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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

On discreteness of spectrum and positivity of the Green’s function for a second order functional-differential operator on semiaxis

Sergey Labovskiy1* and Mário Frengue Getimane2

Author Affiliations

1 Department of Mathematics, Moscow State University of Economics, Statistics and Informatics, Nezhinskaya 7, Moscow, 119501, Russia

2 Instituto Superior de Transportes e Comunicações, Prolong. da Av. Kim Il Sung (IFT/TDM) - Edifício D1, Maputo, 2088, Mozambique

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Boundary Value Problems 2014, 2014:102  doi:10.1186/1687-2770-2014-102

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/102


Received:5 February 2014
Accepted:4 April 2014
Published:7 May 2014

© 2014 Labovskiy and Getimane; 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 credited.

Abstract

We study conditions of discreteness of spectrum of the operator defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M2">View MathML</a>. The operator has two singularities at the ends of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3">View MathML</a>. The second question is positivity of solutions of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4">View MathML</a> under boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6">View MathML</a>. The used abstract scheme is close to the well-known MS Birman’s method in the spectral theory of self-adjoint operators. Conditions for discreteness of spectrum and positivity of the Green’s operator are obtained. The result relates to the MS Birman’s result on the necessary and sufficient condition for discreteness of spectrum of a polar-differential operation. The results may be interesting for researchers in qualitative theory of functional-differential equations and spectral theory of self-adjoint operators.

MSC: 34K08, 34K10, 34K12.

Keywords:
discreteness of spectrum; positive solutions; singular operator

1 Introduction

1.1 Problems and a well-known result

Our first objective is to study the conditions for discreteness of spectruma of the functional-differential operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M7">View MathML</a>

(1.1)

with two singularities: at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M8">View MathML</a> and at infinity. Note that one particular case of the expression (1.1) is the following operator with one deviation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M9">View MathML</a>

The second question is existence of positive solutions of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4">View MathML</a> (see Definition 1.1).

Note the result of Birman [[1], (Chapter 2, Section 29)] for the spectral problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M11">View MathML</a>

(1.2)

where the operation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12">View MathML</a> is called polar-differential operation. This singular spectral problem is usually considered in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a> of functions that are square-integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M14">View MathML</a> with the positive weight ρ. Birman showed that a necessary and sufficient condition of discreteness of spectrum of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M16">View MathML</a>

(1.3)

The singularity at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M8">View MathML</a> is not reflected in this condition. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M18">View MathML</a>, condition (1.3) is not sufficient for discreteness. We impose the second condition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M19">View MathML</a>

(1.4)

The two conditions (1.3) and (1.4) are sufficient for discreteness of the spectrum of (1.2) (Theorem 2.1).b It seems that (1.4) is also necessary one.

Part of this work is a continuation of the research in the articles [2-4].

1.2 Assumptions, notation

Everywhere below, except for the independent appendix, we use the assumptions and notation introduced in this subsection.

The function ρ is assumed to be measurable and positive almost everywhere on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M22">View MathML</a> and satisfying the important condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M23">View MathML</a>

(1.5)

Remark 1.1 Comparing (1.3), (1.4), and (1.5) we see that the key role is played by the properties of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M24">View MathML</a>

(1.6)

Conditions (1.3) and (1.4) can be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M25">View MathML</a>, and condition (1.5) is boundedness of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M26">View MathML</a>. The latter is sufficient (Lemma 4.3) for the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M27">View MathML</a> and for boundedness of the operator T (see, below in this subsection). It is close to a necessary condition: if, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M28">View MathML</a>, then T is unbounded (see Remark 4.3).

Everywhere below the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M29">View MathML</a> is assumed to satisfy the following conditions: it is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M30">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M31">View MathML</a>, it is measurable in x for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M30">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M33">View MathML</a>

(1.7)

is locally integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3">View MathML</a>. We can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M35">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M31">View MathML</a> (as follows from a property of the Stieltjes integral).

Now, let us introduce the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M37">View MathML</a>

(1.8)

Finally, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a> be the Hilbert space of all square-integrable with positive almost everywhere weight ρ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M22">View MathML</a> functions f, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M40">View MathML</a>, and with scalar product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M41">View MathML</a>

W be the set of all locally absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M14">View MathML</a> functions u satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M43">View MathML</a>

(1.9)

and the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5">View MathML</a>; W is a Hilbert space (Lemma 4.1) with scalar product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M45">View MathML</a>

(1.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M46">View MathML</a> be the operator defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M31">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M27">View MathML</a> under condition (1.5) (Lemma 4.3).

1.3 About domain of the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12">View MathML</a> and ℒ

If λ is a regular value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12">View MathML</a>, then any solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M52">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M53">View MathML</a>. So, we have to assume that the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M54">View MathML</a> of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12">View MathML</a> consists of all solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M56">View MathML</a>

(1.11)

These solutions have locally on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3">View MathML</a> absolutely continuous derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M58">View MathML</a> and have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M59">View MathML</a>

(1.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M60">View MathML</a>. In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M61">View MathML</a> the integral is convergent, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M62">View MathML</a>. From this we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M63">View MathML</a>

We impose the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6">View MathML</a>. Both of these conditions are necessary for the implementation of the variational method in the space of W. Then the domain is determined automatically (see Corollary A.1 to Lemma A.1 and Lemma 4.8).

Definition 1.1 We say that the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M66">View MathML</a>

(1.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M67">View MathML</a>

(1.14)

is positively solvable, if it is uniquely solvable for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68">View MathML</a>, and the implication <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M69">View MathML</a> holds.

Note that positive solvability is equivalent to the positivity of the Green’s function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M70">View MathML</a>, which allows one to represent the solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4">View MathML</a> in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M72">View MathML</a>

(1.15)

or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M73">View MathML</a>, where G is the Green’s operator of ℒ.

2 Discreteness of spectrum

2.1 Operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12">View MathML</a>

Instead of direct investigation of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M75">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68">View MathML</a>, consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M77">View MathML</a>

(2.1)

Equation (2.1) is the result of the variational method, in which the primary object is the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M78">View MathML</a>. Equation (2.1) has the short form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M79">View MathML</a>

defined on the spaces W and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M80">View MathML</a>. Note that W is a Hilbert space (Lemma 4.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M46">View MathML</a> is bounded (Lemma 4.3), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a>. So, we can use the abstract scheme in Appendix 1. According to Corollary A.1 (2.1) is equivalent to an equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M75">View MathML</a>. By virtue of Lemma 4.8 (2.1) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M85">View MathML</a>

(2.2)

Theorem 2.1Suppose conditions (1.3) and (1.4) hold. Then the spectral problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M86">View MathML</a>

(2.3)

has a discrete spectrum.

Proof The discreteness follows from Theorem A.1 and Lemma 4.7. □

Remark 2.1 (Estimate of the greatest lower bound of the spectrum)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87">View MathML</a> be the greatest lower bound of spectrum of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M12">View MathML</a>. Thus the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M89">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M90">View MathML</a> is uniquely resolvable for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68">View MathML</a>, but it is not if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M92">View MathML</a>. Then in view of (A.4) from inequality (4.3) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M93">View MathML</a>

This is an accurate estimate. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M94">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M95">View MathML</a>, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M96">View MathML</a> is a point of the spectrum, as follows from Example 2.1.

Example 2.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M97">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M98">View MathML</a>, and the operator T is bounded, but the two conditions (1.3) and (1.4) are not fulfilled. The spectrum of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M99">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M101">View MathML</a> is the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M102">View MathML</a>.

In fact, the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M103">View MathML</a> is regular (Remark 2.1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M104">View MathML</a>.

By means of the change of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M105">View MathML</a> the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M106">View MathML</a>

can be transformed to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M107">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M108">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M109">View MathML</a>

this equation has to be considered in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M110">View MathML</a>.

The homogeneous equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M111">View MathML</a> has the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M112">View MathML</a> for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M113">View MathML</a>. It is not in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M110">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M115">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M116">View MathML</a>

If, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M118">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M120">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M121">View MathML</a> the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M122">View MathML</a> may be defined arbitrarily), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M123">View MathML</a>. But the corresponding solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M124">View MathML</a>. Thus, λ is not a regular value of the operator.

Since the spectrum is a real closed set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M102">View MathML</a> is the spectrum.

2.2 General operator ℒ

The operator (1.1) can be represented as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M126">View MathML</a>, where Q is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M127">View MathML</a>

(2.4)

The operator Q acts from W to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a> and is bounded under certain conditions (see, for example, (5.6)). Along with the general case, let us consider one special case (deviating operator):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M129">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130">View MathML</a> is assumed to be nonnegative locally integrable function, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M131">View MathML</a> is a measurable function. Note that the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130">View MathML</a> in (2.5) corresponds to the definition (1.7), if represent expression (2.5) in the form (2.4).

Theorem 2.2If conditions (1.3) and (1.4) are fulfilled and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M133">View MathML</a>is bounded, then spectrum ofis discrete. If the functionξdefined by (1.8) is symmetric, i.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M134">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M135">View MathML</a>, then the spectrum is real and the system of eigenfunctions has the orthogonal basis properties in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a>.

Proof Conditions (1.3) and (1.4) are sufficient conditions of compactness of the operator T (Lemma 4.7). The symmetry condition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137">View MathML</a> allows one to show the identity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M138">View MathML</a> (see Section 5). Now we can refer to Theorem A.3. □

Using the estimate (5.6) from Theorem A.3 we have the first main result.

Theorem 2.3Suppose (1.3) and (1.4) hold and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M139">View MathML</a>

(2.6)

Then the spectral problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M140">View MathML</a>

has a discrete spectrum. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M134">View MathML</a>,cthe spectrum is real, and the system of eigenfunctions has orthogonal basis properties in the spacesWand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a>.

Remark 2.2 The spectrum is not real in general, because of the non-symmetry of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137">View MathML</a>.

The obtained estimate works well in the case of one deviation, if Q is defined by (2.5). From (2.6) we have the following.

Corollary 2.1Suppose (1.3) and (1.4) hold and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M145">View MathML</a>

(2.7)

Then the spectral problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M146">View MathML</a>

has a discrete spectrum.

Example 2.2 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M147">View MathML</a> the operator ℒ has the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M148">View MathML</a>

If (1.3) and (1.4) are satisfied and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M149">View MathML</a>

(2.8)

then the operator ℒ has discrete spectrum. In this case (2.6) has the form (2.8). It follows also from (5.4). Note that the inequality (2.8) is satisfied, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130">View MathML</a> is bounded.

3 Positive solvability

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M133">View MathML</a> is bounded. By the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M152">View MathML</a> the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4">View MathML</a> is reduced to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M154">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M155">View MathML</a> is an integral operator with nonnegative kernel,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M156">View MathML</a>

Thus, if spectral radius of K is less than unit (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M157">View MathML</a>), then (3.1) is uniquely resolvable and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M158">View MathML</a>

Sinced<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M160">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M161">View MathML</a> is positive, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M162">View MathML</a>. Thus:

Theorem 3.1SupposeQis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M157">View MathML</a>. Then the boundary value problem (1.13), (1.14) is uniquely resolvable for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68">View MathML</a>and the Green’s operatorGis positive.

Remark 3.1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M165">View MathML</a> the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M166">View MathML</a> is sufficient for positivity of the Green’s operator G.

The second main result is presented in the following statement.

Theorem 3.2If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M167">View MathML</a>

then the boundary value problem (1.13), (1.14) is positively solvable (see Definition 1.1).

Proof See Remark 3.1 to Theorem 3.1 and the estimates (4.2) and (5.6). □

Consider the following particular case.

Corollary 3.1The equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M168">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M169">View MathML</a>) has a positive solution inWfor any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M172">View MathML</a>, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M173">View MathML</a>

The following particular case shows that in this estimate the inequality sign < cannot be replaced by ≤.

Example 3.1 From Theorem 3.1 and (5.4) it follows that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M174">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M175">View MathML</a>) is uniquely resolvable for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68">View MathML</a> in W and has positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M171">View MathML</a>, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M178">View MathML</a>

In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M179">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M181">View MathML</a>. Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M182">View MathML</a>, the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M183">View MathML</a> has unique positive solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M172">View MathML</a>.

But if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M187">View MathML</a>, the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M188">View MathML</a> may not have a solution for some f (this was considered in Example 2.1).

In concluding this section consider one useful assertion. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M189">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M190">View MathML</a>

(3.3)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M191">View MathML</a> (see the equality (5.2)).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87">View MathML</a> be the greatest lower bound of spectrum of the operator ℒ. Then by (A.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M193">View MathML</a>

From Theorem A.5 follows the following.

Theorem 3.3Suppose conditions (1.3) and (1.4) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87">View MathML</a>is the smallest eigenvalue of the problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M196">View MathML</a>, and the following statements are equivalent:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197">View MathML</a>is positive definite,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198">View MathML</a>,

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M199">View MathML</a>.

Remark 3.2 We do not suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M200">View MathML</a> (but if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M199">View MathML</a> it is so).

4 Auxiliaries propositions

Recall that in all assertions below condition (1.5) is assumed to be fulfilled.

Lemma 4.1Wis a Hilbert space.

Proof The relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M202">View MathML</a> establishes a bijection between W and the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M203">View MathML</a>. □

Lemma 4.2The value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M204">View MathML</a>satisfies the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M205">View MathML</a>

(4.1)

Proof This follows from the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M206">View MathML</a>

 □

Remark 4.1 The estimate (4.1) is accurate, since if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M208">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M208">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M211">View MathML</a>

Remark 4.2 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M214">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M215">View MathML</a>

In fact, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M216">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M217">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M219">View MathML</a>.

Lemma 4.3Suppose (1.5) holds. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M220">View MathML</a>, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M46">View MathML</a>is bounded, and the norm ofTsatisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M222">View MathML</a>

(4.2)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M223">View MathML</a>, we can estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M224">View MathML</a>

From this and (4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M225">View MathML</a>

(4.3)

 □

Remark 4.3 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M226">View MathML</a> (see (1.6)), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M227">View MathML</a>. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M228">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M229">View MathML</a>. It is possible to find a nonincreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M230">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M231">View MathML</a>

Now find u such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M232">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M233">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M213">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M58">View MathML</a> is nonincreasing, by Remark 4.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M237">View MathML</a>. But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M238">View MathML</a>.

Lemma 4.4The image<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82">View MathML</a>is dense in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M13">View MathML</a>.

Proof The proof is left to the reader. □

The following theorem [[5], p.318] can be used to show compactness.

Theorem 4.1 (Gelfand)

A setEfrom a separable Banach spaceXis relatively compact if and only if for any sequence of linear continuous functionals that converge to zero at each point

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M241">View MathML</a>

(4.4)

the convergence (4.4) would be uniform on theE.

Lemma 4.5Suppose (1.3) holds. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M242">View MathML</a>is bounded, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M243">View MathML</a>

uniformly on the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M244">View MathML</a>.

Proof Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M245">View MathML</a>

it is sufficient to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M246">View MathML</a>

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M248">View MathML</a>

The first term tends to zero because of the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M249">View MathML</a>

and (1.3). The second term is equal to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M250">View MathML</a>

This tends to zero because of (4.1) and (1.3). □

Lemma 4.6Suppose (1.4) holds. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M242">View MathML</a>is bounded, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M252">View MathML</a>

uniformly on the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M244">View MathML</a>.

Proof Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M254">View MathML</a>

it is sufficient to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M255">View MathML</a>

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M256">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M258">View MathML</a>

Now we refer to (1.4) and (4.1). □

Lemma 4.7If (1.3) and (1.4) hold, thenTis compact.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M259">View MathML</a>. We use the criterium of compactness of Gelfand (see Theorem 4.1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M242">View MathML</a> be a sequence of functionals such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M261">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M262">View MathML</a>. We have to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M263">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M265">View MathML</a>. Using Lemma 4.5 choose N such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M266">View MathML</a>

for all n and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247">View MathML</a>. The same we can do with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M268">View MathML</a> for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M269">View MathML</a>. For this aim we can use Lemma 4.6.

Now we only have to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M270">View MathML</a> uniformly on the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M272">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M273">View MathML</a>, it suffices to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M274">View MathML</a>. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M275">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M276">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M277">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M278">View MathML</a>. The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M279">View MathML</a> is compact. Thus by Theorem 4.1 of Gelfand <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M280">View MathML</a> converges to zero uniformly for these s. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M281">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M282">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M263">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M247">View MathML</a>. □

Remark 4.4 It seems that condition (1.4) is necessary for compactness of T.

Lemma 4.8Equation (2.1) is equivalent to problem (2.2).

Proof Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M285">View MathML</a>. From (2.1) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M286">View MathML</a>

(4.5)

(if we choose v such that the corresponding limits exist). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M119">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M288">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M289">View MathML</a>. From this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M290">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M291">View MathML</a>. Since the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M291">View MathML</a> is arbitrary, the relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M293">View MathML</a> is fulfilled on the whole semiaxis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M295">View MathML</a>. By the first equality in (4.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M296">View MathML</a>

Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M297">View MathML</a>. Choosing v such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M298">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6">View MathML</a>. □

5 Operator Q. Symmetry and estimates of the norm

Here we consider the operator Q defined by (2.4).

5.1 Symmetry of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M300">View MathML</a>

Under the assumptions imposed on the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137">View MathML</a> in Section 1.2, from Lemma B.1 (in Appendix 2) we obtain the following statement.

Lemma 5.1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M303">View MathML</a>

(5.1)

In this case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M306">View MathML</a>.

Using Lemma 5.1 the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M307">View MathML</a>

can be represented in the symmetrical form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M308">View MathML</a>

(5.2)

Hence, this form is symmetric if the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137">View MathML</a> is symmetric: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M310">View MathML</a>.

5.2 Unique deviation

Consider first the special case when the operator Q is defined by (2.5), i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M311">View MathML</a>. Using (4.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M312">View MathML</a>

From this follows the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M313">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M314">View MathML</a>

(5.3)

In particular, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M147">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M316">View MathML</a>

(5.4)

5.3 General operator Q

Here we consider the general case of the operator Q defined by (2.4), i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M317">View MathML</a>. Suppose that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M318">View MathML</a> (see (1.8)) is absolutely continuous in y, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M319">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M320">View MathML</a> does not decrease in x.

In this case

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M321">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M130">View MathML</a> is defined by (1.7). Using Lemma 5.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M323">View MathML</a>

The latter step can be done in the same manner as in the relation (5.4).

From this

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M324">View MathML</a>

(5.5)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M325">View MathML</a>

it may be presented in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M326">View MathML</a>

(5.6)

This estimate only works well if for all x deviation is concentrated around the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M131">View MathML</a>. For example, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M330">View MathML</a> then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M331">View MathML</a>

This estimate coincides with (5.3), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M332">View MathML</a>

In this case

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M333">View MathML</a>

Appendix 1: Abstract scheme

We do not use general spectral theory (see, for example, [6,7]). But the scheme below is close to the scheme in [[7], Chapter 10] except for using a different notation. We find also convenient explicit use of the embeddingT from W to H (see below). This scheme was used also in [2-4].

A.1 Positive form

Let W and H be Hilbert spaces with inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M335">View MathML</a>, respectively. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M336">View MathML</a> be a linear bounded operator. The equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M337">View MathML</a>

(A.1)

has the unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M338">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M339">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340">View MathML</a> is adjoint operator. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M341">View MathML</a>.

Assume that

1. the image <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82">View MathML</a> of the operator T is dense in H,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M343">View MathML</a>.

Lemma A.1If the image<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82">View MathML</a>of the operatorTis dense inH, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340">View MathML</a>is an injection.

Proof Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M346">View MathML</a> for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M339">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M348">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M349">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M82">View MathML</a> is dense in H, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M351">View MathML</a>. □

Corollary A.1 (Euler equation)

The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340">View MathML</a>has an inversedefined on the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M353">View MathML</a>. Equation (A.1) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M354">View MathML</a>

(A.2)

The spectral problem for the operator ℒ we will write in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M355">View MathML</a>

(A.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87">View MathML</a> be the greatest lower bound of the spectrum of ℒ. It is well known (see, for example, [[7], Chapter 6]) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M357">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M358">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M359">View MathML</a>

(A.4)

Theorem A.1The spectrum ofis discrete if and only ifTis compact.

Proof Since (A.3) is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M360">View MathML</a>, discreteness of spectrum of ℒ is equivalent to compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M361">View MathML</a>. But the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M361">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M340">View MathML</a> are compact at the same time [[7], Chapter 10]. □

Theorem A.2SupposeTis compact. Then (A.3) has a nonzero solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364">View MathML</a>only in the case of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M366">View MathML</a> , i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M367">View MathML</a>

The system<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364">View MathML</a>forms an orthogonal basis inW. The sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M369">View MathML</a>forms a nondecreasing sequence of positive numbers,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M370">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M371">View MathML</a>.

Remark A.1 The minimal eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M87">View MathML</a> satisfies the equality (A.4).

A.2 General case

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M373">View MathML</a>

(A.5)

be a symmetric bilinear form, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M374">View MathML</a>. Assume that Q is bounded in both arguments. Moreover, suppose that this form has the representation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M191">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M376">View MathML</a> is bounded.e Then the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M379">View MathML</a>, is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M380">View MathML</a>

(A.6)

and in the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M381">View MathML</a> to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M382">View MathML</a>

(A.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M383">View MathML</a>.

Theorem A.3SupposeTis compact. Then the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M384">View MathML</a>

(A.8)

has a nonzero solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364">View MathML</a>only in the case of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M366">View MathML</a> , i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M388">View MathML</a>

The system<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M364">View MathML</a>can be chosen to form an orthogonal basis in the spaceW.

If the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M390">View MathML</a> is lower semi-bounded, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M391">View MathML</a>

(A.9)

then spectrum of ℒ is semi-bounded, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M392">View MathML</a>

(A.10)

is the greatest lower bound of the spectrum [[7], Chapter 6]. Thus we have the following.

Theorem A.4If (A.9) holds andTis compact then the eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M369">View MathML</a>have a minimum and can be put in increasing order

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M394">View MathML</a>

A.3 Positive definiteness and spectral radius of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M395">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M396">View MathML</a> be the spectral radius of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M395">View MathML</a>. Note that the two operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M395">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M399">View MathML</a> have the same spectral radius. So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M400">View MathML</a>

(A.11)

Lemma A.2The quadratic form<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197">View MathML</a>is positive definite if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402">View MathML</a>.

Proof From (A.11) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M403">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M404">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197">View MathML</a> is positive definite. Conversely from the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M407">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M265">View MathML</a>) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M409">View MathML</a>. □

So if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402">View MathML</a>, then by (A.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M411">View MathML</a>

Conversely, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M413">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M414">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M415">View MathML</a> is nonnegative, then spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M399">View MathML</a> is in the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M417">View MathML</a>, and ρ is a point of spectrum. In this case from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198">View MathML</a> it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M402">View MathML</a>. So, we have the following.

Theorem A.5Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M415">View MathML</a>is nonnegative. The following assertions are equivalent:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M197">View MathML</a>is positive definite,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M198">View MathML</a>,

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M199">View MathML</a>.

Appendix 2: A generalization of the Fubini theorem

We used a change of integration order in an integral, which does not follow from the classic Fubini theorem. The following assertion is taken from the monograph [8]. Note that it was used without proof in [2-4].

Lemma B.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M424">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M425">View MathML</a>be measurable spaces, μbe a measurefon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M424">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M431">View MathML</a>be kernel (i.e. forμ-almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M432">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M433">View MathML</a>is a measure on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M425">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M435">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M436">View MathML</a>isμ-measurable onX). Then

1. the functionνdefined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M437">View MathML</a>by the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M438">View MathML</a>

is measure;

2. if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M439">View MathML</a>isν-measurable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M440">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M441">View MathML</a>

Remark B.1 The function ν is the Lebesgue extension from the set of rectangles

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M442">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by SL. He prepared Sections 1-3 and the Appendix. MFG prepared Sections 4-5. Both authors read and approved the final manuscript

Acknowledgements

The authors are grateful to Isaac V Shragin for pointing to the source [8], containing Lemma B.1 (it was used in [2] and in the other previous articles). The authors thank the reviewers for important comments.

End notes

  1. The spectrum of ℒ is discrete if it consists only of eigenvalues of finite multiplicity.

  2. The condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M20">View MathML</a> is sufficient for (1.4), but it is not necessary (if, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M21">View MathML</a> for x near zero).

  3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M137">View MathML</a> is defined by (1.8).

  4. Here I is the identity operator.

  5. We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M377">View MathML</a> the form associated with the operator Q.

  6. The measure is a nonnegative, σ-additive function defined on a σ-algebra; the product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M426">View MathML</a> is the minimal σ-algebra containing the set of all rectangles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M427">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M428">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M429">View MathML</a>.

References

  1. Glazman, IM: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translation, Jerusalem (1965)

  2. Labovskij, S: On the Sturm-Liouville problem for a linear singular functional-differential equation. Russ. Math.. 40(11), 50–56 (1996) (Russian original: Izv. Vysš. Učebn. Zaved., Mat. 11(414), 48-53 (1996)). Zbl 0909.34070

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