On impulsive Hilfer fractional stochastic differential system driven by Rosenblatt process

Abstract

This manuscript investigates a class of impulsive Hilfer fractional stochastic differential system driven by Rosenblatt process with index which is a special case of a self-similar process, Hermite processes with stationary increments with long-range dependence. The Hermite process of order 1 is fractional Brownian motion (fBm) and of order 2 is the Rosenblatt process. We establish new criteria to guarantee the existence of mild solutions in the stochastic settings by utilizing the Darbo-Sadovskii fixed point theorem, fractional calculus, Hausdorff measure of noncompactness and evolution operators. The derived result in this article is new in the sense that it generalizes many of the existing results in the literature, more precisely for Rosenblatt process and impulsive case of Hilfer derivative in the stochastic settings. Finally, stochastic partial differential equations are provided to validate the applicability of the derived theoretical results.