Global existence of positive and negative solutions for IFDEs via Lyapunov-Razumikhin method

Xipu Xu; Xipu Xu

doi:10.3934/mmc.2021014

Mathematical Modelling and Control

2021, Volume 1, Issue 3: 157-163. doi: 10.3934/mmc.2021014

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Research article

Global existence of positive and negative solutions for IFDEs via Lyapunov-Razumikhin method

Xipu Xu ^,

School of Mechanical & Automotive Engineering, Qingdao University of Technology, Qingdao 266520, China

Received: 18 May 2021 Accepted: 23 August 2021 Published: 31 August 2021

This paper considers the global existence of positive and negative solutions for impulsive functional differential equations (IFDEs). First, we introduce the concept of $ \varepsilon $-unstability to IFDEs and establish some sufficient conditions to guarantee the $ \varepsilon $-unstability via Lyapunov-Razumikhin method. Based on the obtained results, we present some sufficient conditions for the global existence of positive and negative solutions of IFDEs. An example is also given to demonstrate the effectiveness of the results.
- impulsive functional differential equations (IFDEs),
- global existence,
- Lyapunov-Razumikhin method,
- positive solution,
- negative solution
Citation: Xipu Xu. Global existence of positive and negative solutions for IFDEs via Lyapunov-Razumikhin method[J]. Mathematical Modelling and Control, 2021, 1(3): 157-163. doi: 10.3934/mmc.2021014

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Abstract

This paper considers the global existence of positive and negative solutions for impulsive functional differential equations (IFDEs). First, we introduce the concept of $ \varepsilon $-unstability to IFDEs and establish some sufficient conditions to guarantee the $ \varepsilon $-unstability via Lyapunov-Razumikhin method. Based on the obtained results, we present some sufficient conditions for the global existence of positive and negative solutions of IFDEs. An example is also given to demonstrate the effectiveness of the results.

References

[1]	D. Bainov, P. Simeonov, Systems with Impulse Effect, Ellis Horwood, Chichester, 1989.
[2]	D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993.
[3]	V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[4]	V. Lakshmikantham, X. Liu, Stability for impulsive differential systems in terms of two measures, Appl. Math. Comput., 29 (1989), 89–98.
[5]	V. Lakshmikantham, S. Leela, S. Kaul, Comparison principle for impulsive differential equations with variable times and stability theory, Nonlinear. Anal-Theor., 22 (1994), 499–503. doi: 10.1016/0362-546X(94)90170-8
[6]	K. Gopalsamy, B. Zhang, On delay differential equation with impulses, J. Math. Anal. Appl., 139 (1989), 110–122. doi: 10.1016/0022-247X(89)90232-1
[7]	X. Fu, B. Yan, Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, 2005.
[8]	G. Ballinger, X. Liu, Existence and uniqueness results for impulsive delay differential equations, Contin. Discrete Impuls. Systems, 74 (2000), 71–93.
[9]	X. Liu, G. Ballinger, Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear. Anal-Theor., 53 (2003), 1041–1062. doi: 10.1016/S0362-546X(03)00041-5
[10]	X. Liu, Q. Wang, On stability in terms of two measures for impulsive systems of functional differential equations, J. Math. Anal. Appl., 326 (2007), 252–265. doi: 10.1016/j.jmaa.2006.02.059
[11]	Y. Zhang, J. Sun, Stability of impulsive functional differential equations, Nonlinear. Anal-Theor., 68 (2008), 3665–3678. doi: 10.1016/j.na.2007.04.009
[12]	D. Lin, X. Li, D. O'Regan, Stability analysis of generalized impulsive functional differential equations, Math. Comput. Model., 55 (2012), 1682–1690. doi: 10.1016/j.mcm.2011.11.008
[13]	X. Li, J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63–69. doi: 10.1016/j.automatica.2015.10.002
[14]	X. Li, J. Shen, R. Rakkiyappan, Persistent impulsive effects on stability of functional differential ite equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14–22.
[15]	Y. Guo, Q. Zhu, F. Wang, Stability analysis of impulsive stochastic functional differential equations, Commun. Nonlinear. Sci., 82 (2020), 105013. doi: 10.1016/j.cnsns.2019.105013
[16]	J. Shen, J. Yan, Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear. Anal-Theor., 33 (1998), 519–531. doi: 10.1016/S0362-546X(97)00565-8
[17]	J. Shen, Z. Luo, X. Liu, Impulsive stabilization of functional differential equations via Lyapunov functionals, J. Math. Anal. Appl., 240 (1999), 1–15. doi: 10.1006/jmaa.1999.6551
[18]	I. Stamova, G. Stamov, Lyapunov-Razumikhin method for impulsive functional equations and applications to the population dynamics, J. Comput. Appl. Math., 130 (2001), 163–171. doi: 10.1016/S0377-0427(99)00385-4
[19]	X. Fu, X. Li, W-stability theorems of nonlinear impulsive functional differential Systems, J. Comput. Appl. Math., 1 (2008), 33–46.
[20]	X. Fu, X. Li, Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems, J. Comput. Appl. Math., 224 (2009), 1–10. doi: 10.1016/j.cam.2008.03.042
[21]	X. Li, Uniform asymptotic stability and global stability of impulsive infinite delay differential equations, Nonlinear. Anal-Theor., 70 (2009), 1975–1983. doi: 10.1016/j.na.2008.02.096
[22]	M. Stamova, T.Stamov, Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics, J. Comput. Appl. Math, 130 (2001), 163–171. doi: 10.1016/S0377-0427(99)00385-4
[23]	Y. Liu, W. Feng, Razumikhin-Lyapunov functional method for the stability of impulsive switched systems with time delay, Math. Comput. Model., 49 (2009), 249–264. doi: 10.1016/j.mcm.2008.01.004
[24]	X. Li, F. Deng, Razumikhin method for impulsive functional differential equations of neutral type, Chaos Soliton. Fract., 101 (2017), 41–49. doi: 10.1016/j.chaos.2017.05.018
[25]	J. Zhang, D. Efimov, A Lyapunov-Razumikhin Condition of ISS for Switched Time-Delay Systems Under Average Dwell Time Commutation, IFAC-Papers OnLine, 53 (2020), 1986–1991. doi: 10.1016/j.ifacol.2020.12.2568
[26]	W. Cao, Q. Zhu, Razumikhin-type theorem for pth exponential stability of impulsive stochastic functional differential equations based on vector Lyapunov function, Nonlinear. Anal-Hybri., 39 (2021), 100983. doi: 10.1016/j.nahs.2020.100983
[27]	W. Li, H. Huo, Existence and global attractivity of positive periodic solutions of functional differential equations with impulses, Nonlinear Anal-Theor., 59 (2004), 857–877. doi: 10.1016/j.na.2004.07.042
[28]	C. Cuevas, E, Hern$\acute{a}$ndez, M. Rabelo, The existence of solutions for impulsive neutral functional differential equations, Comput. Math. Appl., 58 (2009), 744–757. doi: 10.1016/j.camwa.2009.04.008
[29]	T. Jankowski, Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions, Nonlinear Anal-Theor., 74 (2011), 3775–3785. doi: 10.1016/j.na.2011.03.022
[30]	X. Hao, M. Zuo, L. Liu, Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities, Appl. Math. Lett., 82 (2018), 24–31. doi: 10.1016/j.aml.2018.02.015
[31]	S. Heidarkhani, A. Cabada, G. Afrouzi, S. Moradi, G. Caristi, A variational approach to perturbed impulsive fractional differential equations, J. Comput. Appl. Math., 341 (2018), 42–60. doi: 10.1016/j.cam.2018.02.033

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