A note on positive periodic solutions of delayed differential equations

Abstract

We consider the existence of positive $\omega$-periodic solutions for the periodic equation $x^{\prime }\left(t\right)=a\left(t\right)e^{x\left(t\right)}x\left(t\right)-\lambda b\left(t\right)f\left(x(t-\tau \left(t\right))\right)$, where $a,b\in C(\mathbb{R},[0,\infty ))$ are $\omega$-periodic, ${\int }_0^{\omega }a\left(t\right)dt>0$, ${\int }_0^{\omega }b\left(t\right)dt>0$, $f\in C([0,\infty ),[0,\infty ))$, and $f\left(u\right)>0$ for $u>0$, $\tau \left(t\right)$ is a continuous $\omega$-periodic function.