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Subharmonic solutions for convex nonautonomous Hamiltonian systems. (English) Zbl 0879.34043

Nonautonomous Hamiltonian systems of the form

$I\stackrel{˙}{x}\left(t\right)={H}_{x}\left(x\left(t\right),t\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $I$ is the standard symplectic matrix

$I=\left(\begin{array}{cc}0& {I}_{N}\\ -{I}_{N}& 0\end{array}\right)$

$x=\left({x}_{1},\cdots ,{x}_{2N}\right)\in {ℝ}^{2N}$ and the Hamiltonian $H$ is a $T$-periodic (in the second variable) function, i.e. $H\left(x,t+T\right)=H\left(x,t\right)$, for all $\left(x,t\right)$ is considered. Precisely, for any given integer $p\ge 1$ the existence of multiple periodic (subharmonical) solutions of Eq. (1) $x\left(0\right)=x\left(pT\right)$ is investigated.

It is known, that under rather general assumptions on $H$ for any integer $p\ge 1$ the subharmonic problem (1) was shown to have a distinct solution ${X}_{\left(p\right)}$ and if $H$ has subquadratic or superquadratic growth at the origin and at infinity then the minimal period of ${X}_{\left(p\right)}$ tends towards infinity as $p\to +\infty$.

Some new multiplicity results are obtained for $H$ convex with subquadratic growth at the origin and at infinity. Roughly speaking, it is shown that there exist many subharmonics with exact minimal period $pT$ provided ${S}_{p}$ (the smallest prime factor of $p\right)$ is sufficiently large. Using a dual variational method the author answers the question whether the number of solutions of (1) with minimal period $pT$ tends towards infinity as $p\to +\infty$.

##### MSC:
 34C25 Periodic solutions of ODE 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 34C11 Qualitative theory of solutions of ODE: growth, boundedness