A discrete second-order Hamiltonian system with asymptotically linear conditions

Xiaoxing Chen; Chungen Liu; Jiabin Zuo; Xiaoxing Chen; Chungen Liu; Jiabin Zuo

doi:10.3934/era.2023263

Electronic Research Archive

2023, Volume 31, Issue 9: 5151-5160. doi: 10.3934/era.2023263

Previous Article Next Article

Research article Special Issues

A discrete second-order Hamiltonian system with asymptotically linear conditions

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

Received: 08 June 2023 Revised: 03 July 2023 Accepted: 07 July 2023 Published: 17 July 2023

This paper deals with a non-autonomous discrete second-order Hamiltonian system under asymptotically linear conditions. The existence of a periodic solution is obtained via the saddle point theorem.
- asymptotically linear,
- second-order Hamiltonian system,
- saddle point theorem,
- periodic solution,
- existence
Citation: Xiaoxing Chen, Chungen Liu, Jiabin Zuo. A discrete second-order Hamiltonian system with asymptotically linear conditions[J]. Electronic Research Archive, 2023, 31(9): 5151-5160. doi: 10.3934/era.2023263

Related Papers:

Abstract

This paper deals with a non-autonomous discrete second-order Hamiltonian system under asymptotically linear conditions. The existence of a periodic solution is obtained via the saddle point theorem.

References

[1]	Z. Wang, J. 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
[2]	Q. Zheng, Homoclinic solutions for a second-order nonperiodic asymptotically linear Hamiltonian systems, Abstr. Appl. Anal., Hindawi, 2013 (2013), 417020. https://doi.org/10.1155/2013/417020
[3]	M. Makvand Chaharlang, M. Ragusa, A. Razani, A sequence of radially symmetric weak solutions for some nonlocal elliptic problem in $\mathbb R^{N}$, Mediterr. J. Math., 17 (2020), 53. https://doi.org/10.1007/s00009-020-1492-X doi: 10.1007/s00009-020-1492-X
[4]	Z. Guo, J. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A-Math., 46 (2003), 506–515. https://doi.org/10.1007/BF02884022 doi: 10.1007/BF02884022
[5]	Z. Guo, J. Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear Anal. Theory Methods Appl., 55 (2003), 969–983. https://doi.org/10.1016/j.na.2003.07.019 doi: 10.1016/j.na.2003.07.019
[6]	Z. Guo, J. Yu, The Existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419–430. https://doi.org/10.1112/S0024610703004563 doi: 10.1112/S0024610703004563
[7]	X. Tang, X. Zhang, Periodic solutions for second-order discrete Hamiltonian systems, J. Differ. Equations Appl., 17 (2011), 1413–1430. https://doi.org/10.1080/10236190903555237 doi: 10.1080/10236190903555237
[8]	Z. Wang, J. Zhang, M. Chen, A unified approach to periodic solutions for a class of non-autonomous second order Hamiltonian systems, Nonlinear Anal. Real World Appl., 58 (2021), 103218. https://doi.org/10.1016/j.nonrwa.2020.103218 doi: 10.1016/j.nonrwa.2020.103218
[9]	X. Tang, L. Xiao, Homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), 1140–1152. https://doi.org/10.1016/j.na.2008.11.038 doi: 10.1016/j.na.2008.11.038
[10]	Y. Xue, C. Tang, Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems, Appl. Math. Comput., 196 (2008), 494–500. https://doi.org/10.1016/j.amc.2007.06.015 doi: 10.1016/j.amc.2007.06.015
[11]	Z. Wang, J. Xiao, On periodic solutions of subquadratic second order non-autonomous Hamiltonian systems, Appl. Math. Lett., 40 (2015), 72–77. https://doi.org/10.1016/j.aml.2014.09.014 doi: 10.1016/j.aml.2014.09.014
[12]	C. Tang, X. 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
[13]	Q. Jiang, C. Tang, Periodic and subharmonic solutions of a class of sub-quadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 328 (2007), 380–389. https://doi.org/10.1016/j.jmaa.2006.05.064 doi: 10.1016/j.jmaa.2006.05.064
[14]	J. Xie, J. Li, Z. Luo, Periodic and subharmonic solutions for a class of the second-order Hamiltonian systems with impulsive effects, Boundary Value Probl., 2015 (2015), 52. https://doi.org/10.1186/s13661-015-0313-9 doi: 10.1186/s13661-015-0313-9
[15]	F. Zhao, J. Chen, M. Yang, A periodic solution for a second-order asymptotically linear Hamiltonian system, Nonlinear Anal., 70 (2009), 4021–4026. https://doi.org/10.1016/j.na.2008.08.009 doi: 10.1016/j.na.2008.08.009
[16]	X. 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
[17]	R. Cheng, Periodic solutions for asymptotically linear Hamiltonian system with resonance both at infinity and origin via computation of critical groups, Nonlinear Anal., 89 (2013), 134–145. https://doi.org/10.1016/j.na.2013.05.009 doi: 10.1016/j.na.2013.05.009
[18]	G. Fei, Nontrivial periodic solutions of asymptotically linear Hamiltonian systems, Electron. J. Differ. Equations, 2001 (2001), 1–17.
[19]	X. Tang, J. Jiang, Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl., 59 (2010), 3646–3655. https://doi.org/10.1016/j.camwa.2010.03.039 doi: 10.1016/j.camwa.2010.03.039

Reader Comments

Your name:*

Email:*
© 2023 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)