Problem of periodic solutions for neutral functional differential equation

The problem of periodic solutions for a kind of kth_order linear neutral functional differential equation is studied. By using the theory of Fourier expansions, a sufficient and necessary condition to guarantee the existence and uniqueness of periodic solution is obtained. Further, by applying this result and Schauder's fixed point principle, a kind of kth_order nonlinear neutral functional differential equation is investigated, and some new results on existence of the periodic solutions are given as well. These results improve and extend some known results in recent literature.