Ulam–Hyers Stability for a Class of Caputo-Type Fractional Stochastic System with Delays

Abstract

In this study, we explore the Caputo-type fractional stochastic differential equations (FSDEs) with delays. Firstly, the existence and uniqueness of the solutions for the considered system are obtained under the non-Lipschitz condition by using the Carath $\acute{e}$ odory approximation method with the aid of Fatous Lemma, It $\widehat{o}$ isometry, Cauchy–Schwarz inequality, and Gronwall’s inequality. Secondly, we study the Ulam–Hyers (U–H) stability of the addressed equations with the aid of the Cauchy–Schwarz inequality and Gronwall’s inequality. Finally, an example with numerical simulation is presented to support the validity of our theory.