Existence of periodic solutions of a particular type of super-quadratic Hamiltonian systems (Q743223)

The authors consider the Hamiltonian system \[ \dot z = JH'_z(t,z), \] where \(z=(p,q)\in\mathbb{R}^{2n}\), \(J\) is the standard symplectic matrix and \(H\) is \(T\)-periodic in \(t\) and superquadratic. The superquadraticity assumption is weaker than the commonly used ones (in particular, weaker than the Ambrosetti-Rabinowitz condition). It is shown that this system has a nontrivial \(T\)-periodic solution. The proof is effected by employing a linking theorem due to \textit{P. L. Felmer} [J. Differ. Equations 102, No. 1, 188--207 (1993; Zbl 0781.34034)]. The results extend those contained in the paper mentioned above and in [the second author, Electron. J. Differ. Equ. 2002, Paper No. 08, 12 p. (2002; Zbl 0999.37039)].