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 ¨q − Vq (t, q) = f (t), 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 f :R→Rn is continuous, bounded, square integrable and f = 0. We will show that there exists a solution q0 such that q0(t)→0 and ˙q0(t)→0, 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. © 2007 Elsevier Inc. All rights reserved