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Journal of Applied Mathematics
Volume 2015, Article ID 981383, 6 pages
http://dx.doi.org/10.1155/2015/981383
Research Article

Exponential Decay for a System of Equations with Distributed Delays

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 1 June 2015; Accepted 3 August 2015

Academic Editor: Qiankun Song

Copyright © 2015 Nasser-Eddine Tatar. 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. A. Bouzerdoum and T. R. Pattison, “Neural network for quadratic optimization with bound constraints,” IEEE Transactions on Neural Networks, vol. 4, no. 2, pp. 293–304, 1993. View at Publisher · View at Google Scholar · View at Scopus
  2. J. Cao, K. Yuan, and H.-X. Li, “Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays,” IEEE Transactions on Neural Networks, vol. 17, no. 6, pp. 1646–1651, 2006. View at Publisher · View at Google Scholar · View at Scopus
  3. L. O. Chua and T. Roska, “Stability of a class of nonreciprocal cellular neural networks,” IEEE Transactions on Circuits and Systems, vol. 37, no. 12, pp. 1520–1527, 1990. View at Publisher · View at Google Scholar · View at Scopus
  4. C. Feng and R. Plamondon, “On the stability analysis of delayed neural networks systems,” Neural Networks, vol. 14, no. 9, pp. 1181–1188, 2001. View at Publisher · View at Google Scholar · View at Scopus
  5. M. Forti, M. Grazzini, P. Nistri, and L. Pancioni, “Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations,” Physica D: Nonlinear Phenomena, vol. 214, no. 1, pp. 88–99, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. M. Forti and A. Tesi, “New conditions for global stability of neural networks with application to linear and quadratic programming problems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 42, no. 7, pp. 354–366, 1995. View at Publisher · View at Google Scholar · View at Scopus
  7. J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proceedings of the National Academy of Sciences of the United States of America, vol. 79, no. 8, pp. 2554–2558, 1982. View at Publisher · View at Google Scholar · View at Scopus
  8. J. J. Hopfield and D. W. Tank, “Computing with neural circuits: a model,” Science, vol. 233, no. 4764, pp. 625–633, 1986. View at Publisher · View at Google Scholar · View at Scopus
  9. L. Huang, J. Wang, and X. Zhou, “Existence and global stability of periodic solutions for Hopfield neural networks with discontinuous activations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1651–1661, 2009. View at Google Scholar
  10. J.-I. Inoue, “Retrieval phase diagrams of non-monotonic Hopfield networks,” Journal of Physics A: Mathematical and General, vol. 29, no. 16, pp. 4815–4826, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. M. P. Kennedy and L. O. Chua, “Neural networks for non-linear programming,” IEEE Transactions on Circuits and Systems I, vol. 35, pp. 554–562, 1998. View at Google Scholar
  12. X. Liu and N. Jiang, “Robust stability analysis of generalized neural networks with multiple discrete delays and multiple distributed delays,” Neurocomputing, vol. 72, no. 7–9, pp. 1789–1796, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Mohamad, “Exponential stability in Hopfield-type neural networks with impulses,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 456–467, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. S. Mohamad, K. Gopalsamy, and H. Akça, “Exponential stability of artificial neural networks with distributed delays and large impulses,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 872–888, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. J. H. Park, “On global stability criterion for neural networks with discrete and distributed delays,” Chaos, Solitons & Fractals, vol. 30, no. 4, pp. 897–902, 2006. View at Publisher · View at Google Scholar · View at Scopus
  16. J. H. Park, “On global stability criterion of neural networks with continuously distributed delays,” Chaos, Solitons and Fractals, vol. 37, no. 2, pp. 444–449, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. Q. Zhang, M. A. Run-Nian, and X. Jin, “Global exponential convergence analysis of Hopfield neural networks with continuously distributed delays,” Communications in Theoretical Physics, vol. 39, no. 3, pp. 381–384, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. H. Qiao, J. Peng, and Z.-B. Xu, “Nonlinear measures: a new approach to exponential stability analysis for Hopfield-type neural networks,” IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 360–370, 2001. View at Publisher · View at Google Scholar · View at Scopus
  19. Q. Song, “Novel criteria for global exponential periodicity and stability of recurrent neural networks with time-varying delays,” Chaos, Solitons and Fractals, vol. 36, no. 3, pp. 720–728, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. S. I. Sudharsanan and M. K. Sundareshan, “Exponential stability and a systematic synthesis of a neural network for quadratic minimization,” Neural Networks, vol. 4, no. 5, pp. 599–613, 1991. View at Publisher · View at Google Scholar · View at Scopus
  21. P. van den Driessche and X. Zou, “Global attractivity in delayed hopfield neural network models,” SIAM Journal on Applied Mathematics, vol. 58, no. 6, pp. 1878–1890, 1998. View at Publisher · View at Google Scholar · View at Scopus
  22. H. Wu, “Global exponential stability of Hopfield neural networks with delays and inverse Lipschitz neuron activations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2297–2306, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. H. Wu, F. Tao, L. Qin, R. Shi, and L. He, “Robust exponential stability for interval neural networks with delays and non-Lipschitz activation functions,” Nonlinear Dynamics, vol. 66, no. 4, pp. 479–487, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. H. Wu and X. Xue, “Stability analysis for neural networks with inverse Lip-schitzian neuron activations and impulses,” Applied Mathematical Modelling, vol. 32, no. 11, pp. 2347–2359, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. Q. Zhang, X. P. Wei, and J. Xu, “Global exponential stability of Hopfield neural networks with continuously distributed delays,” Physics Letters A, vol. 315, no. 6, pp. 431–436, 2003. View at Publisher · View at Google Scholar · View at Scopus
  26. H. Y. Zhao, “Global stability of neural networks with distributed delays,” Physical Review E, vol. 68, no. 5, Article ID 051909, 7 pages, 2003. View at Google Scholar · View at Scopus
  27. H. Zhao, “Global asymptotic stability of Hopfield neural network involving distributed delays,” Neural Networks, vol. 17, no. 1, pp. 47–53, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. J. Zhou, S. Y. Li, and Z. G. Yang, “Global exponential stability of Hopfield neural networks with distributed delays,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1513–1520, 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. B. Kosko, Neural Network and Fuzzy System—A Dynamical System Approach to Machine Intelligence, Prentice-Hall of India, New Delhi, India, 1991.
  30. D. Bainov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its Applications, Kluwer Academic Publishers, London, UK, 1992.