-posedness and stability of fractional stochastic integro-differential equations with general memory effects[J]. AIMS Mathematics, 2025, 10(9): 22265-22293. doi: 10.3934/math.2025992

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Abstract

Most existing research on well-posedness and stability has focused on fractional stochastic differential equations, with relatively fewer studies addressing fractional stochastic integro-differential equations (FSIDEs). In this work, we address this gap by establishing theoretical results on the well-posedness of FSIDEs. In particular, we derive a generalized Grönwall inequality and present results on Ulam-Hyers stability (UHS). Moreover, we extend existing findings by incorporating both the $ \Phi $-Caputo fractional derivative and the $ \mathrm{p} $th moment, thereby unifying and generalizing current results in the literature.

References

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