Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay
Volume 33, Issue 1, pp 17--26
http://dx.doi.org/10.22436/jmcs.033.01.02
Publication Date: November 16, 2023
Submission Date: July 12, 2023
Revision Date: August 08, 2023
Accteptance Date: September 21, 2023
Authors
A. M. A. El-Sayed
- Faculty of Science, Alexandria University, Alexandria, Egypt.
M. Abdurahman
- College of Science, Taibah University, Al-Madinah, Saudi Arabia.
H. A. Fouad
- Faculty of Science, Alexandria University, Alexandria, Egypt.
- College of Science, Taibah University, Al-Madinah, Saudi Arabia.
Abstract
In this paper, we are concerning with the existence of the solution \( \mathfrak{V} \in L_1([0,\tau],L_2(\Omega))\) of the singular stochastic fractional-order integral equation with delay \(\varrho(.) \),
\[
\mathfrak{V}(t) = B(t) t^{\alpha - 1} + \lambda ~ \mathfrak{I}^{\beta} \mathfrak{G}(t,\mathfrak{V}(\varrho (t))), ~~~t\in (0,\tau],
\]
where \(B(t)\) is a given second order mean square stochastic process, \( \lambda \) is a parameter, \(\varrho (t) \leq t\), and \(\mathfrak{G}(t,\mathfrak{V}) \) is a measurable function in \(t \in (0,\tau]\) and satisfies Lipschitz condition on the second argument.
The Hyers-Ulam and generalized Hyers-Ulam-Rassias stability will be proved. Moreover, the continuous dependence of the solution on the process \(B(t)\) and \(\lambda\) will be studied. As applications,
some nonlocal, weighted and nonlocal-weighted integral problems of stochastic fractional-order differential equations will be studied.
Share and Cite
ISRP Style
A. M. A. El-Sayed, M. Abdurahman, H. A. Fouad, Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay, Journal of Mathematics and Computer Science, 33 (2024), no. 1, 17--26
AMA Style
El-Sayed A. M. A., Abdurahman M., Fouad H. A., Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay. J Math Comput SCI-JM. (2024); 33(1):17--26
Chicago/Turabian Style
El-Sayed, A. M. A., Abdurahman, M., Fouad, H. A.. "Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay." Journal of Mathematics and Computer Science, 33, no. 1 (2024): 17--26
Keywords
- Stochastic fractional calculus
- singular stochastic integral equation
- stochastic fractional-order differential equations
- existence of integrable solution
- continuous dependence
- Hyers-Ulam stability
MSC
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