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 α, $\alpha\in(0,1)$ . 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 criterion 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 (J. Phys. A, Math. Gen. 38:3775-3797, 2005; J. Phys. A, Math. Gen. 39:8747, 2006).