Abstract: In this paper,by using the theory of Fourier series,some necessary and sufficient conditions of existence and uniqueness of periodic solutions of a class of higher order neutral type equations are obtained.The main results by Shi Jian guo in “Discussion on the periodic solutions for linear equation of neutral type with constant coefficients” are improved,i.e.,the condition|b₀|≠1 inst ead of the condition |b₀|<1/2 of Theorem 1 by Shi Jian guo is given.Other theorems by Shi are rebuilt and improved according to the new assumption.

Key words: periodic solution, neutral type equation, Fourier series