Ground state periodic solutions of second order Hamiltonian systems without spectrum 0

In this paper, we consider the second order Hamiltonian system $left{ begin{gathered} u''(t) + A(t)u(t) + nabla H(t,u(t)) = 0,t in R, hfill u(0) = u(T),u'(0) = u'(T),T > 0. hfill end{gathered} right.$ Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= ?d ²/dt ² ?A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct.