In this paper, we first gave the Green's function of a sixth-order linear differential equation with variable coefficients, and then proved the existence of positive periodic solutions for a sixth-order differential equation with a singularity of repulsive type via Schauder's fixed point theorem.

By application of Leray-Schauder alternative principle and Krasnoselskii's fixed point theorem on compression and expansion of cones, existence results of positive periodic solutions are presented for a second-order neutral differential equation with singularity of repulsive type.