Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter April 14, 2021

Global exponential periodicity and stability of neural network models with generalized piecewise constant delay

  • Kuo-Shou Chiu EMAIL logo and Fernando Córdova-Lepe
From the journal Mathematica Slovaca

Abstract

In this paper, the global exponential stability and periodicity are investigated for delayed neural network models with continuous coefficients and piecewise constant delay of generalized type. The sufficient condition for the existence and uniqueness of periodic solutions of the model is established by applying Banach’s fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.


This research was in part supported by FGI 10-18 DIUMCE.


  1. (Communicated by Michal Fečkan)

References

[1] Akhmet, M. U.—Yimaz, E.: Hopfield-type neural networks systems with piecewise constant argument, J. Qual. Theory Differ. Equ. Appl. 3 (2009), 8–14.10.1016/j.nonrwa.2009.09.003Search in Google Scholar

[2] Akhmet, M. U.—Arugaslan, D.—Yimaz, E.: Stability in cellular neural networks with a piecewise constant argument, J. Comput. Appl. Math. 233 (2010), 2365–2373.10.1016/j.cam.2009.10.021Search in Google Scholar

[3] Akhmet, M. U.: Nonlinear Hybrid Continuous/Discrete-Time Models, Atlantis Press, Paris, 2011.10.2991/978-94-91216-03-9Search in Google Scholar

[4] Barone, E.—Tebaldi, C.: Stability of equilibria in a neural network model, Math. Models Appl. Sci. 3 (2000), 1179–1193.10.1002/1099-1476(20000910)23:13<1179::AID-MMA158>3.0.CO;2-6Search in Google Scholar

[5] Busenberg, S.—Cooke, K.: Vertically Transmitted Diseases: Models and Dynamics. Biomathematics, vol. 23, Springer-Verlag, Berlin, 1993.10.1007/978-3-642-75301-5Search in Google Scholar

[6] Cao, J.: Global asymptotic stability of neural networks with transmission delays, Internat. J. Systems Sci. 31 (2000), 1313–1316.10.1080/00207720050165807Search in Google Scholar

[7] Cooke, K. L.—Wiener, J.: Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984), 265–297.10.1016/0022-247X(84)90248-8Search in Google Scholar

[8] Cooke, K. L.—Wiener, J.: An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc. 99 (1987), 726–732.10.1090/S0002-9939-1987-0877047-8Search in Google Scholar

[9] Chen, A.—Cao, J.: Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients, Appl. Math. Comput. 134 (2003), 125–140.Search in Google Scholar

[10] Chiu, K.-S.—Pinto, M.: Oscillatory and periodic solutions in alternately advanced and delayed differential equations, Carpathian J. Math. 29(2) (2013), 149–158.10.37193/CJM.2013.02.15Search in Google Scholar

[11] Chiu, K.-S.: Stability of oscillatory solutions of differential equations with a general piecewise constant argument, Electron. J. Qual. Theory Differ. Equ. 88 (2011), 1–15.10.14232/ejqtde.2011.1.88Search in Google Scholar

[12] Chiu, K.-S.—Pinto, M.: Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument, Acta Math. Appl. Sin. Engl. Ser. 27(4) (2011), 561–568.10.1007/s10255-011-0107-5Search in Google Scholar

[13] Chiu, K.-S.—Pinto, M.: Periodic solutions of differential equations with a general piecewise constant argument and applications, Electron. J. Qual. Theory Differ. Equ. 46 (2010), 1–19.10.14232/ejqtde.2010.1.46Search in Google Scholar

[14] Chiu, K.-S.—Pinto, M.—Jeng, J.-Ch.: Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument, Acta Appl. Math. 133 (2014), 133–152.10.1007/s10440-013-9863-ySearch in Google Scholar

[15] Chiu, K.-S.: Periodic solutions for nonlinear integro-differential systems with piecewise constant argument, The Scientific World Journal 2014 (2014), Art. ID 514854.10.1155/2014/514854Search in Google Scholar

[16] Chiu, K.-S.: Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument, Abstract and Applied Analysis 2013 (2013), Art. ID 196139.10.1155/2013/196139Search in Google Scholar

[17] Chiu, K.-S.—Jeng, J.-Ch.: Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Math. Nachr. 288 (2015), 1085–1097.10.1002/mana.201300127Search in Google Scholar

[18] Chiu, K.-S.: Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument, Acta Appl. Math. 151 (2017), 199-226.10.1007/s10440-017-0108-3Search in Google Scholar

[19] Chiu, K.-S.: Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments, Acta Math. Sci. 38 (2018), 220–236.10.1016/S0252-9602(17)30128-5Search in Google Scholar

[20] Chiu, K.-S.—Li, T.: Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr. 292 (2019), 2153–2164.10.1002/mana.201800053Search in Google Scholar

[21] Chua, L. O.—Yang, L.: Cellular neural networks: Theory, EEE Trans. Circuits Syst. 35 (1988), 1257–1272.10.1109/31.7600Search in Google Scholar

[22] Hua, C.—Yang, X.—Yan, J.—Guan, X.: New stability criteria for neural networks with time-varying delays, Appl. Math. Comput. 218 (2012), 5035–5042.Search in Google Scholar

[23] Huang, Z. K.—Wang, X. H.—Gao, F.: The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks, Phys. Lett. A. 350 (2006), 182–191.10.1016/j.physleta.2005.10.022Search in Google Scholar

[24] Kwon, O. M.—Lee, S. M.—Park, J. H.—Cha, E. J.: New approaches on stability criteria for neural networks with interval time-varying delays, Appl. Math. Comput. 218 (2012), 9953–9964.10.1016/j.amc.2012.03.082Search in Google Scholar

[25] Li, T.—Yao, X.—Wu, L.—Li, J.: Improved delay-dependent stability results of recurrent neural networks, Appl. Math. Comput. 218 (2012), 9983–9991.10.1016/j.amc.2012.03.013Search in Google Scholar

[26] Liu, Z.—Liao, L.: Existence and global exponential stability of periodic solutions of cellular neural networks with time-varying delays, J. Math. Anal. Appl. 290 (2004), 247–262.10.1016/j.jmaa.2003.09.052Search in Google Scholar

[27] Lou, X. Y.—Cui, B. T.: Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays, J. Math. Anal. Appl. 330 (2007), 144–158.10.1016/j.jmaa.2006.07.058Search in Google Scholar

[28] Mohamad, S.—Gopalsamy, K.: Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput. 135 (2003), 17–38.10.1016/S0096-3003(01)00299-5Search in Google Scholar

[29] Park, J. H.: Global exponential stability of cellular neural networks with variable delays, Appl. Math. Comput. 183 (2006), 1214–1219.10.1016/j.amc.2006.06.046Search in Google Scholar

[30] Pinto, M.: Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments, Math. and Comp. Model. 49 (2009), 1750–1758.10.1016/j.mcm.2008.10.001Search in Google Scholar

[31] Pinto, M.: Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equ. Appl. 17(2) (2011), 235–254.10.1080/10236198.2010.549003Search in Google Scholar

[32] Shah, S. M.—Wiener, J.: Advanced differential equations with piecewise constant argument deviations, Internat. J. Math. and Math. Sci. 6 (1983), 671–703.10.1155/S0161171283000599Search in Google Scholar

[33] Wang, B.—Zhong, S.—Liu, X.: Asymptotical stability criterion on neural networks with multiple time-varying delays, Appl. Math. Comput. 195 (2008), 809–818.10.1016/j.amc.2007.05.027Search in Google Scholar

[34] Wiener, J.: Differential equations with piecewise constant delays. In: Trends in the Theory and Practice of Nonlinear Differential Equations (V. Lakshmikantham, ed.), Marcel Dekker, New York, 1983, pp. 547–580.Search in Google Scholar

[35] Wiener, J.: Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993.10.1142/1860Search in Google Scholar

[36] Xu, S.—Chu, Y.—Lu, J.: New results on global exponential stability of recurrent neural networks with time-varying delays, Phys. Lett. A 352 (2006), 371–379.10.1016/j.physleta.2005.12.031Search in Google Scholar

[37] Xu, B.—Liu, X.—Liao, X.: Global exponential stability of high order Hopfield type neural networks, Appl. Math. Comput. 174 (2006), 98–116.10.1016/j.amc.2005.03.020Search in Google Scholar

[38] Yu, T. H.—Cao, D. Q.—Liu, S. Q.—Chen, H. T.: Stability analysis of neural networks with periodic coefficients and piecewise constant arguments, J. Franklin Inst. 353 (2016), 409–425.10.1016/j.jfranklin.2015.11.010Search in Google Scholar

[39] Zhou, L.—Hu, G.: Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput. 195 (2008), 402–411.10.1016/j.amc.2007.04.114Search in Google Scholar

[40] Zhang, Y.—Yue, D.—Tian, E.: New stability criteria of neural networks with interval time-varying delay: A piecewise delay method, Appl. Math. Comput. 208 (2009), 249–259.10.1016/j.amc.2008.11.046Search in Google Scholar

Received: 2019-12-10
Accepted: 2020-06-06
Published Online: 2021-04-14
Published in Print: 2021-04-27

© 2021 Mathematical Institute Slovak Academy of Sciences

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/ms-2017-0483/html
Scroll to top button