Abstract: This paper deals with the existence of positive ω-periodic solutions for the higher-order ordinary differential equation with delayed derivative terms in nonlinearityu⁽ⁿ⁾(t)+a(t)u(t)=f(t, u(t-τ₀(t)), u'(t-τ₁(t)),…, u^(n-1)(t-τ_n-1(t))), t∈R,where n≥2, a:R→(0,∞)is a continuous function which is ω-periodic, f:R×［0,∞)×R^n-1→［0,∞)is a continuous function which is ω-periodic on t, and τ_k:R→［0,∞)is a continuous function which is ω-periodic, k=0,1,…,n-1. By using the perturbation method of positive operator and fixed point index theory in cones, we obtain the existence results of positive ω-periodic solutions for the equation.

Key words: differential equation with delay, positive periodic solution, cone, fixed point index