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.


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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