Open Access Research

Boundary value problems with eigenvalue-dependent boundary and transmission conditions

Kadriye Aydemir

Author Affiliations

Department of Mathematics, Faculty of Arts and Science, Gaziosmanpaşa University, Tokat, 60250, Turkey

Boundary Value Problems 2014, 2014:131  doi:10.1186/1687-2770-2014-131

Published: 22 May 2014

Abstract

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.

Keywords:
Sturm-Liouville problems; eigenparameter-dependent boundary and transmission conditions; Green’s function; resolvent operator