## 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)

### MSC:

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