Spektral properties of many point boundary value problem with discontinuous coefficient

In this PhD thesis, a boundary value problem consisting of a second-order differential equation with discontinuous coefficients and boundary conditions is considered. The differential equation of the problem has an eigenvalue parameter and the coefficients in the equation are the functions that are discontinuous at two internal points of the domain. Therefore, the under consideration problem which has six boundary conditions with transmission conditions of discontinuity points in the domain and the endpoints of the domain becomes three-interval boundary value problem. In addition, all boundary conditions also contain conditions for the the interior points of intervals. Therefore, the problem solvability of which is being investigated is a boundary value problem with discontinuous coefficients, multiple points, eigenvalue parameters and transmission conditions. The properties of the differential operator which is produced by the problem and whose properties have mentioned above are the main focus of the this thesis. Furthermore, the differential operator that considered is defined on the direct sum space of the Sobolev spaces of the intervals. It is evaluated in the direct sum space of Sobolev spaces, which is obtained by adding six-dimensional complex spaces where boundary conditions are evaluated to the same direct sum space. This thesis consists of three main parts. In the first part the literature research is given. In the second part general concepts and definitions are given. In the third part the working stages of considered new problem and the proofs of the theorems about the coercivity of the differential operator and the Fredholm operator are given.