The Solvability and Ulam–Hyers Stability of Fractional Non-autonomous Differential Equations Governed by Mixed Brownian Motion

Abstract

The study of the qualitative attributes of fractional non-autonomous stochastic differential systems is rarely available in the literature. This research work investigates the solvability and Ulam–Hyers stability of non-autonomous fractional differential equations involving standard Brownian motion and fractional Brownian motion (fBm). We describe a mild solution of the considered fractional non-autonomous system through the operators generated by a family of closed linear operators and the probability density function. The existence of a mild solution is established by utilizing the successive approximation approach. Furthermore, we investigate the Ulam–Hyers stability of the considered equation. Our findings are established under Lipschitz conditions on the system parameters, leveraging Borel-Cantelli’s Lemma, Gronwall’s, and Chebyshev’s inequalities. Finally, we present an example to validate our results.