Variational approaches for the existence of multiple periodic solutions of differential delay equations (Q963195)

In this paper, the existence of \(4r\)-periodic solutions of a class of equations having the form \(x'(t) =- f(x(t-r))\) is investigated, where \(r\) is a positive constant and \(f\) is a continuous function from \(\mathbb R\) to \(\mathbb R\). Under the assumption that \(f\) is odd and both \(\lim_{x\to 0}f(x)/x\) and \(\lim_{x\to\infty}f(x)/x\) exist, the existence of such solutions is proved by directly applying variational methods rather than using the well-established Kaplan and Yorke's reduction techniques. Although this is not the first work using this method, the arguments are claimed to be different from those appearing in the references.