A generalized discontinuous Sturm–Liouville problem with boundary conditions rationally dependent on the eigenparameter

Abstract

The present paper deals with a class of generalized discontinuous Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. Using operator theoretic formulation under the new inner product based on boundary conditions rationally dependent on the eigenparameter and general transmission conditions, the self-adjointness of this operator is proved. We also obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problem: the coefficients, the boundary conditions, the general transmission conditions, as well as the endpoints. Moreover, we find the differential expressions for each parameters, respectively.