One Boundary-Value Problem Perturbed by Abstract Linear Operator

The investigation of regular boundary value problems for which the eigenvalue parameter appears in both the ordinary di®erential equation and the boundary conditions originates from the Birkho®'s work [3]. In recent years, more and more researchers are interested in the discontinuous Sturm-Liouville problems. Various physics applications of this kind of problems are found in many literatures (see [1], [2], [6]). 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 di®erential equation but also in the boundary conditions. Morever, the "di®erential equation" contained also an abstract linear operator (unbounded in general) in the Hilbert space L2(¡1; 0) © L2(0; 1). We apply a di®erent approach for investigation some spectral properties of this problem.