On Fractional q-Sturm--Liouville problems

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 $\alpha >1\mathrm{/}2$. 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.