Subharmonic solutions for subquadratic Hamiltonian systems. (English) Zbl 0814.34025

The article is concerned with the existence of distinct \(kT\)-periodic \((k\in \mathbb{N})\) solutions (known as subharmonics) for the Hamiltonian system \(dz/dt= JH_ z(t, z)\), \(z\in \mathbb{R}^{2N}\), with \(H\in C^ 1(\mathbb{R}\times \mathbb{R}^{2N},\mathbb{R})\) a \(T\)-periodic in \(t\) function which is subquadratic in the sense \(\lim_{| z|\to \infty} | H_ z(t, z)|/| z|= 0\) uniformly in \(t\). For the main abstract tool, a new version of the saddle point theorem is employed. We may outline two main results of this well-written article. First: assuming moreover that \[ \lim_{| z|\to \infty} | H_ z(t, z)\cdot z- 2H(t, z)| \] is equal either to \(\infty\) or to \(-\infty\), then there exist \(kT\)-periodic solutions \(z_ k\) with \(\| z_ k\|_ \infty\to \infty\) as \(k\to\infty\) for all \(k\in \mathbb{N}\).
Second: assuming \(H(t, z)\to 0\), \(H_ z(t, z)\to 0\) as \(| z|\to \infty\) uniformly in \(t\), then there exists a sequence \(z_ j\) of \(k_ j T\)-periodic solutions where \(k_ j\to \infty\).
Reviewer: J.Chrastina (Brno)


34C25 Periodic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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