Abstract
In this article, we study Sturm-Liouville Equations (SLEs) in the frame of fractional operators with exponential kernels. We formulate some Fractional Sturm-Liouville Problems (FSLPs) with the differential part containing the left and right sided derivatives with exponential kernels. We investigate the self-adjointness, eigenvalue and eigenfunction properties of the corresponding Fractional Sturm-Liouville Operators (FSLOs) by using fractional integration by parts formulas. The nabla discrete version of our results are also established. Finally, an example is analyzed to illustrate the method of solution.