Regular Fractional Sturm-Liouville Problem with Generalized Derivatives of Order in (0,1)

In this paper, we define a Generalized Fractional Sturm-Liouville Operator (GFSLO) and introduce a regular Generalized Fractional Sturm-Liouville Problem. In the construction of the operator and the problem, we apply a generalized fractional derivatives built using a general kernel. We investigate the properties of the eigenfunctions and the eigenvalues of the GFSLO, and demonstrate that these properties are analogous to those for classical Sturm-Liouville Operator dependent on first-order derivatives. As an example, we study the case when the integrals and the resulting derivatives are built using the generalized Mittag-Leffler function as a kernel.