Existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions ✩

This paper deals with 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) + n−1 i=0 aiu(i)(t) is an nth-order linear differential operator and f : [0, 2π] × R→R is continuous. We present some spectral conditions for the nonlinearity f (t,u) to guarantee the existence and uniqueness. These spectral conditions are the generalization for nonresonance condition of Duffing equation. © 2005 Elsevier Inc. All rights reserved