Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance

Shiwang Ma

doi:10.3934/cpaa.2013.12.2361

Article Contents

2013, Volume 12, Issue 6: 2361-2380. Doi: 10.3934/cpaa.2013.12.2361

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Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance

Shiwang Ma^1,

1.
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071

Received: June 30, 2011

Revised: April 30, 2012

Published: October 2013

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Abstract

In this paper, we consider the existence of nontrivial $1$-periodic solutions of the following Hamiltonian systems \begin{eqnarray} -J\dot{z}=H'(t,z), z\in R^{2N}, \end{eqnarray} where $J$ is the standard symplectic matrix of $2N\times 2N$, $H\in C^2 ( [0,1] \times R^{2N}, R)$ is $1$-periodic in its first variable and $H'(t,z)$ denotes the gradient of $H$ with respect to the variable $z$. Furthermore, $H'(t,z)$ is asymptotically linear both at origin and at infinity. Based on the precise computations of the critical groups, Maslov-type index theory and Galerkin approximation procedure, we obtain some existence results for nontrivial $1$-periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature.

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Mathematics Subject Classification: Primary: 34B15, 34C25, 37J45.

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