We study the minimal period problem of Hamiltonian systems which may not be strictly convex. For the second order Hamiltonian systems, we prove that Rabinowitz's conjecture is true if the potential function V ( x ) is even and V ''( x ) is semi-positive definite. For the first order Hamiltonian systems, we obtain estimates on the minimal period of the corresponding nonconstant periodic solutions. We prove that for any positive T > 0, the corresponding Hamiltonian system has a periodic solution with minimal period not smaller than T /(2 N ) provided the Hamiltonian function H satisfies the condition that H ''( x ) is semi-positive definite.
Finally, we study the existence of nontrival periodic solutions of the asymptotically linear second order Hamiltonian systems in the general case that the action function f may not satisfy the (PS) condition. By using the Galerkin approximation method and the Conley index theory, we establish the existence of periodic solutions and obtain an estimate of the number of periodic solutions without symmetric conditions on the potential function V .
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