This paper devotes to the asymptotic behavior of all solutions of nth order impulsive differential equations. Based on impulsive differential inequality, boundedness and zero tendency of every solution for nth order impulsive differential equations are obtained. In addition, we derive globally uniformly exponential stability of every solution under Lyapunov function and impulsivetechnique, and these results are extend to nth order differential equations with periodic coefficient and periodic impulse. Meanwhile, an example with simulations are provided to verify the conclusion.
| Published in | Science Journal of Applied Mathematics and Statistics (Volume 12, Issue 3) |
| DOI | 10.11648/j.sjams.20241203.12 |
| Page(s) | 43-47 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Asymptotic, nth Order, Impulsive Differential Equation
| [1] | Lakshmikantham. V., Bainov. D., Simeonov. P. Theory of Impulsive Differential Equations. World Scientific. Singapor; 1989. |
| [2] | Bainov. D., Simeonov. P. Oscillation Theorem of Impulsive Differential Equations. International Publications; 1998. |
| [3] | Zhang. W., Tang. Y., Zheng. Z., Liu. Y Stability of time-varying systems with delayed impulsive effects. International Journal of Robust and Nonlinear Control. 2021, 31, 7825-7843. |
| [4] | Li. X., Ding. Y Razumikhin-type theorems for time- delay systems with persistent impulses. Systems Control Letters. 2017, 107, 22-27. |
| [5] | Wang., Duan. H. S., Li. C., Wang. L., Huang. T Globally exponential stability of delayed impulsive functional differential systems with impulse time windows. Nonlinear Dynamics. 2016, 84, 1-11. |
| [6] | Liu. B., Hill. D. Impulsive consensus for complex dynamical networks with nonidentical nodes and coupling time-delays. SIAM J. Control Optim. 2011, 49, 315-338. |
| [7] | Lu. J., Ho. D., Cao. J A unified synchronization criterion for impulsive dynamical networks. Automatica,. 2010, 46, 1215-1221. |
| [8] | Chen. Y., Feng. W Oscillations of second order nonlinear ODE with impulses. J. Math. Anal. Appl. 1997, 210, 150-169. |
| [9] | Wen. L., Chen. Y Razumikhin type theorems for functional differential equations with impulses. Dynamics of Continuous, Discrete and Impulsive Systems. 1999, 6, 389-400. |
| [10] | Jiao. J., Chen. L., Li. L Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations. J. Math. Anal. Appl. 2008, 337, 458-463. |
| [11] | Wang. Q Oscillation and asymptotics for second order half linear differential equation. Appl. Math. Comput. 2001, 112, 253-266. |
| [12] | Wu. X., Chen.S., Ji. H Oscillation of a class of second-order nonlinear ODE with impulses. Appl. Math. Comput. 2003, 138, 181-188. |
| [13] | Chen. Y., Feng. W Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations. Journal of south normal university (Natural science). 2000, 1, 13-17. |
| [14] | Pan. L., Hu. J., Cao. J Razumikhin and Krasovskii stability of impulsive stochastic delay systems via uniformly stable function method. Nonlinear Analysis: Modelling and Control. 2023, 28, 1161-1181. |
| [15] | Mariton. M Oscillation properties of higher oeder linear impulsive delay differential equations. Differential Equations Applications. 2015, 7, 43-55. |
| [16] | Ren. W., Xiong. J Stability analysis of impulsive stochastic nonlinear systems. IEEE Trans. Autom. Control. 2017, 62, 4791-4797. |
APA Style
Zheng, Y., Pan, L. (2024). Asymptotic Behavior of nth Order Impulsive Differential Equations. Science Journal of Applied Mathematics and Statistics, 12(3), 43-47. https://doi.org/10.11648/j.sjams.20241203.12
ACS Style
Zheng, Y.; Pan, L. Asymptotic Behavior of nth Order Impulsive Differential Equations. Sci. J. Appl. Math. Stat. 2024, 12(3), 43-47. doi: 10.11648/j.sjams.20241203.12
AMA Style
Zheng Y, Pan L. Asymptotic Behavior of nth Order Impulsive Differential Equations. Sci J Appl Math Stat. 2024;12(3):43-47. doi: 10.11648/j.sjams.20241203.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.sjams.20241203.12,
author = {Yuxin Zheng and Lijun Pan},
title = {Asymptotic Behavior of nth Order Impulsive Differential Equations},
journal = {Science Journal of Applied Mathematics and Statistics},
volume = {12},
number = {3},
pages = {43-47},
doi = {10.11648/j.sjams.20241203.12},
url = {https://doi.org/10.11648/j.sjams.20241203.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20241203.12},
abstract = {This paper devotes to the asymptotic behavior of all solutions of nth order impulsive differential equations. Based on impulsive differential inequality, boundedness and zero tendency of every solution for nth order impulsive differential equations are obtained. In addition, we derive globally uniformly exponential stability of every solution under Lyapunov function and impulsivetechnique, and these results are extend to nth order differential equations with periodic coefficient and periodic impulse. Meanwhile, an example with simulations are provided to verify the conclusion.},
year = {2024}
}
TY - JOUR T1 - Asymptotic Behavior of nth Order Impulsive Differential Equations AU - Yuxin Zheng AU - Lijun Pan Y1 - 2024/06/14 PY - 2024 N1 - https://doi.org/10.11648/j.sjams.20241203.12 DO - 10.11648/j.sjams.20241203.12 T2 - Science Journal of Applied Mathematics and Statistics JF - Science Journal of Applied Mathematics and Statistics JO - Science Journal of Applied Mathematics and Statistics SP - 43 EP - 47 PB - Science Publishing Group SN - 2376-9513 UR - https://doi.org/10.11648/j.sjams.20241203.12 AB - This paper devotes to the asymptotic behavior of all solutions of nth order impulsive differential equations. Based on impulsive differential inequality, boundedness and zero tendency of every solution for nth order impulsive differential equations are obtained. In addition, we derive globally uniformly exponential stability of every solution under Lyapunov function and impulsivetechnique, and these results are extend to nth order differential equations with periodic coefficient and periodic impulse. Meanwhile, an example with simulations are provided to verify the conclusion. VL - 12 IS - 3 ER -
School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, China
School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, China
Information