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

Let $n \in N^{\star}$, and $N \geq n$ be an integer. We study the spectrum of discrete linear $2n$-th order eigenvalue problems \[ \begin{cases} \Sigma_{k=0}^{n}(-1)^{k} \Delta^{2k} u(t-k) = \lambda{}u(t), & t \in[1,N]_{\mathbb{Z}} \\ \Delta^{i}u(-(n-1)) = \Delta^{i}u(N-(n-1)), & i \in[0, 2n-1]_{\mathbb{Z}} \end{cases} \] where A 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.