In this paper, we introduce a Cohen-Grossberg neural networks model with piecewise alternately advanced and retarded argument. Some sufficient conditions are established for the existence and global exponential stability of periodic solutions. The approaches are based on employing Brouwer's fixed-point theorem and an integral inequality of Gronwall type with deviating argument. The criteria given are easily verifiable, possess many adjustable parameters, and depend on piecewise constant argument deviations, which provide flexibility for the design and analysis of Cohen-Grossberg neural networks model. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.
Universidad Metropolitana de Ciencias de la Educación
Project Number
PGI 03-2020 DIUMCE
References
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argument of generalized type, Nonlinear Anal. Theory Methods Appl. 66, 367–383,
2007.
[3] M.U. Akhmet, D. Arugaslan, M. Tleubergenova and Z. Nugayeva, Unpredictable oscillations
for Hopfield-type neural networks with delayed and advanced arguments,
Mathematics, 9, 571, 2021. https://doi.org/10.3390/math9050571
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Networks, 33, 32–41, 2012.
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differential equations with several non-monotone retarded arguments, Georgian
Math. J. 27 (3), 341–350, 2020.
[9] H. Bereketoglu, G. Seyhan and F. Karakoc, On a second order differential equation
with piecewise constant mixed arguments, Carpathian J. Math. 27, 1–12, 2011.
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constant argument, The Scientific World Journal 2014, Article ID 514854, 14 pages,
2014.
[11] K.-S. Chiu, On generalized impulsive piecewise constant delay differential equations,
Science China Mathematics, 58, 1981–2002, 2015.
[12] K.-S. Chiu, Asymptotic equivalence of alternately advanced and delayed differential
systems with piecewise constant generalized arguments, Acta Math. Sci. 38, 220–236,
2018.
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networks model with piecewise alternately advanced and retarded argument, Comp.
Appl. Math. 40, 263, 2021.
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generalized piecewise constant delays, Discrete and Continuous Dynamical Systems -
B, 27 (2), 659–689, 2022. doi: 10.3934/dcdsb.2021060
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network models with generalized piecewise constant delay, Math. Slovaca, 71, 491–512,
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with general piecewise constant arguments of mixed type, Math. Nachr. 288, 1085–
1097, 2015.
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piecewise constant generalized mixed arguments, Math. Nachr. 292, 2153–2164, 2019.
[18] K.-S. Chiu and T. Li, New stability results for bidirectional associative memory neural
networks model involving generalized piecewise constant delay, Math. Comput. Simul.
194, 719–743, 2022.
[19] K.-S. Chiu and M. Pinto, Periodic solutions of differential equations with a general
piecewise constant argument and applications,Electron. J. Qual. Theory Differ. Equ.
46, 1–19, 2010.
[20] K.-S. Chiu, M. Pinto and J.-C. 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, 133–152, 2014.
[21] M. Cohen and S. Grossberg, Absolute stability and global pattern formation and parallel
memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybernet,
SMC 13, 815–826, 1983.
[22] B. Cui and W. Wu, Global exponential stability of Cohen-Grossberg neural networks
with distributed delays, Neurocomputing, 72, 386–391, 2008.
[23] L. Dai, Nonlinear Dynamics of Piecewise of Constant Systems and Implememtation
of Piecewise Constants Arguments, World Scientific, Singapore, 2008.
[24] S. Esteves and J. Oliveira, Global asymptotic stability of nonautonomous Cohen-
Grossberg neural network models with infinite delays, Appl. Math. Comput. 265,
333–346, 2015.
[25] S. Gao, R. Shen and T. Chen, Periodic solutions for discrete-time Cohen-Grossberg
neural networks with delays, Physics Letters A, 383, 414–420, 2019.
[26] L.V. Hien, T.T. Loan, B.T. Huyen Trang and H. Trinh, Existence and global asymptotic
stability of positive periodic solution of delayed Cohen-Grossberg neural networks,
Appl. Math. Comput. 240, 200–212, 2014.
[27] S. Kakutani A generalization of Brouwer’s fixed-point theorem, Duke Math. J. 8,
457–459, 1941.
[28] H. Kang, X. Fu and Z. Sun, Global exponential stability of periodic solutions for impulsive
Cohen-Grossberg neural networks with delays, Appl. Math. Model. 39, 1526–1535,
2015.
[29] F. Karakoc, Asymptotic behavior of a Lasota-Wazewska model under impulse effect,
Dyn. Syst. Appl. 29(12), 3381–3394, 2020.
[30] Y. Li and X. Fan, 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.
[31] B. Li and Q. Song, Some new results on periodic solution of Cohen-Grossberg neural
network with impulses, Neurocomputing , 177, 401–408, 2016.
[32] B. Li and D. Xu, Existence and exponential stability of periodic solution for impulsive
Cohen-Grossberg neural networks with time-varying delays, Appl. Math. Comput.
219, 2506–2520, 2012.
[33] X. Liao, J. Yang and S. Guo, Exponential stability of Cohen-Grossberg neural networks
with delays, Commun. Nonlinear Sci. Numer. Simu. 13, 1767–1775, 2008.
[34] B. Lisena, Dynamical behavior of impulsive and periodic Cohen-Grossberg neural networks,
Nonlinear Anal. 74, 4511–4519, 2011.
[35] F. Meng, K. Li, Q. Song, Y. Liu and Fuad E. Alsaadi, Periodicity of Cohen-Grossbergtype
fuzzy neural networks with impulses and time-varying delays, Neurocomputing,
325, 254–259, 2019.
[36] F. Meng, K. Li, Zh. Zhao, Q. Song, Y. Liu and Fuad E. Alsaadi, Periodicity of impulsive
Cohen-Grossberg-type fuzzy neural networks with hybrid delays, Neurocomputing,
368, 153–162, 2019.
[37] G.S. Oztepe, F. Karakoc and H. Bereketoglu, Oscillation and periodicity of a second
order impulsive delay differential equation with a piecewise constant argument,
Commun. Math. 25, 89–98, 2017.
[38] G.S. Oztepe, Convergence of solutions of an impulsive differential system with a piecewise
constant argument, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66(2),
115–129, 2017.
[39] S.M. Shah and J. Wiener, Advanced differential equations with piecewise constant
argument deviations, Int. J. Math. Math. Sci. 6, 671–703, 1983.
[40] Y. Shi and J. Cao, Finite-time synchronization of memristive Cohen-Grossberg neural
networks with time delays, Neurocomputing 377, 159–167, 2020.
Year 2022,
Volume: 51 Issue: 5, 1219 - 1236, 01.10.2022
[1] M.U. Akhmet, Integral manifolds of differential equations with piecewise constant
argument of generalized type, Nonlinear Anal. Theory Methods Appl. 66, 367–383,
2007.
[3] M.U. Akhmet, D. Arugaslan, M. Tleubergenova and Z. Nugayeva, Unpredictable oscillations
for Hopfield-type neural networks with delayed and advanced arguments,
Mathematics, 9, 571, 2021. https://doi.org/10.3390/math9050571
[4] C. Aouiti and F. Dridi, New results on impulsive Cohen-Grossberg neural networks,
Neural Process Lett. 49, 1459–1483, 2019.
[5] D. Arugaslan and N. Cengiz, Existence of periodic solutions for a mechanical system
with piecewise constant forces, Hacet. J. Math. Stat. 47 (3), 521–538, 2018.
[6] D. Arugaslan and L. Guzel, Stability of the logistic population model with generalized
piecewise constant delays, Adv. Differ. Equ. 2015, 173, 2015.
[7] G. Bao, S. Wen and Z. Zeng, Robust stability analysis of interval fuzzy Cohen-
Grossberg neural networks with piecewise constant argument of generalized type, Neural
Networks, 33, 32–41, 2012.
[8] H. Bereketoglu, F. Karakoc, G.S. Oztepe and I. P. Stavroulakis, Oscillation of firstorder
differential equations with several non-monotone retarded arguments, Georgian
Math. J. 27 (3), 341–350, 2020.
[9] H. Bereketoglu, G. Seyhan and F. Karakoc, On a second order differential equation
with piecewise constant mixed arguments, Carpathian J. Math. 27, 1–12, 2011.
[10] K.-S. Chiu, Periodic solutions for nonlinear integro-differential systems with piecewise
constant argument, The Scientific World Journal 2014, Article ID 514854, 14 pages,
2014.
[11] K.-S. Chiu, On generalized impulsive piecewise constant delay differential equations,
Science China Mathematics, 58, 1981–2002, 2015.
[12] K.-S. Chiu, Asymptotic equivalence of alternately advanced and delayed differential
systems with piecewise constant generalized arguments, Acta Math. Sci. 38, 220–236,
2018.
[13] K.-S. Chiu, Global exponential stability of bidirectional associative memory neural
networks model with piecewise alternately advanced and retarded argument, Comp.
Appl. Math. 40, 263, 2021.
[14] K.-S. Chiu, Periodicity and stability analysis of impulsive neural network models with
generalized piecewise constant delays, Discrete and Continuous Dynamical Systems -
B, 27 (2), 659–689, 2022. doi: 10.3934/dcdsb.2021060
[15] K.-S. Chiu and F. Córdova-Lepe, Global exponential periodicity and stability of neural
network models with generalized piecewise constant delay, Math. Slovaca, 71, 491–512,
2021.
[16] K.-S. Chiu and J.-C. Jeng, Stability of oscillatory solutions of differential equations
with general piecewise constant arguments of mixed type, Math. Nachr. 288, 1085–
1097, 2015.
[17] K.-S. Chiu and T. Li,Oscillatory and periodic solutions of differential equations with
piecewise constant generalized mixed arguments, Math. Nachr. 292, 2153–2164, 2019.
[18] K.-S. Chiu and T. Li, New stability results for bidirectional associative memory neural
networks model involving generalized piecewise constant delay, Math. Comput. Simul.
194, 719–743, 2022.
[19] K.-S. Chiu and M. Pinto, Periodic solutions of differential equations with a general
piecewise constant argument and applications,Electron. J. Qual. Theory Differ. Equ.
46, 1–19, 2010.
[20] K.-S. Chiu, M. Pinto and J.-C. 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, 133–152, 2014.
[21] M. Cohen and S. Grossberg, Absolute stability and global pattern formation and parallel
memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybernet,
SMC 13, 815–826, 1983.
[22] B. Cui and W. Wu, Global exponential stability of Cohen-Grossberg neural networks
with distributed delays, Neurocomputing, 72, 386–391, 2008.
[23] L. Dai, Nonlinear Dynamics of Piecewise of Constant Systems and Implememtation
of Piecewise Constants Arguments, World Scientific, Singapore, 2008.
[24] S. Esteves and J. Oliveira, Global asymptotic stability of nonautonomous Cohen-
Grossberg neural network models with infinite delays, Appl. Math. Comput. 265,
333–346, 2015.
[25] S. Gao, R. Shen and T. Chen, Periodic solutions for discrete-time Cohen-Grossberg
neural networks with delays, Physics Letters A, 383, 414–420, 2019.
[26] L.V. Hien, T.T. Loan, B.T. Huyen Trang and H. Trinh, Existence and global asymptotic
stability of positive periodic solution of delayed Cohen-Grossberg neural networks,
Appl. Math. Comput. 240, 200–212, 2014.
[27] S. Kakutani A generalization of Brouwer’s fixed-point theorem, Duke Math. J. 8,
457–459, 1941.
[28] H. Kang, X. Fu and Z. Sun, Global exponential stability of periodic solutions for impulsive
Cohen-Grossberg neural networks with delays, Appl. Math. Model. 39, 1526–1535,
2015.
[29] F. Karakoc, Asymptotic behavior of a Lasota-Wazewska model under impulse effect,
Dyn. Syst. Appl. 29(12), 3381–3394, 2020.
[30] Y. Li and X. Fan, 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.
[31] B. Li and Q. Song, Some new results on periodic solution of Cohen-Grossberg neural
network with impulses, Neurocomputing , 177, 401–408, 2016.
[32] B. Li and D. Xu, Existence and exponential stability of periodic solution for impulsive
Cohen-Grossberg neural networks with time-varying delays, Appl. Math. Comput.
219, 2506–2520, 2012.
[33] X. Liao, J. Yang and S. Guo, Exponential stability of Cohen-Grossberg neural networks
with delays, Commun. Nonlinear Sci. Numer. Simu. 13, 1767–1775, 2008.
[34] B. Lisena, Dynamical behavior of impulsive and periodic Cohen-Grossberg neural networks,
Nonlinear Anal. 74, 4511–4519, 2011.
[35] F. Meng, K. Li, Q. Song, Y. Liu and Fuad E. Alsaadi, Periodicity of Cohen-Grossbergtype
fuzzy neural networks with impulses and time-varying delays, Neurocomputing,
325, 254–259, 2019.
[36] F. Meng, K. Li, Zh. Zhao, Q. Song, Y. Liu and Fuad E. Alsaadi, Periodicity of impulsive
Cohen-Grossberg-type fuzzy neural networks with hybrid delays, Neurocomputing,
368, 153–162, 2019.
[37] G.S. Oztepe, F. Karakoc and H. Bereketoglu, Oscillation and periodicity of a second
order impulsive delay differential equation with a piecewise constant argument,
Commun. Math. 25, 89–98, 2017.
[38] G.S. Oztepe, Convergence of solutions of an impulsive differential system with a piecewise
constant argument, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66(2),
115–129, 2017.
[39] S.M. Shah and J. Wiener, Advanced differential equations with piecewise constant
argument deviations, Int. J. Math. Math. Sci. 6, 671–703, 1983.
[40] Y. Shi and J. Cao, Finite-time synchronization of memristive Cohen-Grossberg neural
networks with time delays, Neurocomputing 377, 159–167, 2020.
Chiu, K.-s. (2022). Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument. Hacettepe Journal of Mathematics and Statistics, 51(5), 1219-1236. https://doi.org/10.15672/hujms.1001754
AMA
Chiu Ks. Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. October 2022;51(5):1219-1236. doi:10.15672/hujms.1001754
Chicago
Chiu, Kuo-shou. “Existence and Global Exponential Stability of Periodic Solution for Cohen-Grossberg Neural Networks Model With Piecewise Constant Argument”. Hacettepe Journal of Mathematics and Statistics 51, no. 5 (October 2022): 1219-36. https://doi.org/10.15672/hujms.1001754.
EndNote
Chiu K-s (October 1, 2022) Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument. Hacettepe Journal of Mathematics and Statistics 51 5 1219–1236.
IEEE
K.-s. Chiu, “Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 5, pp. 1219–1236, 2022, doi: 10.15672/hujms.1001754.
ISNAD
Chiu, Kuo-shou. “Existence and Global Exponential Stability of Periodic Solution for Cohen-Grossberg Neural Networks Model With Piecewise Constant Argument”. Hacettepe Journal of Mathematics and Statistics 51/5 (October 2022), 1219-1236. https://doi.org/10.15672/hujms.1001754.
JAMA
Chiu K-s. Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. 2022;51:1219–1236.
MLA
Chiu, Kuo-shou. “Existence and Global Exponential Stability of Periodic Solution for Cohen-Grossberg Neural Networks Model With Piecewise Constant Argument”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 5, 2022, pp. 1219-36, doi:10.15672/hujms.1001754.
Vancouver
Chiu K-s. Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. 2022;51(5):1219-36.