scholarly journals Stability of Nonlinear Neutral Stochastic Functional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-26 ◽  
Author(s):  
Minggao Xue ◽  
Shaobo Zhou ◽  
Shigeng Hu
Keyword(s):  
Differential Equations ◽  
Growth Condition ◽  
Linear Growth ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Linear Growth Condition ◽  
Highly Nonlinear ◽  
Functional Differential ◽  
New Criteria ◽  
Stochastic Functional

Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

scholarly journals The Almost Sure Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Polynomial Growth Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Lichao Feng ◽  
Shoumei Li
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Growth Condition ◽  
Polynomial Growth ◽  
Linear Growth ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Linear Growth Condition ◽  
Functional Differential ◽  
Stochastic Functional

Stability and boundedness are two of the most important topics in the study of stochastic functional differential equations (SFDEs). This paper mainly discusses the almost sure asymptotic stability and the boundedness of nonlinear SFDEs satisfying the local Lipschitz condition but not the linear growth condition. Here we assume that the coefficients of SFDEs are polynomial or dominated by polynomial functions. We give sufficient criteria on the almost sure asymptotic stability and the boundedness for this kind of nonlinear SFDEs. Some nontrivial examples are provided to illustrate our results.

scholarly journals The Boundedness and Exponential Stability Criterions for Nonlinear Hybrid Neutral Stochastic Functional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaofeng Zong ◽  
Fuke Wu ◽  
Chengming Huang
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Linear Growth ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Neutral Differential Equations ◽  
Linear Growth Condition ◽  
Functional Differential ◽  
Theoretical Results ◽  
Stochastic Functional

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results.

scholarly journals Numerical Solutions of Stochastic Functional Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 141-161 ◽  
Author(s):  
Xuerong Mao
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Convergence Theory ◽  
Stochastic Functional Differential Equations ◽  
True Solution ◽  
Linear Growth Condition ◽  
Mean Square Convergence ◽  
Functional Differential ◽  
Stochastic Functional

AbstractIn this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

scholarly journals New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Pham Huu Anh Ngoc
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Mean Square ◽  
Stochastic Functional Differential Equations ◽  
Novel Approach ◽  
Functional Differential ◽  
New Criteria ◽  
Stochastic Functional

<p style='text-indent:20px;'>Stochastic functional differential equations with infinite delay are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of general stochastic functional differential equations with infinite delay are presented. Illustrative examples are given.</p>

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