Abstract: We consider the existence of positive 2π-periodic solutions of the third-order ordinary differential equations with multiple delays,
where coefficient function a: R→(0,+∞) is  continuous, and the  period of  t is 2π, nonlinear term f: R×［0,+∞)ⁿ→［0,+∞) is  continuous, and the   period of  t is 2π, andτ₁,τ₂,…,τ_n are positive constants. Using fixed point index theory in cones, we give the existence results of positive 2π-periodic solutions of the problem under the inequality conditions that allow the nonlinear term f(t,x₁,x₂,…,x_n) to grow superlinearly or sublinearly on x₁,x₂,…,x_n.

Key words: delay, third-order ordinary differential equation, positive periodic solution, cone, fixed point index