Some spectral properties of the Sturm-Liouville problem with discontinuity point and transition conditions

In this Master's thesis, a boundary value problem consisting of a second-order nonhomogeneous differential equation and nonhomogeneous boundary conditions is considered. The problem has a second-order eigenvalue parameter in its differential equation and the coefficients in the equation are functions that are discontinuous at an interior point of the definition range. Since the definition range of the problem under consideration has one discontinuity point, the problem is in the form of a two-interval boundary value problem. The boundary conditions of the problem consist of the boundary conditions considered at the end points of the definition interval and the conditions considered at the discontinuity point. The conditions considered at the discontinuity point are called transmission conditions. By adding the transmission conditions to the boundary conditions of the problem, the problem has become a problem with four boundary conditions. In addition, there are linear functionals in the boundary conditions of the problem. When all these properties and conditions are taken into account, the problem studied is a second-order, discontinuous coefficient, eigenvalue parameter, functional, transmission condition boundary value problem. The essence of this study is to define the differential operator generated by the problem whose properties and conditions are stated above and to investigate some of its properties. The differential operator in question is defined in the direct total space of Sobolev spaces belonging to the definition intervals of the problem. This differential operator is evaluated in the direct total space of Sobolev spaces formed by adding the four-dimensional complex spaces where the boundary conditions are evaluated to the same direct total space. This thesis consists of the following sections; Introduction, Summary of Literature, Measurable Spaces and Measurable Functions, Normed Spaces, Linear Operators, Boundary Value Problems and Coercivity and Isomorphism of the Discontinuous Sturm-Liouville Problem with Transmission Conditions. The last section is the original part of this thesis and in this section, the coercivity, isomorphism and Fredholm operator properties of the differential operator belonging to the problem are investigated.