Open Access Research

Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition

Khanlar R Mamedov* and F Ayca Cetinkaya

Author Affiliations

Department of Mathematics, Science and Letters Faculty, Mersin University, Mersin, 33343, Turkey

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Boundary Value Problems 2013, 2013:183  doi:10.1186/1687-2770-2013-183


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/183


Received:5 April 2013
Accepted:29 July 2013
Published:14 August 2013

© 2013 Mamedov and Cetinkaya; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This work aims to examine a Sturm-Liouville operator with a piece-wise continuous coefficient and a spectral parameter in boundary condition. The orthogonality of the eigenfunctions, realness and simplicity of the eigenvalues are investigated. The asymptotic formula of the eigenvalues is found, and the resolvent operator is constructed. It is shown that the eigenfunctions form a complete system and the expansion formula with respect to eigenfunctions is obtained. Also, the evolution of the Weyl solution and Weyl function is discussed. Uniqueness theorems for the solution of the inverse problem with Weyl function and spectral data are proved.

MSC: 34L10, 34L40, 34A55.

Keywords:
Sturm-Liouville operator; expansion formula; inverse problem; Weyl function

1 Introduction

In recent years, there has been a growing interest in physical applications of boundary value problems with a spectral parameter, contained in the boundary conditions. The relationship between diffusion processes and Sturm-Liouville problem with eigen-parameter in the boundary conditions has been shown in [1]. Another example of this relationship between the same problem and the wave equation has been examined in [2,3]. Sturm-Liouville problems with a discontinuous coefficient arise upon non-homogeneous material properties.

In a finite interval, inverse problems for the Sturm-Liouville operator with spectral parameter, contained in the boundary conditions, have been investigated, and the uniqueness of the solution of these problems has been shown in [4-9]. The inverse problem has been analyzed by zeros of the eigenfunctions in [6], by numerical methods in [9] and by two spectra, consisting of sequences of eigenvalues and the normed constants in [10]. In [11,12], eigenvalue-dependent inverse problem with the discontinuities inside the interval was examined by the Weyl function. In a finite interval, discontinuous and no eigenvalue parameter containing direct problem and inverse problem with the Weyl function were discussed in [13,14]. The similar problem was investigated in the half line by scattering data in [15,16].

We consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M1">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M2">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M3">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M4">View MathML</a> is a real valued function, λ is a complex parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M8">View MathML</a> are real numbers and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M10">View MathML</a>.

2 Special solutions

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12">View MathML</a> be the solutions of equation (1) satisfying the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M13">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M14">View MathML</a>

(5)

For the solution of equation (1), the following integral representation is obtained in [13] for all λ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M15">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M16">View MathML</a>.

The kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M17">View MathML</a> has the partial derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M18">View MathML</a> belonging to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M19">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M20">View MathML</a>, and the properties below hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M21">View MathML</a>

Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M22">View MathML</a> is differentiable, then the following are valid

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M23">View MathML</a>

Using the representation of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M24">View MathML</a> and formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M25">View MathML</a>

we obtain the integral representation of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M27">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M28">View MathML</a>. The kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M29">View MathML</a> can be represented with the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M22">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M32">View MathML</a>

With the help of equation (6), we have a representation for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M34">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M35">View MathML</a> is a real function.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M36">View MathML</a>

(8)

which is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M37">View MathML</a>. Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M39">View MathML</a> into (8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M40">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41">View MathML</a> is an entire function of λ and is called the characteristic function of the boundary value problem (1)-(3).

3 Some spectral properties

Lemma 1The square values of the roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M42">View MathML</a>of the characteristic function coincide with the eigenvalues of the boundary value problem (1)-(3), and for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43">View MathML</a>, there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M45">View MathML</a>

(9)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M47">View MathML</a>are the eigenfunctions of the boundary value problem (1)-(3), corresponding to the eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43">View MathML</a>.

Proof The proof can be done in a similar way to [8]. Indeed, let us assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M49">View MathML</a> is an eigenvalue of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M51">View MathML</a>

holds, i.e., the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M53">View MathML</a> are linearly dependent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M54">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M55">View MathML</a>), and they satisfy the boundary conditions (2), (3). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M56">View MathML</a> is an eigenvalue, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M52">View MathML</a> are eigenfunctions, related to this eigenvalue. Conversely, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M56">View MathML</a> be an eigenvalue of the operator A, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M61">View MathML</a> be the corresponding eigenfunctions. Then the boundary conditions (2), (3) hold both for the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M61">View MathML</a>. Additionally, if the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M61">View MathML</a> satisfy the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M67">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M69">View MathML</a>. According to boundary conditions (2), (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M70">View MathML</a>

Similarly, if we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M72">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M74">View MathML</a>. Again from the boundary conditions (2), (3), it is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M75">View MathML</a>

Therefore, we have proved that for each eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M56">View MathML</a>, there exists only one (up to a multiplicative constant) eigenfunction. □

In the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M77">View MathML</a> an inner product defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M78">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M79">View MathML</a>

Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M80">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M81">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M82">View MathML</a>

The boundary value problem (1)-(3) is equivalent to the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M83">View MathML</a>.

The eigenfunctions of operator A are in the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M84">View MathML</a>

Lemma 2The eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M86">View MathML</a>, corresponding to different eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M87">View MathML</a>, are orthogonal.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M86">View MathML</a> are the solutions of the boundary value problem (1)-(3), the equations below are valid

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M90">View MathML</a>

Multiplying the first equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M91">View MathML</a> and the second equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M92">View MathML</a> and adding together, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M93">View MathML</a>

Integrating it from 0 to π, and using the boundary condition (3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M94">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M87">View MathML</a>, the lemma is proved. □

Corollary 3The eigenvalues of the boundary value problem (1)-(3) are real.

The values

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M96">View MathML</a>

(10)

are called the norming constants of the boundary value problem (1)-(3).

Now, let us agree to denote differentiation with respect to λ with a dot <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M97">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M98">View MathML</a>).

Lemma 4The following equality holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M99">View MathML</a>

(11)

Proof Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M100">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M101">View MathML</a>

With the help of (2) and (3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M102">View MathML</a>

Taking into consideration (9) and (10), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M103">View MathML</a>, we arrive (11). □

Corollary 5All zeros of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41">View MathML</a>are simple, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M105">View MathML</a>.

4 Asymptotic formulas of eigenvalues

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M106">View MathML</a> be the solution of equation (1) satisfying the initial conditions (4) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M107">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M108">View MathML</a>

(12)

The eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M109">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M110">View MathML</a>) of the boundary value problem (1)-(3) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M107">View MathML</a> can be found using the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M112">View MathML</a>

from [17] (see also [18]) and can be represented in the following way

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M113">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M114">View MathML</a>.

Lemma 6Roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M109">View MathML</a>of the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M116">View MathML</a>are separated, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M117">View MathML</a>

Proof Assume the contrary. Then there are sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M119">View MathML</a> of zeros of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M120">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M121">View MathML</a>

Since the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M123">View MathML</a> are orthogonal, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M124">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M125">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M126">View MathML</a>

(14)

Let us prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M127">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M128">View MathML</a>. In fact, (12) implies the following estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M129">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M130">View MathML</a> holds uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M131">View MathML</a>. Now, passing to the limit in equality (14), as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M132">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M133">View MathML</a>. This is a contradiction, and it proves the validity of lemma’s statement. □

Lemma 7The eigenvalues of the boundary value problem (1)-(3) are in the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M134">View MathML</a>

(15)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M135">View MathML</a>is a bounded sequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M136">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M137">View MathML</a>.

Proof From (6), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M138">View MathML</a>

(16)

Lemma 1.3.1 in [19] and from [17], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M139">View MathML</a>

(17)

Therefore, for sufficiently large n, on the contours

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M140">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M141">View MathML</a>

By the Rouche theorem, we obtain that the number of zeros of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M142">View MathML</a> inside the contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M143">View MathML</a> coincides with the number of zeros of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M116">View MathML</a>. Furthermore, applying the Rouche theorem to the circle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M145">View MathML</a>, we get that, for sufficiently large n there exists only one zero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43">View MathML</a> of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M148">View MathML</a>. Owing to the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M149">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M150">View MathML</a>

(18)

Substituting (18) into (16), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M151">View MathML</a>

Hence, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M152">View MathML</a>, taking into account the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M153">View MathML</a> and relations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M155">View MathML</a>, integrating by parts and using the properties of the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M29">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M157">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M158">View MathML</a>

Let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M159">View MathML</a>. It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M160">View MathML</a>

can be reduced to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M161">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M162">View MathML</a>. Now, take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M163">View MathML</a>

It is clear from [19] (p.66) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M164">View MathML</a>. By virtue of this we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M165">View MathML</a> (see [20,21]). Therefore, as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M166">View MathML</a>

the validity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M167">View MathML</a> can be seen directly. Lemma is proved. □

5 Expansion formula

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M168">View MathML</a> is not a spectrum point of operator A. Then, there exists resolvent operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M169">View MathML</a>. Let us find the expression of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M170">View MathML</a>.

Lemma 8The resolvent<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M170">View MathML</a>is the integral operator with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M172">View MathML</a>

(19)

Proof To construct the resolvent operator of A, we need to solve the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M173">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M174">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M175">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M176">View MathML</a>. By applying the method of variation of constants, we seek the solution of the problem (20)-(22) in the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M177">View MathML</a>

(23)

and we get the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M179">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M180">View MathML</a>

(24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M181">View MathML</a>

(25)

Substituting (24) and (25) into (23) and taking into account the boundary conditions (21) and (22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M182">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M183">View MathML</a> is as in (19). □

Theorem 9The eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M184">View MathML</a>of the boundary value problem (1)-(3) form a complete system in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M185">View MathML</a>.

Proof With the help of (9) and (11), we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M186">View MathML</a>

(27)

Using (19) and (26), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M187">View MathML</a>

(28)

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M188">View MathML</a> and assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M189">View MathML</a>

(29)

Then from (28), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M190">View MathML</a>. Consequently, for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M191">View MathML</a> the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M192">View MathML</a> is entire with respect to λ.

Let us denote that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M193">View MathML</a>

where δ is a sufficiently small positive number. (16) is valid from Theorem 12.4 in [22] for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194">View MathML</a>.

On the other hand, we can say from Lemma 1.3.1 in [19] that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M195">View MathML</a>, the following relation holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M196">View MathML</a>

(30)

Also, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M197">View MathML</a>, the relations below hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M198">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M199">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M200">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M201">View MathML</a>

(34)

From the estimates (31)-(34), it is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M202">View MathML</a>

(35)

From (26), it follows that for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M149">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M204">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M205">View MathML</a>

Using maximum principle for module of analytic functions and Liouville theorem, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M206">View MathML</a>. From this and the expression of the boundary value problem (20)-(22), we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M207">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M208">View MathML</a>. Thus, we reach the completeness of the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M47">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M210">View MathML</a>. □

Theorem 10If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M176">View MathML</a>, then the expansion formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M212">View MathML</a>

(36)

is valid, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M213">View MathML</a>

and the series converges uniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M20">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M215">View MathML</a>, the series converges in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M216">View MathML</a>, moreover, the Parseval equality holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M217">View MathML</a>

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12">View MathML</a> are the solutions of the boundary value problem (1)-(3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M220">View MathML</a>

(37)

Integrating by parts and taking into account the boundary conditions (2), (3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M221">View MathML</a>

(38)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M222">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M223">View MathML</a>. Let us consider the following contour integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M224">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M225">View MathML</a> is a contour oriented counter-clockwise, and n is a sufficiently large natural number. With the help of Residue theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M226">View MathML</a>

(39)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M227">View MathML</a>

On the other hand, taking into account (38), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M228">View MathML</a>

(40)

Comparing (39) and (40), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M229">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M230">View MathML</a>

Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M231">View MathML</a>

(41)

Now, let us show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M232">View MathML</a>

(42)

From estimates (31)-(34) of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12">View MathML</a> and the inequality (35) for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41">View MathML</a>, it follows, for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M149">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M204">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M238">View MathML</a>

(43)

Let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M239">View MathML</a>. If we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M240">View MathML</a>, and by then integrate by parts the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M241">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M242">View MathML</a>

Hence, similar to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M243">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M244">View MathML</a>

(44)

In general case, let us take an arbitrary fixed number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M245">View MathML</a> and assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M246">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M247">View MathML</a>. Then we can find a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M248">View MathML</a> that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M250">View MathML</a>. Also, using the equation below,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M251">View MathML</a>

and with the help of the estimates of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M255">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M257">View MathML</a>. Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M258">View MathML</a>

From the arbitrariness of ϵ, we reach

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M259">View MathML</a>

(45)

The validity of (42) can be easily seen from (43) and (44). Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M260">View MathML</a>

If we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M261">View MathML</a>

the last equation gives us the expansion formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M262">View MathML</a>

Since the system of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M263">View MathML</a> is complete and orthogonal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M264">View MathML</a>, the Parseval equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M265">View MathML</a>

holds. Extension of the Parseval equality to an arbitrary vector-function of the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M266">View MathML</a> can be carried out by usual methods. □

6 Weyl solution, Weyl function

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M267">View MathML</a> be the solution of equation (1) that satisfies the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M268">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M269">View MathML</a> the solution of equation (1), which satisfies the initial conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M271">View MathML</a>. Then the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M12">View MathML</a> can be represented as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M273">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M274">View MathML</a>

(46)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M275">View MathML</a>

(47)

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M276">View MathML</a>

(48)

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M267">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M278">View MathML</a> are respectively called the Weyl solution and the Weyl function of the boundary value problem (1)-(3). The Weyl function is a meromorphic function having simple poles at points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43">View MathML</a> eigenvalues of boundary value problem (1)-(3). Relations (46), (48) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M280">View MathML</a>

(49)

It can be shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M281">View MathML</a>

(50)

Let us take into consideration a boundary value problem with the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M282">View MathML</a> similar to (1)-(3) and assume that if an element α belongs to boundary value problem (1)-(3), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M283">View MathML</a> belongs to one with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M282">View MathML</a>.

Validity of the equation below can be shown analogously to [8]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M285">View MathML</a>

(51)

Theorem 11The boundary value problem (1)-(3) is identically denoted by the Weyl function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M286">View MathML</a>. (If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M287">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M288">View MathML</a>.)

Proof Let us identify the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M289">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M290">View MathML</a>

(52)

From (50) and (52), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M291">View MathML</a>

(53)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M292">View MathML</a>

(54)

Taking equation (49) into consideration in (54), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M293">View MathML</a>

(55)

Now, from the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M294">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M295">View MathML</a>

we have from equation (55)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M296">View MathML</a>

(56)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M194">View MathML</a>. Now, if we take consideration equation (48) into (53), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M298">View MathML</a>

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M287">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M300">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M301">View MathML</a> are entire functions for every fixed x. It can easily be seen from equation (56) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M303">View MathML</a>. Consequently, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M305">View MathML</a> for every x and λ. Hence, we arrive at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M306">View MathML</a>. □

Theorem 12The spectral data identically define the boundary value problem (1)-(3).

Proof From (51), it is clear that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M286">View MathML</a> can be constructed by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M43">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M309">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M310">View MathML</a>, from Theorem 10, we can say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M287">View MathML</a>. Then from Theorem 11, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/183/mathml/M312">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).

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