%0 Journal Article
%T Existence for a Second-Order Impulsive Neutral Stochastic Integrodifferential Equations with Nonlocal Conditions and Infinite Delay
%A Dang Huan Diem
%J Chinese Journal of Mathematics
%D 2014
%R 10.1155/2014/143860
%X The current paper is concerned with the existence of mild solutions for a class of second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays in a Hilbert space. A sufficient condition for the existence results is obtained by using the Krasnoselskii-Schaefer-type fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. Finally, an application to the stochastic nonlinear wave equation with infinite delay is given. 1. Introduction The theory of impulsive neutral differential equations has been emerging as an important area of investigation in recent years, stimulated by their numerous applications to problems from physics, mechanics, electrical engineering, medicine biology, ecology, and so on. Ordinary differential equations of first and second order with impulses have been treated in several works and we refer the reader to the monographs of Lakshmikantham et al. [1], the papers [2¨C5], and the references therein related to this matter. Besides, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of great significance to import the stochastic effects into the investigation of impulsive neutral differential equations. As the generalization of classic impulsive neutral differential equations, impulsive neutral stochastic integrodifferential equations with infinite delays have attracted the researchersĄŻ great interest. There are few publications on well-posedness of solutions for these equations (e.g., see, [6¨C8] and the references therein). Recently, in [9], Cui and Yan proved sufficient conditions for the existence of fractional neutral stochastic integrodifferential equations with infinite delay of the form where and denotes the Caputo fractional derivative operator of order by means of Sadovskii's fixed point theorem. And very recently, also thanks to the Sadovskii fixed point theorem combined with a noncompact condition on the cosine family of operators, Arthi and Balachandran [10] established the controllability of the following damped second-order impulsive neutral functional differential systems with infinite delay: where is a bounded linear operator on a Banach space with . On the other hand, there has not been very much study of second-order impulsive neutral stochastic functional differential equations with infinite delays, while these have begun to gain attention recently. To be more precise, in [11], Balasubramaniam and Muthukumar discussed on approximate
%U http://www.hindawi.com/journals/cjm/2014/143860/