Variational methods for 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 $\alpha$, $\alpha\in (0,1)$\mathrm{.}Weintroducetheessential$q$-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when $\alpha >1\mathrm{/}2$ 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].