Existence and multiplicity of homoclinic orbits of a second-orderdifferential difference equation via variational methods

Abstract:

This paper is concerned with the existence of homoclinic orbits of the second order differential difference equation q(t)−Kq(t,q(t))+ f (t,q(t +τ),q(t),q(t −τ)) = h(t). By using critical point theory and variational methods, a nontrivial homoclinic orbit is obtained as a limit of a certain sequence of periodic solutions of an appropriate equation. As a result, we generalize the results obtained by Smets and Willem. Also, by applying a Symmetric Mountain Pass Lemma, we obtain infinitely many homoclinic orbits of the above equation.