• Li, Dingshi ;
  • Xu, Daoyi
  • Received : 2011.03.24
  • Published : 2013.11.01


In this paper, we consider a class of periodic It$\hat{o}$ stochastic delay differential equations by using the properties of periodic Markov processes, and some sufficient conditions for the existence of periodic solution of the delay equations are given. These existence theorems improve the results obtained by It$\hat{o}$ et al. [6], Bainov et al. [1] and Xu et al. [15]. As applications, we study the existence of periodic solution of periodic stochastic logistic equation and periodic stochastic neural networks with infinite delays, respectively. The theorem for the existence of periodic solution of periodic stochastic logistic equation improve the result obtained by Jiang et al. [7].


stochastic differential equations;periodic solution;infinite delay;logistic equation;neural networks


  1. D. D. Bainov and V. B. Kolmanovskii, Periodic solution of stochastic functional differ- ential equations, Math. J. Toyama Univ. 14 (1991), 1-39.
  2. L. E. Bertram and P. E. Sarachik, Stability of Circuits with randomly time-varying parameters, IRE. Trans. Circuit Theory, CT-6, Special supplement, 1959, 260-270.
  3. R. Z. Has'minskii, On the dissipativity of random processes defined by differential equations, Problemy Peredaci Informacii 1 (1965), no. 1, 88-104.
  4. R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Maryland, 1980.
  5. K. Ito, On stochastic differential equations, Mem. Amer. Math. Soc. (1951), no. 4, 51 pp.
  6. K. Ito and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ. 4 (1964), no. 1, 1-75.
  7. D. Q. Jiang and N. Z. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005), no. 1, 164-172.
  8. D. Q. Jiang, N. Z. Shi, and X. Y. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 340 (2008), no. 1, 588-597.
  9. V. B. Kolmanovskii and A. Myshkis, Introduction to the Theory and Application of Functional Differential Equations, London, 1999.
  10. X. X. Liao and X. R. Mao, Exponential stability and instability of stochastic neural networks, Stochastic Anal. Appl. 14 (1996), no. 2, 165-185.
  11. K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations 236 (2007), no. 2, 460-492.
  12. X. R. Mao, Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994.
  13. X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997.
  14. L. Y. Teng, L. Xiang, and D. Y. Xu, Existence-uniqueness of the solution for neutral stochastic functional differential equations, Rocky Mountain Journal of Mathematics (in press).
  15. D. Y. Xu, Y. M. Huang, and Z. G. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst. 24 (2009), no. 3, 1005-1023.
  16. D. Y. Xu and L. G. Xu, New results for studying a certain class of nonlinear delay differential systems, IEEE Trans. Automat. Control 55 (2010), no. 7, 1641-1645.
  17. B. G. Zhang, On the periodic solution of n−dimensional stochastic population models, Stochastic Anal. Appl. 18 (2000), no. 2, 323-331.

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Supported by : National Natural Science Foundation of China