## Nontrivial periodic solutions for strong resonance Hamiltonian systems.(English)Zbl 0881.34061

A nonautonomous Hamiltonian system of the form $-J{dx\over dt}= H_x(t,x),$ where $$H\in C^2([0,1]\times \mathbb{R}^{2n},\mathbb{R})$$ is 1-periodic in $$t$$ and $J=\begin{pmatrix} 0 & -I_n\\ I_n & 0\end{pmatrix}$ is considered. Under the assumption that the Hamiltonian has an asymptotically linear gradient and the linearized operators have distinct Maslov indices at $$0$$ and at infinity it is proved the existence of nontrivial periodic solutions.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70H05 Hamilton’s equations 70K30 Nonlinear resonances for nonlinear problems in mechanics
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### References:

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