Qualitative study of Caputo Erdélyi-Kober stochastic fractional delay differential equations

Wedad Albalawi; Muhammad Imran Liaqat; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty; Wedad Albalawi; Muhammad Imran Liaqat; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty

doi:10.3934/math.2025381

AIMS Mathematics

2025, Volume 10, Issue 4: 8277-8305. doi: 10.3934/math.2025381

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Research article

Qualitative study of Caputo Erdélyi-Kober stochastic fractional delay differential equations

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
3.
Department of Mathematics, College of Science and Humanities in Al Kharj, Prince Sattam Bin Abdulaziz University, Al Kharj 11942, Saudi Arabia
4.
Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia

Received: 29 January 2025 Revised: 22 March 2025 Accepted: 27 March 2025 Published: 11 April 2025
MSC : 34A07, 34A08, 60G22

We present new results on the well-posedness and time regularity of solutions to stochastic fractional delay differential equations (SFDDEs) using the Caputo-Erdélyi-Kober fractional derivative. Additionally, we prove the averaging principle. We establish all results in the $ \mathfrak{p} $th moment, which generalizes the case $ \mathfrak{p} = 2 $. First, by applying fixed-point theory (FPT), we prove that the solution exists, is unique, and continuously depends on the initial values as well as the fractional derivative. Second, we establish a smoothness theorem for the solution and demonstrate that the solution of the original system converges to the averaged system in the $ \mathfrak{p} $th moment. Finally, to support our theoretical findings, we present illustrative examples.
- Caputo Erdélyi-Kober derivatives,
- stochastic processes,
- existence and uniqueness,
- inequalities
Citation: Wedad Albalawi, Muhammad Imran Liaqat, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty. Qualitative study of Caputo Erdélyi-Kober stochastic fractional delay differential equations[J]. AIMS Mathematics, 2025, 10(4): 8277-8305. doi: 10.3934/math.2025381

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Abstract

We present new results on the well-posedness and time regularity of solutions to stochastic fractional delay differential equations (SFDDEs) using the Caputo-Erdélyi-Kober fractional derivative. Additionally, we prove the averaging principle. We establish all results in the $ \mathfrak{p} $th moment, which generalizes the case $ \mathfrak{p} = 2 $. First, by applying fixed-point theory (FPT), we prove that the solution exists, is unique, and continuously depends on the initial values as well as the fractional derivative. Second, we establish a smoothness theorem for the solution and demonstrate that the solution of the original system converges to the averaged system in the $ \mathfrak{p} $th moment. Finally, to support our theoretical findings, we present illustrative examples.

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