On the Existence of Homoclinic Type Solutions of a Class of Inhomogenous Second Order Hamiltonian Systems

Abstract

We show the existence of homoclinic type solutions of second order Hamiltonian systems of the type ddot{q}(t)+nabla _{q}V(t,q(t))=f(t), where tin mathbb {R}, the C^1-smooth potential V:mathbb {R}times mathbb {R}^nrightarrow mathbb {R} satisfies a relaxed superquadratic growth condition, its gradient is bounded in the time variable, and the forcing term f:mathbb {R}rightarrow mathbb {R}^n is sufficiently small in the space of square integrable functions. The idea of our proof is to approximate the original system by time-periodic ones, with larger and larger time-periods. We prove that the latter systems admit periodic solutions of mountain-pass type, and obtain homoclinic type solutions of the original system from them by passing to the limit (in the topology of almost uniform convergence) when the periods go to infinity.