In this paper, we study the existence of positive periodic solutions for a particular class of first-order delay differential equations, where the linearized system may be degenerate. It is known that most methods may fail to guarantee the existence of periodic solution under the degenerate case. To alleviate this issue, this paper presents an efficient algorithm based on Schauder's fixed point theorem. Theorems which list the conditions for the existence of periodic solutions are presented. Simulation studies show the correctness of the improved method.