Asymptotic behavior, attracting and quasi-invariant sets for impulsive neutral SPFDE driven by Lévy noise

Abstract

By establishing two new impulsive-integral inequalities, the attracting and quasi-invariant sets of the mild solution for impulsive neutral stochastic partial functional differential equations driven by Lévy noise are obtained, respectively. Moreover, we shall derive some sufficient conditions to ensure stability of this mild solution in the sense of both moment exponential stability and almost surely exponential stability.

Asymptotic behavior of attracting and quasi-invariant sets of impulsive stochastic partial integrodifferential equations with delays and Poisson jumps

“…If Ψ Q < ∞, then Ψ is known as Q-Hilbert Schmidt operator. For more details on concepts and theory on SDEs, one can refer to the articles [1,4,9,13,21] and references therein.…”

“…Li [17] discussed the global attracting set and quasi-invariant set of impulsive neutral stochastic functional partial differential equations driven by fractional Brownian motion (fBm). In very recent years, a few works have been done on the qualitative properties of solutions for attracting set and quasi-invariant set of SDEs system, readers are referred to [6,10,13,22].…”

“…On the other hand, the Poisson jumps have become a very popular in recent years, the Poisson jumps are generally based on the Poisson random measure in aspects of applications in many real life phenomena such as biology, finance and other field of science, see [14,21]. Hence the analysis of the qualitative behavior of SDEs with Poisson jumps has attracted many researchers in recent years, see [1,12,13]. That is to say, there is no study on global attracting set and exponential stability of impulsive SDEs with Poisson jumps in the existing literature.…”

Controllability of impulsive stochastic functional integrodifferential equations driven by Rosenblatt process and Lévy noise

“…Lakhel et al [33] showed the existence, uniqueness, and asymptotic behavior of mild solutions for a family of neutral functional stochastic differential equations with fBm and Poisson jumps. Huan and Agarwal [25] found attractive and quasi-invariant sets of the mild solution for impulsive neutral stochastic PDEs driven by Levy noise. Sakthivel and Ren [56], addressed the complete controllability of stochastic evolution equations with jumps in a separable Hilbert space, while in, Ren et al [53] studied the approximate controllability of stochastic differential systems driven by Teugels martingales coupled with a Levy process.…”