Nonlinear elliptic Dirichlet boundary value problems on time scales

TL;DR

This paper develops a unified approach to solving nonlinear elliptic boundary value problems on time scales, covering continuous, discrete, and hybrid cases, by establishing existence, uniqueness, and spectral theory results.

Contribution

It introduces a spectral theory for the Dirichlet Laplacian on time scales and applies fixed point theorems to prove existence and uniqueness of solutions.

Findings

Established spectral properties of the Dirichlet Laplacian on time scales.

Proved existence and uniqueness results under Lipschitz and growth conditions.

Developed a framework unifying continuous, discrete, and hybrid boundary value problems.

Abstract

We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings. Under a Lipschitz condition on the nonlinearity bounded by the first eigenvalue, we prove existence and uniqueness using the contraction mapping theorem. Under a weaker one-sided growth condition, we establish existence using the Leray--Schauder fixed point theorem. To apply these functional analytic methods, we reformulate the problem as an operator equation, which requires developing the spectral theory for the Dirichlet Laplacian with mixed nabla-delta derivatives. We establish self-adjointness, positivity, and completeness of eigenfunctions, and the product eigenfunctions form a complete orthonormal basis in the n-dimensional setting.