The Strict Stability of Impulsive Stochastic Functional Differential Equations with Markovian Switching

Authors: Dezhi Liu Guiyuan Yang Wei Zhang

Abstract:

Strict stability can present the rate of decay of the solution, so more and more investigators are beginning to study the topic and some results have been obtained. However, there are few results about strict stability of stochastic differential equations. In this paper, using Lyapunov functions and Razumikhin technique, we have gotten some criteria for the strict stability of impulsive stochastic functional differential equations with markovian switching.

Keywords: Impulsive; Stochastic functional differential equation; Strict stability; Razumikhin technique.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1085439

References:

[1] X.Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chischester, UK, 1997.
[2] V.Lakshmikantham, S.Leea, Differential and Integral Inequalities, Academic Press, New York, 1969.
[3] V.Lakshmikantham, R.N.Mohapatra, Strict stability of differential equations, Nonlinear Anal., 46(2001)915-921.
[4] V.Lakshmikantham, D.Bainov, P.Simeonov, theory of impulsive differential equations, World Scientific, Singapore, 1989.
[5] Yu Zhang, Jitao Sun, Strict stability of impulsive functional differential equations, J.Math.Anal.Appl., 301(2005)237-248.
[6] Zhiguo Yang, Daoyi Xu, Li Xiang, exponential p-stability of impulsive stochastic differential equations with delay, Physics Leters A. 359(2006)129-137.
[7] Shijin Wu, Dong Han, Xianzhang Meng, p-moment stability of stochastic differential equations with jumps, Applied Mathematics and computation, 152(2004)505-519.
[8] X.Mao, Stability of stochastic differential equations with Markovian switching, Stoc.Proc.Appl. 79(1)(1999)45-67.
[9] Yumin Zhang, Yi Shen, Xiaoxin Liao, expenential stability of uncertain hopfied neutral networks with discrete and distributed delays, Physics Leters A., 354(4)(2006)288-297.
[10] Y.Zhang, J.Sun, Stability of impulsive functional differential equations, Nonlinear Analysis 68(12)(2007),3665-3678.
[11] S.R.Bernfeld, C.Corduneanu, A.O.Ignatyev, On the stability of invariant sets of functional equations, Nonlinear Analysis 55(2003)641-656.
[12] A.A. Martynyuk, Matrix-valued functionals approach for stability analysis of functional differential equations, Nonlinear Analysis 56(2004)793- 802.
[13] V.Lakshmikantham, Uniform asymptotic stability criteria for functional differential equations in terms of two measures, Nonlinear Analysis 34(1)(1998)1-6.
[14] JH Shen, Razumikhin techniques in impulsive functional differential equations, Nonlinear Analysis 36(1)(1999)119-130.
[15] Z Luo,J Shen, New Razumikhin type theorems for impulsive functional differential equations, Appl.Math.Comput. 125(2002)375-386.