Exploring mild solution and Ulam–Hyers stability for non-autonomous fractional stochastic differential systems with non-instataneous impulses

In this study, we deal with the non-autonomous fractional impulsive stochastic differential equation for the first time. This article examines the existence of a mild solution and the Ulam–Hyers stability of non-autonomous stochastic fractional differential system driven by fractional Brownian motion involving non-instantaneous impulses. The notion of mild solution for the considered system is explored via the operators induced by the closed linear operator and the probability density function. The existence of solution is investigated by the successive approximation method. We additionally study the Ulam–Hyers stability for the aforementioned system. The results are derived under the Lipschitz condition by using Borel–Cantelli’s lemma, Gronwall’s, and Chebyshev’s inequalities. Finally, two examples are discussed to illustrate the obtained results.