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

Joanna Janczewska¹

¹ Faculty of Technical Physics and Applied Mathematics Gdańsk University of Technology Narutowicza 11/12 80-233 Gdańsk, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Annales Polonici Mathematici, Tome 100 (2011) no. 1, pp. 13-24.

Résumé

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.

Détail

DOI : 10.4064/ap100-1-2

Keywords: shall concerned existence almost homoclinic solutions class second order functional differential equations mixed type ddot t t t where mathbb mathbb fixed positive number almost homoclinic solution mean joins itself equiv may stationary point assume t periodic respect time variable smooth continuous moreover non zero bounded continuous square integrable main result provides certain approximative scheme finding almost homoclinic solution