Existence of positive periodic solutions for higher-order ordinary differential equations

Abstract

This paper deals with the existence of positive periodic solutions for the n th-order ordinary differential equation u ( n ) ( t ) = f ( t , u ( t ) , u ′ ( t ) , … , u ( n − 1 ) ( t ) ) , where n ≥ 2 , f : R × [ 0 , ∞ ) × R n − 1 → R is a continuous function and f ( t , x 0 , x 1 , … , x n − 1 ) is 2 π -periodic in t . Some existence results of positive 2 π -periodic solutions are obtained assuming f satisfies some superlinear or sublinear growth conditions on x 0 , x 1 , … , x n − 1 . The discussion is based on the fixed point index theory in cones.