On the existence and uniqueness for higher order periodic boundary value problems

This paper discusses the existence and uniqueness for the nth-order periodic boundary value problem Lnu(t)=f(t,u(t)),0≤t≤2π,u(i)(0)=u(i)(2π),i=0,1,…,n−1, where Lnu(t)=u(n)(t)+∑i=0n−1aiu(i)(t) is an nth-order linear differential operator, n≥2, and f:[0,2π]×R→R is continuous. In the case that Ln has an even order derivative, we present some new spectral conditions for the nonlinearity f(t,u) to guarantee the existence and uniqueness. These spectral conditions allow f(t,u) to be a superlinear growth, and are the extension for the spectral separation condition presented recently in [Y. Li, Existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions, J. Math. Anal. Appl. 322 (2) (2006) 530-539].