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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 $${\Cal J}\dot u- A(t)u+\nabla H(t,u)= 0,\quad u\in\bbfR^{2N},\quad t\in\bbfR.$$ 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 $$0<\mu H(t, u)\le u\cdot\nabla H(t,u),\quad \mu> 2,\quad |u|\ge R> 0.$$ This result extends theorems given by {\it S. J. Li} and {\it M. Willem} [J. Math. Anal. Appl. 189, 6--32 (1995; Zbl 0820.58012)] and {\it S. J. Li} and {\it A. Szulkin} [J. Differ. Equations 112, 226--238 (1994; Zbl 0807.58040)].

34C25Periodic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory
Full Text: DOI
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