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Neural Processing Letters

, Volume 33, Issue 1, pp 61–81 | Cite as

Global Exponential Stability and Existence of Periodic Solution of Impulsive Cohen–Grossberg Neural Networks with Distributed Delays on Time Scales

  • Yongkun Li
  • Lili Zhao
  • Tianwei Zhang
Article

Abstract

On time scales, by using the continuation theorem of coincidence degree theory, M-matrix theory and constructing some suitable Lyapunov functions, some sufficient conditions are obtained for the existence and exponential stability of periodic solutions of impulsive Cohen–Grossberg neural networks with distributed delays, which are new and complement of previously known results. Finally, an example is given to illustrate the effectiveness of our main results.

Keywords

Cohen–Grossberg neural networks Stability Periodic solutions Time scales 

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References

  1. 1.
    Cohen M, Grossberg S (1983) Abusolute Stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybern 13: 815–826zbMATHMathSciNetGoogle Scholar
  2. 2.
    Arik S, Orman Z (2005) Global stability analysis of Cohen–Grossberg neural networks with time varying delays. Phys Lett A 341: 410–421zbMATHCrossRefGoogle Scholar
  3. 3.
    Cao J, Liang J (2004) Boundedness and stability for Cohen–Grossberg neural network with time-varying delays. J Math Anal Appl 296: 665–685zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Huang T, Chen A, Huang Y, Cao J (2007) Stability of Cohen–Grossberg neural networks with time-varying delays. Neural Netw 20: 868–873zbMATHCrossRefGoogle Scholar
  5. 5.
    Huang T, Li C, Chen G (2007) Stability of Cohen–Grossberg neural networks with unbounded distributed delays. Chaos Soliton Fract 34: 992–996zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Jing M, Shen Y, Luo X (2006) Boundeness and globall exponential stability for generalized Cohen–Grossberg neural networks with variable delays. Appl Math Comput 172: 379–393CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chen Z, Ruan J (2005) Global stability analysis of impulsive Cohen–Grossberg neural networks with delay. Phys Lett A 345: 101–111zbMATHCrossRefGoogle Scholar
  8. 8.
    Gopalsamy K, Leung I (1996) Delay induced periodicity in a neural netlet of excitation and inhibition. Physica D 89: 395–426zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Liu Z, Chen A, Cao J, Huang L (2003) Existence and global exponential stability of almost periodic solutions of BAM neural networks with continuously distributed delays. Phys Lett A 319: 305–316zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer-Verlag, New YorkzbMATHGoogle Scholar
  11. 11.
    Gopalsamy K, He X (1994) Delay-independent stability in bidirectional associative memory networks. IEEE Trans Neural Netw 5: 998–1002CrossRefGoogle Scholar
  12. 12.
    Gopalsamy K (2004) Stability of artificial neural networks with impulses. Appl Math Comput 154: 783–813zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Guan Z, James L, Chen G (2000) On impulsive auto-associative neural networks. Neural Netw 13: 63–69CrossRefGoogle Scholar
  14. 14.
    Li Y (2005) Global exponential stability of BAM neural networks with delays and impulses. Chaos Solitons Fract 24: 279–285zbMATHGoogle Scholar
  15. 15.
    Li Y, Yang C (2006) Global exponential stability analysis on impulsive BAM neural networks with distributed delays. J Math Anal Appl 324: 1125–1139zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Li Y, Lu L (2004) Global exponential stability and existence of periodic solutions of Hopfield-type neural networks with impulses. Phys Lett A 333: 62–71zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Bohner M, Peterson A (2001) Dynamic equations on time scales, an introducation with applications. Birkhauser, BostonGoogle Scholar
  18. 18.
    Bohner M, Peterson A (2003) Advances in dynamic equations on time scales. Birkhauser, BostonzbMATHGoogle Scholar
  19. 19.
    Li Y, Gao S (2010) Global exponential stability for impulsive BAM neural networks with distributed delays on time scales. Neural Process Lett 31: 65–91CrossRefGoogle Scholar
  20. 20.
    Li Y, Hua Y, Fei Y (2009) Global exponential stability of delayed Cohen–Grossberg BAM neural networks with impulses on time scales. J Inequal Appl, Volume 2009, Article ID 491268, 17 pGoogle Scholar
  21. 21.
    Li Y, Chen X, Zhao L (2009) Stability and existence of periodic solutions to delayed Cohen–Grossberg BAM neural networks with impulses on time scales. Neurocomputing 72: 1621–1630CrossRefGoogle Scholar
  22. 22.
    Li Y (2004) Existence and stability of periodic solutions for Cohen–Grossberg neural networks with multiple delays. Chaos Solitons and Fract 20: 459–466zbMATHCrossRefGoogle Scholar
  23. 23.
    Sun JH, Wan L (2005) Global exponential stability and periodic solutions of Cohen–Grossberg neural networks with continuously distributed delays. Physica D 208: 1–20zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wu C, Ruan J, Lin W (2006) On the existence and stability of the periodic solution in the Cohen–Grossberg neural network with time delay and high-order terms. Appl Math Comput 177: 194–210zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Li CH, Yang SY (2009) Existence and attractivity of periodic solutions to non-autonomous Cohen–Grossberg neural networks with time delays. Chaos Solitons Fract 41: 1235–1244zbMATHCrossRefGoogle Scholar
  26. 26.
    Bi L, Bohner M, Fan M (2008) Periodic solutions of functional dynamic equations with infinite delay. Nonlinear Anal 68: 1226–1245zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lakshmikantham V, Vatsala AS (2002) Hybrid systems on time scales. J Comput Appl Math 141: 227–235zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Kaufmann ER, Raffoul YN (2006) Periodic solutions for a neutral nonlinear dynamic equation on a time scale. J Math Anal Appl 319: 315–325zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Bohner M, Fan M, Zhang J (2006) Existence of periodic solutions in predator-prey and competition dynamic systems. Nonlinear Anal Real World Appl 7: 1193–1204zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Wang FH, Yeh CC, Yu SL, Hong CH (2005) Youngs inequality and related results on time scales. Appl Math Lett 18: 983–988CrossRefMathSciNetGoogle Scholar
  31. 31.
    Xing Y, Han M, Zheng G (2005) Initial value problem for first-order intrgro-differential equation of Volterra type on time scale. Nonlinear Anal 60: 429–442zbMATHMathSciNetGoogle Scholar
  32. 32.
    Mawhin JL (1979) Topological degree methods in noninear boundary value problems. In: CBMS regional conference series in mathematics No. 40. American Mathematical Society, Provedence, RIGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of MathematicsYunnan UniversityKunmingPeople’s Republic of China

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