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Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays. (English) Zbl 1218.34096

For the system

${\stackrel{˙}{x}}_{i}\left(t\right)=-{a}_{i}\left(t\right){x}_{i}\left(t\right)+{f}_{i}\left({x}_{1}\left(t\right),\cdots ,{x}_{n}\left(t\right),{x}_{1}\left(t-{\tau }_{i1}\right),\cdots ,{x}_{n}\left(t-{\tau }_{in}\right)\right),\phantom{\rule{1.em}{0ex}}t\ne {t}_{k},$
${\Delta }{x}_{i}\left({t}_{k}\right)={x}_{i}\left({t}_{k}^{+}\right)-{x}_{i}\left({t}_{k}^{-}\right)={J}_{k}\left({x}_{i}\left({t}_{k}\right)\right)$

existence and exponential stability of a periodic solution are established.

The obtained results are compared with some known ones.

##### MSC:
 34K45 Functional-differential equations with impulses 34K20 Stability theory of functional-differential equations 34K13 Periodic solutions of functional differential equations 47N20 Applications of operator theory to differential and integral equations
##### References:
 [1] Zhao, H. Y.: Global stability of bidirectional associative memory neural networks with distributed delays, Phys. lett. A 297, 182-190 (2002) · Zbl 0995.92002 · doi:10.1016/S0375-9601(02)00434-6 [2] Liao, X. F.; Wong, K. W.; Yang, S. Z.: Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays, Phys. lett. A 316, 55-64 (2003) · Zbl 1038.92001 · doi:10.1016/S0375-9601(03)01113-7 [3] Li, Y. K.; Liu, P.: Existence and stability of positive periodic solution for BAM neural networks with delays, Math. comput. Modelling 40, 757-770 (2004) · Zbl 1197.34125 · doi:10.1016/j.mcm.2004.10.007 [4] Zhao, H.: Global stability of associative memory neural networks with distributed delays, Phys. lett. A 13, 182-190 (2003) · Zbl 0995.92002 · doi:10.1016/S0375-9601(02)00434-6 [5] Sun, C.; Feng, C.: Exponential periodicity and stability of delayed neural networks, Math. comput. Simulation 66, 469-478 (2004) · Zbl 1057.34097 · doi:10.1016/j.matcom.2004.03.001 [6] Wu, R. C.: Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays, Nonlinear anal. RWA 11, No. 1, 562-573 (2010) · Zbl 1186.34121 · doi:10.1016/j.nonrwa.2009.02.003 [7] Xia, Y. H.; Wong, P. J. Y.: Global exponential stability of a class of retarded impulsive differential equations with applications, Chaos solitons fractals 39, 440-453 (2009) · Zbl 1197.34146 · doi:10.1016/j.chaos.2007.04.005 [8] Li, Y. K.: Global exponential stability of BAM neural networks with delays and impulses, Chaos solitons fractals 24, 279-285 (2005) · Zbl 1099.68085 · doi:10.1016/j.chaos.2004.09.027 [9] Gui, Z.; Yang, X.; Ge, W.: Existence and global exponential stability of periodic solutions of recurrent cellular neural networks with impulses and delays, Math. comput. Simulation 79, 14-29 (2008) · Zbl 1144.92002 · doi:10.1016/j.matcom.2007.09.001 [10] Wang, H.; Liao, X. F.; Li, C. D.: Existence and exponential stability of periodic solution of BAM neural networks with impulses and time-varying delay, Chaos solitons fractals 33, 1028-1039 (2007) · Zbl 1148.34049 · doi:10.1016/j.chaos.2006.01.112 [11] Wu, H. Q.; Shan, C. H.: Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses, Appl. math. Model. 33, 2564-2574 (2009) · Zbl 1205.34017 · doi:10.1016/j.apm.2008.07.022 [12] Samidurai, R.; Sakthivel, R.; Anthoni, S. M.: Global asymptotic stability of BAM neural networks with mixed delays and impulses, Appl. math. Comput. 212, 113-119 (2009) · Zbl 1173.34346 · doi:10.1016/j.amc.2009.02.002 [13] Zhou, Q. H.: Global exponential stability of BAM neural networks with distributed delays and impulses, Nonlinear anal. RWA 10, 144-153 (2009) · Zbl 1154.34391 · doi:10.1016/j.nonrwa.2007.08.019 [14] Mohamad, S.; Gopalsamy, K.; Acka, H.: Exponential stability of artificial neural networks with distributed delays and large impulses, Nonlinear anal. RWA 9, 872-888 (2008) · Zbl 1154.34042 · doi:10.1016/j.nonrwa.2007.01.011 [15] Bai, C. Z.: Global exponential stability and existence of periodic solution of Cohen–Grossberg type neural networks with delays and impulses, Nonlinear anal. RWA 9, 747-761 (2008) · Zbl 1151.34062 · doi:10.1016/j.nonrwa.2006.12.007 [16] Yang, X. S.: Existence and global exponential stability of periodic solution for Cohen–Grossberg shunting inhibitory cellular neural networks with delays and impulses, Neurocomputing 72, 2219-2226 (2009) [17] Gu, H. B.; Jiang, H. J.; Teng, Z. D.: Existence and globally exponential stability of periodic solution of BAM neural networks with impulses and recent-history distributed delays, Neurocomputing 71, 813-822 (2008) [18] Li, Y. T.; Wang, J. Y.: 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 · doi:10.1016/j.camwa.2008.03.048 [19] Tan, M. J.; Tan, Y.: Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays, Appl. math. Model. 33, 373-385 (2009) · Zbl 1167.34373 · doi:10.1016/j.apm.2007.11.010 [20] Huang, L.; Huang, C.; Liu, B.: Dynamics of a class of celluar neural networks with time-varying delays, Phys. lett. A 345, 330-344 (2005) [21] Li, Y.; Zhu, L.; Liu, P.: Existence and stability of periodic solutions of delayed cellular neural networks, Nonlinear anal. RWA 7, 225-234 (2006) · Zbl 1086.92002 · doi:10.1016/j.nonrwa.2005.02.004 [22] Liu, B.; Huang, L.: Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, Phys. lett. A 349, 274-283 (2006) · Zbl 1195.34061 · doi:10.1016/j.physleta.2005.09.064 [23] Liu, Z.; Liao, L.: Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, J. math. Anal. appl. 290, 247-262 (2004) · Zbl 1055.34135 · doi:10.1016/j.jmaa.2003.09.052 [24] Li, Y. K.; Fan, X. L.: Existence and globally exponential stability of almost periodic solution for Cohen–Grossberg BAM neural networks with variable coefficients, Appl. math. Model. 33, 2114-2120 (2009) · Zbl 1205.34086 · doi:10.1016/j.apm.2008.05.013 [25] Cao, J. D.; Song, Q.: Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity 19, 1601-1617 (2006) · Zbl 1118.37038 · doi:10.1088/0951-7715/19/7/008 [26] Zhou, Q.; Wan, L.; Sun, J.: Exponential stability of reaction–diffusion generalized Cohen–Grossberg neural networks with time-varying delays, Chaos solitons fractals 32, 129-142 (2007) [27] Zhang, Y.; Pheng, H.; Vadakkepat, P.: Absolute periodicity and absolute stability of delayed neural networks, IEEE trans. Circuits syst. 49, 256-261 (2002) [28] Xia, Y.; Han, M. A.: New conditions on the existence and stability of periodic solution in Lotka–Volterra population system, SIAM J. Appl. math. 69, 1580-1597 (2009) · Zbl 1181.92084 · doi:10.1137/070702485 [29] Huang, Z.; Xia, Y.: Exponential periodic attractor of impulsive BAM networks with finite distributed delays, Chaos solitons fractals 39, 373-384 (2009) · Zbl 1197.34124 · doi:10.1016/j.chaos.2007.04.014 [30] Horn, R. A.; Johnson, C. R.: Topics in matrix analysis, (1991) [31] Fang, B.; Zhou, J.; Li, Y.: Matrix theory, (2004) [32] Lu, S.; Ge, W.: Periodic solutions for a kind of second order differential equation with multiple deviating arguments, Appl. math. Comput. 146, 195-209 (2003) · Zbl 1037.34065 · doi:10.1016/S0096-3003(02)00536-2 [33] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equation, (1977) [34] Rudin, W.: Real and complex analysis, (1987) · Zbl 0925.00005