Odd periodic solutions for [formula omitted]th-order ordinary differential equations

Abstract

This paper deals with the existence of periodic solutions for the 2 n th-order ordinary differential equation u ( 2 n ) ( t ) = f ( t , u ( t ) , u ″ ( t ) , … , u ( 2 n − 2 ) ( t ) ) , where the nonlinear term f : R × R n → R is a continuous odd function and f ( t , x 0 , x 1 , … , x n − 1 ) is 2 π -periodic in t . Some existence results for odd 2 π -periodic solutions are obtained under the condition that f satisfies some linear, superlinear or sublinear growth conditions on x 0 , x 1 , … , x n − 1 .