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

Abstract

We shall be concerned with the existence of homoclinic solutions for the second order Hamiltonian system q ¨ − V q ( t , q ) = f ( t ) , where t ∈ R and q ∈ R n . A potential V ∈ C 1 ( R × R n , 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 → R n is continuous, bounded, square integrable and f ≠ 0 . We will show that there exists a solution q 0 such that q 0 ( t ) → 0 and q ˙ 0 ( t ) → 0 , as t → ± ∞ . Although q ≡ 0 is not a solution of our system, we are to call q 0 a homoclinic solution. It is obtained as a limit of 2 k T -periodic orbits of a sequence of the second order differential equations.