On fractional q-Sturm–Liouville problems

Abstract

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 a fixed point theorem to introduce a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions when $$\alpha >1/2$$ . These results are a generalization of the integer regular q-Sturm–Liouville problem introduced by Annaby and Mansour (J Phys A Math Gen 39:8747, 2005). An example for a qFSLP whose eigenfunctions are little q-Jacobi polynomials is introduced.