Variational methods for fractional $q$-Sturm--Liouville Problems

TL;DR

This paper develops a variational framework for solving fractional q-Sturm--Liouville problems involving q-fractional derivatives, establishing eigenvalue existence, orthogonality, and criteria for the first eigenvalue, generalizing classical results.

Contribution

It introduces a variational approach to fractional q-Sturm--Liouville problems, proving eigenvalue existence and properties for the first time in this context.

Findings

Existence of a countable set of real eigenvalues and orthogonal eigenfunctions.

Criteria established for the first eigenvalue.

Generalization of classical q-Sturm--Liouville problems to fractional derivatives.

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 $\alpha$, $\alpha\in (0,1)$. We introduce the essential $q$-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when $\alpha>1/2$ associated with the boundary condition $y(0)=y(a)=0$. A criteria for the first eigenvalue is proved. Examples are included. These results are a generalization of the integer regular $q$-Sturm--Liouville problem introduced by Annaby and Mansour in [1].