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January
2022, 21(1): 337-354.
doi: 10.3934/cpaa.2021180
Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth
| 1. | School of Mathematical Sciences, Soochow University, Suzhou 215006, China |
| 2. | Department of Fundamental Courses, Wuxi Institute of Technology, Wuxi 214121, China |
We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.
Keywords:
Hamiltonian systems,
Coupled system,
Subharmonic solutions,
sub-linear growth,
Poincaré-Birkhoff theorem.
Mathematics Subject Classification:
Primary: 34C25; Secondary: 34C15, 46N20.
Citation:
Fanfan Chen, Dingbian Qian, Xiying Sun, Yinyin Wu. Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth. Communications on Pure and Applied Analysis,
2022, 21
(1)
: 337-354.
doi: 10.3934/cpaa.2021180
![]()
References:
| [1] |
C. Bereanu and J. Mawhin,
Existence and multiplicity results for some nonlinear problems with singular-Laplacian, J. Differ. Equ., 243 (2007), 536-557.
doi: 10.1016/j.jde.2007.05.014. |
| [2] |
A. Boscaggin and R. Ortega,
Monotone twist maps and periodic solutions of systems of Duffing type, Math. Proc. Cambridge Philos. Soc., 157 (2014), 279-296.
doi: 10.1017/S0305004114000310. |
| [3] |
A. Calamai and A. Sfecci,
Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations, Nonlinear Differ. Equ. Appl., 24 (2017), 1-17.
doi: 10.1007/s00030-016-0427-5. |
| [4] |
T. Ding and F. Zanolin,
Periodic solutions of Duffing's equations with superquadratic potential, J. Differ. Equ., 97 (1992), 328-378.
doi: 10.1016/0022-0396(92)90076-Y. |
| [5] |
T. Ding and F. Zanolin,
Subharmonic solutions of second order nonlinear equations: A time-map approach, Nonlinear Anal., 20 (1993), 509-532.
doi: 10.1016/0362-546X(93)90036-R. |
| [6] |
A. Fonda,
Positively homogeneous Hamiltonian systems in the plane, J. Differ. Equ., 200 (2004), 162-184.
doi: 10.1016/j.jde.2004.02.001. |
| [7] |
A. Fonda, M. Garrione and P. Gidoni,
Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal., 5 (2016), 367-382.
|
| [8] |
A. Fonda and P. Gidoni,
An avoiding cones condition for the Poincaré-Birkhoff theorem, J. Differ. Equ., 262 (2017), 1064-1084.
doi: 10.1016/j.jde.2016.10.002. |
| [9] |
A. Fonda and M. Ramos,
Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities, J. Differ. Equ., 109 (1994), 354-372.
doi: 10.1006/jdeq.1994.1055. |
| [10] |
A. Fonda, Z. Schneider and F. Zanolin,
Periodic oscillations for a nonlinear suspension bridge model, J. Comput. Appl. Math., 52 (1994), 113-140.
|
| [11] |
A. Fonda and A. Sfecci,
Multiple periodic solutions of Hamiltonian systems confined in a box, Discrete Contin. Dyn. Syst., 37 (2017), 297-301.
doi: 10.3934/dcds.2017059. |
| [12] |
A. Fonda and A. Sfecci,
Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differ. Integral Equ., 25 (2012), 993-1010.
|
| [13] |
A. Fonda and A. Sfecci,
Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260 (2016), 2150-2162.
doi: 10.1016/j.jde.2015.09.056. |
| [14] |
A. Fonda and R. Toader,
Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc, 140 (2012), 1331-1341.
|
| [15] |
A. Fonda and R. Toader,
Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583-602.
doi: 10.1515/anona-2017-0040. |
| [16] |
A. Fonda and A. J. Ureña,
A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.
doi: 10.1016/j.anihpc.2016.04.002. |
| [17] |
M. Garcia-Huidobro, R. Manásevich and F. Zanolin,
Strongly nonlinear second-order ODEs with unilateral conditions, Differ. Integral Equ., 8 (1993), 1057-1078.
|
| [18] |
P. Hartman,
On boundary value problems for second order differential equations, J. Differ. Equ., 26 (1977), 37-53.
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 (1976), 37-52.
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.
doi: 10.1007/978-3-642-32906-7_3. |
| [21] |
J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Institute of Mathematical Sciences, New York University, 2005.
doi: 10.1090/cln/012. |
| [22] |
D. Qian, L. Chen and X. Sun,
Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differ. Equ., 258 (2015), 3088-3106.
doi: 10.1016/j.jde.2015.01.003. |
| [23] |
D. Qian and P. J. Torres,
Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.
doi: 10.1137/S003614100343771X. |
| [24] |
D. Qian, P. J. Torres and P. Wang,
Periodic solutions of second order equations via rotation numbers, J. Differ. Equ., 266 (2019), 4746-4768.
doi: 10.1016/j.jde.2018.10.010. |
| [25] |
C. Rebelo and F. Zanolin,
Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities, Trans. Amer. Math. Soc., 348 (1996), 2349-2389.
doi: 10.1090/S0002-9947-96-01580-2. |
| [26] |
X. Sun, Q. Liu, D. Qian and N. Zhao,
Infinitely many subharmonic solutions for nonlinear equations with singular $\phi$-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 279-292.
doi: 10.3934/cpaa.20200015. |
show all references
References:
| [1] |
C. Bereanu and J. Mawhin,
Existence and multiplicity results for some nonlinear problems with singular-Laplacian, J. Differ. Equ., 243 (2007), 536-557.
doi: 10.1016/j.jde.2007.05.014. |
| [2] |
A. Boscaggin and R. Ortega,
Monotone twist maps and periodic solutions of systems of Duffing type, Math. Proc. Cambridge Philos. Soc., 157 (2014), 279-296.
doi: 10.1017/S0305004114000310. |
| [3] |
A. Calamai and A. Sfecci,
Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations, Nonlinear Differ. Equ. Appl., 24 (2017), 1-17.
doi: 10.1007/s00030-016-0427-5. |
| [4] |
T. Ding and F. Zanolin,
Periodic solutions of Duffing's equations with superquadratic potential, J. Differ. Equ., 97 (1992), 328-378.
doi: 10.1016/0022-0396(92)90076-Y. |
| [5] |
T. Ding and F. Zanolin,
Subharmonic solutions of second order nonlinear equations: A time-map approach, Nonlinear Anal., 20 (1993), 509-532.
doi: 10.1016/0362-546X(93)90036-R. |
| [6] |
A. Fonda,
Positively homogeneous Hamiltonian systems in the plane, J. Differ. Equ., 200 (2004), 162-184.
doi: 10.1016/j.jde.2004.02.001. |
| [7] |
A. Fonda, M. Garrione and P. Gidoni,
Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal., 5 (2016), 367-382.
|
| [8] |
A. Fonda and P. Gidoni,
An avoiding cones condition for the Poincaré-Birkhoff theorem, J. Differ. Equ., 262 (2017), 1064-1084.
doi: 10.1016/j.jde.2016.10.002. |
| [9] |
A. Fonda and M. Ramos,
Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities, J. Differ. Equ., 109 (1994), 354-372.
doi: 10.1006/jdeq.1994.1055. |
| [10] |
A. Fonda, Z. Schneider and F. Zanolin,
Periodic oscillations for a nonlinear suspension bridge model, J. Comput. Appl. Math., 52 (1994), 113-140.
|
| [11] |
A. Fonda and A. Sfecci,
Multiple periodic solutions of Hamiltonian systems confined in a box, Discrete Contin. Dyn. Syst., 37 (2017), 297-301.
doi: 10.3934/dcds.2017059. |
| [12] |
A. Fonda and A. Sfecci,
Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differ. Integral Equ., 25 (2012), 993-1010.
|
| [13] |
A. Fonda and A. Sfecci,
Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260 (2016), 2150-2162.
doi: 10.1016/j.jde.2015.09.056. |
| [14] |
A. Fonda and R. Toader,
Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc, 140 (2012), 1331-1341.
|
| [15] |
A. Fonda and R. Toader,
Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583-602.
doi: 10.1515/anona-2017-0040. |
| [16] |
A. Fonda and A. J. Ureña,
A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.
doi: 10.1016/j.anihpc.2016.04.002. |
| [17] |
M. Garcia-Huidobro, R. Manásevich and F. Zanolin,
Strongly nonlinear second-order ODEs with unilateral conditions, Differ. Integral Equ., 8 (1993), 1057-1078.
|
| [18] |
P. Hartman,
On boundary value problems for second order differential equations, J. Differ. Equ., 26 (1977), 37-53.
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 (1976), 37-52.
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.
doi: 10.1007/978-3-642-32906-7_3. |
| [21] |
J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Institute of Mathematical Sciences, New York University, 2005.
doi: 10.1090/cln/012. |
| [22] |
D. Qian, L. Chen and X. Sun,
Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differ. Equ., 258 (2015), 3088-3106.
doi: 10.1016/j.jde.2015.01.003. |
| [23] |
D. Qian and P. J. Torres,
Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.
doi: 10.1137/S003614100343771X. |
| [24] |
D. Qian, P. J. Torres and P. Wang,
Periodic solutions of second order equations via rotation numbers, J. Differ. Equ., 266 (2019), 4746-4768.
doi: 10.1016/j.jde.2018.10.010. |
| [25] |
C. Rebelo and F. Zanolin,
Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities, Trans. Amer. Math. Soc., 348 (1996), 2349-2389.
doi: 10.1090/S0002-9947-96-01580-2. |
| [26] |
X. Sun, Q. Liu, D. Qian and N. Zhao,
Infinitely many subharmonic solutions for nonlinear equations with singular $\phi$-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 279-292.
doi: 10.3934/cpaa.20200015. |
Figure 2.
$ z_k(t)\in \mathcal{D}_{2} $
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