Comparison Criteria for Discrete Fractional Sturm-Liouville Equations

TL;DR

This paper establishes Sturm comparison theorems for discrete fractional Sturm-Liouville equations in Riemann-Liouville and Grünwald-Letnikov senses, extending classical results to fractional discrete settings.

Contribution

It introduces comparison theorems for discrete fractional Sturm-Liouville equations, a novel extension of classical Sturm theory to fractional discrete operators.

Findings

Sturm comparison theorems for Riemann-Liouville and Grünwald-Letnikov fractional differences

Properties of zeros of solutions to discrete fractional Sturm-Liouville equations

Extension of classical Sturm theory to fractional discrete equations

Abstract

In this study, we give the Sturm comparison theorems for discrete fractional Sturm-Liouville (DFSL) equations within Riemann-Liouville and Gr\"unwald-Letnikov sense. The emergence of Sturm-Liouville equations began as one dimensional Schr\"odinger equation in quantum mechanics and one of the most important results is Sturm comparison theorems [27]. These theorems give information about the properties of zeros of two equations having different potentials.