Spectral Problem and Initial Value Problem of a Nonlocal Sturm-Liouville Equation

In this paper, we considered the spectral problem and initial value problem of a nonlocal Sturm-Liouville equation with fractional integrals and fractional derivatives. We proved that the fractional operator associated to the nonlocal Sturm-Liouville equation is self-adjoint in Hilbert space. And then, we derived the corresponding spectral problem consists of countable number of real eigenvalues, and the algebraic multiplicity of each eigenvalue is simple. We also discussed the orthogonal completeness of the corresponding eigenfunction system in the Hilbert spaces. Furthermore, we obtained asymptotic properties of eigenvalues and the number of zeros of eigenfunctions by using the perturbation theory for linear operators. Finally, we studied the uniqueness of solutions for the nonlocal Sturm-Liouville equation under some initial value conditions.