Uniqueness and existence of positive periodic solutions of functional differential equations

In this paper, some new findings on the uniqueness and existence of positive periodic solutions to first-order functional differential equations are presented. These equations have wide applications in a variety of fields. The most important feature of our argument is that we use the theory of Hilbert's metric to prove the uniqueness of the positive periodic solution when $q=-1$ and $-1<q<0$. In addition, we also investigate the existence results of positive periodic solutions by applying a fixed point theorem for completely continuous maps in a cone. Two examples demonstrate our findings.