Existence of periodic solutions of higher-order differential equations

We established some sufficient conditions for the existence of solutions of high-order periodic boundary value problem for n ≥ 2 ( E ) u ( n ) ( t ) + D ( t , u ( t ) , … , u ( n − 2 ) ( t ) ) u ( n − 1 ) ( t ) + g ( t , u ( t ) , … , u ( n − 2 ) ( t ) ) = h ( t ) , for t ∈ [ 0 , T ] , ( BC ) { u ( i ) ( 0 ) = 0 , i = 0 , 1 , 2 , … , n − 3 , ( BVP ) u ( n − 2 ) ( 0 ) = u ( n − 2 ) ( T ) , u ( n − 1 ) ( 0 ) = u ( n − 1 ) ( T ) , h ε L1(0,T), D ε C([0,T]× Rn−1,R), and g:[0,T] × Rn−1 → R be a Carathéodory function, T-periodic in the first variable.