Araştırma Makalesi

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.

Cohen-Grossberg neural networks model piecewise constant argument periodic solutions global exponential stability Gronwall integral inequality

Universidad Metropolitana de Ciencias de la Educación

PGI 03-2020 DIUMCE

- [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.
- [2] M.U. Akhmet, Nonlinear Hybrid Continuous/Discrete-Time Models, Atlantis Press, Paris, 2011.
- [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.

Yıl 2022,
Cilt: 51 Sayı: 5, 1219 - 1236, 01.10.2022
### Öz

### Proje Numarası

### Kaynakça

PGI 03-2020 DIUMCE

- [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.
- [2] M.U. Akhmet, Nonlinear Hybrid Continuous/Discrete-Time Models, Atlantis Press, Paris, 2011.
- [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.

Toplam 40 adet kaynakça vardır.

Birincil Dil | İngilizce |
---|---|

Konular | Matematik |

Bölüm | Matematik |

Yazarlar | |

Proje Numarası | PGI 03-2020 DIUMCE |

Yayımlanma Tarihi | 1 Ekim 2022 |

Yayımlandığı Sayı | Yıl 2022 Cilt: 51 Sayı: 5 |