On the existence of periodic solutions for a class of non-autonomous differential delay equations

Abstract

This paper considers the existence of periodic solutions for a class of non-autonomous differential delay equations \begin{equation} x'(t)=-\sum_{i=1}^{n-1}f(t,x(t-i\tau)), \tag{$*$} \end{equation} where $\tau> 0$ is a given constant. It is shown that under some conditions on $f$ and by using symplectic transformations, Floquet theory and some results in critical point theory, the existence of single periodic solution of the differential delay equation $(*)$ is obtained. These results generalize previous results on the cases that the equations are autonomous.