Existence of solution for an impulsive differential system with improved boundary value conditions

Bing Hu; Yingting Qiu; Weiting Zhou; Luyao Zhu; Bing Hu; Yingting Qiu; Weiting Zhou; Luyao Zhu

doi:10.3934/math.2023878

AIMS Mathematics

2023, Volume 8, Issue 7: 17197-17207. doi: 10.3934/math.2023878

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Existence of solution for an impulsive differential system with improved boundary value conditions

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China

Received: 15 March 2023 Revised: 07 April 2023 Accepted: 26 April 2023 Published: 17 May 2023
MSC : 34B37, 34B05

In this paper, we study the existence of solution for a class of impulsive integro-differential equations. Different from traditional periodic and anti-periodic boundary value problems, a more general boundary condition introduced in this new system. First, we obtain some new comparison principles. Then, we obtain the expression of the solution for a class of linearized systems. Finally, the existence of extremal solutions for the new boundary value system are obtained by using the monotone iterative technique. The theoretical results obtained have wider applications in practical fields.
- boundary value problem,
- integro-differential equation,
- extremal solutions,
- monotone iterative technique
Citation: Bing Hu, Yingting Qiu, Weiting Zhou, Luyao Zhu. Existence of solution for an impulsive differential system with improved boundary value conditions[J]. AIMS Mathematics, 2023, 8(7): 17197-17207. doi: 10.3934/math.2023878

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Abstract

In this paper, we study the existence of solution for a class of impulsive integro-differential equations. Different from traditional periodic and anti-periodic boundary value problems, a more general boundary condition introduced in this new system. First, we obtain some new comparison principles. Then, we obtain the expression of the solution for a class of linearized systems. Finally, the existence of extremal solutions for the new boundary value system are obtained by using the monotone iterative technique. The theoretical results obtained have wider applications in practical fields.

References

[1]	B. Hu, M. B.Xu, Z. Z.Wang, J. H. Lin, L. Y. Zhu, D. J. Wang, Existence of solutions of an impulsive integro-differential equation with a general boundary value condition, Math. Biosci. Eng., 19 (2022), 4166–4177. https://doi.org/10.3934/mbe.2022192 doi: 10.3934/mbe.2022192
[2]	V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, World Scientific, 1989. https://doi.org/10.1142/0906
[3]	X. D. Li, M. Bohner, C. K. Wang, Impulsive differential equations: periodic solutions and applications, Automatica, 52 (2015), 173–178. https://doi.org/10.1016/j.automatica.2014.11.009 doi: 10.1016/j.automatica.2014.11.009
[4]	J. J. Nieto, D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680–690. https://doi.org/10.1016/j.nonrwa.2007.10.022 doi: 10.1016/j.nonrwa.2007.10.022
[5]	G. C. Wu, D. Q. Zeng, D. Baleanu, Fractional impulsive differential equations: exact solutions, integral equations and short memory case, Fract. Calc. Appl. Anal., 22 (2019), 180–192. https://doi.org/10.1515/fca-2019-0012 doi: 10.1515/fca-2019-0012
[6]	S. S. Santra, K. M. Khedher, K. Nonlaopon, H. Ahmad, New results on qualitative behavior of second order nonlinear neutral impulsive differential systems with canonical and non-canonical conditions, Symmetry, 13 (2021), 1–13. https://doi.org/10.3390/sym13060934 doi: 10.3390/sym13060934
[7]	I. F. Pinheiro, L. A. Sphaier, L. S. de B. Alves, Integral transform solution of integro-differential equations in conduction-radiation problems, Numer. Heat Transfer. Part A Appl., 73 (2018), 94–114. https://doi.org/10.1080/10407782.2017.1421359 doi: 10.1080/10407782.2017.1421359
[8]	S. Santra, J. Mohapatra, A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type, J. Comput. Appl. Math., 400 (2022), 113746. https://doi.org/10.1016/j.cam.2021.113746 doi: 10.1016/j.cam.2021.113746
[9]	S. Aggarwal, S. D. Sharma, A. Vyas, Application of Sawi transform for solving convolution type Volterra integro-differential equations of first kind, Int. J. Latest Techno. Eng. Manag. Appl. Sci., IX (2020), 13–19.
[10]	P. Agarwal, U. Baltaeva, Y. Alikulov, Solvability of the boundary-value problem for a linear loaded integro-differential equation in an infinite three-dimensional domain, Chaos Solitons Fract., 140 (2020), 110108. https://doi.org/10.1016/j.chaos.2020.110108 doi: 10.1016/j.chaos.2020.110108
[11]	T. K. Yuldashev, On a boundary-value problem for Boussinesq type nonlinear integro-differential equation with reflecting argument, Lobachevskii J. Math., 41 (2020), 111–123. https://doi.org/10.1134/S1995080220010151 doi: 10.1134/S1995080220010151
[12]	P. Das, S. Rana, H. Ramos, On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis, J. Comput. Appl. Math., 404 (2022), 113116. https://doi.org/10.1016/j.cam.2020.113116 doi: 10.1016/j.cam.2020.113116
[13]	S. Asawasamrit, S. K. Ntouyas, P. Thiramanus, J. Tariboon, Periodic boundary value problems for impulsive conformable fractional integro-differential equations, Bound. Value Probl., 2016 (2016), 1–18. https://doi.org/10.1186/s13661-016-0629-0 doi: 10.1186/s13661-016-0629-0
[14]	M. J. Mardanov, Y. A. Sharifov, F. M. Zeynalli, Existence and uniqueness of the solutions to impulsive nonlinear integro-differential equations with nonlocal boundary conditions, Proc. Inst. Math. Mech., 45 (2019), 222–233. https://doi.org/10.29228/proc.6 doi: 10.29228/proc.6
[15]	P. G. Wang, C. R. Li, J. Zhang, T. X. Li, Quasilinearization method for first-order impulsive integro-differential equations, Electron. J. Differ. Equ., 2019 (2019), 1–14.
[16]	K. Kaliraj, K. S. Viswanath, K. Logeswari, C. Ravichandran, Analysis of fractional integro-differential equation with Robin boundary conditions using topological degree method, Int. J. Appl. Comput. Math., 8 (2022), 176. https://doi.org/10.1007/s40819-022-01379-1 doi: 10.1007/s40819-022-01379-1
[17]	F. Wan, X. P. Liu, M. Jia, Ulam-Hyers stability for conformable fractional integro-differential impulsive equations with the antiperiodic boundary conditions, AIMS Math., 7 (2022), 6066–6083. https://doi.org/10.3934/math.2022338 doi: 10.3934/math.2022338
[18]	M. Y. Zuo, X. Hao, L. S. Liu, Y. J. Cui, Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions, Bound. Value Probl., 2017 (2017), 1–15. https://doi.org/10.1186/s13661-017-0892-8 doi: 10.1186/s13661-017-0892-8
[19]	Z. G. Luo, J. J. Nieto, New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear Anal., 70 (2009), 2248–2260. https://doi.org/10.1016/j.na.2008.03.004 doi: 10.1016/j.na.2008.03.004
[20]	Z. M. He, X. M. 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
[21]	Y. X. Li, Z. Liu, Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces, Nonlinear Anal., 66 (2007), 83–92. https://doi.org/10.1016/j.na.2005.11.013 doi: 10.1016/j.na.2005.11.013
[22]	J. L. Li, J. J. Nieto, J. H. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 325 (2007), 226–236. https://doi.org/10.1016/j.jmaa.2005.04.005 doi: 10.1016/j.jmaa.2005.04.005
[23]	J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl., 205 (1997), 423–433. https://doi.org/10.1006/jmaa.1997.5207 doi: 10.1006/jmaa.1997.5207

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