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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.
Reviewer: Reviewer (Berlin)

##### MSC:
 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
##### Keywords:
time dependent Hamiltonian system
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##### References:
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