A differential-delay equation arising in optics and physiology

In recent papers the authors have studied differential-delay equations $E_\varepsilon $ of the form $\varepsilon \dot x(t) = - x(t) + f(x(t - 1))$. For functions like $f(x) = \mu _1 + \mu _2 \sin (\mu _3 x + \mu _4 )$, such equations arise in optics, while for choices like $f(x) = \mu x^\nu e^{ - x} $ and $f(x) = \mu x^\nu (1 + x^\lambda )^{ - 1} $ and for $x \geqq 0$, the equation has been suggested in physiological models. Under varying hypotheses on f (labeled (I), (II), and (III) below), previous work has given theorems concerning existence and asymptotic properties as $\varepsilon \to 0^ + $ of periodic solutions of $E_\varepsilon $, which oscillate about a value a such that $f(\alpha ) = \alpha $. However, verifying (I), (II), or (III) for specific examples can be difficult. This paper gives general principles that help in verifying (I), (II), or (III), and then applies these results to specific classes of functions of interest.