By using Schaeffer's theorem and Lyapunov functional, sufficient conditions of the existence and globally exponential stability of positive periodic solution to an impulsive neural network with time-varying delays are established. Applications, examples, and numerical analysis are given to illustrate the effectiveness of the main results.

## References

M. Tan and Y. Tan, “Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays,”

*Applied Mathematical Modelling*, vol. 33, no. 1, pp. 373–385, 2009. 1167.34373 MR2458515 10.1016/j.apm.2007.11.010 M. Tan and Y. Tan, “Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays,”*Applied Mathematical Modelling*, vol. 33, no. 1, pp. 373–385, 2009. 1167.34373 MR2458515 10.1016/j.apm.2007.11.010 R. Wu, “Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays,”

*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 562–573, 2010. 1186.34121 MR2570576 10.1016/j.nonrwa.2009.02.003 R. Wu, “Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays,”*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 562–573, 2010. 1186.34121 MR2570576 10.1016/j.nonrwa.2009.02.003 I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,”

*Abstract and Applied Analysis*, vol. 2011, Article ID 920412, 14 pages, 2011. 1217.93150 MR2800069 10.1155/2011/920412 I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,”*Abstract and Applied Analysis*, vol. 2011, Article ID 920412, 14 pages, 2011. 1217.93150 MR2800069 10.1155/2011/920412 K. Gopalsamy and Sariyasa, “Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons,”

*IEEE Transactions on Neural Networks*, vol. 13, no. 3, pp. 551–563, 2002. K. Gopalsamy and Sariyasa, “Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons,”*IEEE Transactions on Neural Networks*, vol. 13, no. 3, pp. 551–563, 2002. L. Huang, C. Huang, and B. Liu, “Dynamics of a class of cellular neural networks with time-varying delays,”

*Physics Letters, Section A*, vol. 345, no. 4–6, pp. 330–344, 2005. L. Huang, C. Huang, and B. Liu, “Dynamics of a class of cellular neural networks with time-varying delays,”*Physics Letters, Section A*, vol. 345, no. 4–6, pp. 330–344, 2005. Y. Li, L. Zhu, and P. Liu, “Existence and stability of periodic solutions of delayed cellular neural networks,”

*Nonlinear Analysis: Real World Applications*, vol. 7, no. 2, pp. 225–234, 2006. 1086.92002 MR2192599 10.1016/j.nonrwa.2005.02.004 Y. Li, L. Zhu, and P. Liu, “Existence and stability of periodic solutions of delayed cellular neural networks,”*Nonlinear Analysis: Real World Applications*, vol. 7, no. 2, pp. 225–234, 2006. 1086.92002 MR2192599 10.1016/j.nonrwa.2005.02.004 B. Liu and L. Huang, “Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays,”

*Physics Letters, Section A*, vol. 349, no. 6, pp. 474–483, 2006. B. Liu and L. Huang, “Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays,”*Physics Letters, Section A*, vol. 349, no. 6, pp. 474–483, 2006. Y. Shao and Y. Zhou, “Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 4, pp. 1107–1118, 2011. 1218.34096 MR2746792 Y. Shao and Y. Zhou, “Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 4, pp. 1107–1118, 2011. 1218.34096 MR2746792 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*, vol. 9, no. 3, pp. 747–761, 2008. 1151.34062 MR2392372 10.1016/j.nonrwa.2006.12.007 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*, vol. 9, no. 3, pp. 747–761, 2008. 1151.34062 MR2392372 10.1016/j.nonrwa.2006.12.007 Y. Shao and B. Dai, “The existence of exponential periodic attractor of impulsive BAM neural network with periodic coefficients and distributed delays,”

*Neurocomputing*, vol. 73, no. 16–18, pp. 3123–3131, 2010. Y. Shao and B. Dai, “The existence of exponential periodic attractor of impulsive BAM neural network with periodic coefficients and distributed delays,”*Neurocomputing*, vol. 73, no. 16–18, pp. 3123–3131, 2010. Q. Zhou, “Global exponential stability of BAM neural networks with distributed delays and impulses,”

*Nonlinear Analysis: Real World Applications*, vol. 10, no. 1, pp. 144–153, 2009. 1154.34391 MR2451697 10.1016/j.nonrwa.2007.08.019 Q. Zhou, “Global exponential stability of BAM neural networks with distributed delays and impulses,”*Nonlinear Analysis: Real World Applications*, vol. 10, no. 1, pp. 144–153, 2009. 1154.34391 MR2451697 10.1016/j.nonrwa.2007.08.019 Y. Xia and P. J. Y. Wong, “Global exponential stability of a class of retarded impulsive differential equations with applications,”

*Chaos, Solitons and Fractals*, vol. 39, no. 1, pp. 440–453, 2009. 1197.34146 MR2504578 Y. Xia and P. J. Y. Wong, “Global exponential stability of a class of retarded impulsive differential equations with applications,”*Chaos, Solitons and Fractals*, vol. 39, no. 1, pp. 440–453, 2009. 1197.34146 MR2504578 Y. Yang and J. Cao, “Stability and periodicity in delayed cellular neural networks with impulsive effects,”

*Nonlinear Analysis: Real World Applications*, vol. 8, no. 1, pp. 362–374, 2007. 1115.34072 MR2268091 10.1016/j.nonrwa.2005.11.004 Y. Yang and J. Cao, “Stability and periodicity in delayed cellular neural networks with impulsive effects,”*Nonlinear Analysis: Real World Applications*, vol. 8, no. 1, pp. 362–374, 2007. 1115.34072 MR2268091 10.1016/j.nonrwa.2005.11.004 X. Li, X. Zhang, and D. Jiang, “A new existence theory for positive periodic solutions to functional differential equations with impulse effects,”

*Computers & Mathematics with Applications*, vol. 51, no. 12, pp. 1761–1772, 2006. 1156.34053 MR2245704 X. Li, X. Zhang, and D. Jiang, “A new existence theory for positive periodic solutions to functional differential equations with impulse effects,”*Computers & Mathematics with Applications*, vol. 51, no. 12, pp. 1761–1772, 2006. 1156.34053 MR2245704 J. J. Nieto, “Impulsive resonance periodic problems of first order,”

*Applied Mathematics Letters*, vol. 15, no. 4, pp. 489–493, 2002. 1022.34025 MR1902284 10.1016/S0893-9659(01)00163-X J. J. Nieto, “Impulsive resonance periodic problems of first order,”*Applied Mathematics Letters*, vol. 15, no. 4, pp. 489–493, 2002. 1022.34025 MR1902284 10.1016/S0893-9659(01)00163-X X. Yang, “Existence and global exponential stability of periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks with delays and impulses,”

*Neurocomputing*, vol. 72, no. 10–12, pp. 2219–2226, 2009. X. Yang, “Existence and global exponential stability of periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks with delays and impulses,”*Neurocomputing*, vol. 72, no. 10–12, pp. 2219–2226, 2009. J. Diblík, D. Ya. Khusainov, I. V. Grytsay, and Z. Šmarda, “Stability of nonlinear autonomous quadratic discrete systems in the critical case,”

*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 539087, 23 pages, 2010. 1200.39005 MR2660702 J. Diblík, D. Ya. Khusainov, I. V. Grytsay, and Z. Šmarda, “Stability of nonlinear autonomous quadratic discrete systems in the critical case,”*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 539087, 23 pages, 2010. 1200.39005 MR2660702 R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects,”

*Journal of Optimization Theory and Applications*, vol. 147, no. 3, pp. 583–596, 2010. 1211.93108 MR2733995 10.1007/s10957-010-9728-8 R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects,”*Journal of Optimization Theory and Applications*, vol. 147, no. 3, pp. 583–596, 2010. 1211.93108 MR2733995 10.1007/s10957-010-9728-8 R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Exponential stability for stochastic neural networks of neural type with impulsive effects,”

*Modern Physics Letters B*, vol. 24, no. 11, pp. 1099–1110, 2010. 1195.82066 MR2650898 10.1142/S0217984910023141 R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Exponential stability for stochastic neural networks of neural type with impulsive effects,”*Modern Physics Letters B*, vol. 24, no. 11, pp. 1099–1110, 2010. 1195.82066 MR2650898 10.1142/S0217984910023141 J. Baštinec, J. Diblík, D. Ya. Khusainov, and A. Ryvolová, “Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients,”

*Boundary Value Problems*, vol. 2010, Article ID 956121, 20 pages, 2010. 1214.34059 MR2739194 J. Baštinec, J. Diblík, D. Ya. Khusainov, and A. Ryvolová, “Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients,”*Boundary Value Problems*, vol. 2010, Article ID 956121, 20 pages, 2010. 1214.34059 MR2739194 R. Samidurai, R. Sakthivel, and S. M. Anthoni, “Global asymptotic stability of BAM neural networks with mixed delays and impulses,”

*Applied Mathematics and Computation*, vol. 212, no. 1, pp. 113–119, 2009. 1173.34346 MR2519265 10.1016/j.amc.2009.02.002 R. Samidurai, R. Sakthivel, and S. M. Anthoni, “Global asymptotic stability of BAM neural networks with mixed delays and impulses,”*Applied Mathematics and Computation*, vol. 212, no. 1, pp. 113–119, 2009. 1173.34346 MR2519265 10.1016/j.amc.2009.02.002 R. Raja, R. Sakthivel, S. M. Anthoni, and H. Kim, “Stability of impulsive Hopfield neural networks with Markovian switching and time-varying delays,”

*International Journal of Applied Mathematics and Computer Science*, vol. 21, no. 1, pp. 127–135, 2011. 1221.93265 MR2814467 10.2478/v10006-011-0009-y R. Raja, R. Sakthivel, S. M. Anthoni, and H. Kim, “Stability of impulsive Hopfield neural networks with Markovian switching and time-varying delays,”*International Journal of Applied Mathematics and Computer Science*, vol. 21, no. 1, pp. 127–135, 2011. 1221.93265 MR2814467 10.2478/v10006-011-0009-y Y. Shao, “Exponential stability of periodic neural networks with impulsive effects and time-varying delays,”

*Applied Mathematics and Computation*, vol. 217, no. 16, pp. 6893–6899, 2011. 1216.34072 MR2775680 10.1016/j.amc.2011.01.068 Y. Shao, “Exponential stability of periodic neural networks with impulsive effects and time-varying delays,”*Applied Mathematics and Computation*, vol. 217, no. 16, pp. 6893–6899, 2011. 1216.34072 MR2775680 10.1016/j.amc.2011.01.068 J. Chen and B. Cui, “Impulsive effects on global asymptotic stability of delay BAM neural networks,”

*Chaos, Solitons and Fractals*, vol. 38, no. 4, pp. 1115–1125, 2008. 1152.34386 MR2435607 10.1016/j.chaos.2007.01.042 J. Chen and B. Cui, “Impulsive effects on global asymptotic stability of delay BAM neural networks,”*Chaos, Solitons and Fractals*, vol. 38, no. 4, pp. 1115–1125, 2008. 1152.34386 MR2435607 10.1016/j.chaos.2007.01.042 H. Wu and C. Shan, “Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses,”

*Applied Mathematical Modelling*, vol. 33, no. 6, pp. 2564–2574, 2009. 1205.34017 MR2502132 10.1016/j.apm.2008.07.022 H. Wu and C. Shan, “Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses,”*Applied Mathematical Modelling*, vol. 33, no. 6, pp. 2564–2574, 2009. 1205.34017 MR2502132 10.1016/j.apm.2008.07.022 D. Bainov and P. Simeonov,

*Impulsive Defferential Equatios: Periodic Solution and Applications*, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 66, 1993. D. Bainov and P. Simeonov,*Impulsive Defferential Equatios: Periodic Solution and Applications*, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 66, 1993. S. Mohamad and K. Gopalsamy, “Dynamics of a class of discrete-time neural networks and their continuous-time counterparts,”

*Mathematics and Computers in Simulation*, vol. 53, no. 1-2, pp. 1–39, 2000. MR1777734 10.1016/S0378-4754(00)00168-3 S. Mohamad and K. Gopalsamy, “Dynamics of a class of discrete-time neural networks and their continuous-time counterparts,”*Mathematics and Computers in Simulation*, vol. 53, no. 1-2, pp. 1–39, 2000. MR1777734 10.1016/S0378-4754(00)00168-3