%0 Journal Article
%T A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition
%A Erdo£żan £żen
%J Physics Research International
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/159243
%X We study a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We give an operator-theoretic formulation, construct fundamental solutions, investigate some properties of the eigenvalues and corresponding eigenfunctions of the discontinuous Sturm-Liouville problem and then obtain asymptotic formulas for the eigenvalues and eigenfunctions and find Green function of the discontinuous Sturm-Liouville problem. 1. Introduction In recent years, more and more researchers are interested in the discontinuous Sturm-Liouville problem for its application in physics (see [1¨C16]). Such problems are connected with discontinuous material properties, such as heat and mass transfer, varied assortment of physical transfer problems, vibrating string problems when the string loaded additionally with point masses, and diffraction problems. Moreover, there has been a growing interest in Sturm-Liouville problems with eigenparameter-dependent boundary conditions; that is, the eigenparameter appears not only in the differential equations but also in the boundary conditions of the problems (see [1¨C16] and corresponding bibliography). In this paper, following [8] we consider the boundary value problem for the differential equation for (i.e., belongs to , but the two inner points are and ), where is a real-valued function, continuous in , , and with the finite limits , ; is a discontinuous weight function such that for , for , and for , together with the standard boundary condition at the spectral parameter-dependent boundary condition at and the four transmission conditions at the points of discontinuity and in the Hilbert space , where is a complex spectral parameter; and all coefficients of the boundary and transmission conditions are real constants. We assume that and . Moreover, we will assume that . Some special cases of this problem arise after application of the method of separation of variables to the diverse assortment of physical problems, heat and mass transfer problems (e.g., see [11]), vibrating string problems when the string was loaded additionally with point masses (e.g., see [11]), and a thermal conduction problem for a thin laminated plate. 2. Operator-Theoretic Formulation of the Problem In this section, we introduce a special inner product in the Hilbert space and define a linear operator in it so that the problem (1)¨C(5) can be interpreted as the eigenvalue problem for . To this end, we define a new Hilbert space inner product on by for and . For convenience, we will use the notations In
%U http://www.hindawi.com/journals/physri/2013/159243/