An approximative scheme of finding almost homoclinic solutions for a class of Newtonian systems

Abstract

In this work the problem of the existence of almost homoclinic solutions for a Newtonian system $\ddot{q}+V_{q}(t,q)=f(t)$, where $t\in\mathbb{R}$ and $q\in\mathbb{R}^n$, is considered. It is assumed that a potential $V\colon\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ is $C^{1}$-smooth with respect to all variables and $T$-periodic in a time variable $t$. Moreover, $f\colon\mathbb{R}\to\mathbb{R}^{n}$ is a continuous bounded square integrable function and $f\neq 0$. This system may not have a trivial solution. However, we show that under some additional conditions there exists a solution emanating from $0$ and terminating to $0$. We are to call such a solution almost homoclinic to $0$.