Periodic solutions for a class of non-autonomous Hamiltonian systems

We consider the existence of nontrivial periodic solutions for a superlinear Hamiltonian system: ( H ) J u ˙ - A ( t ) u + ∇ H ( t , u ) = 0 , u ∈ R 2 N , t ∈ 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 works in the literature, the new deformation theorem is constructed under the 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 μ H ( t , u ) ⩽ u · ∇ H ( t , u ) , μ > 2 , | u | ⩾ R > 0 . This result extends theorems given by Li and Willem (J. Math. Anal. Appl. 189 (1995) 6–32) and Li and Szulkin (J. Differential Equations 112 (1994) 226–238).