Significant results in the $ \mathrm{p} $th moment for Hilfer fractional stochastic delay differential equations

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

doi:10.3934/math.2025451

AIMS Mathematics

2025, Volume 10, Issue 4: 9852-9881. doi: 10.3934/math.2025451

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Significant results in the $ \mathrm{p} $th moment for Hilfer fractional stochastic 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 Decision Sciences, SK-Hub-Oxford Business College, OX1 2EP, Oxford, UK
4.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
5.
Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia

Received: 14 January 2025 Revised: 26 March 2025 Accepted: 15 April 2025 Published: 27 April 2025
MSC : 34A07, 34A08, 60G22

Well-posedness is crucial in studying fractional stochastic differential equations, as it ensures that solutions are mathematically sound and applicable to practical situations. A well-formulated model satisfies the essential requirements for solutions, such as existence, uniqueness, and stability concerning various parameters. Using fixed-point theory, we prove that the solution to stochastic fractional delay differential equations with the Hilfer fractional operator exists, is unique, and continuously depends on the initial values and the fractional derivative. Additionally, we establish a smoothness theorem for the solution and demonstrate that the solution of the original system converges to the averaged system in the $ \mathrm{p} $th moment. Last, to support our theoretical findings, we provide examples and graphical illustrations. The primary tools used in our proofs include the Burkholder-Davis-Gundy inequality, Jensen's inequality, and Hölder's inequality.
- averaging principle,
- Hilfer fractional derivative,
- well-posedness,
- contraction mapping principle,
- regularity
Citation: Wedad Albalawi, Muhammad Imran Liaqat, Fahim Ud Din, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty. Significant results in the $ \mathrm{p} $th moment for Hilfer fractional stochastic delay differential equations[J]. AIMS Mathematics, 2025, 10(4): 9852-9881. doi: 10.3934/math.2025451

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Abstract

Well-posedness is crucial in studying fractional stochastic differential equations, as it ensures that solutions are mathematically sound and applicable to practical situations. A well-formulated model satisfies the essential requirements for solutions, such as existence, uniqueness, and stability concerning various parameters. Using fixed-point theory, we prove that the solution to stochastic fractional delay differential equations with the Hilfer fractional operator exists, is unique, and continuously depends on the initial values and the fractional derivative. Additionally, we establish a smoothness theorem for the solution and demonstrate that the solution of the original system converges to the averaged system in the $ \mathrm{p} $th moment. Last, to support our theoretical findings, we provide examples and graphical illustrations. The primary tools used in our proofs include the Burkholder-Davis-Gundy inequality, Jensen's inequality, and Hölder's inequality.

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