Opuscula Math. 41, no. 4 (2021), 489-507
https://doi.org/10.7494/OpMath.2021.41.4.489

Opuscula Mathematica

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

Abdelrachid El Amrouss
Omar Hammouti

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.

Keywords: discrete boundary value problems, 2n-th order, variational methods, critical point theory.

Mathematics Subject Classification: 39A10, 34B08, 34B15, 58E30.

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• Communicated by Jean Mawhin.
• Received: 2020-10-31.
• Revised: 2021-04-02.
• Accepted: 2021-04-04.
• Published online: 2021-07-09.

Cite this article as:
Abdelrachid El Amrouss, Omar Hammouti, Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications, Opuscula Math. 41, no. 4 (2021), 489-507, https://doi.org/10.7494/OpMath.2021.41.4.489

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