Second-order Neutral Stochastic Evoltion Equations With Delay

Differential equations are well known to describe many sophisticated dynamical sys-tems in economics, control theory and bioengineering et al. Taking the environmental disturbances, the delay phenomenon and the abrupt change at certain moments of time et al. into account, the second-order neutral stochastic evolution equations with delay is the right model. The study of abstract deterministic second-order evolution equations governed by the generator of a strongly continuous cosine family was initiated by Travis and Webb in 1978 and subsequently studied by Fattorini et al. in 1985. The study of second-order stochastic differential equations is comparatively late. In 2003, McKibben studied the second-order damped functional stochastic evolution equations, established the existence and uniqueness of mild solutions by means of fixed point theorem under uniform Lispchitz condition and established the existence of mild solutions by means of Schaefer's theorem under sublinear growth condition. Subsequently, McKibben (2004) established the existence and uniqueness of mild solutions for a class of second-order neutral stochastic evolution equations with finite delay under Lipshitz condition and global Caratheodory condition. Very recently(2009), Balasubramaniam and Muthuku-mar proved the existence, uniqueness and the controllability of mild solutions for the second-order neutral stochastic evolution equations with infinite delay by means of the fixed point theorem under Lipschitz condition. Moreover, impulsive effects exist in many evolution processes in which states are changed abruptly at certain moments of time and it has an important influence to the system's behavior. However, so far no work has been reported in the literature about second-order neutral impulsive stochastic evolution equations with delay and second-order neutral stochastic evolution equations with infi-nite delay under Caratheodory conditions. Motivated by the above work, we devoted to study these two kinds of equations.This paper is made up of three chapters:In chapter 1, we briefly review the background, the organization and some prelimi-naries.In Chapter 2, we study the second-order neutral stochastic evolution equations with impulsive effect and delay (SNSEEIDs). We establish the existence and uniqueness of mild solutions for SNSEEIDs under non-Lipschitz condition by means of the successive approximation. Furthermore, we give the continuous dependence of solutions on the ini-tial data by means of corollary of the Bihari inequality. An application to the stochastic nonlinear wave equation with impulsive effect and delay is given to illustrate the the-ory. Especially, the results of McKibben (2004) are generalized. The results have been published in an international journal named Journal of Mathematical Physics.In Chapter 3, we study the second-order neutral stochastic evolution equations with infinite delay (SNSEEIs) under Caratheodory conditions. We establish the existence and uniqueness of mild solutions for SNSEEIs under the global and local Caratheodory conditions by means of the successive approximation. An application to the stochastic nonlinear wave equations with infinite delay is given to illustrate the theory. This part has been accepted by Journal of Optimization Theory and Applications (SCI).