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Abstract

In this work we will be concerned with the existence of almost homoclinic solutions for a Newtonian system $\ddot{q}+\nabla_{q}V(t,q)=f(t)$, where $t\in\mathbb R$, $q\in\mathbb R^n$. It is assumed that a potential $V:\mathbb R\times\mathbb R^n\to\mathbb R$ is $C^1$-smooth and its gradient map $\nabla_{q}V:\mathbb R\times\mathbb R^n\to\mathbb R^n$ is bounded with respect to $t$. Moreover, a forcing term $f:\mathbb R\to\mathbb R^n$ is continuous, bounded and square integrable. We will show that the approximative scheme due to J. Janczewska (see [J2]) for a time periodic potential extends to our case.