Attracting and Invariant Sets of Nonlinear Stochastic Neutral Differential Equations with Delays

Abstract

In this paper, a class of nonlinear stochastic neutral differential equations with delays is investigated. By using the properties of $${\mathcal{M}}$$ -matrix, a differential-difference inequality is established. Basing on the differential-difference inequality, we develop a $${\mathcal{L}}$$ -operator-difference inequality such that it is effective for stochastic neutral differential equations. By using the $${\mathcal{L}}$$ -operator-difference inequality, we obtain the global attracting and invariant sets of nonlinear stochastic neutral differential equations with delays. In addition, we derive the sufficient condition ensuring the exponential p-stability of the zero solution of nonlinear stochastic neutral differential equations with delays. One example is presented to illustrate the effectiveness of our conclusion.