Differential delay equations that have periodic solutions of long period

Abstract:

If $f:{\mathbf {R}} \to {\mathbf {R}}$ is a continuous odd function satisfying $xf(x) > 0$, $x \ne 0$, and $f(x) = o({x^{ - 2}})$ as $x \to \infty$, then so-called periodic solutions of long period seem to play a prominent role in the dynamics of $({\ast })$ \[ x’(t) = - \alpha f(x(t - 1)),\qquad \alpha > 0.\] In this paper we prove the existence of long-period periodic solutions of $({\ast })$ for a class of nonodd functions that decay "rapidly" to $0$ at infinity and satisfy $xf(x) \geqslant 0$. These solutions have quite different qualitative features than in the odd case.