Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i _?1 of linear Hamiltonian systems. As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive, even, and superquadratic at zero and infinity. This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption. We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive, even, and superquadratic at zero and infinity which was proved by Fei, Kim and Wang in 2001.