Abstract
The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1972)
Arutunian, N.H., Kolmanovskii, V.B.: Greep Theory of Nonhomogeneous Bodies. Nauka, Moscow (1983)
Atkinson, F.V., Haddock, J.R.: On determining phase space for functional differential equations. Funkc. Ekvacioj 31, 331–347 (1988)
Chen, F.: Global asymptotic stability in n-species nonautonomous Lotka-Volterra competitive systems with infinite delays and feedback control. Appl. Math. Comput. 170, 1452–1468 (2005)
Cuevas, C., Pinto, M.: Asymptotic properties of solutions for a nonautonomous Volterra difference systems with infinite delay. Nonlinear Anal. 44, 671–685 (2001)
Drozdov, A.D., Kolmanovskii, V.B.: Stability in Viscoelasticity. North-Holland, Amsterdam (1994)
Drozdov, A.D.: Stability of equations with aftereffect describing dynamics of viscoelastic bodies. Differ. Equ. 28(2), 187–277 (1992)
Drozdov, A.D., Kolmanovskii, V.B.: Stochastic stability of viscoelastic bars. Stoch. Anal. Appl. 10(3), 265–276 (1992)
Friedman, A.: Stochastic Differential Equations and Application, vol. 2. Academic Press, New York (1976)
Haddock, J.R., Hornor, W.E.: Precompactness and convergence on norm of positive orbits in a certain fading memory space. Funkc. Ekvacioj 31, 349–361 (1988)
Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkc. Ekvacioj 21, 11–41 (1978)
Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen aan den Rijin (1981)
He, X.: The Lyapunov functionals for delay Lotka-Volterra-type models. SIAM J. Appl. Math. 58(4), 1222–1236 (1998)
Kato, J.: Stability problem in functional differential equations with infinite delay. Funkc. Ekvacioj 21, 63–80 (1978)
Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Kluwer Academic, New York (1986)
Kolmanovskii, V.B., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic, Dordrecht (1992)
Kuang, Y.: Global stability for infinite delay Lotka-Volterra type systems. J. Differ. Equ. 103, 473–490 (1989)
Mao, X.: Exponential Stability of Stochastic Differential Equations. Dekker, New York (1994)
Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations. Stochastic Process. Appl. 65, 233–250 (1996)
Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)
Mohammed, S.E.A.: Stochastic Functional Differential Equations. Longmans, Green, New York (1986)
Wei, F., Wang, K.: The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J. Math. Anal. Appl. 331, 516–531 (2007)
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Yang, Z., Zhu, E., Xu, Y. et al. Razumikhin-type Theorems on Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay. Acta Appl Math 111, 219–231 (2010). https://doi.org/10.1007/s10440-009-9542-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-009-9542-1