Stochastic Volterra Integro-differential Equations Driven by a Fractional Brownian Motion with Delayed Impulses

Abstract

In this paper, the problem of existence of mild solutions for a stochastic Volterra integro-differential equation with delayed impulses and driven by a fractional Brownian motion (Hurst parameter H belong to (o.5,1) ) is investigated. Here, we assume that the delayed impulses are linear and impulsive transients depend on not only their current but also historical states of the system. Utilizing the fixed point theorem combine with fractional power of operators and the semigroup theory, sufficient conditions that guarantee the existence and uniqueness of mild solutions for such equations are obtained. Finally, an example is presented to demonstrate the effectiveness of the proposed results.