Exponential Stability of Higher Order Fractional Neutral Stochastic Differential Equation Via Integral Contractors

The well-posedness results for mild solutions to the fractional neutral stochastic differential system with Rosenblatt process with Hurst index $Ĥ\in \left(\frac{1}{2},1\right)$ is discussed in this article. To demonstrate the results, the concept of bounded integral contractors is combined with the stochastic result and sequencing technique. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the stability results, and the relevant nonlinear function does not have to meet the Lipschitz condition. Furthermore, exponential stability results for neutral stochastic differential systems with Poisson jump have been established. Finally, an application to demonstrate the acquired results is discussed. We demonstrate the fractional Zener model for wave equation obeying the viscoelastic materials as a practical application of the system studied, which is a generalization of classical wave equation.