Hayati Olğar

Transmission problems for the Sturm-Liouville equation involving an abstract linear operator

The goal of this paper is to study a Sturm-Liouville problem with discontinuities in the case when an eigenparameter appears not only in the differential equation but also in one of the transmission conditions. Moreover, the equation contains an abstract linear operator. We establish such properties as isomorphism, coercive solvability and prove the discreteness of the spectrum. Finally we investigate an asymptotic behavior of the eigenvalues.

Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem

We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piece-wise continuous potential together with eigenparameter-dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular it is shown that the generalized eigen-functions form a Riesz basis of the adequate Hilbert space.

Generalized eigenfunctions of one Sturm-Liouville system with symmetric jump conditions

The aim of this study is to investigate one discontinuous Sturm-Liouville problem with eigenparameter appearing in the boundary-transmission conditions. We esthabilish some spectral properties for the considered problem in suitable Sobolev spaces. The main result of this study state that the system of generalized eigenfunctions form a Riesz basis of the appropriate Hilbert space.

Riesz basis property of weak eigenfunctions for boundary-value problem with discontinuities at two interior points

We investigate one discontinuous boundary value problem which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter-dependent boundary conditions and four supplementary transmission conditions. We establish some spectral properties of the considered problem. For the problem under consideration we define a new concept so-called weak eigenfunctions which is an extension of a classical eigenfunction and prove that the system of weak eigen-functions form a Riesz basis of the appropriate Hilbert space for the modified Lebesgue space.

Coercive solvability of two-interval Sturm-Liouville problems with abstract linear operator

In this paper we focus our attention on a new type nonhomogeneous Sturm-Liouville systems with abstract linear operator contained in the equation. A different approach is used here for investigation such important properties as topological isomorphism and coercive solvability. Moreover we prove that the corresponding resolvent operator is compact in a suitable Hilbert space.

Positiveness of second order differential operators with interior singularity

The main goal of this study is to provide an operator-theoretic framework for the investigation of discontinuous Sturm-Liouville problems with eigenparameter appearing in the boundary conditions. We introduce some self-adjoint compact operators in suitable Sobolev spaces such a way that the considered problem can be reduced to an operator-pencil equation. We define a new concept so-called generalized eigenfunctions and prove positiveness of operator-pencil corresponding to the considered problem.

Spectrum of one BVP with discontinuities and spectral parameter in the boundary conditions

The aim this of paper is to investigate the spectral problem for the equation −(pu′)′(x) + q(x)u(x) = λu(x), under eigen-dependent boundary conditions and supplementary transmission conditions at finite number interior points. By modifying some techniques of classical Sturm-Liouville theory and suggesting own approaches we esthabilish some properties of the eigenvalues and eigenfunction.