×

Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth. (English) Zbl 1490.34048

The authors study the existence and multiplicity of subharmonic solutions for some Hamiltonian systems in \(\mathbb{R}^{2N}\) which can be considered as \(N\) weakly coupled planar systems. However, improving some previous results, they do not need to assume the Hamiltonian function to depend on some small parameters. The nonlinearities are assumed to satisfy Landesman-Lazer conditions, and to have some kind of sublinear behavior at infinity. The proof is based on a higher dimensional version of the Poincaré-Birkhoff Theorem proved by A. Fonda and A. J. Ureña [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34, No. 3, 679–698 (2017; Zbl 1442.37076)].

MSC:

34C25 Periodic solutions to ordinary differential equations
37J46 Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems

Citations:

Zbl 1442.37076
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Bereanu; J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular-Laplacian, J. Differ. Equ., 243, 536-557 (2007) · Zbl 1148.34013 · doi:10.1016/j.jde.2007.05.014
[2] A. Boscaggin; R. Ortega, Monotone twist maps and periodic solutions of systems of Duffing type, Math. Proc. Cambridge Philos. Soc., 157, 279-296 (2014) · Zbl 1331.34070 · doi:10.1017/S0305004114000310
[3] A. Calamai; A. Sfecci, Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations, Nonlinear Differ. Equ. Appl., 24, 1-17 (2017) · Zbl 1366.34057 · doi:10.1007/s00030-016-0427-5
[4] T. Ding; F. Zanolin, Periodic solutions of Duffing’s equations with superquadratic potential, J. Differ. Equ., 97, 328-378 (1992) · Zbl 0763.34030 · doi:10.1016/0022-0396(92)90076-Y
[5] T. Ding; F. Zanolin, Subharmonic solutions of second order nonlinear equations: A time-map approach, Nonlinear Anal., 20, 509-532 (1993) · Zbl 0778.34027 · doi:10.1016/0362-546X(93)90036-R
[6] A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differ. Equ., 200, 162-184 (2004) · Zbl 1068.34035 · doi:10.1016/j.jde.2004.02.001
[7] A. Fonda; M. Garrione; P. Gidoni, Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal., 5, 367-382 (2016) · Zbl 1353.37124
[8] A. Fonda; P. Gidoni, An avoiding cones condition for the Poincaré-Birkhoff theorem, J. Differ. Equ., 262, 1064-1084 (2017) · Zbl 1369.37064 · doi:10.1016/j.jde.2016.10.002
[9] A. Fonda; M. Ramos, Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities, J. Differ. Equ., 109, 354-372 (1994) · Zbl 0798.34048 · doi:10.1006/jdeq.1994.1055
[10] A. Fonda; Z. Schneider; F. Zanolin, Periodic oscillations for a nonlinear suspension bridge model, J. Comput. Appl. Math., 52, 113-140 (1994) · Zbl 0810.73030
[11] A. Fonda; A. Sfecci, Multiple periodic solutions of Hamiltonian systems confined in a box, Discrete Contin. Dyn. Syst., 37, 297-301 (2017) · Zbl 1373.37147 · doi:10.3934/dcds.2017059
[12] A. Fonda; A. Sfecci, Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differ. Integral Equ., 25, 993-1010 (2012) · Zbl 1274.34125
[13] A. Fonda; A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260, 2150-2162 (2016) · Zbl 1351.34046 · doi:10.1016/j.jde.2015.09.056
[14] A. Fonda; R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc, 140, 1331-1341 (2012) · Zbl 1248.34051
[15] A. Fonda; R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8, 583-602 (2019) · Zbl 1419.37059 · doi:10.1515/anona-2017-0040
[16] A. Fonda; A. J. Ureña, A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34, 679-698 (2017) · Zbl 1442.37076 · doi:10.1016/j.anihpc.2016.04.002
[17] M. Garcia-Huidobro; R. Manásevich; F. Zanolin, Strongly nonlinear second-order ODEs with unilateral conditions, Differ. Integral Equ., 8, 1057-1078 (1993) · Zbl 0785.34023
[18] P. Hartman, On boundary value problems for second order differential equations, J. Differ. Equ., 26, 37-53 (1977) · Zbl 0365.34032 · doi:10.1016/0022-0396(77)90097-3
[19] H. Jacobowitz, Periodic solutions of \(x''+f(x, t) = 0\) via the Poincaré-Birkhoff theorem, J. Differ. Equ., 20, 37-52 (1976) · Zbl 0285.34028 · doi:10.1016/0022-0396(76)90094-2
[20] J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations, in Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Mathematics, 2065, Springer Verlag, 2013. · Zbl 1402.34026
[21] J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Institute of Mathematical Sciences, New York University, 2005. · Zbl 1087.37001
[22] D. Qian; L. Chen; X. Sun, Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differ. Equ., 258, 3088-3106 (2015) · Zbl 1348.34041 · doi:10.1016/j.jde.2015.01.003
[23] D. Qian; P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36, 1707-1725 (2005) · Zbl 1092.34019 · doi:10.1137/S003614100343771X
[24] D. Qian; P. J. Torres; P. Wang, Periodic solutions of second order equations via rotation numbers, J. Differ. Equ., 266, 4746-4768 (2019) · Zbl 1410.34125 · doi:10.1016/j.jde.2018.10.010
[25] C. Rebelo; F. Zanolin, Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities, Trans. Amer. Math. Soc., 348, 2349-2389 (1996) · Zbl 0856.34051 · doi:10.1090/S0002-9947-96-01580-2
[26] X. Sun; Q. Liu; D. Qian; N. Zhao, Infinitely many subharmonic solutions for nonlinear equations with singular \(\phi \)-Laplacian, Commun. Pure Appl. Anal., 19, 279-292 (2020) · Zbl 1443.34044 · doi:10.3934/cpaa.20200015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.