Analyzing stability of equilibrium points in impulsive neural network models involving generalized piecewise alternately advanced and retarded argument
Main Article Content
In this paper, we investigate the models of the impulsive cellular neural network with piecewise alternately advanced and retarded argument of generalized argument (in short IDEPCAG).
To ensure the existence, uniqueness and global exponential stability of the equilibrium state, several new sufficient conditions are obtained.
The method is based on utilizing Banach's fixed point theorem and a new IDEPCAG's Gronwall inequality.
The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems.
Typical numerical simulation examples are used to show the validity and effectiveness of proposed results.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
M. Akhmet and E. Yimaz, Impulsive Hopfield-type neural network system with piecewise constant argument, Nonlinear Anal. Real World Appl. 11 (2010) 2584-2593.
E. Barone and C. Tebaldi, Stability of equilibria in a neural network model, Math. Meth. Appl. Sci., 23 (2000) 1179-1193.
S. Busenberg and K. Cooke, Vertically transmitted diseases: models and dynamics in Biomathematics, vol. 23, Springer-Verlag, Berlin, 1993.
E. Caglioti, M. Infusino, and T. Kuna, Translation invariant realizability problem on the d-dimensional lattice: an explicit construction, Electron. Commun. Probab., 21 (2016), Art. 45, pp. 1-9.
J. Cao, Global asymptotic stability of neural networks with transmission delays, Int. J. Syst. Sci. 31 (2000) 1313-1316.
S. Castillo, M. Pinto, R. Torres, Asymptotic formulae for solutions to impulsive differential equations with piecewise constant argument of generalized type, Electron. J. Differential Equations, Vol. 2019 (2019), No. 40, pp. 1-22.
T. Chen, Global exponential stability of delayed Hopfield neural networks, Neural Networks 14 (2001) 977-980.
K.-S. Chiu and M. Pinto, Periodic solutions of differential equations with a general piecewise constant argument and applications, E. J. Qualitative Theory of Diff. Equ., 46. (2010) 1-19.
K.-S. Chiu, M. Pinto and J.-Ch. Jeng, 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.
K.-S. Chiu, Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument, Abstract and Applied Analysis, vol. 2013, Article ID 196139, 13 pages, 2013. doi:10.1155/2013/196139
K.-S. Chiu, Periodic solutions for nonlinear integro-differential systems with piecewise constant argument, The Scientific World Journal, vol. 2014, Article ID 514854, 14 pages, 2014. doi:10.1155/2014/514854
K.-S. Chiu, On generalized impulsive piecewise constant delay differential equations, Science China Mathematics 58 (2015) 1981-2002.
K.-S. Chiu and J.-Ch. Jeng, Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Math. Nachr. 288 (2015) 1085-1097.
K.-S. Chiu, Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument, Acta Appl. Math. 151 (2017) 199-226.
K.-S. Chiu, Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments, Acta Math. Sci. 38 (2018) 220-236.
K.-S. Chiu and T. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr. 292 (2019) 2153-2164.
K.-S. Chiu, Green’s function for periodic solutions in alternately advanced and delayed differential systems, Acta Math. Appl. Sin. Engl. Ser. 36 (4) (2020) 936-951.
K.-S. Chiu, Green’s function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 70 (1) (2021) 15-37.
K.-S. Chiu, Existence and global exponential stability of equilibrium for impulsive neural network models with generalized piecewise constant delay, Asian-European Journal of Mathematics, https://doi.org/10.1142/S1793557122500012
K.-S. Chiu and F. Córdova-Lepe, Global exponential periodicity and stability of neural network models with generalized piecewise constant delay, Math. Slovaca 71 (2) (2021) 491-512.
L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988) 1257-1272.
K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput. 154 (2004) 783-813.
Z. K. Huang, X. H. Wang and F. Gao, The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks, Phys. Lett. A, 350 (2006) 182-191.
M. Infusino, T. Kuna, The full moment problem on subsets of probabilities and point configurations, J. Math. Anal. Appl., 483(1) (2020), Art. 123551, pp. 1-29.
O. M. Kwona, S. M. Lee, Ju H. Park and E. J. Cha , New approaches on stability criteria for neural networks with interval time-varying delays, Appl. Math. Comput. 218 (2012) 9953-9964.
T. Li, X. Yao, L. Wu and J. Li, Improved delay-dependent stability results of recurrent neural networks, Appl. Math. Comput. 218 (2012) 9983-9991.
T. Li, N. Pintus, and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70 (3) (2019), Art. 86, pp. 1-18.
T. Li and G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differential Integral Equations, 34 (5-6) (2021), 315-336.
Z. Liu, and L. Liao, Existence and global exponential stability of periodic solutions of cellular neural networks with time-varying delays, J. Math. Anal. Appl. 290 (2004) 247- 262.
X. Y. Lou and B. T. Cui, Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays, J. Math. Anal. Appl. 330 (2007) 144-158.
S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discretetime cellular neural networks with delays, Appl. Math. Comput. 135 (2003) 17-38.
J. H. Park, Global exponential stability of cellular neural networks with variable delays, Appl. Math. Comput. 183 (2006) 1214-1219.
M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equ. Appl. 17 (2) (2011) 235-254.
S. M. Shah and J. Wiener, Advanced differential equations with piecewise constant argument deviations, Internat. J. Math.and Math. Sci. 6 (1983) 671-703.
I. M. Stamova and G. T. Stamov, Applied Impulsive Mathematical Models, 1st ed.; Springer: Cham, Switzerland, 2016.
G. Viglialoro and T. E. Woolley, Solvability of a Keller-Segel system with signaldependent sensitivity and essentially sublinear production, Appl. Anal., 99 (14) (2020), 2507-2525.
B. Wang, S. Zhong and X. Liu, Asymptotical stability criterion on neural networks with multiple time-varying delays, Appl. Math. Comput. 195 (2008) 809-818.
J. Wiener, Differential equations with piecewise constant delays, V. Lakshmikantham, Trends in the Theory and Practice of Nonlinear Differential Equations, Marcel Dekker, New York, 1983, 547-580.
J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993.
J. Wiener and V. Lakshmikantham, Differential equations with piecewise constant argument and impulsive equations, Nonlinear Stud., 7 (2000) 60-69.
B. Xu, X. Liu and X. Liao, Global exponential stability of high order Hopfield type neural networks, Appl. Math. Comput. 174 (2006) 98-116.
L. Zhou and G. Hu, Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput. 195 (2008) 402-411.
Y. Zhang, D. Yue and E. Tian, New stability criteria of neural networks with interval timevarying delay: A piecewise delay method, Appl. Math. Comput. 208 (2009) 249-259.