A note on periodic solutions of the delay differential equation $x’(t)=-f(x(t-1))$

Abstract:

Consider the delay differential equation $x’(t)=-f(x(t-1))$, where $f\in C(\mathbb {R}, \mathbb {R})$ is odd and satisfies $xf(x)>0$ for $x\ne 0$. When $\alpha =\lim _{x\to 0}\frac {f(x)}{x}$ and $\beta =\lim _{x\to \infty }\frac {f(x)}{x}$ exist, there is at least one Kaplan-Yorke periodic solution with period $4$ if $\min \{\alpha ,\beta \}<\frac {\pi }{2}<\max \{\alpha ,\beta \}$. When this condition is not satisfied, we present several sufficient conditions on the existence/nonexistence of such periodic solutions. It is worthy of mention that some results are on the existence of at least two Kaplan-Yorke periodic solutions with period $4$ and in some cases we do not require the limits $\alpha$ and/or $\beta$ to exist. Hence our results not only greatly improve but also complement existing ones. Moreover, some of the theoretical results are illustrated with examples.