In this paper, we investigate the controllability for a class of nonlocal second-order impulsive neutral stochastic integro-differential equations with infinite delay in Hilbert spaces. More precisely, a set of sufficient conditions for the controllability results of nonlocal second-order impulsive neutral stochastic integro-differential equations with infinite delay are derived by means of the Banach fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. As an application, an example is provided to illustrate the obtained theory.

a Communicated by T. WannerIn this study, we investigate the existence of mild solutions for a class of impulsive neutral stochastic integro-differential equations with infinite delays, using the Krasnoselskii-Schaefer type fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result.

The current paper is concerned with the controllability of nonlocal secondorder impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. Using the theory of a strongly continuous cosine family of bounded linear operators, stochastic analysis theory and with the help of the Banach fixed point theorem, we derive a new set of sufficient conditions for the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps. Finally, an application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given.

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.