Existence and Hyers–Ulam Stability of Stochastic Delay Systems Governed by the Rosenblatt Process

Abstract

Under the effect of the Rosenblatt process, time-delay systems of nonlinear stochastic delay differential equations are considered. Utilizing the delayed matrix functions and exact solutions for these systems, the existence and Hyers–Ulam stability results are derived. First, depending on the fixed point theory, the existence and uniqueness of solutions are proven. Next, sufficient criteria for the Hyers–Ulam stability are established. Ultimately, to illustrate the importance of the results, an example is provided.

Keywords

• Hyers–Ulam stability
• stochastic delay system
• Rosenblatt process
• delayed matrix function
• Krasnoselskii’s fixed point theorem