Lyapunov-type inequalities for the modified discrete Helmholtz operator on $ \mathbb{Z} $

Bessem Samet; Bessem Samet

doi:10.3934/math.2026687

AIMS Mathematics

2026, Volume 11, Issue 6: 16735-16762. doi: 10.3934/math.2026687

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Lyapunov-type inequalities for the modified discrete Helmholtz operator on $ \mathbb{Z} $

Bessem Samet ^,

Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received: 28 March 2026 Revised: 15 May 2026 Accepted: 02 June 2026 Published: 11 June 2026
MSC : 39A06, 39A27, 05C50, 05A20

We establish Lyapunov-type inequalities for Dirichlet boundary value problems driven by the modified discrete Helmholtz operator on finite balls of the integer lattice $ \mathbb{Z} $. This yields weighted inequalities for scalar problems and a spectral-radius criterion for coupled systems, as well as sharpness results. As an application, we study a weighted eigenvalue problem and obtain explicit two-sided bounds for the first eigenvalue by combining a Lyapunov-type lower estimate with a variational upper bound. Numerical illustrations are provided for localized weights.
- Lyapunov-type inequality,
- difference equations,
- discrete Helmholtz operator,
- normalized discrete Laplacian,
- Dirichlet boundary value problem,
- eigenvalue bounds
Citation: Bessem Samet. Lyapunov-type inequalities for the modified discrete Helmholtz operator on $ \mathbb{Z} $[J]. AIMS Mathematics, 2026, 11(6): 16735-16762. doi: 10.3934/math.2026687

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Abstract

We establish Lyapunov-type inequalities for Dirichlet boundary value problems driven by the modified discrete Helmholtz operator on finite balls of the integer lattice $ \mathbb{Z} $. This yields weighted inequalities for scalar problems and a spectral-radius criterion for coupled systems, as well as sharpness results. As an application, we study a weighted eigenvalue problem and obtain explicit two-sided bounds for the first eigenvalue by combining a Lyapunov-type lower estimate with a variational upper bound. Numerical illustrations are provided for localized weights.

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