Periodic solutions for a class of symmetric and subquadratic hamiltonian systems

This paper is concerned with the existence of periodic solutions for the following Hamiltonian system: ż = J Hz(t, z), (HS) where H(t, z) ϵ C1(R × Br̄, R), Br̄ = z ϵ R2N;|z| ≤ r̄ for some r̄ > 0 is T-periodic in t, even in z, and subquadratic at z = 0. We prove that (HS) has a sequence of T-periodic solutions zm satisfying zm → 0 in Wloc1,2(R, R2N) as m → ∞ under some mild assumptions. Moreover, we show that if further H(t, z) = B(t)W(z) with B(·) = B(· + T) and having T as its minimal period, then T is the minimal period of zm for each m.