Periodic solutions for non-autonomous Hamiltonian systems possessing super-quadratic potentials

We consider the non-autonomous Hamiltonian system Ju̇+∇H (t; u) = 0; (t; u)∈ ST × R ; where H ∈C1(R×R2N ;R) is T -periodic in t-variable, and J= ( 0 −IN IN 0 ) denotes the standard symplectic matrix and ∇ denotes the gradient with respect to the u-variable. We are interested in the existence of T -periodic solutions of (1). For the autonomous case, i.e. H is independent of t, in his pioneer work [8] Rabinowitz :rst proved the existence of at least one periodic solution for (1). Many works have been done on this topics, such as [2–13]. We refer to [2,3,11] for further references. At this paper, we :rst consider the case that H is symmetric in the u-variable. In [2] when H is super-quadratic at in:nitely and satis:es (H2)p There are constants c; d¿ 0 and p∈ (1; 2), such that |∇H (t; u)|p 6 c(∇H (t; u); u) + d; ∀u∈R2N ;