A note on the minimal periodic solutions of nonconvex superlinear Hamiltonian system

In this paper, we consider the multiplicity result of minimal periodic solutions for the first order Hamiltonian system x˙(t)=J∇H(x(t)). We point out a mistake in the literatures [M. Girardi, M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear Anal. T.M.A. 10 (4) (1986) 371–382] and [Tianqing An, On the minimal periodic solutions of nonconvex superlinear Hamiltonian systems, J. Math. Anal. Appl. 329 (2007) 1273–1284]. When the potential is even, under some stronger restricted conditions, we give the correct proof and obtain more distinct nonconstant periodic solutions with minimal period T.