@article{Samet2026, 
author = {Bessem Samet},
title = {Lyapunov-type inequalities for the modified discrete Helmholtz operator on        Z},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {16735-16762},
keywords = {Lyapunov-type inequality, difference equations, discrete Helmholtz operator, normalized discrete Laplacian, Dirichlet boundary value problem, eigenvalue bounds},
url = {https://www.sciopen.com/article/10.3934/math.2026687},
doi = {10.3934/math.2026687},
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        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.}
}