Open Access Research

Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions

Sergey A Buterin1, Chung-Tsun Shieh2* and Vjacheslav A Yurko1

Author Affiliations

1 Department of Mathematics, Saratov State University, Astrakhanskaya str. 83, Saratov, 410012, Russia

2 Department of Mathematics, Tamkang University, Tamsui, New Taipei, 25137, Taiwan

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Boundary Value Problems 2013, 2013:180  doi:10.1186/1687-2770-2013-180


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/180


Received:26 June 2013
Accepted:23 July 2013
Published:6 August 2013

© 2013 Buterin et al.; 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

We study the inverse problem for non-selfadjoint Sturm-Liouville operators on a finite interval with possibly multiple spectra. We prove the uniqueness theorem and obtain constructive procedures for solving the inverse problem along with the necessary and sufficient conditions of its solvability and also prove the stability of the solution.

MSC: 34A55, 34B24, 47E05.

Keywords:
non-selfadjoint Sturm-Liouville operators; inverse spectral problems; method of spectral mappings; generalized spectral data; generalized weight numbers

1 Introduction

Inverse spectral problems consist of recovering operators from given spectral characteristics. Such problems play an important role in mathematics and have many applications in natural sciences and engineering (see, for example, monographs [1-7] and the references therein). We study the inverse problem for the Sturm-Liouville operator corresponding to the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M1">View MathML</a> of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M2">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M3">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M4">View MathML</a> is a complex-valued function. The results for the non-selfadjoint operator (1), (2) that we obtain in this paper are crucial in studying inverse problems for Sturm-Liouville operators on graphs with cycles. Here also lies the main reason of considering the case of Dirichlet boundary conditions (2) and arbitrary length T of the interval.

For the selfadjoint case, i.e., when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a> is a real-valued function, the inverse problem of recovering L from its spectral characteristics was investigated fairly completely. As the most fundamental works in this direction we mention [8,9], which gave rise to the so-called transformation operator method having become an important tool for studying inverse problems for selfadjoint Sturm-Liouville operators. The inverse problems for non-selfadjoint operators are more difficult for investigation. Some aspects of the inverse problem theory for non-selfadjoint Sturm-Liouville operators were studied in [10-14] and other papers.

In the present paper, we use the method of spectral mappings [7], which is effective for a wide class of differential and difference operators including non-selfadjoint ones. The method of spectral mappings is connected with the idea of the contour integration method and reduces the inverse problem to the so-called main equation of the inverse problem, which is a linear equation in the Banach space of bounded sequences. We prove the uniqueness theorem of the inverse problem, obtain algorithms for constructing its solution together with the necessary and sufficient conditions of its solvability. In general, by sufficiency one should require solvability of the main equation. Therefore, we also study those cases when the solvability of the main equation can be proved or easily checked, namely, selfadjoint case, the case of finite perturbations of the spectral data and the case of small perturbations. The study of the latter case allows us to prove also the stability of the inverse problem.

In the next section, we introduce the spectral data, study their properties and give the formulation of the inverse problem. In Section 3, we prove the uniqueness theorem. In Section 4 we derive the main equation and prove its solvability. Further, using the solution of the main equation, we provide an algorithm for solving the inverse problem. In Section 5, we obtain another algorithm, which we use in Section 6 for obtaining necessary and sufficient conditions of solvability of the inverse problem and for proving its stability.

2 Generalized spectral data. Inverse problem

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6">View MathML</a> be the spectrum of the boundary value problem (1), (2). In the self-adjoint case, the potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a> is determined uniquely by specifying the classical discrete spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9">View MathML</a> are weight numbers determined by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M10">View MathML</a>

(3)

while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11">View MathML</a> is a solution of equation (1) satisfying the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M12">View MathML</a>

(4)

In the non-selfadjoint case, there may be a finite number of multiple eigenvalues and, hence, for unique determination of the Sturm-Liouville operator, one should specify some additional information. In the present section, we introduce the so-called generalized weight numbers, as was done for the case of operator (1) with Robin boundary conditions (see [11,12]) and study the properties of the generalized spectral data.

Let the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M13">View MathML</a> be a solution of equation (1) under the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M14">View MathML</a>

(5)

For every fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M17">View MathML</a> and their derivatives with respect to x are entire in λ. The eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a> of the problem L coincide with the zeros of its characteristic function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M20">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M21">View MathML</a>. It is known (see, e.g., [2]) that the spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6">View MathML</a> has the asymptotics

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M23">View MathML</a>

(7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M24">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M25">View MathML</a> the multiplicity of the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M27">View MathML</a>) and put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M28">View MathML</a>. Note that by virtue of (7) for sufficiently large n, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M29">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M30">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M31">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M32">View MathML</a>

(8)

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33">View MathML</a> formula (6) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M34">View MathML</a>

(9)

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M35">View MathML</a>

(10)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M37">View MathML</a> are complete systems of eigen- and the associated functions of the boundary value problem L. Together with the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18">View MathML</a> we consider generalized weight numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, determined in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M41">View MathML</a>

(11)

We note that the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9">View MathML</a> for sufficiently large n coincide with the classical weight numbers (4) for the selfadjoint Sturm-Liouville operator.

Definition 1 The numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43">View MathML</a> are called the generalized spectral data of L.

Consider the following inverse problem.

Inverse Problem 1 Given the generalized spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M44">View MathML</a>, find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a>.

Let the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M47">View MathML</a> be solutions of equation (1) under the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M48">View MathML</a>

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M50">View MathML</a> are called the Weyl solution and the Weyl function for L, respectively. According to (6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M51">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M52">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M53">View MathML</a>

(14)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M54">View MathML</a> is the characteristic function of the boundary value problem for the equation (1) with the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M55">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M56">View MathML</a> be its spectrum. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M57">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M58">View MathML</a> is a meromorphic function with poles in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, and zeros in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M62">View MathML</a>. Moreover, (see, e.g., [2])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M63">View MathML</a>

(15)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M64">View MathML</a> and put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M65">View MathML</a>. Using the known method (see, e.g., [3]), one can prove the following asymptotics.

Lemma 1 (i) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M66">View MathML</a>, the following asymptotics holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M67">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M68">View MathML</a>

(17)

uniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>.

(ii) Fix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M70">View MathML</a>. Then for sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M71">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M72">View MathML</a>

(18)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M73">View MathML</a>.

Using (7), (10), (11) and (16), one can calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M74">View MathML</a>

(19)

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M75">View MathML</a>. According to (14), the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M76">View MathML</a> has a representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M77">View MathML</a>

(20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M78">View MathML</a>, and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M79">View MathML</a> is regular in a vicinity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M80">View MathML</a>. The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M81">View MathML</a> is called the Weyl sequence for L. By virtue of (14), (17) and (18), the following estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M82">View MathML</a>

(21)

Moreover, according to (6), (14), (16) and (17), for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M83">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M84">View MathML</a>

(22)

The maximum modulus principle together with (7), (20) and (21) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M85">View MathML</a>

(23)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M86">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M87">View MathML</a> and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M88">View MathML</a>

According to (7) and (23), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M89">View MathML</a>

and hence the series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M90">View MathML</a>

(24)

converges absolutely and uniformly in λ on bounded sets.

Theorem 1The following representation holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M91">View MathML</a>

(25)

Proof Consider the contour integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M92">View MathML</a>

where the contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M94">View MathML</a> has the counterclockwise circuit. According to (7), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M95">View MathML</a> for sufficiently large N and sufficiently small fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M70">View MathML</a>. By virtue of (21), we obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M97">View MathML</a>

uniformly with respect to λ in bounded subsets of ℂ, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M98">View MathML</a>

(26)

On the other hand, using the residue theorem [15], we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M99">View MathML</a>

(27)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M100">View MathML</a>

Further, we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M101">View MathML</a>

Substituting this into (27) and using (26), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M102">View MathML</a>. By virtue of (22), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M103">View MathML</a> and arrive at (25). □

Theorem 2The coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104">View MathML</a>and the generalized weight numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M9">View MathML</a>determine each other uniquely by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M106">View MathML</a>

(28)

Proof Using (10), (14) and (20), one can calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M107">View MathML</a>

(29)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M108">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33">View MathML</a>. Moreover, by virtue of (8) and (10), induction gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M111">View MathML</a>

(30)

Further, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M112">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M113">View MathML</a>

and (4), (5) and (6) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M114">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M115">View MathML</a>

and we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M116">View MathML</a>

Using (8) and (10) and integrating by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M117">View MathML</a>

(31)

Substituting (30) in (31) and taking (11) into account, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M118">View MathML</a>

(32)

Finally, substituting (32) in (29), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M119">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33">View MathML</a>, by induction we obtain (28). □

According to (19) and (28), we have the asymptotics

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M122">View MathML</a>

(33)

Consider the following inverse problems.

Inverse Problem 2 Given the spectra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M124">View MathML</a>, construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a>.

Inverse Problem 3 Given the Weyl function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M76">View MathML</a>, construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a>.

Remark 1 According to (14), (15), (24), (25) and (28), inverse Problems 1-3 are equivalent. The numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M128">View MathML</a> can also be used as spectral data.

3 The uniqueness theorem

We agree that together with L we consider a boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M129">View MathML</a> of the same form but with another potential. If a certain symbol γ denotes an object related to L, then this symbol with tilde <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M130">View MathML</a> denotes the analogous object related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M132">View MathML</a>.

Theorem 3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M135">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M136">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M138">View MathML</a>a.e. on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M139">View MathML</a>. Thus, the specification of the generalized spectral data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M140">View MathML</a>determines the potential uniquely.

Proof By virtue of (7), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M137">View MathML</a>. According to Remark 1, it is sufficient to prove that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M142">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M136">View MathML</a>. Define the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M144">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M145">View MathML</a>

(34)

Using (13) and (34), we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M146">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M147">View MathML</a>

(36)

It follows from (12) and (35) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M148">View MathML</a>

By virtue of (16)-(18), this yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M149">View MathML</a>

(37)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M150">View MathML</a>

(38)

uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>. On the other hand, according to (12) and (35), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M152">View MathML</a>

Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M153">View MathML</a>, then for each fixed x, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M155">View MathML</a> are entire in λ. Together with (37) this yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M157">View MathML</a>. Substituting into (36), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M158">View MathML</a> and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M136">View MathML</a>. □

4 Main equation. Solution of the inverse problem

Let the spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M1">View MathML</a> be given. We choose an arbitrary model boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M162">View MathML</a> (e.g., one can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M163">View MathML</a>). Introduce the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a> by the formulae

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M166">View MathML</a>

(39)

According to (7) and (33), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M167">View MathML</a>

(40)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M168">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M170">View MathML</a> put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M171">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M173">View MathML</a>. Analogously, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M169">View MathML</a>, replacing S with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M180">View MathML</a> in the definitions above.

By the same way as in [2], using (7), (16), (33) and Schwarz’s lemma [15], we get the such estimates as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M184">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M185">View MathML</a>

(41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M186">View MathML</a>

(42)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M187">View MathML</a>

(43)

The analogous estimates are also valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177">View MathML</a>.

Lemma 2The following relation holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M191">View MathML</a>

(44)

where the series converges absolutely and uniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>.

Proof Let real numbers a, b be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196">View MathML</a>. In the λ-plane consider a closed contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M197">View MathML</a> (with a counterclockwise circuit), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M198">View MathML</a>. By the standard method (see [2]), using (12), (35)-(37) and Cauchy’s integral formula [15], we obtain the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M199">View MathML</a>

(45)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M200">View MathML</a>

uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a> and λ on bounded sets. Calculating the integral in (45) by the residue theorem and using (20), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M202">View MathML</a>

for sufficiently large N. Taking the limit in (45) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M203">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M204">View MathML</a>

(46)

Differentiating this with respect to λ, the corresponding number of times and then taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M205">View MathML</a>, we arrive at (44). □

Analogously to (46), one can obtain the following relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M206">View MathML</a>

(47)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M207">View MathML</a>

For each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the relation (44) can be considered as a system of linear equations with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196">View MathML</a>. But the series in (44) converges only with brackets, i.e., the terms in them cannot be dissociated. Therefore, it is inconvenient to use (44) as a main equation of the inverse problem. Below, we will transfer (44) to a linear equation in the Banach space of bounded sequences (see (53)).

Let w be the set of indices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196">View MathML</a>. For each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, we define the vector

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M216">View MathML</a>

(where T is the sign for transposition) by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M217">View MathML</a>

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M219">View MathML</a> are given, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M221">View MathML</a> can be found by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M222">View MathML</a>

(48)

Consider also a block-matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M223">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M224">View MathML</a>

Analogously, we introduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228">View MathML</a> by the replacement of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M230">View MathML</a> in the preceding definitions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177">View MathML</a>, respectively. Using (41) and (43), we get the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M233">View MathML</a>

(49)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M234">View MathML</a>

(50)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M235">View MathML</a>

(51)

Consider the Banach space B of bounded sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M236">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M237">View MathML</a>. It follows from (49) and (50) that for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M238">View MathML</a>, the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228">View MathML</a>, acting from B to B, are linear bounded ones and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M241">View MathML</a>

(52)

Theorem 4For each fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the vector<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M243">View MathML</a>satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M244">View MathML</a>

(53)

in the Banach spaceB, whereIis the identity operator.

Proof We rewrite (44) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M245">View MathML</a>

Substituting here (48), and taking into account our notations, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M246">View MathML</a>

which is equivalent to (53). □

For each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the relation (53) can be considered as a linear equation with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M248">View MathML</a>. This equation is called the main equation of the inverse problem. Thus, the nonlinear inverse problem is reduced to the solution of the linear equation. Let us prove the unique solvability of the main equation.

Theorem 5For each fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M250">View MathML</a>has a bounded inverse operator, namely<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M251">View MathML</a>, i.e., the main equation (53) is uniquely solvable.

Proof Acting in the same way as in Lemma 2 and using (37) and (38), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M252">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M253">View MathML</a>

uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a> and λ, μ on bounded sets. Calculating the integral by the residue theorem and passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M203">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M256">View MathML</a>

According to the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177">View MathML</a>, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M259">View MathML</a>

Further, taking the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M227">View MathML</a> into account, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M262">View MathML</a>

which is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M263">View MathML</a>. Symmetrically, one gets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M264">View MathML</a>

Hence the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M265">View MathML</a> exists, and it is a linear bounded operator. □

Using the solution of the main equation, one can construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a>. Thus, we obtain the following algorithm for solving the inverse problem.

Algorithm 1Let the spectral data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M267">View MathML</a>be given. Then

(i) construct<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, by solving the linear systems (28);

(ii) choose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a>and calculate<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228">View MathML</a>;

(iii) find<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M248">View MathML</a>by solving equation (53);

(iv) choose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33">View MathML</a> (e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M275">View MathML</a>) and construct<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a>by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M277">View MathML</a>

Remark 2 In the particular case, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M134">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M280">View MathML</a> (let for definiteness <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M281">View MathML</a>) according to (44) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M177">View MathML</a>, the main equation becomes the linear algebraic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M284">View MathML</a>

(54)

whose determinant does not vanish for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a> by virtue of Theorem 5.

In the next section for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286">View MathML</a>, we give another algorithm, which is used in Section 6 for obtaining the necessary and sufficient conditions for the solvability of the inverse problem.

5 Algorithm 2

Here and in the sequel, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286">View MathML</a>. It is known that then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M288">View MathML</a> in formulae (7), (19) and (33). We agree that in the sequel one and the same symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M289">View MathML</a> denotes different sequences in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M290">View MathML</a>. Let us choose the model boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M291">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M292">View MathML</a> (for example, one can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M293">View MathML</a>). Then besides (40), according to (7), (33) and (39), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M294">View MathML</a>

(55)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M295">View MathML</a>

(56)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M296">View MathML</a>

(57)

It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M297">View MathML</a>

(58)

Lemma 3The series in (57) converges absolutely and uniformly on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M298">View MathML</a>and allows termwise differentiation. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M299">View MathML</a>is absolutely continuous, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M300">View MathML</a>.

Proof It is sufficient to prove for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196">View MathML</a>. We rewrite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M299">View MathML</a> to the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M305">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M306">View MathML</a>

(59)

It follows from (33), (41) and (55) that the series in (59) converges absolutely and uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M298">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M308">View MathML</a>

Furthermore, using the asymptotic formulae (7), (16) and (33), we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M309">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M310">View MathML</a>. Similarly, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M311">View MathML</a>, and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M312">View MathML</a>. □

Lemma 4The following relation holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M313">View MathML</a>

(60)

Proof Differentiating (46) twice with respect to x and using (57) and (58), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M314">View MathML</a>

Using (1) and (8), we replace here the second derivatives, and then replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11">View MathML</a> using (46). This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M316">View MathML</a>

(61)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M317">View MathML</a>

Using (1) and (8) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M320">View MathML</a>, we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M321">View MathML</a>

Applying this relation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M322">View MathML</a>

which together with (57) and (61) gives (60). □

Thus, we obtain the following algorithm for solving the inverse problem.

Algorithm 2Let the spectral data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M267">View MathML</a>be given. Then

(i) construct<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, by solving the linear systems (28);

(ii) choose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a>so that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M327">View MathML</a>and calculate<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228">View MathML</a>;

(iii) find<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M248">View MathML</a>by solving equation (53), and calculate<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M331">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M318">View MathML</a>, by (48);

(iv) calculate<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a>by formulae (56), (57) and (60).

6 Necessary and sufficient conditions

In the present section, we obtain necessary and sufficient conditions for the solvability of the inverse problem. In the general non-selfadjoint case, they must include the requirement of the solvability of the main equation. In Section 7, some important cases will be considered when the solvability of the main equation can be proved by sufficiency, namely, the selfadjoint case, the case of finite-dimensional perturbations of the spectral data and the case of small perturbations.

Theorem 6For complex numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M335">View MathML</a>to be the spectral data of a certain boundary value problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286">View MathML</a>, it is necessary and sufficient that

(i) the relations (7) and (19) hold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M288">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M339">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33">View MathML</a>;

(iii) (Condition S) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the linear bounded operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M250">View MathML</a>, acting fromBtoB, has a bounded inverse one. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a>is chosen so that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M292">View MathML</a>.

The boundary value problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M1">View MathML</a>can be constructed by Algorithms 1 and 2.

The necessity part of the theorem was proved above; here, we prove the sufficiency. We note that sufficiency condition (ii) of the theorem allows to solve linear systems (28) for finding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a>, which are used for constructing the main equation. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M348">View MathML</a>

(62)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M349">View MathML</a> be the solution of the main equation (53). Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M350">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M351">View MathML</a>

(63)

Similarly to Lemma 1.6.7 in [2] using (51) and (53), one can prove the following assertion.

Lemma 5For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M353">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the following relations hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M355">View MathML</a>

(64)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M356">View MathML</a>

(65)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M357">View MathML</a>

(66)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M358">View MathML</a>

(67)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M359">View MathML</a>

(68)

where Ω is defined in (55) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M360">View MathML</a>

We define the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M209">View MathML</a> by formulae (48), and according to (64), we get (41). Then (44) is also valid. By virtue of (48), (65) and Lemma 5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M362">View MathML</a>

(69)

Furthermore, we construct the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M364">View MathML</a> via (46) and (47) and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a> by formulae (56), (57) and (60). Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M366">View MathML</a>

(70)

Analogously to Lemma 1.6.8 in [2] using (41) and (69), one can prove the following assertion.

Lemma 6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286">View MathML</a>.

Lemma 7For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M370">View MathML</a>the following relations hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M371">View MathML</a>

(71)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M372">View MathML</a>

(72)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M373">View MathML</a>

(73)

Proof (1) According to the estimates (42), the series in (46) is termwise differentiable with respect to x, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M374">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M375">View MathML</a>. By virtue of (70), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M376">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M377">View MathML</a>. Thus, formula (47) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M378">View MathML</a>.

(2) In order to prove (71) and (72), we first assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M379">View MathML</a>

(74)

Differentiating (63) twice, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M380">View MathML</a>

(75)

It follows from (50), (66) and (67) that the series in (75) converges absolutely and uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M382">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M383">View MathML</a>

(76)

Solving the main equation (53), we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M384">View MathML</a>

(77)

According to (50) and (66), the series in (77) converges absolutely and uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>. Further, using (77), we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M386">View MathML</a>

where according to (50), (67), (68) and (76), the series converges absolutely and uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M388">View MathML</a>

On the other hand, it follows from the proof of Lemma 3 and from (74) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M389">View MathML</a>; hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M390">View MathML</a>

Together with (48) this implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M391">View MathML</a>

Using (44), (57) and (60), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M392">View MathML</a>

(78)

Similarly, using (46) and (47), we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M393">View MathML</a>

(79)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M394">View MathML</a>

(80)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M196">View MathML</a>, it follows from (78) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M397">View MathML</a>

and, consequently, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M398">View MathML</a>

(81)

where for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M400">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M401">View MathML</a>

Using (81), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M402">View MathML</a>

(82)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M403">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M404">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M405">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M406">View MathML</a>

(83)

It follows from (50), (74), (82) and (83) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M407">View MathML</a>. Then, by virtue of Condition S in Theorem 6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M408">View MathML</a>, and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M409">View MathML</a>. Thus, we obtain (71).

Furthermore, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M410">View MathML</a>

formula (79) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M411">View MathML</a>

From this, by virtue of (46), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M412">View MathML</a>. Analogously, using (80) we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M413">View MathML</a>. Thus, (71) and (72) are proved for the case when (74) is fulfilled.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M414">View MathML</a>. It follows from (46) and (47) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M415">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M416">View MathML</a>

(84)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M417">View MathML</a>

(85)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M418">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M419">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M173">View MathML</a>. Differentiating (84) with respect to λ an appropriate number of times and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M421">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M422">View MathML</a>

(86)

Let us show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M423">View MathML</a>

(87)

Indeed, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M424">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M425">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M426">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M427">View MathML</a>

(88)

Moreover, according to (9), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M428">View MathML</a>, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M429">View MathML</a>

(89)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M430">View MathML</a> for negative α or β. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M431">View MathML</a> solving the system (89), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M432">View MathML</a>, which together with (88) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M433">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M434">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M435">View MathML</a>, then (11) and (89) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M436">View MathML</a>

(90)

According to (88) and (90), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M437">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M438">View MathML</a>. Moreover, using (88), (90) and (28), we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M439">View MathML</a>

and arrive at (87). Using (86) and (87), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M440">View MathML</a>

Then, by virtue of Condition S, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M441">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M442">View MathML</a>. Substituting this into (85) and using the relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M443">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M172">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M445">View MathML</a>.

(3) Let us now consider the general case when instead of (74) only (55) holds. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M446">View MathML</a>

We agree that if the symbol γ denotes an object constructed with the help of the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M447">View MathML</a>, then the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M448">View MathML</a> denotes the corresponding object, constructed with the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M449">View MathML</a>. Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M450">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M451">View MathML</a>

For each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M450">View MathML</a>, we solve the corresponding main equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M453">View MathML</a>

and construct the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M454">View MathML</a> and the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M455">View MathML</a>. Using Lemma 1.5.1 in [2], one can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M456">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M457">View MathML</a> the solution of equation (1) under the initial conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M458">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M459">View MathML</a>. According to Lemma 1.5.3 in [2], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M460">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M461">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M412">View MathML</a>. Similarly, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M463">View MathML</a>.

Notice that we additionally proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M464">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M465">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M467">View MathML</a> is a spectrum of L. □

Proof of Theorem 6 According to (72) and (73), the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M364">View MathML</a> is the Weyl function for the constructed boundary value problem L. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M469">View MathML</a> so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M470">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M471">View MathML</a>, and put. Differentiating (47) with respect to x and then substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M472">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M473">View MathML</a>

(91)

where the series converges uniformly with respect to λ in bounded sets. From (62) and (91), it follows that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M474">View MathML</a>, the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M18">View MathML</a> is a pole of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M76">View MathML</a> of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M25">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M6">View MathML</a> is the spectrum, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M81">View MathML</a> is the Weyl sequence of L. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8">View MathML</a> are the spectral data of L. □

7 Spacial cases and stability of the solution

The requirement that the main equation is uniquely solvable (Condition S in Theorem 6) is essential and cannot be omitted (see Example 1.6.1 in [2]). Condition S is difficult to check in the general case. We point out three cases, for which the unique solvability of the main equation can be proved or checked.

(1) The selfadjoint case. It is known that in the selfadjoint case, i.e., when the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a> is real-valued, the spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8">View MathML</a> are real numbers, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M483">View MathML</a>

(92)

Let real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43">View MathML</a> having the asymptotics (7) and (19) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M288">View MathML</a> and satisfying (92) be given. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a>, construct <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M228">View MathML</a> and consider the equation (53). Similarly to Lemma 1.6.6 in [2], one can prove the following assertion.

Lemma 8For each fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M250">View MathML</a>, acting fromBtoB, has a bounded inverse operator. Thus, the main equation (53) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M491">View MathML</a>.

By virtue of Theorem 6 and Lemma 8, the following theorem holds.

Theorem 7For real numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M335">View MathML</a>to be the spectral data of a certain selfadjoint boundary value problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286">View MathML</a>, it is necessary and sufficient to satisfy the asymptotics (7) and (19) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M495">View MathML</a>and condition (92).

(2) Finite-dimensional perturbations of the spectral data. Let a model boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a> with the spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M497">View MathML</a> be given. We change a finite subset of these numbers. In other words, we consider numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M280">View MathML</a> for certain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M502">View MathML</a> and arbitrary in the rest. Then for such spectral data, the main equation becomes the linear algebraic system (54), and Condition S is equivalent to the condition that the determinant of this system does not equal zero for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>. Such perturbations are very popular in applications. We note that for the selfadjoint case the determinant of the system (54) is always nonzero.

(3) Local solvability of the main equation. For small perturbations of the spectral data, Condition S is fulfilled automatically. Let us for simplicity consider the case of simple spectra, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M504">View MathML</a>. The following theorem is valid.

Theorem 8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M162">View MathML</a>be given. There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M70">View MathML</a> (which depends on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a>) such that if complex numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43">View MathML</a>satisfy the condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M509">View MathML</a>, then there exists a unique boundary value problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286">View MathML</a>, for which the numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M267">View MathML</a>are the spectral data, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M513">View MathML</a>

(93)

whereCdepends only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a>.

Proof Let C denote various constants, which depend only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M131">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M516">View MathML</a>, the asymptotical formulae (7) and (19) are fulfilled. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M517">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M518">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M33">View MathML</a>. According to (52), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M521">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M522">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M523">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M524">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M15">View MathML</a>. In this case, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M265">View MathML</a>. Thus, all conditions of Theorem 6 are fulfilled, and hence there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M286">View MathML</a>, such that the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M43">View MathML</a> are the spectral data of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M336">View MathML</a>. Moreover, (41) and (69) are valid. Using (57), one can get (93). □

Theorem 8 gives the stability of Inverse Problem 1. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M530">View MathML</a>

where the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M531">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M19">View MathML</a> are determined by the formulae

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M533">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M534">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M535">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M536">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M537">View MathML</a> for other n. According to (7), (19) and (28), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M538">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M539">View MathML</a>, and hence (93) is equivalent to the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M540">View MathML</a>

Similarly to [2], one can obtain the stability of the solution in the uniform norm and also the necessary and sufficient conditions of the solvability for the inverse problem, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M5">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M542">View MathML</a> or in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/180/mathml/M543">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

This research was supported in part by the Russian Foundation for Basic Research (project 13-01-00134) and Taiwan National Science Council (project 99-2923-M-032-001-MY3).

References

  1. Chadan, K, Colton, D, Paivarinta, L, Rundell, W: An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM, Philadelphia (1997)

  2. Freiling, G, Yurko, VA: Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York (2001)

  3. Levitan, BM: Inverse Sturm-Liouville Problems, Nauka, Moscow (1984) English transl., VNU Sci. Press, Utrecht (1987)

  4. Marchenko, VA: Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev (1977) English transl., Birkhäuser (1986)

  5. Pöschel, J: Trubowitz E. Inverse Spectral Theory, Academic Press, New York (1987)

  6. Yurko, VA: Inverse Spectral Problems for Linear Differential Operators and Their Applications, Gordon & Breach, New York (2000)

  7. Yurko, V: Method of Spectral Mappings in the Inverse Problem Theory, VSP, Utrecht (2002)

  8. Gelfand, IM, Levitan, BM: On the determination of a DE from its spectral function. Izv. Akad. Nauk SSSR, Ser. Mat.. 15, 309–360 English transl. in AMS Transl. (2) 1 (1955) (1951)

  9. Marchenko, VA: Some problems in the theory of a second-order differential operator. Dokl. Akad. Nauk SSSR. 72, 457–460 (1950)

  10. Albeverio, S, Hryniv, R, Mykytyuk, Y: On spectra of non-self-adjoint Sturm-Liouville operators. Sel. Math. New Ser.. 13, 571–599 (2008). Publisher Full Text OpenURL

  11. Buterin, SA: On the reconstruction of a non-selfadjoint Sturm-Liouville operator. Matematika. Mekhanika, pp. 10–13. Saratov University, Saratov (2000) (in Russian)

  12. Buterin, SA: On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval. J. Math. Anal. Appl.. 335, 739–749 (2007). Publisher Full Text OpenURL

  13. Karaseva, TM: On the inverse Sturm-Liouville problem for a non-Hermitian operator. Mat. Sb.. 32(74), 477–484 (in Russian) (1953)

  14. Tkachenko, V: Non-selfadjoint Sturm-Liouville operators with multiple spectra. Interpolation Theory, Systems Theory and Related Topics, pp. 403–414. Birkhäuser, Basel (2002)

  15. Conway, JB: Functions of One Complex Variable, Springer, New York (1995)