Long time behavior of fractional impulsive stochastic differential equations with infinite delay

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both K-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Fixed points and exponential stability for stochastic partial integro–differential equations with delays

In this paper, we study the existence and asymptotic stability in the pth-moment of mild solutions of nonlinear impulsive stochastic partial functional integro-differential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given by Grimmer in [9] , and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is used to achieve the required result. An example is provided to illustrate the abstract results in this work