Fractional Sturm-Liouville Problem in Terms of Riesz Derivatives

In the paper, a regular fractional Sturm-Liouville problem on a bounded domain is formulated in terms of Riesz derivatives. The considered case includes vanishing Dirichlet boundary conditions and we prove that its eigenvalues are real, eigenfunctions are continuous and form orthogonal sets of functions in the respective Hilbert spaces. In addition, a boundedness results for eigenvalues are derived and a connection between the discussed fractional Sturm-Liouville equations and Euler-Lagrange equations for the corresponding action functionals is established.