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Periodic solutions for a class of non-autonomous Hamiltonian systems. (English) Zbl 1071.34039

Summary: We consider the existence of nontrivial periodic solutions for the superlinear Hamiltonian system

$𝒥\stackrel{˙}{u}-A\left(t\right)u+\nabla H\left(t,u\right)=0,\phantom{\rule{1.em}{0ex}}u\in {ℝ}^{2N},\phantom{\rule{1.em}{0ex}}t\in ℝ·$

We prove an abstract result on the existence of a critical point for a real-valued functional on a Hilbert space via a new deformation theorem. Different from the work in the literature, the new deformation theorem is constructed under a Cerami-type condition instead of Palais-Smale-type condition. In addition, the main assumption here is weaker than the usual Ambrosetti-Rabinowitz-type condition

$0<\mu H\left(t,u\right)\le u·\nabla H\left(t,u\right),\phantom{\rule{1.em}{0ex}}\mu >2,\phantom{\rule{1.em}{0ex}}|u|\ge R>0·$

This result extends theorems given by S. J. Li and M. Willem [J. Math. Anal. Appl. 189, 6–32 (1995; Zbl 0820.58012)] and S. J. Li and A. Szulkin [J. Differ. Equations 112, 226–238 (1994; Zbl 0807.58040)].

##### MSC:
 34C25 Periodic solutions of ODE 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 47J30 Variational methods (nonlinear operator equations) 58E05 Abstract critical point theory