Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential

We shall be concerned with the existence of homoclinic solutions for the second order Hamiltonian system , where t∈R and q∈Rn. A potential V∈C1(R×Rn,R) is T-periodic in t, coercive in q and the integral of V(⋅,0) over [0,T] is equal to 0. A function is continuous, bounded, square integrable and f≠0. We will show that there exists a solution q0 such that q0(t)→0 and , as t→±∞. Although q≡0 is not a solution of our system, we are to call q0 a homoclinic solution. It is obtained as a limit of 2kT-periodic orbits of a sequence of the second order differential equations.