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

• Main Authors: Abdelrachid El Amrouss, Omar Hammouti
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2021-07-01
• Series: Opuscula Mathematica
• Subjects: discrete boundary value problems; 2n-th order; variational methods; critical point theory
• Online Access: https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4124.pdf

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.