Almost homoclinic solutions for the second order Hamiltonian systems - Janczewska, Joanna (Topological Methods in Nonlinear Analysis, 2008)
The second order Hamiltonian system q+V_q(t,q)=f(t), where tR and qR^n, is considered. We assume that a potential VC^1(R×R^n,R) is of the form V(t,q)=-K(t,q)+W(t,q), where K satisfies the pinching condition and W_q(t,q)=o(|q|), as |q|→ 0 uniformly with respect to t. It is also assumed that fC(R,R^n) is non-zero and sufficiently small in L^2(R,R^n). In this case q0 is not a solution. Therefore there are no orbits homoclinic to 0 in a classical sense. However, we show that there is a solution emanating from 0 and terminating at 0. We are to call such a solution almost homoclinic to 0. It is obtained here as a weak limit in W^1,2(R,R^n) of a sequence of almost critical points.