Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument. (English) Zbl 1298.34136
Summary: We introduce impulsive cellular neural network models with piecewise alternately advanced and retarded argument (in short IDEPCA). The model with the advanced argument is system with strong anticipation. Some sufficient conditions are established for the existence and global exponential stability of a unique equilibrium. The approaches are based on employing Banach’s fixed point theorem and a new IDEPCA integral inequality of Gronwall type. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses and piecewise constant argument deviations, which provides exibility for the design and analysis of cellular neural network models. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 34K45 | Functional-differential equations with impulses |
| 34K21 | Stationary solutions of functional-differential equations |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 47N20 | Applications of operator theory to differential and integral equations |
References:
| [1] | Chua, L. O.; Yang, L., Cellular neural networks: theory, IEEE Transactions on Circuits and Systems, 35, 10, 1257-1272 (1988) · Zbl 0663.94022 · doi:10.1109/31.7600 |
| [2] | Wolfram, S., Theory and Applications of Cellular Automata. Theory and Applications of Cellular Automata, Advanced Series on Complex Systems, 1 (1986), Singapore: World Scientific, Singapore · Zbl 0609.68043 |
| [3] | Chua, L. O.; Yang, L., Cellular neural networks: applications, IEEE Transactions on Circuits and Systems, 35, 10, 1273-1290 (1988) · doi:10.1109/31.7601 |
| [4] | Civalleri, P. P.; Gilli, M.; Pandolfi, L., On stability of cellular neural networks with delay, IEEE Transactions on Circuits and Systems I, 40, 3, 157-165 (1993) · Zbl 0792.68115 · doi:10.1109/81.222796 |
| [5] | Cao, J., A set of stability criteria for delayed cellular neural networks, IEEE Transactions on Circuits and Systems I, 48, 4, 494-498 (2001) · Zbl 0994.82066 · doi:10.1109/81.917987 |
| [6] | Chua, L. O.; Roska, T., Stability of a class of nonreciprocal cellular neural networks, IEEE Transactions on Circuits and Systems I, 37, 12, 1520-1527 (1990) · doi:10.1109/31.101272 |
| [7] | Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations. Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6 (1989), Singapore: World Scientific, Singapore · Zbl 0719.34002 |
| [8] | Akhmet, M. U.; Yılmaz, E., Impulsive Hopfield-type neural network system with piecewise constant argument, Nonlinear Analysis: Real World Applications, 11, 4, 2584-2593 (2010) · Zbl 1202.92001 · doi:10.1016/j.nonrwa.2009.09.003 |
| [9] | Shao, Y. F.; Xu, C. J.; Zhang, Q. H., Globally exponential stability of periodic solutions to impulsive neural networks with time-varying delays, Abstract and Applied Analysis (2012) · Zbl 1242.93104 · doi:10.1155/2012/358362 |
| [10] | Sun, G.; Li, X., A new criterion to global exponential stability for impulsive neural networks with continuously distributed delays, Mathematical Methods in the Applied Sciences, 33, 17, 2107-2117 (2010) · Zbl 1213.34085 · doi:10.1002/mma.1322 |
| [11] | Wu, B.; Liu, Y.; Lu, J., New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays, Mathematical and Computer Modelling, 55, 3-4, 837-843 (2012) · Zbl 1255.93103 · doi:10.1016/j.mcm.2011.09.009 |
| [12] | Xi, Q., Global exponential stability for a class of generalized delayed neural networks with impulses, Mathematical Methods in the Applied Sciences, 34, 11, 1414-1420 (2011) · Zbl 1219.92002 · doi:10.1002/mma.1451 |
| [13] | Xia, Y.; Wong, P. J. Y., Global exponential stability of a class of retarded impulsive differential equations with applications, Chaos, Solitons & Fractals, 39, 1, 440-453 (2009) · Zbl 1197.34146 · doi:10.1016/j.chaos.2007.04.005 |
| [14] | Xiang, L., Dynamical analysis of impulsive neural networks with time-varying delays, Mathematical Methods in the Applied Sciences, 35, 13, 1564-1569 (2012) · Zbl 1253.34068 · doi:10.1002/mma.2542 |
| [15] | Xu, D.; Yang, Z., Impulsive delay differential inequality and stability of neural networks, Journal of Mathematical Analysis and Applications, 305, 1, 107-120 (2005) · Zbl 1091.34046 · doi:10.1016/j.jmaa.2004.10.040 |
| [16] | Yang, Z.; Xu, D., Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays, Applied Mathematics and Computation, 177, 1, 63-78 (2006) · Zbl 1103.34067 · doi:10.1016/j.amc.2005.10.032 |
| [17] | Zhang, Q.; Shao, Y.; Liu, J., Analysis of stability for impulsive fuzzy Cohen-Grossberg BAM neural networks with delays, Mathematical Methods in the Applied Sciences, 36, 7, 773-779 (2013) · Zbl 1271.34080 · doi:10.1002/mma.2624 |
| [18] | Zhang, Y., Robust exponential stability of discrete-time uncertain impulsive neural networks with time-varying delay, Mathematical Methods in the Applied Sciences, 35, 11, 1287-1299 (2012) · Zbl 1250.93099 · doi:10.1002/mma.2531 |
| [19] | Barone, E.; Tebaldi, C., Stability of equilibria in a neural network model, Mathematical Methods in the Applied Sciences, 23, 13, 1179-1193 (2000) · Zbl 0959.92002 · doi:10.1002/1099-1476(20000910)23:13<1179::AID-MMA158>3.3.CO;2-Y |
| [20] | Cao, J., Global asymptotic stability of neural networks with transmission delays, International Journal of Systems Science, 31, 10, 1313-1316 (2000) · Zbl 1080.93517 · doi:10.1080/00207720050165807 |
| [21] | Cao, J.; Yuan, K.; Li, H. X., Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays, IEEE Transactions on Neural Networks, 17, 6, 1646-1651 (2006) · doi:10.1109/TNN.2006.881488 |
| [22] | Du, Y.; Xu, R.; Liu, Q., Stability and bifurcation analysis for a neural network model with discrete and distributed delays, Mathematical Methods in the Applied Sciences, 36, 1, 49-59 (2013) · Zbl 1275.34102 · doi:10.1002/mma.2568 |
| [23] | Liu, Q. S.; Cao, J., Improved global exponential stability criteria of cellular neural networks with time-varying delays, Mathematical and Computer Modelling, 43, 3-4, 423-432 (2006) · Zbl 1157.34055 · doi:10.1016/j.mcm.2005.11.007 |
| [24] | Liu, X. Z.; Dickson, R., Stability analysis of Hopfield neural networks with uncertainty, Mathematical and Computer Modelling, 34, 3-4, 353-363 (2001) · Zbl 0999.34052 · doi:10.1016/S0895-7177(01)00067-X |
| [25] | Oliveira, J. J., Global asymptotic stability for neural network models with distributed delays, Mathematical and Computer Modelling, 50, 1-2, 81-91 (2009) · Zbl 1185.34107 · doi:10.1016/j.mcm.2009.02.002 |
| [26] | Qiu, J.; Cao, J., Delay-dependent robust stability of neutral-type neural networks with time delays, Journal of Mathematical Control Science and Applications, 1, 179-188 (2007) · Zbl 1170.93364 |
| [27] | Wei, X.; Zhou, D.; Zhang, Q., On asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks, Computers & Mathematics with Applications, 57, 11-12, 1938-1942 (2009) · Zbl 1186.39028 · doi:10.1016/j.camwa.2008.10.031 |
| [28] | Shah, S. M.; Wiener, J., Advanced differential equations with piecewise constant argument deviations, International Journal of Mathematics and Mathematical Sciences, 6, 4, 671-703 (1983) · Zbl 0534.34067 · doi:10.1155/S0161171283000599 |
| [29] | Cooke, K. L.; Wiener, J., Retarded differential equations with piecewise constant delays, Journal of Mathematical Analysis and Applications, 99, 1, 265-297 (1984) · Zbl 0557.34059 · doi:10.1016/0022-247X(84)90248-8 |
| [30] | Wiener, J., Generalized Solutions of Functional-Differential Equations (1993), Singapore: World Scientific, Singapore · Zbl 0874.34054 |
| [31] | Chiu, K.-S.; Pinto, M., Oscillatory and periodic solutions in alternately advanced and delayed differential equations, Carpathian Journal of Mathematics, 29, 2, 149-158 (2013) · Zbl 1299.34225 |
| [32] | Chiu, K.-S., Stability of oscillatory solutions of differential equations with a general piecewise constant argument, Electronic Journal of Qualitative Theory of Differential Equations, 88, 1-15 (2011) · Zbl 1340.34240 |
| [33] | Chiu, K.-S.; Pinto, M., Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument, Acta Mathematicae Applicatae Sinica (English Series), 27, 4, 561-568 (2011) · Zbl 1270.34169 · doi:10.1007/s10255-011-0107-5 |
| [34] | Chiu, K.-S.; Pinto, M., Periodic solutions of differential equations with a general piecewise constant argument and applications, Electronic Journal of Qualitative Theory of Differential Equations, 46, 1-19 (2010) · Zbl 1211.34082 |
| [35] | Chiu, K.-S.; Pinto, M., Stability of periodic solutions for neural networks with a general piecewise constant argument, Proceedings of the 1st Joint International Meeting AMS-SOMACHI |
| [36] | Chiu, K.-S., Periodic solutions for nonlinear integro-differential systems with piecewise constant argument |
| [37] | Pinto, M., Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Mathematical and Computer Modelling, 49, 9-10, 1750-1758 (2009) · Zbl 1171.34321 · doi:10.1016/j.mcm.2008.10.001 |
| [38] | Busenberg, S.; Cooke, K.; Lakshmikantham, V., Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, 179-187 (1982), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0512.92018 |
| [39] | Bereketoglu, H.; Seyhan, G.; Ogun, A., Advanced impulsive differential equations with piecewise constant arguments, Mathematical Modelling and Analysis, 15, 2, 175-187 (2010) · Zbl 1218.34095 · doi:10.3846/1392-6292.2010.15.175-187 |
| [40] | Karakoc, F.; Bereketoglu, H.; Seyhan, G., Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument, Acta Applicandae Mathematicae, 110, 1, 499-510 (2010) · Zbl 1196.34104 · doi:10.1007/s10440-009-9458-9 |
| [41] | Huang, Z. K.; Wang, X. H.; Gao, F., The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks, Physics Letters A, 350, 3-4, 182-191 (2006) · Zbl 1195.34066 · doi:10.1016/j.physleta.2005.10.022 |
| [42] | Cooke, K. L.; Wiener, J., An equation alternately of retarded and advanced type, Proceedings of the American Mathematical Society, 99, 4, 726-732 (1987) · Zbl 0628.34074 · doi:10.2307/2046483 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
