Abstract: By using the monotone iterative method of upper and lower solution and establishing a new maximum principle, we constructed 
a monotone iterative procedure for the ω-periodic solutions of nth order delay differential equation -u⁽ⁿ⁾(t)=f(t,u(t),u(t-τ₁),u(t-τ₂),…,u(t-τ_n)),t∈R, and proved existence and uniqueness of its ω-periodic solutions, where f: R×Rⁿ⁺¹→R was a continuous function which was ω-periodic on t, τ₁,τ₂,…,τ_n were positive constants.

Key words: delay differential equation, periodic solution, maximum principle