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 problemLnu(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 n  th-order linear differential operator and f:[0,2π]×R→R is continuous. We present some spectral conditions for the nonlinearity f(t,u)f(t,u) to guarantee the existence and uniqueness. These spectral conditions are the generalization for nonresonance condition of Duffing equation.