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

DOI: http://dx.doi.org/10.12775/TMNA.2009.012

#### 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 $0$.

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 $0$.

#### Keywords

Almost homoclinic solution; periodic orbit; action functional; Newtonian system

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