{"title":"The Strict Stability of Impulsive Stochastic Functional Differential Equations with Markovian Switching","authors":"Dezhi Liu Guiyuan Yang Wei Zhang","volume":56,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1438,"pagesEnd":1444,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/15649","abstract":"Strict stability can present the rate of decay of the\r\nsolution, so more and more investigators are beginning to study the\r\ntopic and some results have been obtained. However, there are few\r\nresults about strict stability of stochastic differential equations. In\r\nthis paper, using Lyapunov functions and Razumikhin technique, we\r\nhave gotten some criteria for the strict stability of impulsive stochastic\r\nfunctional differential equations with markovian switching.","references":"[1] X.Mao, Stochastic Differential Equations and Applications, Ellis Horwood,\r\nChischester, UK, 1997.\r\n[2] V.Lakshmikantham, S.Leea, Differential and Integral Inequalities, Academic\r\nPress, New York, 1969.\r\n[3] V.Lakshmikantham, R.N.Mohapatra, Strict stability of differential equations,\r\nNonlinear Anal., 46(2001)915-921.\r\n[4] V.Lakshmikantham, D.Bainov, P.Simeonov, theory of impulsive differential\r\nequations, World Scientific, Singapore, 1989.\r\n[5] Yu Zhang, Jitao Sun, Strict stability of impulsive functional differential\r\nequations, J.Math.Anal.Appl., 301(2005)237-248.\r\n[6] Zhiguo Yang, Daoyi Xu, Li Xiang, exponential p-stability of impulsive\r\nstochastic differential equations with delay, Physics Leters A.\r\n359(2006)129-137.\r\n[7] Shijin Wu, Dong Han, Xianzhang Meng, p-moment stability of stochastic\r\ndifferential equations with jumps, Applied Mathematics and computation,\r\n152(2004)505-519.\r\n[8] X.Mao, Stability of stochastic differential equations with Markovian\r\nswitching, Stoc.Proc.Appl. 79(1)(1999)45-67.\r\n[9] Yumin Zhang, Yi Shen, Xiaoxin Liao, expenential stability of uncertain\r\nhopfied neutral networks with discrete and distributed delays, Physics\r\nLeters A., 354(4)(2006)288-297.\r\n[10] Y.Zhang, J.Sun, Stability of impulsive functional differential equations,\r\nNonlinear Analysis 68(12)(2007),3665-3678.\r\n[11] S.R.Bernfeld, C.Corduneanu, A.O.Ignatyev, On the stability of invariant\r\nsets of functional equations, Nonlinear Analysis 55(2003)641-656.\r\n[12] A.A. Martynyuk, Matrix-valued functionals approach for stability analysis\r\nof functional differential equations, Nonlinear Analysis 56(2004)793-\r\n802.\r\n[13] V.Lakshmikantham, Uniform asymptotic stability criteria for functional\r\ndifferential equations in terms of two measures, Nonlinear Analysis\r\n34(1)(1998)1-6.\r\n[14] JH Shen, Razumikhin techniques in impulsive functional differential\r\nequations, Nonlinear Analysis 36(1)(1999)119-130.\r\n[15] Z Luo,J Shen, New Razumikhin type theorems for impulsive functional\r\ndifferential equations, Appl.Math.Comput. 125(2002)375-386.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 56, 2011"}