Variational Approaches for the Existence of Multiple Periodic Solutions of Differential Delay Equations

Abstract

The existence of multiple periodic solutions of the following differential delay equation $x^{\prime }\left(t\right)=-f\left(x\left(t-r\right)\right)$ is established by applying variational approaches directly, where $x\in ℝ$, $f\in C\left(ℝ,ℝ\right)$ and $r>0$ is a given constant. This means that we do not need to use Kaplan and Yorke's reduction technique to reduce the existence problem of the above equation to an existence problem for a related coupled system. Such a reduction method introduced first by Kaplan and Yorke in (1974) is often employed in previous papers to study the existence of periodic solutions for the above equation and its similar ones by variational approaches.