Infinitely many periodic solutions for a class of second-order Hamiltonian systems. (English) Zbl 1347.37110

In this paper, the authors study the following second-order Hamiltonian systems \[ \begin{cases} \ddot{u}(t)+A(t)u(t)+\nabla F(t,u(t))=0,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \end{cases}\tag{PS} \] where \(A \in C ([0, T],\mathbb{R}^{N^2})\) is a symmetric matrix valued function, \(F : [0, T ]\times \mathbb{R}^N \rightarrow \mathbb{R}\) is subquadratic, even in \(u\) and satisfies the Ahmad-Lazer-Paul condition. By using the dual fountain theorem due to T. Bartsch and M. Willem [Proc. Am. Math. Soc. 123, No. 11, 3555–3561 (1995; Zbl 0848.35039)], the authors obtain the existence of infinitely many periodic solutions for systems (PS).


37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C25 Periodic solutions to ordinary differential equations
70H05 Hamilton’s equations


Zbl 0848.35039
Full Text: DOI


[1] Bartsch, T.; Willem, M., On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123, 3555-3561, (1995) · Zbl 0848.35039
[2] Fei, G., On periodic solutions of superquadratic Hamiltonian systems, Electronic Journal of Differential Equations, 2002, 1-12, (2002) · Zbl 0999.37039
[3] Han, Z., 2p-periodic solutions to ordinary differential systems at resonance, Acta Math. Sinica, 43, 639-644, (2000) · Zbl 1027.34050
[4] Mawhin, J. Willem, M. Critical Point Theory and Hamiltonian Systems. Springer, New York, 1989 · Zbl 0676.58017
[5] Meng, F.; Zhang, F., Periodic solutions for some second order systems, Nonlinear Anal., 68, 3388-3396, (2008) · Zbl 1169.34322
[6] Rabinowitz, P.H., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31, 157-184, (1978) · Zbl 0358.70014
[7] Silva, E., Subharmonic solutions for subquadratic Hamiltonian systems, J. Differential Equations, 115, 120-145, (1995) · Zbl 0814.34025
[8] Tang, C., Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126, 3263-3270, (1998) · Zbl 0902.34036
[9] Willem, M. Minimax Theorems. Birkh√§user, Boston, 1996 · Zbl 0856.49001
[10] Zhao, F.; Chen, C.; Yang, M., A periodic solution for a second-order asymptotically linear Hamiltonian system, Nonlinear Anal., 70, 4021-4026, (2009) · Zbl 1167.34345
[11] Zou, W.; Li, S., Infinitely many solutions for Hamiltonian systems, J. Differential Equations, 186, 141-164, (2002)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.