A q-fractional approach to the regular Sturm-Liouville problems

Abstract

In this article, we study the regular $q$-fractional Sturm-Liouville problems that include the right-sided Caputo q-fractional derivative and the left-sided Riemann-Liouville q-fractional derivative of the same order, $\alpha \in (0,1)$. We prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. We use a fixed point theorem for proving the existence and uniqueness of the eigenfunctions. We also present an example involving little q-Legendre polynomials.

Keywords

• Boundary value problems
• eigenvalues and eigenfunctions
• left and right sided Riemann-Liouville and Caputo q-fractional derivatives