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 L(n)u(t) = f(t, u(t)), 0 <= t <= 2 pi, u((i)) (0) = u((i)) (2 pi), i = 0,1 ,..., n - 1, where L(n)u(t) = u((n))(t) + Sigma(n-1)(i=0) a(i)u((i)) (t) is an nth-order linear differential operator, n >= 2, and f : [0, 2 pi] x R -> R is continuous. In the case that L,, 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]. (C) 2008 Elsevier Ltd. All rights reserved.