Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

PERIODIC SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS AND APPLICATIONS TO LOGISTIC EQUATION AND NEURAL NETWORKS

  • Li, Dingshi (School of Mathematics Southwest Jiaotong University) ;
  • Xu, Daoyi (Yangtze Center of Mathematics Sichuan University)
  • Received : 2011.03.24
  • Published : 2013.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

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].

Keywords

References

  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. https://doi.org/10.1215/kjm/1250524705
  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. https://doi.org/10.1016/j.jmaa.2004.08.027
  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. https://doi.org/10.1016/j.jmaa.2007.08.014
  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. https://doi.org/10.1080/07362999608809432
  11. K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations 236 (2007), no. 2, 460-492. https://doi.org/10.1016/j.jde.2006.09.024
  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. https://doi.org/10.3934/dcds.2009.24.1005
  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. https://doi.org/10.1109/TAC.2010.2048939
  17. B. G. Zhang, On the periodic solution of n−dimensional stochastic population models, Stochastic Anal. Appl. 18 (2000), no. 2, 323-331. https://doi.org/10.1080/07362990008809671

Cited by

  1. Stationary distribution and periodic solutions for stochastic Holling–Leslie predator–prey systems vol.460, 2016, https://doi.org/10.1016/j.physa.2016.04.037
  2. Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay vol.22, pp.6, 2017, https://doi.org/10.3934/dcdsb.2017127
  3. Stochastic periodic solution for a perturbed non-autonomous predator–prey model with generalized nonlinear harvesting and impulses vol.486, 2017, https://doi.org/10.1016/j.physa.2017.05.058
  4. Periodic solution for a stochastic non-autonomous competitive Lotka–Volterra model in a polluted environment vol.471, 2017, https://doi.org/10.1016/j.physa.2016.12.008
  5. Periodic solution and stationary distribution of stochastic S-DI-A epidemic models 2018, https://doi.org/10.1080/00036811.2016.1257123
  6. Stationary distribution and periodic solution for stochastic predator-prey systems with nonlinear predator harvesting vol.36, 2016, https://doi.org/10.1016/j.cnsns.2015.11.014
  7. Periodic solutions and stationary distribution of mutualism models in random environments vol.460, 2016, https://doi.org/10.1016/j.physa.2016.05.015
  8. The periodic solutions of a stochastic chemostat model with periodic washout rate vol.37, 2016, https://doi.org/10.1016/j.cnsns.2016.01.002
  9. Stochastic periodic solution of a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation vol.44, 2017, https://doi.org/10.1016/j.cnsns.2016.08.013
  10. Stochastic periodic solutions of stochastic differential equations driven by Lévy process vol.430, pp.1, 2015, https://doi.org/10.1016/j.jmaa.2015.04.090
  11. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients vol.37, pp.4, 2016, https://doi.org/10.3934/dcds.2017093
  12. Periodic solutions for a stochastic non-autonomous Holling–Tanner predator–prey system with impulses vol.22, 2016, https://doi.org/10.1016/j.nahs.2016.03.004
  13. Periodic solutions of stochastic coupled systems on networks with periodic coefficients vol.205, 2016, https://doi.org/10.1016/j.neucom.2016.04.031
  14. Stochastic Periodic Solution of a Susceptible-Infective Epidemic Model in a Polluted Environment under Environmental Fluctuation vol.2018, pp.1748-6718, 2018, https://doi.org/10.1155/2018/7360685