Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 5 (2010), 1 -- 16

NEW RESULT OF EXISTENCE OF PERIODIC SOLUTION FOR A HOPFIELD NEURAL NETWORKS WITH NEUTRAL TIME-VARYING DELAYS

Chuanzhi Bai and Chunhong Li

Abstract. In this paper, a Hopfield neural network with neutral time-varying delays is investigated by using the continuation theorem of Mawhin's coincidence degree theory and some analysis technique. Without assuming the continuous differentiability of time-varying delays, sufficient conditions for the existence of the periodic solutions are given. The result of this paper is new and extends previous known result.

2000 Mathematics Subject Classification: 34K13; 92B20
Keywords: Hopfield neural networks; Neutral delay; Coincidence degree theory; Periodic solution

Full text

References

  1. 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 9(2008), 747-761. MR2392372 (2009a:34122). Zbl 1151.34062.

  2. J. Cao, New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. Lett. A 307(2003), 136-147. MR1974596(2004a:62176). Zbl 1006.68107.

  3. S. Guo, L. Huang, Periodic oscillation for a class of neural networks with variable coefficients, Nonlinear Anal. Real World Appl. 6 (2005), 545-561. MR2129564(2006h:34139). Zbl 1080.34051.

  4. Y. Li, Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays, Chaos, Solotons & Fractals, 20(2004), 459-466. MR2024869(2004i:34184). Zbl 1048.34118.

  5. B. Liu, L. Huang, Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays, Phys. Lett. A 349(2006), 474-483. Zbl 1171.82329.

  6. Z. Liu, L. Liao, Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, J. Math. Anal. Appl. 290 (2004), 247-262. MR2032238(2004j:34161). Zbl 1055.34135.

  7. J.H. Park, O.M. Kwon, Lee, S. M, LMI optimization approach on stability for delayed neural networks of neutral-type, Appl. Math. Comput. 196 (2008), 236-244. MR2382607. Zbl 1157.34056.

  8. S. Xu, J. Lam, D.W.C. Ho, Y. Zou, Delay-dependent exponential stability for a class of neural networks with time delays, J. Comput. Appl. Math. 183 (2005), 16-28. MR2156097(2006d:34171). Zbl 1097.34057.

  9. C. Bai, Global stability of almost periodic solutions of Hopfield neural networks with neutral time-varying delays, Appl. Math. Comput. 203 (2008), 72-79. MR2451540. Zbl 1173.34344.

  10. O.M. Kwon, J.H. Park, S.M. Lee, On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays, Appl. Math. Comput. 197 (2008), 864-873. MR2400710. Zbl 1144.34052.

  11. Z. Gui, W. Ge, X. Yang, Periodic oscillation for a Hopfield neural networks with neutral delays, Phys. Lett. A 364 (2007), 267-273.

  12. J. Chen, X. Chen, Special matrices, Tsinghua Univ. Press, Beijing, 2001.

  13. R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag, 1977. MR0637067 (58 #30551). Zbl 0339.47031.

  14. A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Computer Science and Applied Mathematics. New York, San Francisco, London: Academic Press. XVIII, 1979. MR0544666 (82b:15013). Zbl 0484.15016.



Chuanzhi Bai Chunhong Li
Department of Mathematics, Department of Mathematics,
Huaiyin Normal University, Huaiyin Normal University,
Huaian, Jiangsu 223300, P. R. China. Huaian, Jiangsu 223300, P. R. China.
e-mail: czbai@hytc.edu.cn e-mail: lichshy2006@126.com

http://www.utgjiu.ro/math/sma