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Zbl 1071.34039
Luan, Shixia; Mao, Anmin
Periodic solutions for a class of non-autonomous Hamiltonian systems.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 61, No. 8, A, 1413-1426 (2005). ISSN 0362-546X

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)].
MSC 2000:
*34C25 Periodic solutions of ODE
37J45 Periodic, homoclinic and heteroclinic orbits, etc.
47J30 Variational methods
58E05 Abstract critical point theory

Keywords: Hamiltonian system; Periodic solutions; Cerami condition; Local linking

Citations: Zbl 0820.58012; Zbl 0807.58040

Cited in: Zbl 1122.37048

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