Periodic solutions of a class of non-autonomous second-order discrete Hamiltonian systems

Huiting He; Chungen Liu; Jiabin Zuo; Huiting He; Chungen Liu; Jiabin Zuo

doi:10.3934/math.2024161

AIMS Mathematics

2024, Volume 9, Issue 2: 3303-3319. doi: 10.3934/math.2024161

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Research article

Periodic solutions of a class of non-autonomous second-order discrete Hamiltonian systems

School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

Received: 19 October 2023 Revised: 14 December 2023 Accepted: 25 December 2023 Published: 04 January 2024
MSC : 35B10, 39A23

In this paper, in the view of control functions, the existence of periodic solutions of the following second-order discrete Hamiltonian system

$ \bigtriangleup ^{2} u(n-1) = \nabla F(n, u(n)), \ \ n\in \mathbb{Z} $

with a generalized sublinear condition is further explored.
- periodic solution,
- control function,
- the least action principle,
- saddle point theorem
Citation: Huiting He, Chungen Liu, Jiabin Zuo. Periodic solutions of a class of non-autonomous second-order discrete Hamiltonian systems[J]. AIMS Mathematics, 2024, 9(2): 3303-3319. doi: 10.3934/math.2024161

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Abstract

In this paper, in the view of control functions, the existence of periodic solutions of the following second-order discrete Hamiltonian system

$ \bigtriangleup ^{2} u(n-1) = \nabla F(n, u(n)), \ \ n\in \mathbb{Z} $

with a generalized sublinear condition is further explored.

References

[1]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun. Pur. Appl. Math., 31 (1978), 157–184. https://doi.org/10.1002/cpa.3160310203 doi: 10.1002/cpa.3160310203
[2]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Mathematical Society, 1986.
[3]	J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, New York: Springer-Verlag, 1989.
[4]	C. L. Tang, Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, P. Amer. Math. Soc., 126 (1998), 3263–3270. https://doi.org/10.1090/s0002-9939-98-04706-6 doi: 10.1090/s0002-9939-98-04706-6
[5]	X. H. Tang, Q. Meng, Solutions of a second-order Hamiltonian system with periodic boundary conditions, Nonlinear Anal. Real., 11 (2010), 3722–3733. https://doi.org/10.1016/j.nonrwa.2010.02.002 doi: 10.1016/j.nonrwa.2010.02.002
[6]	X. P. Wu, C. L. Tang, Periodic solutions of a class of non-autonomous second-order systems, J. Math. Anal. Appl., 236 (1999), 227–235. https://doi.org/10.1006/jmaa.1999.6408 doi: 10.1006/jmaa.1999.6408
[7]	F. K. Zhao, X. Wu, Periodic solutions for a class of non-autonomous second order systems, J. Math. Anal. Appl., 296 (2004), 422–434. https://doi.org/10.1016/j.jmaa.2004.01.041 doi: 10.1016/j.jmaa.2004.01.041
[8]	X. F. Chen, F. Guo, Existence and multiplicity of periodic solutions for nonautonomous second order Hamiltonian systems, Bound. Value Probl., 2016 (2016), 1–10. https://doi.org/10.1186/s13661-016-0647-y doi: 10.1186/s13661-016-0647-y
[9]	C. L. Tang, X. P. Wu, Periodic solutions for a class of new superquadratic second order Hamiltonian systems, Appl. Math. Lett., 34 (2014), 65–71. https://doi.org/10.1016/j.aml.2014.04.001 doi: 10.1016/j.aml.2014.04.001
[10]	Z. Y. Wang, J. H. Zhang, New existence results on periodic solutions of non-autonomous second order Hamiltonian systems, Appl. Math. Lett., 79 (2018), 43–50. https://doi.org/10.1016/j.aml.2017.11.016 doi: 10.1016/j.aml.2017.11.016
[11]	X. F. Chen, F. Guo, P. Liu, Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions, Front. Math. China, 13 (2018), 1313–1323. https://doi.org/10.1007/s11464-018-0736-6 doi: 10.1007/s11464-018-0736-6
[12]	Z. Y. Wang, J. H. Zhang, Periodic solutions of a class of second order non-autonomous Hamiltonian systems, Nonlinear Anal. Theor., 72 (2010), 4480–4487. https://doi.org/10.1016/j.na.2010.02.023 doi: 10.1016/j.na.2010.02.023
[13]	Z. Y. Wang, J. H. Zhang, Periodic solutions for nonautonomous second order Hamiltonian systems with sublinear nonlinearity, Bound. Value Probl., 2011 (2011), 1–14. https://doi.org/10.1186/1687-2770-2011-23 doi: 10.1186/1687-2770-2011-23
[14]	Z. Y. Wang, J. H. Zhang, M. X. Chen, A unified approach to periodic solutions for a class of non-autonomous second order Hamiltonian systems, Nonlinear Anal. Real., 58 (2021), 103218. https://doi.org/10.1016/j.nonrwa.2020.103218 doi: 10.1016/j.nonrwa.2020.103218
[15]	Z. M. Guo, J. S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506–515. https://doi.org/10.1007/BF02884022 doi: 10.1007/BF02884022
[16]	Z. M. Guo, J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. Lond. Math. Soc., 68 (2003), 419–430. https://doi.org/10.1112/S0024610703004563 doi: 10.1112/S0024610703004563
[17]	X. X. Chen, C. G. Liu, J. B. Zuo, A discrete second-order Hamiltonian system with asymptotically linear conditions, Electron. Res. Arch., 31 (2023), 5151–5160. https://doi.org/10.3934/era.2023263 doi: 10.3934/era.2023263
[18]	Y. F. Xue, C. L. Tang, Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system, Nonlinear Anal. Theor., 67 (2007), 2072–2080. https://doi.org/10.1016/j.na.2006.08.038 doi: 10.1016/j.na.2006.08.038
[19]	X. H. Tang, X. Y. Zhang, Periodic solutions for second-order discrete Hamiltonian systems, J. Differ. Equ. Appl., 17 (2011), 1413–1430. https://doi.org/10.1080/10236190903555237 doi: 10.1080/10236190903555237

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