About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 358362, 14 pages
http://dx.doi.org/10.1155/2012/358362
Research Article

Globally Exponential Stability of Periodic Solutions to Impulsive Neural Networks with Time-Varying Delays

1College of Science, Guilin University of Technology, Guangxi, Guilin 541004, China
2Guizhou Key Laboratory of Economics System, Guizhou College of Finance and Economics, Guizhou, Guiyang 550004, China

Received 12 January 2012; Accepted 26 February 2012

Academic Editor: Josef Diblík

Copyright © 2012 Yuanfu Shao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Tan and Y. Tan, “Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 373–385, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. R. Wu, “Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 562–573, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,” Abstract and Applied Analysis, vol. 2011, Article ID 920412, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. K. Gopalsamy and Sariyasa, “Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons,” IEEE Transactions on Neural Networks, vol. 13, no. 3, pp. 551–563, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. L. Huang, C. Huang, and B. Liu, “Dynamics of a class of cellular neural networks with time-varying delays,” Physics Letters, Section A, vol. 345, no. 4–6, pp. 330–344, 2005. View at Publisher · View at Google Scholar · View at Scopus
  6. Y. Li, L. Zhu, and P. Liu, “Existence and stability of periodic solutions of delayed cellular neural networks,” Nonlinear Analysis: Real World Applications, vol. 7, no. 2, pp. 225–234, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. B. Liu and L. Huang, “Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays,” Physics Letters, Section A, vol. 349, no. 6, pp. 474–483, 2006. View at Publisher · View at Google Scholar · View at Scopus
  8. Y. Shao and Y. Zhou, “Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 4, pp. 1107–1118, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. C. Bai, “Global exponential stability and existence of periodic solution of Cohen-Grossberg type neural networks with delays and impulses,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 747–761, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Y. Shao and B. Dai, “The existence of exponential periodic attractor of impulsive BAM neural network with periodic coefficients and distributed delays,” Neurocomputing, vol. 73, no. 16–18, pp. 3123–3131, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. Q. Zhou, “Global exponential stability of BAM neural networks with distributed delays and impulses,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 144–153, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Xia and P. J. Y. Wong, “Global exponential stability of a class of retarded impulsive differential equations with applications,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 440–453, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y. Yang and J. Cao, “Stability and periodicity in delayed cellular neural networks with impulsive effects,” Nonlinear Analysis: Real World Applications, vol. 8, no. 1, pp. 362–374, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. X. Li, X. Zhang, and D. Jiang, “A new existence theory for positive periodic solutions to functional differential equations with impulse effects,” Computers & Mathematics with Applications, vol. 51, no. 12, pp. 1761–1772, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. J. Nieto, “Impulsive resonance periodic problems of first order,” Applied Mathematics Letters, vol. 15, no. 4, pp. 489–493, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. X. Yang, “Existence and global exponential stability of periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks with delays and impulses,” Neurocomputing, vol. 72, no. 10–12, pp. 2219–2226, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. J. Diblík, D. Ya. Khusainov, I. V. Grytsay, and Z. Šmarda, “Stability of nonlinear autonomous quadratic discrete systems in the critical case,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 539087, 23 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects,” Journal of Optimization Theory and Applications, vol. 147, no. 3, pp. 583–596, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Exponential stability for stochastic neural networks of neural type with impulsive effects,” Modern Physics Letters B, vol. 24, no. 11, pp. 1099–1110, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. J. Baštinec, J. Diblík, D. Ya. Khusainov, and A. Ryvolová, “Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients,” Boundary Value Problems, vol. 2010, Article ID 956121, 20 pages, 2010. View at Zentralblatt MATH
  21. R. Samidurai, R. Sakthivel, and S. M. Anthoni, “Global asymptotic stability of BAM neural networks with mixed delays and impulses,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 113–119, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. R. Raja, R. Sakthivel, S. M. Anthoni, and H. Kim, “Stability of impulsive Hopfield neural networks with Markovian switching and time-varying delays,” International Journal of Applied Mathematics and Computer Science, vol. 21, no. 1, pp. 127–135, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. Y. Shao, “Exponential stability of periodic neural networks with impulsive effects and time-varying delays,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6893–6899, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. Chen and B. Cui, “Impulsive effects on global asymptotic stability of delay BAM neural networks,” Chaos, Solitons and Fractals, vol. 38, no. 4, pp. 1115–1125, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. H. Wu and C. Shan, “Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses,” Applied Mathematical Modelling, vol. 33, no. 6, pp. 2564–2574, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. D. Bainov and P. Simeonov, Impulsive Defferential Equatios: Periodic Solution and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 66, 1993.
  27. D. R. Smart, Fixed Point Theorems, Cambridge University Press, London, UK, 1974.
  28. S. Mohamad and K. Gopalsamy, “Dynamics of a class of discrete-time neural networks and their continuous-time counterparts,” Mathematics and Computers in Simulation, vol. 53, no. 1-2, pp. 1–39, 2000. View at Publisher · View at Google Scholar