Almost homoclinic solutions for a certain class of mixed type functional differential equations

Volume 100 / 2011

Joanna Janczewska Annales Polonici Mathematici 100 (2011), 13-24 MSC: Primary 34C37; Secondary 34K13, 58E50. DOI: 10.4064/ap100-1-2


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.


  • Joanna JanczewskaFaculty of Technical Physics and Applied Mathematics
    Gdańsk University of Technology
    Narutowicza 11/12
    80-233 Gdańsk, Poland
    Institute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland

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