Anti-periodic solution for delayed cellular neural networks with impulsive effects. (English) Zbl 1231.34121

Summary: We discuss anti-periodic solution for delayed cellular neural networks with impulsive effects. By means of contraction mapping principle and Krasnoselski’s fixed point theorem, we obtain the existence of anti-periodic solution for neural networks. By establishing a new impulsive differential inequality, using Lyapunov functions and inequality techniques, some new results for exponential stability of anti-periodic solution are obtained. Meanwhile, an example and numerical simulations are given to show that impulses may change the exponentially stable behavior of anti-periodic solution.


34K13 Periodic solutions to functional-differential equations
34K45 Functional-differential equations with impulses
Full Text: DOI


[1] T. Poska, T. Boros, P. Thirn, L.O. Chua, Detecting simple motion using cellular neural networks, in: Proceedings of the 1990 IEEE International Workshop on Cellular Neural Networks and Their Applications, pp. 127-138.
[2] Chua, L.O.; Yang, L., Cellular neural networks, IEEE trans. circuits syst., 35, 1257-1290, (1998)
[3] Venetianer, P.L.; Roska, T., Image compression by cellular neural networks, IEEE trans. circuits syst., 45, 205-215, (1998)
[4] Wang, L.; Zou, X., Exponential stability of cohen – grossberg neural networks, Neural netw., 15, 415-422, (2002)
[5] Feng, C.; Plamondon, R., Stability analysis of bidirectional associative memory networks with time delays, IEEE trans. neural netw., 14, 1560-1565, (2003)
[6] Cao, J., New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. lett. A, 307, 136-147, (2003) · Zbl 1006.68107
[7] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[8] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[9] Akhmet, M.U., On the general problem of stability for impulsive differential equations, J. math. anal. appl., 288, 182-196, (2003) · Zbl 1047.34094
[10] Gopalsamy, K., Stability of artificial neural networks with impulses, Appl. math. comput., 154, 783-813, (2004) · Zbl 1058.34008
[11] Guan, Z.; Chen, G., On delayed impulsive Hopfield neural networks, Neural netw., 12, 273-280, (1999)
[12] Zhou, Q., Global exponential stability of BAM neural networks with distributed delays and impulses, Nonlinear anal. RWA, 10, 144-153, (2009) · Zbl 1154.34391
[13] Xu, D.; Yang, Z., Impulsive delay differential inequality and stability of neural networks, J. math. anal. appl., 305, 107-120, (2005) · Zbl 1091.34046
[14] Yang, Y.Q.; Cao, J.D., Stability and periodicity in delayed cellular neural networks with impulsive effects, Nonlinear anal. RWA, 8, 362-374, (2007) · Zbl 1115.34072
[15] Li, Y.; Wang, J., An analysis on the global exponential stability and the existence of periodic solutions for non-autonomous hybrid BAM neural networks with distributed delays and impulses, Comput. math. appl., 56, 2256-2267, (2008) · Zbl 1165.34410
[16] Zhang, J.; Gui, Z., Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays, J. comput. appl. math., 224, 602-613, (2009) · Zbl 1168.34042
[17] Ou, C., Anti-periodic solution for high-order Hopfield neural networks, Comput. math. appl., 56, 1838-1844, (2008) · Zbl 1152.34378
[18] Li, Y.K.; Yang, L., Anti-periodic solutions for cohen – grossberg neural networks with bounded and unbounded delays, Commun. nonlinear sci. numer. simul., 14, 3134-3140, (2009) · Zbl 1221.34167
[19] Peng, G.Q.; Huang, L.H., Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays, Nonlinear anal. RWA, 10, 2434-2440, (2009) · Zbl 1163.92306
[20] Shao, J.Y., An anti-periodic solution for a class of recurrent neural networks, J. comput. appl. math., 228, 231-237, (2009) · Zbl 1175.34089
[21] Shi, P.L.; Dong, L.Z., Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses, Appl. math. comput., 216, 623-630, (2010) · Zbl 1200.34046
[22] M.A. Krasnosel’Skii, Positive Solutions of Operator Equations, Groningen, Netherlands. 1964.
[23] Halanay, A., Differential equations, (1996), Academic Press New York
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