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

The existence of multiple periodic solutions of the following differential delay equation is established by applying variational approaches directly, where , and 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.