Subharmonic solutions of Hamiltonian systems and the Maslov-type index theory

This paper deals with the subharmonic solutions of Hamiltonian systems(H)x˙=J∇H(t,x),t∈R1,x∈R2n. Where H(t,x)=12〈Bˆ(t)x,x〉+Hˆ(t,x) is T-periodic in t, Bˆ(t) is a T-periodic semipositive symmetric matrix. We prove that for each positive integer k, (H) has a nonconstant kT-periodic solution xk such that xj and xpj are geometrically distinct if p satisfies the certain conditions.