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Existence and exponential stability of mild solutions for second-order neutral stochastic functional differential equation with random impulses
Linxin Shu,Xiao-Bao Shu,Quanxin Zhu,Fei Xu
Keywords:Existence of mild solution, Exponential stability, Second-order neutral stochastic functional differential equation, Random impulsive, It^o integrals, Measurement of noncompact.
Abstract:
      In this paper, we consider the existence and exponential stability in mean square of mild solutions to second-order neutral stochastic functional differential equations with random impulses in Hilbert space. Firstly, the existence of mild solutions to the equations is proved by using the noncompact measurement strategy and the M"onch fixed point theorem. Then, the mean square exponential stability for the mild solution of the considered equations is obtained by establishing integral inequality. Finally, an example is given to illustrate our results.