Kadriye Aydemir

Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point

Abstract The purpose of this article is to extend some spectral properties of regular Sturm-Liouville problems to the special type discontinuous boundary-value problem, which consists of a Sturm-Liouville equation together with eigenparameter-dependent boundary conditions and two supplementary transmission conditions. We construct the resolvent operator and Greens function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space L 2 [a, b].

Green's Function Method for Self-Adjoint Realization of Boundary-Value Problems with Interior Singularities

The purpose of this paper is to investigate some spectral properties of Sturm-Liouville type problems with interior singularities. Some of the mathematical aspects necessary for developing our own technique are presented. By applying this technique we construct some special solutions of the homogeneous equation and present a formula and the existence conditions of Greens function. Furthermore, based on these results and introducing operator treatment in adequate Hilbert space, we derive the resolvent operator and prove self-adjointness of the considered problem.

Boundary value problems with eigenvalue-dependent boundary and transmission conditions

In this paper the operator-theoretical method to investigate a new type boundary value problems consisting of a two-interval Sturm-Liouville equation together with boundary and transmission conditions dependent on eigenparameter is developed. By suggesting our own approach, we construct modified Hilbert spaces and a linear operator in them in such a way that the considered problem can be interpreted as a spectral problem for this operator. Then we introduce so-called left- and right-definite solutions and give a representation of solution of the corresponding nonhomogeneous problem in terms of these one-hand solutions. Finally, we construct Green’s vector-function and investigate some important properties of the resolvent operator by using this Green’s vector-function.

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.

Class of Sturm–Liouville Problems with Eigenparameter Dependent Transmission Conditions

ABSTRACT The objective of this paper is the investigation of a new class Sturm–Liouville problems on two disjoint intervals when the eigenparameter appears not only in the equation but also in the boundary and transmission conditions. We concerned with the problem of justifying some spectral properties of the eigenvalues and corresponding eigenfunctions. Particularly, we examined asymptotic behavior of the eigenvalues and corresponding eigenfunctions.

Spectrum and Green’s Function of a Many-Interval Sturm–Liouville Problem

Abstract This article considers a Sturm–Liouville-type problem on a finite number disjoint intervals together with transmission conditions at the points of interaction. We introduce a new operator-theoretic formulation in such a way that the problem under consideration can be interpreted as a spectral problem for a suitable self-adjoint operator. We investigate some principal properties of eigenvalues, eigenfunctions, and resolvent operator. Particularly, by applying Green’s function method, it is shown that the problem has only point spectrum, and the set of eigenfunctions form a basis of the adequate Hilbert 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.

Solvability of boundary value transmission problems by Green’s function method

In this study we give a comprehensive treatment for a class BVP’s together with transmission (impulsive, jump or interface) conditions at the one interior singular point. A self-adjoint linear operator ℛ is defined in a suitable Hilbert space ℋ such that the eigenvalues of such a problem coincide with those of ℋ and showed that it has a compact resolvent. We construct fundamental solutions and discuss some properties of spectrum. Moreover, we obtain the representation of Green’s function for the considered problem.