Positive solutions of higher-order periodic boundary value problems

Abstract

This paper deals with the existence of positive solutions for n th-order periodic boundary value problem L n u ( t ) = f ( t , u ( t ) ) , 0 ≤ t ≤ 2 π , u ( i ) ( 0 ) = u ( i ) ( 2 π ) , i = 0 , 1 , … , n − 1 , where L n u(t) = u (n)(t) + Σ i=0 n−1 a i u (i)(t) is an n th-order linear differential operator and ƒ :[0, 2π] × ℝ + → ℝ + is continuous. We obtain a sufficient condition that operator La satisfies the maximum principle in periodic boundary condition. Using this maximum principle and fixed-point index theory in cones, we obtain existence results of positive solutions.