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),fort@?[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), where h @e L^1(0,T), D @e C([0,T]x R^n^-^1,R), and g:[0,T] x R^n^-^1 -> R be a Caratheodory function, T-periodic in the first variable.