Periodic solutions of prescribed minimal period for Hamiltonian systems: An extension of a theorem by Ekeland and Hofer to the non convex case (Q2639396)

A theorem concerning the existence of periodic solutions of prescribed minimal period is proved for the autonomous Hamiltonian system. It is assumed (besides others) that the Hamiltonian function H is even satisfying \[ <H''(z)\zeta,\zeta >\geq -c| z|^{\beta -2}| \zeta |^ 2\quad \forall z,\zeta \in {\mathbb{R}}^{2N},\quad \zeta \neq 0, \] where c is a positive constant and \(\beta >2\) is the superquadracity exponent of H. The main theorem extends the result of \textit{I. Ekeland} and \textit{H. Hofer} [Ann. Math., II. Ser. to appear] dealing with the case that H is a convex superquadratic function on \({\mathbb{R}}^{2N}\).