Properties of systems of retarded functional differential equations which have periodic solutions

Abstract

In the first part of this note, a periodic system of retarded functional differential equations is considered. It is supposed that this system has a periodic solution. It is clarified the possible relationship between periods of the system and its periodic solution. A non-autonomous system of differential-difference equations with several delays which has a periodic solution is considered in the second part of this paper. The estimation from below of the the value of the period of a periodic solution is obtained. It has been shown that this estimation is sharp. The linear differential equation $x'(t)=c_0x(t)+c_1x(t-h_1)+\dots+c_rx(t-h_r)$ with several delays which has a periodic solution $x(t)$ with the period $T$ is considered as an application. It has been proved that $T\ge\frac{2\pi}{|c_0|+|c_1|+\dots+|c_r|}$.