DISCONTINUOUS FRACTIONAL BOUNDARY VALUE PROBLEMS WITH ORDER $ \alpha\in (1,2) $

This paper focuses on investigating the discontinuous fractional Sturm-Liouville problem equipped with a transmission condition of order $ \alpha \in(1,2) $. Through rigorous analysis, it is demonstrated that the eigenvalues and their corresponding eigenfunctions of this problem coincide with those of the constructed operator in Hilbert space. Furthermore, a necessary and sufficient condition for the existence of eigenvalues is established, providing a theoretical foundation for the spectral characterization of such fractional boundary value problems.