Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 395457, 6 pages
http://dx.doi.org/10.1155/2013/395457
Research Article
A Dirac System with Transmission Condition and
Eigenparameter in Boundary Condition
Abdullah Kablan and Tülay Özden
Department of Mathematics, Faculty of Arts and Sciences, Gaziantep University, 27310 Gaziantep, Turkey
Correspondence should be addressed to Abdullah Kablan; kablan@gantep.edu.tr
Received 5 May 2013; Accepted 11 July 2013
Academic Editor: Ravshan Ashurov
Copyright © 2013 A. Kablan and T.
¨
Ozden. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Tis paper deals with a Dirac system with transmission condition and eigenparameter in boundary condition. We give an operator-
theoretic formulation of the problem then investigate the existence of the solution. Some spectral properties of the problem are
studied.
1. Introduction
Afer Walter [1] had given an operator-theoretic formulation
of eigenvalue problems with eigenvalue parameter in the
boundary conditions, Fulton [2, 3] has carried over the
methods of Titchmarsh [4, chapter 1] to this problem. Ten,
a large amount of the mathematical literature was devoted to
these subjects during the last twenty years. We will mention
some of the papers published at least twenty years ago, but
of course there are many other interesting and important
papers published more recently, which are not referred to
here. Te existence of solution and some spectral properties
of Sturm-Liouville problem with eigenparameter-dependent
boundary conditions and also with transmission conditions
at one or more inner points of considered fnite interval
has been studied by Mukhtarov and Tunc ¸[5]; see also [6,
7]. A Dirac system when the eigenparameter appears in
boundary conditions has been studied by Kerimov [8]. In
[9], an inverse problem for the Dirac system with eigenvalue-
dependent boundary conditions and transmission condition
is investigated.
Te aim of the present paper is to study a Dirac system
with transmission condition and eigenparameter in bound-
ary condition. For this, we follow the method in [5]. We
consider the Dirac system
ℓ()=
()−()()=(), (1)
where
=(
01
−1 0
),
()= (
1
() 0
0
2
()
),
()= (
1
()
2
()
),
(2)
or
2
()−
1
()
1
() =
1
(),
1
()+
2
()
2
() = −
2
(), ∈[,)∪(,],
(3)
with boundary conditions
sin
1
()− cos
2
() = 0, (4)
1
1
()−
1
2
()+ (sin
1
()− cos
2
())=0, (5)
and transmission conditions at the inner point =
1
(−0)=
1
(+0),
2
(−0)=
−1
2
(+0).
(6)
Here and later on, is a complex eigenvalue parameter; the
functions
()( = 1,2) are continuous on [,)∪(,] which
have fnite limits
(±) = lim
→ ±
()( = 1,2).
1
,
1
, are
real numbers and ,∈[0,).