Convergence conditions for extreme solutions of an impulsive differential system

Bing Hu; Yikai Liao; Bing Hu; Yikai Liao

doi:10.3934/math.2025481

AIMS Mathematics

2025, Volume 10, Issue 5: 10591-10604. doi: 10.3934/math.2025481

Previous Article Next Article

Research article Special Issues

Convergence conditions for extreme solutions of an impulsive differential system

Bing Hu ^,,
Yikai Liao

Department of Mathematics, School of Mathematical Sciences, Zhejiang University of Technology, Hangzhou 310023, China

Received: 28 February 2025 Revised: 16 April 2025 Accepted: 25 April 2025 Published: 09 May 2025
MSC : 34B37, 34B05

In this paper, we discuss the existence and convergence conditions of extreme solutions for a first-order differential boundary value system with an impulsive integral condition. By employing quasi-linear and monotone iterative methods, the existence, uniform convergence, and quadratic convergence for solution sequences are obtained.
- functional differential equation,
- impulsive integral condition,
- monotone iterative technique,
- quasilinearization method,
- quadratic convergence
Citation: Bing Hu, Yikai Liao. Convergence conditions for extreme solutions of an impulsive differential system[J]. AIMS Mathematics, 2025, 10(5): 10591-10604. doi: 10.3934/math.2025481

Related Papers:

Abstract

In this paper, we discuss the existence and convergence conditions of extreme solutions for a first-order differential boundary value system with an impulsive integral condition. By employing quasi-linear and monotone iterative methods, the existence, uniform convergence, and quadratic convergence for solution sequences are obtained.

References

[1]	A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, Singapore: World scientific, 1995.
[2]	H. Xu, Q. Zhu, W. X. Zheng, Exponential stability of stochastic nonlinear delay systems subject to multiple periodic impulses, IEEE Trans. Autom. Control, 69 (2024), 2621–2628. https://doi.org/10.1109/TAC.2023.3335005 doi: 10.1109/TAC.2023.3335005
[3]	I. Stamova, G. T. Stamov, Applied impulsive mathematical models, Cham, Switzerland: Springer, 2016.
[4]	Z. Li, Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by fBm, Neurocomputing, 177 (2016), 620–627. https://doi.org/10.1016/j.neucom.2015.11.070 doi: 10.1016/j.neucom.2015.11.070
[5]	W. Hu, Q. Zhu, H. R. Karimi, Some improved Razumikhin stability criteria for impulsive stochastic delay differential systems, IEEE Trans. Autom. Control, 64 (2019), 5207–5213. https://doi.org/10.1109/TAC.2019.2911182 doi: 10.1109/TAC.2019.2911182
[6]	M. Zaczynska, H. Abramovich, C. Bisagni, Parametric studies on the dynamic buckling phenomenon of a composite cylindrical shell under impulsive axial compression, J. Sound. Vib., 482 (2020), 115462. https://doi.org/10.1016/j.jsv.2020.115462 doi: 10.1016/j.jsv.2020.115462
[7]	Z. Wang, J. Sun, J. Chen, Input-to-state stability of impulsive switched nonlinear time-delay systems with two asynchronous switching phenomena, Int. J. Robust. Nonlin., 30 (2020), 4463–4484. https://doi.org/10.1002/rnc.4995 doi: 10.1002/rnc.4995
[8]	Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Le'vy processes, IEEE Trans. Autom. Control, 70 (2025), 1176–1183. https://doi.org/10.1109/TAC.2024.3448128 doi: 10.1109/TAC.2024.3448128
[9]	A. Kumar, H. V. S. Chauhan, C. Ravichandran, K. S. Nisar, D. Baleanu, Existence of solutions of non-autonomous fractional differential equations with integral impulse condition, Adv. Differ. Equ., 2020 (2020), 434. https://doi.org/10.1186/s13662-020-02888-3 doi: 10.1186/s13662-020-02888-3
[10]	A. Zada, S. Ali, Y. Li, Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equ., 2017 (2017), 317. https://doi.org/10.1186/s13662-017-1376-y doi: 10.1186/s13662-017-1376-y
[11]	X. Li, H. Akca, X. Fu, Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions, Appl. Math. Comput., 219 (2013), 7329–7337. https://doi.org/10.1016/j.amc.2012.12.033 doi: 10.1016/j.amc.2012.12.033
[12]	J. Tariboon, Boundary value problems for first order functional differential equations with impulsive integral conditions, J. Comput. Appl. Math., 234 (2010), 2411–2419. https://doi.org/10.1016/j.cam.2010.03.007 doi: 10.1016/j.cam.2010.03.007
[13]	G. Li, Y. Zhang, Y. Guan, W. Li, Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse, Math. Biosci. Eng., 20 (2023), 7020–7041. https://doi.org/10.3934/mbe.2023303 doi: 10.3934/mbe.2023303
[14]	C. Thaiprayoon, S. K. Ntouyas, J. Tariboon, Monotone iterative technique for nonlinear impulsive conformable fractional differential equations with delay, Commun. Math. Appl., 12 (2021), 11. https://doi.org/10.26713/CMA.V12I1.587 doi: 10.26713/CMA.V12I1.587
[15]	B. Li, H. Gou, Monotone iterative method for the periodic boundary value problems of impulsive evolution equations in Banach spaces, Chaos Solitons Fract., 110 (2018), 209–215. https://doi.org/10.1016/j.chaos.2018.03.027 doi: 10.1016/j.chaos.2018.03.027
[16]	Z. He, X. He, Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions, Comput. Math. Appl., 48 (2004), 73–84. https://doi.org/10.1016/j.camwa.2004.01.005 doi: 10.1016/j.camwa.2004.01.005
[17]	Z. Luo, J. J. Nieto, New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear Anal.-Theory, 70 (2009), 2248–2260. https://doi.org/10.1016/j.na.2008.03.004 doi: 10.1016/j.na.2008.03.004
[18]	B. Hu, Z. Wang, M. Xu, D. Wang, Quasilinearization method for an impulsive integro-differential system with delay, Math. Biosci. Eng., 19 (2022), 612–623. https://doi.org/10.3934/mbe.2022027 doi: 10.3934/mbe.2022027
[19]	B. Ahmad, A quasilinearization method for a class of integro-differential equations with mixed nonlinearities, Nonlinear Anal. Real World Appl., 7 (2006), 997–1004. https://doi.org/10.1016/j.nonrwa.2005.09.003 doi: 10.1016/j.nonrwa.2005.09.003
[20]	R. K.Pandey, S. Tomar, An effective scheme for solving a class of nonlinear doubly singular boundary value problems through quasilinearization approach, J. Comput. Appl. Math., 392 (2021), 113411. https://doi.org/10.1016/j.cam.2021.113411 doi: 10.1016/j.cam.2021.113411
[21]	B. Ahmad, J. J. Nieto, Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions, Nonlinear Anal.-Theory, 69 (2008), 3291–3298. https://doi.org/10.1016/j.na.2007.09.018 doi: 10.1016/j.na.2007.09.018
[22]	Z. Liu, J. Han, L. Fang, Integral boundary value problems for first order integro-differential equations with impulsive integral conditions, Comput. Math. Appl., 61 (2011), 3035–3043. https://doi.org/10.1016/j.camwa.2011.03.094 doi: 10.1016/j.camwa.2011.03.094
[23]	V. Lakshmikantham, P. S. Simeonov, Theory of impulsive differential equations, Singapore: World scientific, 1989.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)