Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay

Abstract

In this paper, we are concerning with the existence of the solution \( \V \in L_1([0,\tau],L_2(\Omega))\) of the singular stochastic fractional-order integral equation with delay \(\varrho(.) \), \[ \V(t) = B(t) t^{\alpha - 1} + \lambda ~ \I^{\beta} \G(t,\V(\varrho (t))), ~~~t\in (0,\tau], \] where \(B(t)\) is a given second order mean square stochastic process, \( \lambda \) is a parameter, \(\varrho (t) \leq t\), and \(\G(t,\V) \) is a measurable function in \(t \in (0,\tau]\) and satisfies Lipschitz condition on the second argument. %and \(x\in C_{1-\beta}([0,T],L_2(\Omega)) $ will be proved 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.