SUMMARYIn this paper, we deal with Sturm-Liouville-type problems when the potential of the differential equation may have discontinuity at one inner point and the eigenparameter appears not only in the differential equation, but also in both boundary and transmission conditions. By modifying some techniques of (Twopoint boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edin. 1977; 77A:293-308; Eigenfunction Expenses Associated with Second-Order Differential Equations I (2nd edn). Oxford University Press: London, 1962) we generalize some results of the classic regular Sturm-Liouville problems. In particular, we construct Green's function, and derive asymptotic approximation formulae for Green's function. Further, we introduce a new operator-theoretic formulation in suitable Hilbert space such a way that the considered problem can be interpreted as the eigenvalue problem of this operator and constract the resolvent of this operator in terms of Green's function. Finally, we estimate the norm of resolvent of this operator.

The purpose of this paper is to study a Sturm-Liouville problem with discontinuities in the case when an eigenparameter appears not only in the differential equation but it also appears in both the boundary and transmission conditions.We suggest a new approach for the definition of a suitable Hilbert space and a symmetric linear operator defined in this space in such a way that the considered problem can be interpreted as the eigenvalue problem of this operator and for construction and approximation of a fundamental solution. We apply these results to find asymptotic formulas of eigenvalues and corresponding eigenfunctions. (2000): 34L20. Mathematics Subject Classification

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The main purpose of this study is to investigate a fractional discontinuous Sturm-Liouville problem with transmission conditions. We shall consider a fractional boundary value problem involving an operator with two parts. It is shown that the eigenvalues and corresponding eigenfunctions of the main problem coincide with the eigenvalues and corresponding eigenfunctions of the constructed operator in Hilbert spaces.