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Abstract

We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: $\ddot{q}(t)+V_{q}(t,q(t))+u(t,q(t),q(t-T),q(t+T))=f(t)$, where $t\in\mathbb R$, $q\in\mathbb R^{n}$ and $T>0$ is a fixed positive number. By an almost homoclinic solution (to $0$) we mean one that joins $0$ to itself and $q\equiv 0$ may not be a stationary point. We assume that $V$ and $u$ are $T$-periodic with respect to the time variable, $V$ is $C^{1}$-smooth and $u$ is continuous. Moreover, $f$ is non-zero, bounded, continuous and square-integrable. The main result provides a certain approximative scheme of finding an almost homoclinic solution.