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

This paper discusses the existence and uniqueness for the nnth-order periodic boundary value problem View the MathML sourceLnu(t)=f(t,u(t)),0≤t≤2π, Turn MathJax on View the MathML sourceu(i)(0)=u(i)(2π),i=0,1,…,n−1, Turn MathJax on where View the MathML sourceLnu(t)=u(n)(t)+∑i=0n−1aiu(i)(t) is an nnth-order linear differential operator, n≥2n≥2, and f:[0,2π]×R→Rf:[0,2π]×R→R is continuous. In the case that LnLn has an even order derivative, we present some new spectral conditions for the nonlinearity f(t,u)f(t,u) to guarantee the existence and uniqueness. These spectral conditions allow f(t,u)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)