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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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