# American Institute of Mathematical Sciences

2014, 19(3): 761-773. doi: 10.3934/dcdsb.2014.19.761

## Average criteria for periodic neural networks with delay

 1 Dipartimento di Matematica, Universitá degli studi di Bari, 70125 Bari, Italy

Received  June 2013 Revised  October 2013 Published  February 2014

By using Lyapunov functions and some recent estimates of Halanay type, new criteria are introduced for the global exponential stability of a class of cellular neural networks, with delay and periodic coefficients and inputs. The novelty of those criteria lies in the fact that they are very efficient in presence of oscillating coefficients, because they are given in average form.
Citation: Benedetta Lisena. Average criteria for periodic neural networks with delay. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 761-773. doi: 10.3934/dcdsb.2014.19.761
##### References:
 [1] S. Ahmad and I. M. Stamova, Global exponential stability for impulsive cellular neural networks with time-delays,, Nonlinear Anal., 69 (2008), 786. doi: 10.1016/j.na.2008.02.067. [2] H. Gu, H. Jiang and Z. Teng, Stability and periodicity in high-order neural networks with impulsive effects,, Nonlinear Anal., 68 (2008), 3186. doi: 10.1016/j.na.2007.03.024. [3] H. Jiang, Z. Li and Z. Teng, Boundedness and stability for nonautonomous cellular networks with delays,, Phys. Lett. A, 306 (2003), 313. doi: 10.1016/S0375-9601(02)01608-0. [4] B. Li and D. Xu, Existence and exponential stability of periodic solution for impulsive Cohen-Grossberg neural networks with time varying delays,, Appl. Math. Comput., 219 (2012), 2506. doi: 10.1016/j.amc.2012.08.086. [5] B. Lisena, Exponential stability of Hopfield neural networks with impulses,, Nonlinear Anal. Real World Appl., 12 (2011), 1923. doi: 10.1016/j.nonrwa.2010.12.008. [6] B. Lisena, Dynamical behavior of impulsive and periodic Cohen-Grossberg neural networks,, Nonlinear Anal., 74 (2011), 4511. doi: 10.1016/j.na.2011.04.015. [7] B. Lisena, Asymptotic properties in a delay differential inequality with periodic coefficients,, Mediterr. J. Math., 10 (2013), 1717. doi: 10.1007/s00009-013-0261-5. [8] B. Liu and L. Huang, Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays,, Phys. Lett. A, 349 (2006), 474. [9] H. Liu and L. Wang, Globally exponential stability and periodic solutions of CNNs with variable coefficients and variable delays,, Chaos Solitons Fractals, 29 (2006), 1137. doi: 10.1016/j.chaos.2005.08.120. [10] S. Long and D. Xu, Delay-dependent stability analysis for impulsive neural networks with time varying delays,, Neurocomputing, 71 (2008), 1705. doi: 10.1016/j.neucom.2007.03.010. [11] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays,, Appl. Math. Comput., 135 (2003), 17. doi: 10.1016/S0096-3003(01)00299-5. [12] Y. Shao, Exponential stability of periodic neural networks with impulsive effects and time-varying delays,, Appl. Math. Comput., 217 (2011), 6893. doi: 10.1016/j.amc.2011.01.068. [13] I. M. Stamova and R. Ilarionov, On global exponential stability for impulsive cellular neural networks with time-varying delays,, Comput. Math. Appl., 59 (2010), 3508. doi: 10.1016/j.camwa.2010.03.043. [14] M. Tan and Y. Tan, Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays,, Appl. Math. Model., 33 (2009), 373. doi: 10.1016/j.apm.2007.11.010. [15] H. Wang, C. Li and H. Xu, Existence and global exponential stability of periodic solution of cellular neural networks with delay and impulses,, Results Math., 58 (2010), 191. doi: 10.1007/s00025-010-0048-y. [16] Z. Yuan and L. Yuan, Existence and global convergence of periodic solution of delayed neural networks,, Math. Comput. Modelling, 48 (2008), 101. doi: 10.1016/j.mcm.2007.08.010.

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##### References:
 [1] S. Ahmad and I. M. Stamova, Global exponential stability for impulsive cellular neural networks with time-delays,, Nonlinear Anal., 69 (2008), 786. doi: 10.1016/j.na.2008.02.067. [2] H. Gu, H. Jiang and Z. Teng, Stability and periodicity in high-order neural networks with impulsive effects,, Nonlinear Anal., 68 (2008), 3186. doi: 10.1016/j.na.2007.03.024. [3] H. Jiang, Z. Li and Z. Teng, Boundedness and stability for nonautonomous cellular networks with delays,, Phys. Lett. A, 306 (2003), 313. doi: 10.1016/S0375-9601(02)01608-0. [4] B. Li and D. Xu, Existence and exponential stability of periodic solution for impulsive Cohen-Grossberg neural networks with time varying delays,, Appl. Math. Comput., 219 (2012), 2506. doi: 10.1016/j.amc.2012.08.086. [5] B. Lisena, Exponential stability of Hopfield neural networks with impulses,, Nonlinear Anal. Real World Appl., 12 (2011), 1923. doi: 10.1016/j.nonrwa.2010.12.008. [6] B. Lisena, Dynamical behavior of impulsive and periodic Cohen-Grossberg neural networks,, Nonlinear Anal., 74 (2011), 4511. doi: 10.1016/j.na.2011.04.015. [7] B. Lisena, Asymptotic properties in a delay differential inequality with periodic coefficients,, Mediterr. J. Math., 10 (2013), 1717. doi: 10.1007/s00009-013-0261-5. [8] B. Liu and L. Huang, Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays,, Phys. Lett. A, 349 (2006), 474. [9] H. Liu and L. Wang, Globally exponential stability and periodic solutions of CNNs with variable coefficients and variable delays,, Chaos Solitons Fractals, 29 (2006), 1137. doi: 10.1016/j.chaos.2005.08.120. [10] S. Long and D. Xu, Delay-dependent stability analysis for impulsive neural networks with time varying delays,, Neurocomputing, 71 (2008), 1705. doi: 10.1016/j.neucom.2007.03.010. [11] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays,, Appl. Math. Comput., 135 (2003), 17. doi: 10.1016/S0096-3003(01)00299-5. [12] Y. Shao, Exponential stability of periodic neural networks with impulsive effects and time-varying delays,, Appl. Math. Comput., 217 (2011), 6893. doi: 10.1016/j.amc.2011.01.068. [13] I. M. Stamova and R. Ilarionov, On global exponential stability for impulsive cellular neural networks with time-varying delays,, Comput. Math. Appl., 59 (2010), 3508. doi: 10.1016/j.camwa.2010.03.043. [14] M. Tan and Y. Tan, Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays,, Appl. Math. Model., 33 (2009), 373. doi: 10.1016/j.apm.2007.11.010. [15] H. Wang, C. Li and H. Xu, Existence and global exponential stability of periodic solution of cellular neural networks with delay and impulses,, Results Math., 58 (2010), 191. doi: 10.1007/s00025-010-0048-y. [16] Z. Yuan and L. Yuan, Existence and global convergence of periodic solution of delayed neural networks,, Math. Comput. Modelling, 48 (2008), 101. doi: 10.1016/j.mcm.2007.08.010.
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