Existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions

Song Wang; Xiao-Bao Shu; Linxin Shu; Song Wang; Xiao-Bao Shu; Linxin Shu

doi:10.3934/math.2022431

AIMS Mathematics

2022, Volume 7, Issue 5: 7685-7705. doi: 10.3934/math.2022431

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Research article

Existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions

School of Mathematics, Hunan University, Changsha, Hunan 410082, China

Received: 09 August 2021 Revised: 17 January 2022 Accepted: 24 January 2022 Published: 16 February 2022
MSC : 34B37, 37H10, 37J51

In this paper, we study sufficient conditions for the existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions. By using variational method we first obtain the corresponding energy functional. Then the existence of critical points are obtained by using Mountain pass lemma and Minimax principle. Finally we assert the critical point of enery functional is the mild solution of damped random impulsive differential equations.
- damped random impulsive differential equations,
- mild solution,
- mountain pass lemma,
- minimax principle,
- theoery of critical point,
- energy functional
Citation: Song Wang, Xiao-Bao Shu, Linxin Shu. Existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions[J]. AIMS Mathematics, 2022, 7(5): 7685-7705. doi: 10.3934/math.2022431

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Abstract

In this paper, we study sufficient conditions for the existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions. By using variational method we first obtain the corresponding energy functional. Then the existence of critical points are obtained by using Mountain pass lemma and Minimax principle. Finally we assert the critical point of enery functional is the mild solution of damped random impulsive differential equations.

References

[1]	T. E. Carter, Optimal impulsive space trajectories based on linear equations, J. Optim. Theory Appl. 70 (1991), 277–297. https://doi.org/10.1007/BF00940627 doi: 10.1007/BF00940627
[2]	T. E. Carter, Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dynam. Control., 10 (2000), 219–227.
[3]	P. Chen, X. H. Tang, Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems, Appl. Math. Comput., 218 (2012), 11775–11789. https://doi.org/10.1016/j.amc.2012.05.027 doi: 10.1016/j.amc.2012.05.027
[4]	S. Deng, X. Shu, J. Mao, Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Monch fixed point, J. Math. Anal. Appl., 467 (2018), 398–420. https://doi.org/10.1016/j.jmaa.2018.07.002 doi: 10.1016/j.jmaa.2018.07.002
[5]	W. Ding, D. B. Qian, Periodic solutions for sublinear systems via variational approach, Nonlinear Anal-Real., 11 (2010), 2603–2609. https://doi.org/10.1016/j.nonrwa.2009.09.007 doi: 10.1016/j.nonrwa.2009.09.007
[6]	C. J. Guo, D. O'Regan, Y. T. Xu, Existence of homoclinic orbits for a class of first-order differential difference equations, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 1077–1094. https://doi.org/10.1016/S0252-9602(15)30041-2 doi: 10.1016/S0252-9602(15)30041-2
[7]	P. K. George, A. K. Nandakumaran, A. Arapostathis, A note on controllability of impulsive systems, J. Math. Anal. Appl., 241 (2000), 276–283. https://doi.org/10.1006/jmaa.1999.6632 doi: 10.1006/jmaa.1999.6632
[8]	Z. Guan, G. Chen, T. Ueta, On impulsive control of a periodically forced chaotic pendulum system, IEEE Trans. Automat. Control, 45 (2000), 1724–1727. https://doi.org/10.1109/9.880633 doi: 10.1109/9.880633
[9]	Y. Guo, X. B. Shu, Y. Li, F. Xu, The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order $1 < \beta < 2$, Bound. Value Probl., 2019 (2019), 1–18.
[10]	M. N. Huang, C. J. Guo, J. M. Liu, The existence of periodic solutions for three-order neutral differential equations, J. Appl. Anal. Comput., 10 (2020), 457–473. https://doi.org/10.11948/20180139 doi: 10.11948/20180139
[11]	J. H. Kuang, Z. M. Guo, Heteroclinic solutions for a class of p-Laplacian difference equations with a parameter, Appl. Math. Lett., 100 (2020), 106034, 6 pp. https://doi.org/10.1016/j.aml.2019.106034 doi: 10.1016/j.aml.2019.106034
[12]	J. H. Kuang, Z. M. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 89 (2013), 208–218. https://doi.org/10.1016/j.na.2013.05.012 doi: 10.1016/j.na.2013.05.012
[13]	Y. H. Lee, X. Z. Liu, Study of singular boundary value problems for second order impulsive differential equations, J. Math. Anal. Appl., 331 (2007), 159–176. https://doi.org/10.1016/j.jmaa.2006.07.106 doi: 10.1016/j.jmaa.2006.07.106
[14]	S. Li, L. Shu, X. Shu, F. Xu, Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics, 91 (2019), 857–872. https://doi.org/10.1080/17442508.2018.1551400 doi: 10.1080/17442508.2018.1551400
[15]	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
[16]	X. N. Lin, D. Q. Jiang, Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations, J. Math. Anal. Appl., 321 (2006), 501–514. https://doi.org/10.1016/j.jmaa.2005.07.076 doi: 10.1016/j.jmaa.2005.07.076
[17]	X. Liu, A. R. Willms, Impulsive controllability of linear dynamical systems with applications to maneuvers of Spacecraft, Math. Problems Eng., 2 (1996), 277–299. https://doi.org/10.1155/S1024123X9600035X doi: 10.1155/S1024123X9600035X
[18]	J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, Berlin, 1989.
[19]	J. J. Nieto, Variational formulation of a damped Dirichlet impulsive problem, Appl. Math. Lett., 23 (2010), 940–942. https://doi.org/10.1016/j.aml.2010.04.015 doi: 10.1016/j.aml.2010.04.015
[20]	J. J. Nietoa, D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal-Real., 10 (2009), 680–690. https://doi.org/10.1016/j.nonrwa.2007.10.022 doi: 10.1016/j.nonrwa.2007.10.022
[21]	P. P Niu, X. B. Shu, Y. J. Li, The Existence and Hyers-Ulam stability for second order random impulsive differential equation, Dynam. Syst. Appl., 28 (2019), 673–690.
[22]	M. Onodera, Variational Methods in Differential Equations, A Mathematical Approach to Research Problems of Science and Technology, Springer, Tokyo, 2014.
[23]	A. F. B. A. Prado, Bi-impulsive control to build a satellite constellation, Nonlinear Dyn. Syst. Theory, 5 (2005), 169–175.
[24]	X. B. Shu, Y. Z. Lai, F. Xu, Existence of subharmonic periodic solutions to a class of second-order non-autonomous neutral functional differential equations, Abstr. Appl. Anal., (2012), Art. ID 404928, 26 pp. https://doi.org/10.1155/2012/404928
[25]	X. B. Shu, Y. Z. Lai, F. Xu, Existence of infinitely many periodic subharmonic solutions for nonlinear non-autonomous neutral differential equations, Electron. J. Differential Equations, 2013 (2013), 21.
[26]	X. B. Shu, Y. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465–476. https://doi.org/10.1016/j.amc.2015.10.020 doi: 10.1016/j.amc.2015.10.020
[27]	L. X. Shu, X. B. Shu, Existence and exponential stability of mild solutions for second-order neutral stochastic functional differential equation with random impulses, J. Appl. Anal. Comput., 11 (2021), 1–22.
[28]	X. B. Shu, F. Xu, Upper and lower solution method for fractional evolution equations with order $1 < \alpha < 2$, J. Korean Math. Soc., 51 (2014), 1123–1139. https://doi.org/10.4134/JKMS.2014.51.6.1123 doi: 10.4134/JKMS.2014.51.6.1123
[29]	Y. Tian, W. G. Ge, Applications of variational methods to boundary value problem for impulsive differential equations, Proc. Edinburgh Math. Soc., 51 (2008), 509–527. https://doi.org/10.1017/S0013091506001532 doi: 10.1017/S0013091506001532
[30]	S. J. Wu, X. L. Guo, S. Q. Lin, Existence and uniqueness of solutions to random impulsive differential systems, Acta Mathe. Appl. Sinica, English Series, 22 (2006), 627–632. https://doi.org/10.1007/s10255-006-0336-1 doi: 10.1007/s10255-006-0336-1
[31]	S. J. Wu, Y. R. Duan, Oscillation stability and boundedness of second-order differential systems with random impulses, Comput. Math. Appl., 49 (2005), 1375–1386. https://doi.org/10.1016/j.camwa.2004.12.009 doi: 10.1016/j.camwa.2004.12.009
[32]	S. J. Wu, X. Z, Meng, Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Math. Appl. Sinica, English Series, 20 (2004), 147–154. https://doi.org/10.1007/s10255-004-0157-z doi: 10.1007/s10255-004-0157-z
[33]	X. Xian, D. O'Regan, R. P. Agarwa, Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions, Hindawi Publishing Corporation Boundary Value Problems, Vol. 2008, Article ID 197205, 21 pages. https://doi.org/10.1155/2008/197205
[34]	Z. H. Zhang, R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal-Real., 11 (2010), 155–162. https://doi.org/10.1016/j.nonrwa.2008.10.044 doi: 10.1016/j.nonrwa.2008.10.044
[35]	H. Zhang, Z. X. Li, Variational approach to impulsive differential equations with periodic boundary conditions, Nonlinear Anal-Real., 11 (2010), 67–78. https://doi.org/10.1016/j.nonrwa.2008.10.016 doi: 10.1016/j.nonrwa.2008.10.016
[36]	Z. Li, X. Shu, F. Xu, The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem, AIMS Math., 5 (2020), 6189–6210. https://doi.org/10.3934/math.2020398 doi: 10.3934/math.2020398
[37]	W. D. Lu, Means of Variations in Ordinary Differential Equations, Scientific Press, 2003 (in Chinese).
[38]	J. C. Cort$\acute{e}$s, Sandra E. Delgadillo-Alem$\acute{a}$n, R. A. K$\acute{u}$-Carrillo, R. J. Villanueva, Full probabilistic analysis of random first-order linear differential equations with Dirac delta impulses appearing in control, Math. Meth. Appl. Sci., (2021), 1–20. https://doi.org/10.1002/mma.7715 doi: 10.1002/mma.7715
[39]	J. C. Cort$\acute{e}$s, S. E. Delgadillo-Alem$\acute{a}$n, R. A. K$\acute{u}$-Carrillo, R. J. Villanueva, Probabilistic analysis of a class of impulsive linear random differential equations via density functions, Appl. Math. Lett., 121 (2021), 107519. https://doi.org/10.1016/j.aml.2021.107519 doi: 10.1016/j.aml.2021.107519

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