Subharmonics for convex nonautonomous Hamiltonian systems. (English) Zbl 0601.58035

Denote by J the standard symplectic matrix in \({\mathbb{R}}^{2n}\) and let \(H\in C^ 2({\mathbb{R}}\times {\mathbb{R}}^{2n}, {\mathbb{R}})\) be T-periodic in the first variable. In this paper we investigate the existence of kT-periodic solutions of the time dependent Hamiltonian system \[ (HS)_ k\quad - J\dot x=H'(t,x),\quad x(0)=x(kT), \] where \(k\in {\mathbb{N}}\). A solution of \((HS)_ k\) for \(k\geq 2\) is called a subharmonic. Clearly, a solution of \((HS)_ k\) will also be a solution of \((HS)_{2k}\), \((HS)_{3k}\), etc. We show under a convexity assumption on H and a suitable asymptotic behaviour of H that for every \(k\in {\mathbb{N}}\) there is a solution \(x_ k\) of \((HS)_ k\) such that the \(x_ k\), \(k\in {\mathbb{N}}\), are pairwise geometrically distinct.


37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: DOI


[1] Clarke, Comm. PAM 33 pp 103– (1980)
[2] Clarke, Archive Rat. Mech. An. 78 pp 315– (1982)
[3] Ekeland, J. Diff. Eq. 34 pp 523– (1973)
[4] An index theory for periodic solutions of convex Hamiltonian systems. Proceedings AMS Summer Institute on Nonlinear Functional Analysis at Berkeley, 1983, (to appear).
[5] Une théorie de Morse pour les systèmes hamiltoniens convexes. Annales IHP ”Analyse non linéaire,” 1, 1984, pp. 19–78.
[6] Ekeland, Inventiones Math. 81 pp 155– (1985)
[7] Hofer, Proceedings of Symposia in Pure Mathematics 45 pp 501– (1986)
[8] Hofer, J. London Math Society 31 pp 566– (1985)
[9] Topological Methods in the Study of Nonlinear Integral Equations. Pergamon Press, 1963.
[10] Mather, Adv. in Math 4 pp 301– (1970)
[11] and The structure of the critical set in the mountain pass theorems. Preprint 1985.
[12] Rabinowitz, Comm. PAM 33 pp 609– (1980)
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.