Abstract
This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.
Highlights
The differential equation is an important tool to describe the law of development and change of things, and random disturbance is inevitable, many practical problems can be modeled by stochastic differential equations
Many researchers have increased their interests in investigating the stability of neutral stochastic functional differential equations (NSFDEs) with delays, for instance, Chen [12] discussed the exponential stability in pth moment and almost surely exponential stability of mild solutions for NSFDEs
Zhang and Ruan [14] studied the existence, uniqueness and exponential stability in the pth moment of mild solution for NSFDEs driven by mixed fractional Brownian motion
Summary
The differential equation is an important tool to describe the law of development and change of things, and random disturbance is inevitable, many practical problems can be modeled by stochastic differential equations. Zhang and Ruan [14] studied the existence, uniqueness and exponential stability in the pth moment of mild solution for NSFDEs driven by mixed fractional Brownian motion. Ma et al [23] discussed the existence, uniqueness and mean-square exponential stability of a mild solution of impulsive neutral stochastic integro-differential equation. As a Gaussian stochastic process, fractional Brownian motion heavily relies on the Hurst index H ∈ (0, 1) introduced by Kolmogorov [27], it is an effective tool in modelling many stochastic systems since its stationary increments and long-range dependence properties. To the best of our knowledge, there is no paper considering the exponential stability of impulsive neutral stochastic integro-differential equation driven by mixed fractional Brownian motion. We consider the following impulsive neutral stochastic integro-differential equation driven by mixed fractional Brownian motion:.
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