In this paper the existence results of positive ω-periodic solutions are obtained for second order ordinary differential equation -u″(t) = f(t, u(t)) (t ∈ R), and also for first order ordinary differential equation u′(t) = f(t, u(t)) (t ∈ R), where f: R × R+ → R is a continuous function which is ω-periodic in t. The discussion is based on the fixed point index theory in cones.