On Fractional q-Sturm--Liouville problems

TL;DR

This paper formulates and analyzes a regular q-fractional Sturm--Liouville problem, exploring eigenvalues, eigenfunctions, and their properties, including a q-fractional Wronskian and conditions for existence and uniqueness.

Contribution

It introduces a generalized q-fractional Sturm--Liouville problem incorporating Riemann--Liouville and Caputo derivatives, extending classical results to fractional q-calculus.

Findings

Eigenvalues and eigenfunctions properties are characterized.

A q-fractional Wronskian is defined and related to eigenfunction simplicity.

Conditions for existence and uniqueness of eigenfunctions are established.

Abstract

In this paper, we formulate a regular $q$-fractional Sturm--Liouville problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo q-fractional derivatives of the same order $\alpha$, $\alpha\in (0,1)$. The properties of the eigenvalues and the eigenfunctions are investigated. A $q$-fractional version of the Wronskian is defined and its relation to the simplicity of the eigenfunctions is verified. We use the fixed point theorem to introduce a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions when $\alpha>1/2$. These results are a generalization of the integer regular $q$-Sturm--Liouville problem introduced by Annaby and Mansour in[1]. An example for a qFSLP whose eigenfunctions are little $q$-Jacobi polynomials is introduced.