Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications

Abstract

Let $n\in\mathbb{N}^{*}$, and $N\geq n$ be an integer. We study the spectrum of discrete linear $2n$-th order eigenvalue problems $\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}$ where $\lambda$ is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear $2n$-th order problems by applying the variational methods and critical point theory.