Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system q¨+Vq(t,q)=f(t), where q∈Rn and V∈C1(R×Rn,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b1|q|2⩽K(t,q)⩽b2|q|2, W is superlinear at the infinity and f is sufficiently small in L2(R,Rn). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.