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

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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.

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Keywords

• Stochastic fractional calculus
• singular stochastic integral equation
• stochastic fractional-order differential equations
• existence of integrable solution
• continuous dependence
• Hyers-Ulam stability

MSC

•  26A33
•  34A08
•  34A60
•  60H10

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