Abstract: Applying Schauder fixedpoint theorem and Fourier’s method, we generalized the results on theexistence and uniqueness of the periodic solutions for a class of second order delay differential equations, and gave the existence and uniqueness of 2πperiodic solutions for (2k)thorder nonlinear delay differential equations on space L²(0,2π). The condition to ensure the theorems is a variety of usual Lazer type one.

Key words: (2k)thorder delay differential equation, periodic solution, Schauder fixedpoint theorem