Existence and Exponential Stability of Solutions to Stochastic Neutral Functional Differential Equations

Ling Hu,Zhangzhi Wei,Zheng Wu,Lianglong Wang

doi:10.1155/2017/2623035

Abstract

In this paper we consider the existence and stability of solutions to stochastic neutral functional differential equations with finite delays. Under suitable conditions, the existence and exponential stability of solutions were obtained by using the semigroup approach and Banach fixed point theorem.

Highlights

IntroductionStochastic phenomena are everywhere. Many stochastic facts exist in biology, chemistry, physics, and economical systems
In natural world, stochastic phenomena are everywhere
Delays appear sometimes to change the results. These facts imply the necessity to study stochastic functional differential equations (SFDEs), there are a lot of papers on the related topics for deterministic partial functional differential equations [1,2,3,4]

Summary

Introduction

Stochastic phenomena are everywhere. Many stochastic facts exist in biology, chemistry, physics, and economical systems. Delays appear sometimes to change the results These facts imply the necessity to study stochastic functional differential equations (SFDEs), there are a lot of papers on the related topics for deterministic partial functional differential equations [1,2,3,4]. Stochastic partial FDEs with finite delays seem very important, the corresponding properties of these systems have not been studied in great detail. We shall discuss the existence and uniqueness of mild solutions to a class of stochastic NFDEs with finite delays, d [X (t) − f (t, Xt)] = [−AX (t) + F (t, Xt)] dt. We obtain the almost sure exponential stability of the solutions

Preliminaries

Existence of Mild Solutions

Pth Moment Exponential Stability