Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process

Source abstract

This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effects, and stochastic perturbations exhibiting long-range dependence. The primary objective is to establish sufficient conditions ensuring the existence and uniqueness of mild solutions for the proposed system. To achieve this objective, the theory of resolvent operators is combined with stochastic analysis techniques and the Banach fixed point theorem. In particular, an appropriate operator is constructed from the mild solution formulation, and suitable conditions are imposed to guarantee its contractive property. Consequently, the existence and uniqueness of a mild solution are established. The obtained results extend and generalize several existing results for fractional differential and stochastic systems to the discrete fractional setting involving Rosenblatt stochastic processes. The proposed framework effectively accounts for the combined influence of fractional memory, delays, impulsive effects, and long-range-dependent stochastic disturbances. Furthermore, an application to a class of impulsive stochastic partial fractional difference equations is presented to demonstrate the applicability and effectiveness of the theoretical results. The application verifies that the established assumptions can be satisfied in a relevant stochastic fractional model. Thus, the results contribute to the qualitative theory of neutral stochastic fractional difference equations and provide a useful framework for the analysis of discrete-time stochastic systems with memory, delay, impulsive phenomena, and long-range dependence. These findings may also serve as a basis for further investigations of more general stochastic fractional difference systems.