On the existence of homoclinic type solutions of a class of inhomogenous second order Hamiltonian systems

TL;DR

This paper proves the existence of homoclinic solutions in certain second order Hamiltonian systems with relaxed growth conditions by approximating with periodic systems and passing to the limit.

Contribution

It introduces a novel approach of using time-periodic approximations to establish homoclinic solutions under relaxed superquadratic conditions.

Findings

Existence of homoclinic solutions under relaxed conditions.

Periodic solutions of mountain-pass type in approximating systems.

Homoclinic solutions obtained via limit process.

Abstract

We show the existence of homoclinic type solutions of second order Hamiltonian systems with a potential satisfying a relaxed superquadratic growth condition and a forcing term that 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.