## Periodic solutions for some nonautonomous second order Hamiltonian systems.(English)Zbl 1142.37023

In this paper, the second order Hamiltonian systems:
$\ddot u(t)= \nabla F(t,u(t))\;\text{a.e. }t\in[0,T], \qquad u(0)-u(T)= \dot u(0)-\dot u(T)=0,$
where $$T>0$$ and $$F:[0,T]\times\mathbb R^N\to\mathbb R$$ is measurable in $$t$$ for every $$x\in\mathbb R^N$$ and continuously differentiable in $$u$$ for a.e. $$t\in [0,T]$$, and there exist $$a\in C(\mathbb R^+,\mathbb R^+)$$, $$b\in L^1(0,T;\mathbb R^+)$$ such that $$|F(t,x)|\leq a(|x|)b(t)$$, $$|\nabla F(t,x)|\leq a(|x|)b(t)$$ for all $$x\in\mathbb R^N$$ and a.e. $$t\in[0,T]$$ is considered. The existence and multiplicity of periodic solutions are obtained for these systems by minimax methods in critical point theory.

### MSC:

 37C60 Nonautonomous smooth dynamical systems 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34C25 Periodic solutions to ordinary differential equations
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