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A “Birkhoff-Lewis” type result for a class of Hamiltonian systems. (English) Zbl 0616.58014
Consider the Hamiltonian system of differetial equations $(1)\quad - J\dot z=H_ z(t,z),$ where $$z=(p,q)\in {\mathbb{R}}^{2N}$$, J denotes the symplectic structure in $${\mathbb{R}}^{2N}$$ and $$H_ z$$ denotes the gradient of the Hamiltonian function H with respect to z. Suppose that H is $$\tau$$-periodic in the t variable and $$H_ z(t,0)=0$$. If H satisfies suitable assumptions it can be proved that (1) possesses a sequence $$z_ k$$ of $$k\cdot \tau$$ periodic solutions (k-prime) converging to zero uniformly in $$C^ 1({\mathbb{R}},{\mathbb{R}}^{2N})$$.

MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37G99 Local and nonlocal bifurcation theory for dynamical systems 58E30 Variational principles in infinite-dimensional spaces
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