Existence of periodic solutions for a class of $ (\phi_{1}, \phi_{2}) $-Laplacian discrete Hamiltonian systems

Hai-yun Deng; Jue-liang Zhou; Yu-bo He; Hai-yun Deng; Jue-liang Zhou; Yu-bo He

doi:10.3934/math.2023537

AIMS Mathematics

2023, Volume 8, Issue 5: 10579-10595. doi: 10.3934/math.2023537

Previous Article Next Article

Research article

Existence of periodic solutions for a class of $ (\phi_{1}, \phi_{2}) $-Laplacian discrete Hamiltonian systems

1.
School of Mathematics and Computational Science, Huaihua University, Hunan 418000, China
2.
Key Laboratory of Intelligent Control Technology for Wuling-Mountain Ecological Agriculture in Hunan Province, Hunan 418000, China

Received: 05 January 2023 Revised: 14 February 2023 Accepted: 19 February 2023 Published: 02 March 2023
MSC : 34C25, 37J45, 58E50

In this paper, we consider the existence of periodic solutions for a class of nonlinear difference systems involving classical $ (\phi_{1}, \phi_{2}) $-Laplacian. By using the least action principle, we obtain that the system with classical $ (\phi_{1}, \phi_{2}) $-Laplacian has at least one periodic solution when potential function is $ (p, q) $-sublinear growth condition, subconvex condition. The results obtained generalize and extend some known works.
- $ (\phi_{1}, \phi_{2}) $-Laplacian,
- discrete systems,
- periodic solutions,
- the least action principle
Citation: Hai-yun Deng, Jue-liang Zhou, Yu-bo He. Existence of periodic solutions for a class of $ (\phi_{1}, \phi_{2}) $-Laplacian discrete Hamiltonian systems[J]. AIMS Mathematics, 2023, 8(5): 10579-10595. doi: 10.3934/math.2023537

Related Papers:

Abstract

In this paper, we consider the existence of periodic solutions for a class of nonlinear difference systems involving classical $ (\phi_{1}, \phi_{2}) $-Laplacian. By using the least action principle, we obtain that the system with classical $ (\phi_{1}, \phi_{2}) $-Laplacian has at least one periodic solution when potential function is $ (p, q) $-sublinear growth condition, subconvex condition. The results obtained generalize and extend some known works.

References

[1]	J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: a variational approach, Nonlinear Anal., 75 (2012), 4672–4687. https://doi.org/10.1016/j.na.2011.11.018 doi: 10.1016/j.na.2011.11.018
[2]	J. Mawhin, M. Willem, Critical point theory and Hamilotonian systems, New York: Springer, 1989. https://doi.org/10.1007/978-1-4757-2061-7
[3]	P. H. Rabinowitz, Minimax methods methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 1986. https://doi.org/10.1090/cbms/065
[4]	Y. H. Ding, Variational methods for strongly indefinite problems, World Scientific Publishing, 2007. https://doi.org/10.1142/6565
[5]	M. Schechter, Minimax systems and critical point theory, Boston: Birkhäuser, 2009. https://doi.org/10.1007/978-0-8176-4902-9
[6]	W. D. Lu, Variational methods in differential equations, Scientific Publishing House in China, 2002. https://doi.org/10.1142/6565
[7]	J. Mawhin, Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential, Discrete Contin. Dyn. Syst., 6 (2013), 1065–1076. https://doi.org/10.3934/dcdss.2013.6.1065 doi: 10.3934/dcdss.2013.6.1065
[8]	Z. M. Guo, J. S. Yu, The existence of periodic and subharmonic solutions to subquadratic second-order difference equations, J. Lond. Math. Soc., 68 (2003), 419–430. https://doi.org/10.1112/S0024610703004563 doi: 10.1112/S0024610703004563
[9]	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
[10]	Y. F. Xue, C. L. Tang, Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system, Nonlinear Anal., 67 (2007), 2072–2080. https://doi.org/10.1016/j.na.2006.08.038 doi: 10.1016/j.na.2006.08.038
[11]	T. He, W. Chen, Periodic solutions of second order convex systems involving the $p$-Laplacian, Appl. Math. Comput., 206 (2008), 124–132. https://doi.org/10.1016/j.amc.2008.08.037 doi: 10.1016/j.amc.2008.08.037
[12]	X. F. He, P. Chen, Homoclinic solutions for second order discrete $p$-Laplacian systems, Adv. Differ. Equ., 57 (2011), 2011. https://doi.org/10.1186/1687-1847-2011-57 doi: 10.1186/1687-1847-2011-57
[13]	X. Y. Lin, X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59–72. https://doi.org/10.1016/j.jmaa.2010.06.008 doi: 10.1016/j.jmaa.2010.06.008
[14]	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
[15]	X. Y. Zhang, X. H. Tang, Existence of solutions for a nonlinear discrete system involing the $p$-Laplacian, Appl. Math., 57 (2012), 11–30. https://doi.org/10.1007/s10492-012-0002-2 doi: 10.1007/s10492-012-0002-2
[16]	X. Y. Zhang, Notes on periodic solutions for a nonlinear discrete system involving the $p$-Laplacian, Bull. Malays. Math. Sci. Soc., 37 (2014), 499–509.
[17]	Q. F. Zhang, X. H. Tang, Q. M. Zhang, Existence of periodic solutions for a class of discrete Hamiltonian systems, Discrete Dyn. Nat. Soc., 2011 (2011), 463480. https://doi.org/10.1155/2011/463480 doi: 10.1155/2011/463480
[18]	M. J. Ma, Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67 (2007), 1737–1745. https://doi.org/10.1016/j.na.2006.08.014 doi: 10.1016/j.na.2006.08.014
[19]	J. Q. Liu, A generalized saddle point theorem, J. Differ. Equations, 82 (1989), 372–385. https://doi.org/10.1016/0022-0396(89)90139-3 doi: 10.1016/0022-0396(89)90139-3
[20]	Y. Wang, X. Y. Zhang, Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded $(\phi_1, \phi_2)$-Laplacian, Adv. Differ. Equ., 2014 (2014), 218. https://doi.org/10.1186/1687-1847-2014-218 doi: 10.1186/1687-1847-2014-218
[21]	X. Y. Zhang, C. Zong, H. Y. Deng, L. B. Wang, Existence and multiplicity of homoclinic solutions for difference systems involving classical $(\phi_1, \phi_2)$-Laplacian and a parameter, Adv. Differ. Equ., 2017 (2017), 380. https://doi.org/10.1186/s13662-017-1419-4 doi: 10.1186/s13662-017-1419-4
[22]	X. Y. Zhang, Y. Wang, Homoclinic solutions for a class of nonlinear difference systems with classical $(\phi_1, \phi_2)$-Laplacian, Adv. Differ. Equ., 2015 (2015), 149. https://doi.org/10.1186/s13662-015-0467-x doi: 10.1186/s13662-015-0467-x
[23]	H. Y. Deng, X. Y. Zhang, H. Fang, Existence of periodic solutions for a class of discrete systems with classical or bounded $(\phi_1, \phi_2)$-Laplacian, J. Nonlinear Sci. Appl., 10 (2017), 535–559. http://doi.org/10.22436/jnsa.010.02.19 doi: 10.22436/jnsa.010.02.19
[24]	L. B. Wang, X. Y. Zhang, H. Fang, Existence and multiplicity of solutions for a class of $(\phi_1, \phi_2)$-Laplacian elliptic system in $\mathbb R^N$ via genus theory, Comput. Math. Appl., 72 (2016), 110–130. https://doi.org/10.1016/j.camwa.2016.04.034 doi: 10.1016/j.camwa.2016.04.034
[25]	D. Pasca, C. L. Tang, Some existence results on periodic solutions of nonautonomous second-order differential systems with $(q, p)$-Laplacian, Appl. Math. Lett., 23 (2010), 246–251. https://doi.org/10.1016/j.aml.2009.10.005 doi: 10.1016/j.aml.2009.10.005
[26]	D. Pasca, C. L. Tang, Some existence results on periodic solutions of ordinary $(q, p)$-Laplacian systems, J. Appl. Math. Inform., 29 (2011), 39–48. https://doi.org/10.14317/jami.2011.29.12.039 doi: 10.14317/jami.2011.29.12.039
[27]	C. Li, Z. Q. Ou, C. L. Tang, Periodic solutions for non-autonomous second-order differential systems with $(q, p)$-Laplacian, Electron. J. Differ. Eq., 2014 (2014), 1–13.
[28]	X. X. Yang, H. B. Chen, Periodic solutions for autonomous $(q, p)$-Laplacian system with implusive effects, J. Appl. Math., 2011 (2011), 378389. https://doi.org/10.1155/2011/378389 doi: 10.1155/2011/378389
[29]	Y. K. Li, T. W. Zhang, Infinitely many periodic solutions for second-order $(q, p)$-Laplacian differential systems, Nonlinear Anal., 74 (2011), 5215–5221. https://doi.org/10.1016/j.na.2011.05.024 doi: 10.1016/j.na.2011.05.024
[30]	X. L. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problem involving the $p(x)$-Laplacian, J. Math. Anal. Appl., 334 (2007), 248–260. https://doi.org/10.1016/j.jmaa.2006.12.055 doi: 10.1016/j.jmaa.2006.12.055
[31]	Q. Jiang, C. L. Tang, Periodic and subharmonic solutions of a class of subquadratic 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
[32]	C. L. Tang, X. P. Wu, Notes on periodic solutions of subquadratic second order systems, J. Math. Anal. Appl., 285 (2003), 8–16. https://doi.org/10.1016/S0022-247X(02)00417-1 doi: 10.1016/S0022-247X(02)00417-1

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)