SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

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

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/131


Received:4 March 2014
Accepted:8 May 2014
Published:22 May 2014

© 2014 Aydemir; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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

1 Introduction

Many important special equations which appear in physics, such as Airy’s equation, Bessel’s equation, wave equation, heat equation, Schrödinger’s equation, Heun’s equation, advection-dispersion equation, etc., are associated with Sturm-Liouville type operators. For instance, the one-dimensional form of the advection-dispersion equation for a nonreactive dissolved solute in a saturated, homogeneous, isotropic porous medium under steady, uniform flow is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M2">View MathML</a> is the concentration of the solute, ν is the average linear groundwater velocity, D is the coefficient of hydrodynamic dispersion, and L is the length of the aquifer. Using the method of separation of variables, the problem can be written in the simplest Sturm-Liouville form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M3">View MathML</a>

This example makes it clear that the Sturm-Liouville problems are of broad interest. There is a well-developed theory for classical Sturm-Liouville problems (see, e.g., [1-5] and the references therein). Details of the derivation of the theory and of related background results can be found in the cited references. Although the subject of Sturm-Liouville problems is over 160 years old, these problems are an intensely active field of research today. The main tool for solvability analysis of such problems is the concept of Green’s function. Green’s functions have played an important role as a theoretical tool in the field of physics, since the possibility of a transition from the problems in mathematical physics to integral equations is based on the fundamental concept of Green’s function. Therefore, the powerful and unifying formalism of Green’s functions finds applications not only in standard physics subjects such as perturbation and scattering theory, bound-state formation, etc., but also at the forefront of current and, most likely, future developments (see [6]). Green’s function transforms the differential equation into the integral equation, which, at times, is more informative. In terms of Green’s function, the BVP with arbitrary data can be solved in a form that shows clearly the dependence of the solution on the data. Namely, Green’s function approach would allow us to have an integral representation of the solution instead of an infinite series. Determination of Green’s functions is also possible using Sturm-Liouville theory. This leads to series representation of Green’s functions (see, e.g., the monograph [1] as well as the recent results in [7] and the references therein).

Sturm-Liouville type problems with transmission conditions have become an important area of research in recent years because of the needs of modern technology, engineering and physics. Many of the mathematical problems encountered in the study of boundary-value-transmission problem cannot be treated with the usual techniques within the standard framework of boundary value problem (see [8-12]). In this study we shall consider a new type of Sturm-Liouville problems consisting of the two-interval Sturm-Liouville equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M4">View MathML</a>

(1)

together with eigenparameter-dependent boundary conditions of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M5">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M6">View MathML</a>

(3)

and eigenparameter-dependent transmission conditions at one interaction point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M7">View MathML</a> of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M8">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M9">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M10">View MathML</a> is a real-valued piecewise constant function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M13">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M14">View MathML</a>, the potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M15">View MathML</a> is a real-valued function continuous in each of the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M17">View MathML</a>, and has finite limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M18">View MathML</a>, μ is a complex spectral parameter, the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M22">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M24">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M25">View MathML</a> are real numbers. This Sturm-Liouville problem is a non-classical eigenvalue problem since the eigenvalue parameter μ appears not only in the differential equation, but also in the boundary and transmission conditions. Moreover, in the differential equation there appears a singularity at one interior point. Because of these reasons the spectral theory of this problem is more complicate. Naturally, eigenfunctions of this problem may have discontinuity at the singular interior point. Some special cases of this problem arise after an application of the Fourier method to a varied assortment of physical problems. For instance, some boundary value problems with transmission conditions arise in heat and mass transfer problems [13], in vibrating string problems when the string is loaded additionally with point masses [14], in diffraction problems [12], in quantum mechanics [15], in thermal conduction problems for a thin laminated plate [16]etc. Such properties as isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel bases property of a system of root functions of some boundary value problems with transmission conditions and its applications to the corresponding initial boundary value problems for parabolic equations have been investigated in [16-19]. For the background and applications of boundary value transmission problems to different areas, we refer the reader to the monographs and some recent contribution [8-11,17,18,20-25].

2 Hilbert space formulation of the problem

In certain cases the boundary value problem can be characterized by means of a uniquely determined unbounded self-adjoint operator. In these cases the eigenvalues and eigenfunctions of the boundary value problem are determined by the eigenvalues and eigenvectors of the corresponding operator; these will be called a self-adjoint case of the boundary value problem. In some cases such a characterization is not possible and these will be referred to as ‘symmetric’ cases in general. In classical point of view, our problem cannot be characterized as ‘self-adjoint case’. For ‘self-adjoint characterization’ of the considered problem (1)-(5), we shall define a new Hilbert space as follows.

Denote the determinant of the ith and jth columns of the matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M26">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M28">View MathML</a>). Throughout the paper we shall assume that the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M29">View MathML</a>

hold. Define a new inner-product space ℋ as a direct sum space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M30">View MathML</a> equipped with the modified inner-product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M31">View MathML</a>

(6)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M32">View MathML</a>. It is easy to see that the relation (6) really defines a new inner product in the direct sum space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M30">View MathML</a>.

Lemma 1is a Hilbert space.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M35">View MathML</a> , be any Cauchy sequence in ℋ. Then by (6) the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M37">View MathML</a> will be Cauchy sequences in the Hilbert spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M39">View MathML</a>, respectively. Therefore they are convergent. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M41">View MathML</a> be limits of these sequences, respectively. Defining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M42">View MathML</a> we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M44">View MathML</a> in ℋ. The proof is complete. □

Let us now define the boundary and transmission functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M50">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M52">View MathML</a> and the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M53">View MathML</a> with the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M54">View MathML</a>

and action low

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M55">View MathML</a>

Then problem (1)-(5) can be written in the operator equation form as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M57">View MathML</a> in the Hilbert space ℋ.

Theorem 1The linear operatoris symmetric in the Hilbert space ℋ.

Proof By applying the method of [22] it is not difficult to prove that the operator ℜ is densely defined in ℋ, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M58">View MathML</a>. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M59">View MathML</a>. By partial integration we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M60">View MathML</a>

(7)

where, as usual, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M61">View MathML</a> denotes the Wronskians of the functions u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M62">View MathML</a>. From the definitions of boundary functionals we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M63">View MathML</a>

(8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M64">View MathML</a>

(9)

Further, taking in view the definition of ℜ and initial conditions (14)-(19) we can derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M65">View MathML</a>

(10)

Finally, substituting (8), (9) and (10) in (7) we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M66">View MathML</a>

The proof is complete. □

Theorem 2The linear operatoris self-adjoint in ℋ.

Proof Since ℜ is symmetric and densely defined on ℋ, it is sufficient to show that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M67">View MathML</a>

(11)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M68">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M70">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M72">View MathML</a>. Writing equality (11) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M73">View MathML</a> by standard Sturm-Liouville theory, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M75">View MathML</a>. Then from equality (11) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M76">View MathML</a>

On the other hand, by two partial integrations we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M77">View MathML</a>

(12)

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M78">View MathML</a>

(13)

From this equality, by applying the technique of Theorem 2.5 in our previous work [11], it can be derived easily that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M85">View MathML</a>. The proof is complete. □

Theorem 3The operatorhas only point spectrum, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M86">View MathML</a>.

Proof It suffices to prove that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87">View MathML</a> is not an eigenvalue of ℜ, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87">View MathML</a> is a regular point of ℜ, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M89">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87">View MathML</a> not be an eigenvalue of ℜ. The resolvent operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M91">View MathML</a> exists and is defined on all of ℋ. By Theorem 2 and the closed graph theorem, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M92">View MathML</a> is bounded. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M89">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M86">View MathML</a>. □

3 Left-definite and right-definite solutions

In this section we shall define two basic solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M96">View MathML</a> on the left interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M16">View MathML</a> (so-called left-definite solutions) and two basic solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M99">View MathML</a> on the right interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M17">View MathML</a> (so-called right-definite solutions) by a special procedure as follows. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M99">View MathML</a> be solutions of equation (1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M17">View MathML</a> satisfying the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M105">View MathML</a>

(14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M106">View MathML</a>

(15)

respectively. By using these solutions we shall define the other solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M96">View MathML</a> by the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M109">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M110">View MathML</a>

(17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M111">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M112">View MathML</a>

(19)

respectively. The existence of these solutions follows from the well-known Cauchy-Picard theorem of ordinary differential equation theory. Moreover, by applying the method of [20], we can prove that each of these solutions are entire functions of the parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M113">View MathML</a> for each fixed x.

4 Construction of Green’s function

In this section we develop the idea of a resolvent operator to solve nonhomogeneous boundary-value transmission problems (BVTP) as follows. Consider the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M114">View MathML</a>

(20)

for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M115">View MathML</a>. This operator equation is equivalent to the following nonhomogeneous BVTP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M116">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M117">View MathML</a>

(22)

Let us define the Wronskians <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M118">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M120">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M121">View MathML</a> and suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M122">View MathML</a>. We shall search the resolvent function of this BVTP in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M123">View MathML</a>

(23)

where the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M125">View MathML</a> are the solutions of the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M126">View MathML</a>

(24)

and the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M128">View MathML</a> are the solutions of the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M129">View MathML</a>

(25)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M131">View MathML</a>, respectively. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M133">View MathML</a>, from (24) and (25) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M134">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M135">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M136">View MathML</a>) are unknown functions depending only on the parameter μ. Substituting into (23) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M137">View MathML</a>

(26)

By differentiating we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M138">View MathML</a>

(27)

By using (26), (27) and conditions (22) we can derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M139">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M140">View MathML</a>

Putting in (26) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M141">View MathML</a>

(28)

Thus we find the needed resolvent function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M142">View MathML</a> in terms of the left- and right-define solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M144">View MathML</a>. By introducing Green’s function as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M145">View MathML</a>

(29)

from (28) and (29) we have that the considered problem (21)-(22) has a unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M146">View MathML</a>

(30)

5 Representations of the resolvent operator in terms of Green’s vector-function

We now shall define Green’s vector-function as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M147">View MathML</a>

(31)

Consequently, for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M148">View MathML</a> of nonhomogeneous operator equation (21), we obtain the following formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M149">View MathML</a>

(32)

Using this, the resolvent function (30) can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M150">View MathML</a>

(33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M151">View MathML</a>. Consequently, we have the following theorem.

Theorem 4For the resolvent operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M152">View MathML</a>, the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M153">View MathML</a>

(34)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M154">View MathML</a>, holds.

Theorem 5The estimation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M155">View MathML</a>

holds for all regular value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M87">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M157">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M115">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M159">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M160">View MathML</a>, taking into account that the operator ℜ is symmetric, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M161">View MathML</a>

Using the well-known Cauchy-Schwarz inequality, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M162">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M163">View MathML</a>

The proof is complete. □

Theorem 6The resolvent operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M92">View MathML</a>is compact in the Hilbert space ℋ.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M165">View MathML</a> be eigenvalues of ℜ and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M166">View MathML</a> be orthogonal projections onto the corresponding eigenspace. Since ℜ is a self-adjoint operator with discrete spectrum, we can write the spectral resolution of the resolvent operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M92">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M168">View MathML</a>

(35)

Similarly to [22] we can easily show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M169">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M170">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M171">View MathML</a>. Consequently, the series (35) is strongly convergent. It is obvious that the orthogonal projections <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/131/mathml/M173">View MathML</a> , are compact operators since each of them are of finite rank. Consequently, the sum of series (35) is also compact in ℋ. The proof is complete. □

Competing interests

The author declares that she has no competing interests.

Acknowledgements

The author is grateful to anonymous referees for their constructive comments and suggestions, which led to the improvement of the original manuscript.

References

  1. Levitan, BM, Sargsyan, IS: Sturm-Liouville and Dirac Operators, Springer, New York (1991)

  2. Pryce, JD: Numerical Solution of Sturm-Liouville Problems, Oxford University Press, New York (1993)

  3. Hinton, D, Schaefer, PW: Spectral Theory and Computational Methods for Sturm-Liouville Problems, Dekker, New York (1997)

  4. Titchmarsh, EC: Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, Oxford University Press, London (1962)

  5. Zettl, A: Sturm-Liouville Theory, Am. Math. Soc., Providence (2005)

  6. Duffy, DG: Green’s Functions with Applications, Chapman & Hall/CRC, Boca Raton (2001).

  7. Stakgold, I, Holst, MJ: Green’s Functions and Boundary Value Problems, Wiley, New York (2011).

  8. Ao, J, Sun, J, Zhang, M: Matrix representations of Sturm-Liouville problems with transmission conditions. Comput. Math. Appl.. 63, 1335–1348 (2012). Publisher Full Text OpenURL

  9. Aydemir, K, Mukhtarov, OS: Green’s function method for self-adjoint realization of boundary-value problems with interior singularities. Abstr. Appl. Anal.. 2013, (2013) Article ID 503267

    Article ID 503267

    Publisher Full Text OpenURL

  10. Kong, Q, Wang, Q: Using time scales to study multi-interval Sturm-Liouville problems with interface conditions. Results Math.. 63, 451–465 (2013). Publisher Full Text OpenURL

  11. Mukhtarov, OS, Aydemir, K: New type Sturm-Liouville problems in associated Hilbert spaces. J. Funct. Spaces Appl.. 2014, (2014) Article ID 606815

  12. Voitovich, NN, Katsenelbaum, BZ, Sivov, AN: Generalized Method of Eigen-Vibration in the Theory of Diffraction, Nauka, Moscow (1997)

  13. Likov, AV, Mikhailov, YA: The Theory of Heat and Mass Transfer, Qosenergaizdat, Moscow (1963)

  14. Tikhonov, AN, Samarskii, AA: Equations of Mathematical Physics, Pergamon, New York (1963)

  15. Albeverio, S, Gesztesy, F, Hoegh Krohn, R, Holden, H: Solvable Models in Quantum Mechanics, AMS Chelsea Publishing, Providence (2005)

  16. Titeux, I, Yakubov, Y: Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients. Math. Models Methods Appl. Sci.. 7, 1035–1050 (1997). Publisher Full Text OpenURL

  17. Mukhtarov, OS, Demir, H: Coerciveness of the discontinuous initial-boundary value problem for parabolic equations. Isr. J. Math.. 114, 239–252 (1999). Publisher Full Text OpenURL

  18. Mukhtarov, OS, Yakubov, S: Problems for ordinary differential equations with transmission conditions. Appl. Anal.. 81, 1033–1064 (2002). Publisher Full Text OpenURL

  19. Rasulov, ML: Methods of Contour Integration, North-Holland, Amsterdam (1967)

  20. Akdoğan, Z, Demirci, M, Mukhtarov, OS: Green function of discontinuous boundary-value problem with transmission conditions. Math. Methods Appl. Sci.. 30, 1719–1738 (2007). Publisher Full Text OpenURL

  21. Bairamov, E: On the characteristic values of the real component of a dissipative boundary value transmission problem. Appl. Math. Comput.. 218, 9657–9663 (2012). Publisher Full Text OpenURL

  22. Mukhtarov, OS, Kadakal, M: Some spectral properties of one Sturm-Liouville type problem with discontinuous weight. Sib. Math. J.. 46, 681–694 (2005). Publisher Full Text OpenURL

  23. Altınışık, N, Mukhtarov, OS, Kadakal, M: Asymptotic formulas for eigenfunctions of the Sturm-Liouville problems with eigenvalue parameter in the boundary conditions. Kuwait J. Sci. Eng.. 39, 1–19 (2012)

  24. Muhtarov, FS, Aydemir, K: Distributions of eigenvalues for Sturm-Liouville problem under jump conditions. J. New Results Sci.. 1, 81–89 (2012)

  25. Uǧurlu, E, Bairamov, E: Dissipative operators with impulsive conditions. J. Math. Chem.. 51, 1670–1680 (2013). Publisher Full Text OpenURL