Variational Methods for the Fractional Sturm--Liouville Problem

TL;DR

This paper investigates the fractional Sturm--Liouville eigenvalue problem using variational methods, establishing the existence of orthogonal solutions and characterizing the lowest eigenvalue as a variational minimum.

Contribution

It introduces a variational framework for the fractional Sturm--Liouville problem and proves the existence of eigenvalues and eigenfunctions within this context.

Findings

Existence of a countable set of orthogonal solutions

The lowest eigenvalue minimizes a specific variational functional

Framework extends classical Sturm--Liouville theory to fractional operators

Abstract

This article is devoted to the regular fractional Sturm--Liouville eigenvalue problem. Applying methods of fractional variational analysis we prove existence of countable set of orthogonal solutions and corresponding eigenvalues. Moreover, we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional.