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 Discontinuous fractional Sturm-Liouville problems with eigen-dependent boundary conditions Jing Fu,Xiaoling Hao,Kun Li,Siqin Yao Keywords:Fractional Sturm-Liouville problem; Fractional eigen-depandent boundary conditions; Fractional transmission conditions Abstract: In this paper, a fractional discontinuous Sturm-Liouville type boundary-value problem with eigenparameter-dependent boundary conditions and with two fractional transmission conditions is investigated. Using operator theory, a new inner product is defined by combining the parameters in the boundary and transmission conditions, then the boundary value transmission problem is transferred to an operator in the new Hilbert space such that the eigenvalues and eigenfunctions of the main problem coincide with those of this operator. Moreover, the fundamental solutions is constructed, and then the characteristic function whose zeros are the eigenvalues of the problem is established.