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

Open Access Research

Fredholm alternative for the second-order singular Dirichlet problem

Alexander Lomtatidze12 and Zdeněk Opluštil2*

Author Affiliations

1 Institute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Žižkova 22, Brno, 616 62, Czech Republic

2 Faculty of Mechanical Engineering, Institute of Mathematics, Brno University of Technology, Technická 2, Brno, 616 69, Czech Republic

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Boundary Value Problems 2014, 2014:13  doi:10.1186/1687-2770-2014-13


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/13


Received:13 September 2013
Accepted:19 November 2013
Published:13 January 2014

© 2014 Lomtatidze and Oplu¿til; licensee Springer.

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

Abstract

Consider the singular Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M2">View MathML</a> are locally Lebesgue integrable functions. It is proved that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M3">View MathML</a>

then the Fredholm alternative remains true.

MSC: 34B05.

Keywords:
singular Dirichlet problem; Fredholm alternative

1 Introduction

Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M4">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M5">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M6">View MathML</a>. We are mainly interested in the case when the functions p and q are not (in general) integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M7">View MathML</a>. In this case, equation (1) as well as problem (1), (2) are said to be singular. It is well known that for singular problem (1), (2), the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M8">View MathML</a>

(3)

guarantees the validity of the Fredholm alternative. More precisely, if (3) holds, then problem (1), (2) is uniquely solvable for any q satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M9">View MathML</a>

(4)

iff the corresponding homogeneous equation

has no nontrivial solution satisfying (2). The above statement plays an important role in the theory of singular problems; however, it does not cover many interesting, even rather simple, equations. For example, consider the Dirichlet problem for the Euler equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M11">View MathML</a>

(5)

where α and β are real constants. By direct calculations, one can easily verify that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M12">View MathML</a>, then the homogeneous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M13">View MathML</a>

has only the trivial solution, while problem (5) is uniquely solvable. However, in this case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M14">View MathML</a> and therefore condition (3) is not satisfied.

The aim of this paper is to show that the Fredholm alternative remains true even in the case when instead of (3) only the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M15">View MathML</a>

(6)

holds. The paper is organized as follows. At the end of this section, we state our main results, the proofs of which one can find in Section 4. In Section 2, we recall some known results in a suitable for us form. Section 3 is devoted to a priori estimates and plays a crucial role in the proofs of the main results.

Throughout the paper we use the following notation.

ℝ is the set of real numbers.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M16">View MathML</a>, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M17">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M18">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M19">View MathML</a> is the set of continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M20">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M21">View MathML</a>, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M22">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M23">View MathML</a> is the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M24">View MathML</a>, which are absolutely continuous together with their first derivative on every closed subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M26">View MathML</a> is the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M27">View MathML</a>, which are Lebesgue integrable on every closed subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25">View MathML</a>.

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M29">View MathML</a> (resp., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M30">View MathML</a>) we denote the right (resp., left) limit of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M31">View MathML</a> at the point a (resp., b).

Under a solution of equation (1) we understand a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M32">View MathML</a> which satisfies it almost everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M33">View MathML</a>. A solution of equation (1) satisfying (2) is said to be a solution of problem (1), (2).

We say that a certain property holds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25">View MathML</a> if it takes place on every closed subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M25">View MathML</a>.

Recall that we consider problem (1), (2), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M6">View MathML</a>. Our main results are the following.

Theorem 1.1Let condition (6) hold. Then problem (1), (2) is uniquely solvable for anyqsatisfying (4) iff homogeneous problem (10), (2) has no nontrivial solution.

Remark 1.1 In Theorem 1.1, condition (4) is essential and cannot be omitted. Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M39">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M40">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M41">View MathML</a>

(7)

Evidently, (6) holds and problem (10), (2) has no nontrivial solution. On the other hand, a general solution of (1) is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M42">View MathML</a>

However, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M43">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M44">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M45">View MathML</a>

Therefore, in view of (7), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M46">View MathML</a> and, consequently, problem (1), (2) has no solution.

Remark 1.2 Theorem 1.1 concerns half homogeneous problem (1), (2) and does not remain true for the fully nonhomogeneous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M47">View MathML</a>

(8)

Let, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M50">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M51">View MathML</a>. It is clear that (6) holds and the corresponding homogeneous problem (10), (2) has no nontrivial solution. On the other hand, a general solution of (1) is of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M52">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M40">View MathML</a> and, therefore, (8) has no solution.

Theorem 1.2Let (6) hold and problem (10), (2) have no nontrivial solution. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M54">View MathML</a>such that for anyqsatisfying (4), the solutionuof problem (1), (2) admits the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M55">View MathML</a>

(9)

Consider now a sequence of equations

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M57">View MathML</a> are such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M58">View MathML</a>

(10)

Let, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M38">View MathML</a> satisfy (4) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M60">View MathML</a>

(11)

Corollary 1.1Let (4), (6) hold and problem (10), (2) have no nontrivial solution. Let, moreover, (11) and (12) be fulfilled. Then the problems (1), (2) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M61">View MathML</a>), (2) have unique solutionsuand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M62">View MathML</a>, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M63">View MathML</a>

(12)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M64">View MathML</a>

(13)

2 Auxiliary statements

In this section, we consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M65">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M66">View MathML</a>, q satisfies (4), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M67">View MathML</a>

(14)

Below we state some known results in a suitable for us form.

Proposition 2.1Let (15) hold. Then the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M68">View MathML</a>

is uniquely solvable for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M69">View MathML</a>andqsatisfying (4) iff the homogeneous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M70">View MathML</a>

has no nontrivial solution.

Proof See, e.g., [[1], Theorem 3.1] or [[2], Theorem 1.1]. □

Proposition 2.2Let (15) hold. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M72">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M73">View MathML</a>satisfying either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M74">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M75">View MathML</a>, the homogeneous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M76">View MathML</a>

(15)

has no nontrivial solution. Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M77">View MathML</a> (where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M73">View MathML</a>are the same as above) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M79">View MathML</a>

the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M80">View MathML</a>

holds.

Proof In view of (15), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M72">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M83">View MathML</a>

Hence, the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M84">View MathML</a>

hold as well. The latter inequalities, by virtue of [[2], Lemma 4.1], imply that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M73">View MathML</a> satisfying either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M74">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M75">View MathML</a>, homogeneous problem (16) has no nontrivial solution.

The second part of the proposition follows easily from the above-proved part and [[2], Lemma 1.3]. □

Proposition 2.3Let (15) hold. Let, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M72">View MathML</a>be from the assertion of Proposition 2.2. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M90">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M91">View MathML</a>and anyqsatisfying (4), the solutionvof the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M92">View MathML</a>

(16)

admits the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M93">View MathML</a>

(17)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M94">View MathML</a>, while the solutionvof the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M95">View MathML</a>

(18)

admits the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M96">View MathML</a>

(19)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M97">View MathML</a>.

Proof By virtue of (15) and [[1], Lemma 2.2], the initial value problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M98">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M99">View MathML</a>

have unique solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M101">View MathML</a>, respectively, and the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M102">View MathML</a>

(20)

are fulfilled, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M103">View MathML</a>

On the other hand, by virtue of Proposition 2.2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M104">View MathML</a>

In view of Propositions 2.1 and 2.2, problem (17) has a unique solution v. By direct calculations, one can easily verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M105">View MathML</a>

(21)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M106">View MathML</a>. Analogously, the (unique) solution v of problem (19) is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M107">View MathML</a>

(22)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M108">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M110">View MathML</a> are solutions of the problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M111">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M112">View MathML</a>

respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M114">View MathML</a>, and the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M115">View MathML</a>

(23)

are fulfilled with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M116">View MathML</a>

Now, it follows from (22) and (23), in view of (21) and (24), that the estimates (18) and (20) hold with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M117">View MathML</a>

 □

3 Lemmas on a priori estimates

Lemma 3.1Let (4) and (6) hold. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M118">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M119">View MathML</a>, every solutionuof equation (1) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M120">View MathML</a>

(24)

admits the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M121">View MathML</a>

(25)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M122">View MathML</a>. Then it is clear that either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M123">View MathML</a>

(26)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M124">View MathML</a>

(27)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M125">View MathML</a>

(28)

Assume that (27) (resp., (28)) holds. Then, in view of (25), there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M126">View MathML</a> (resp., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M127">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M128">View MathML</a>

(29)

Multiplying both sides of (1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M129">View MathML</a> (resp., by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M130">View MathML</a>) and integrating it from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M131">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M132">View MathML</a> (resp., from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M133">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M131">View MathML</a>), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M135">View MathML</a>

Hence, in view of (30), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M136">View MathML</a>

Multiplying both parts of the latter inequality by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M137">View MathML</a> (resp., by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M138">View MathML</a>), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M139">View MathML</a>

(30)

Suppose now that (29) holds. Then either there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M140">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M141">View MathML</a>

(31)

or there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M142">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M143">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M144">View MathML</a>

(33)

If (32) holds, then evidently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M145">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M146">View MathML</a> and, consequently, (31) is fulfilled. On the other hand, if (34) holds, then, by virtue of the above-proved, the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M147">View MathML</a>

are fulfilled, and therefore, in view of (33), inequality (31) holds as well. Thus, estimate (26) is fulfilled. □

Lemma 3.2Let (6) hold. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M149">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M90">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M152">View MathML</a>and anyqsatisfying (4), every solutionuof equation (1) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M153">View MathML</a>

(34)

admits the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M154">View MathML</a>

(35)

while every solutionuof equation (1) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M155">View MathML</a>

(36)

admits the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M156">View MathML</a>

(37)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M158">View MathML</a>, and ϱ be from the assertion of Propositions 2.2 and 2.3 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M159">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M160">View MathML</a>. Let, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M151">View MathML</a> (resp., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M152">View MathML</a>) and u be a solution of problem (1), (35) (resp., (1), (37)). By virtue of Propositions 2.2 and 2.3, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M163">View MathML</a>

(38)

has a unique solution v and, moreover, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M94">View MathML</a> (resp., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M97">View MathML</a>), the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M166">View MathML</a>

(39)

holds. Let us show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M167">View MathML</a>

(40)

Assume the contrary, let (41) be violated. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M168">View MathML</a>

Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M170">View MathML</a> (resp., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M172">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M173">View MathML</a>

(41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M174">View MathML</a>

(42)

In view of (1), (39), and (42), it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M175">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M176">View MathML</a>

Hence, by virtue of (43) and Proposition 2.2, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M177">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M178">View MathML</a>, which contradicts (42). Therefore, (41) is fulfilled. The estimate (36) (resp., (38)) now follows from (40) and (41). □

Lemma 3.3Let (6) hold and problem (10), (2) have no nontrivial solution. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M180">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M181">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M182">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M183">View MathML</a>and anyqsatisfying (4), every solutionuof equation (1) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M184">View MathML</a>

admits the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M185">View MathML</a>

Proof Suppose to the contrary that the lemma is not true. Then there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M188">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M189">View MathML</a> such that (11) holds,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M190">View MathML</a>

(43)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M191">View MathML</a>

(44)

Introduce the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M192">View MathML</a>

Then it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M193">View MathML</a>

(45)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M194">View MathML</a>

(46)

Moreover, it follows from (45) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M195">View MathML</a>

(47)

and, consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M196">View MathML</a>

(48)

By virtue of Lemma 3.1, (46), and (47),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M197">View MathML</a>

Hence, in view of (44) and (48), the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M198">View MathML</a> is uniformly bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M33">View MathML</a> and, therefore, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M200">View MathML</a> is equicontinuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M201">View MathML</a>. Taking, moreover, into account (46), by virtue of the Arzelá-Ascoli lemma, we can assume, without loss of generality, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M202">View MathML</a>

(49)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M203">View MathML</a> and, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M204">View MathML</a>

(50)

By a direct calculation, one can easily verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M205">View MathML</a>

whence, in view of (49)-(51), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M206">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208">View MathML</a> is a solution of equation (10).

Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M149">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M90">View MathML</a> be from the assertion of Lemma 3.2. Assume, without loss of generality, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M213">View MathML</a> for any natural n. Then, by virtue of Lemma 3.2, (46), and (47), the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M214">View MathML</a>

(51)

are fulfilled. Moreover, in view of (48), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M215">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M216">View MathML</a>

Taking, moreover, into account (50), we get from (52) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M217">View MathML</a>

and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208">View MathML</a> satisfies the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M219">View MathML</a>

On account of (44) and (48), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M221">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M222">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M223">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M224">View MathML</a>

Then it follows from (52) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M225">View MathML</a>

Hence, in view of (46), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M226">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M227">View MathML</a>. Taking now into account (50), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M228">View MathML</a>, and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208">View MathML</a> is a nontrivial solution of problem (10), (2). However, this contradicts an assumption of the lemma. □

4 Proofs of the main results

Proof of Theorem 1.1 To prove the theorem, it is sufficient to show that if problem (10), (2) has no nontrivial solution, then problem (1), (2) has at least one solution.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M233">View MathML</a>, ϱ, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M234">View MathML</a> be from the assertions of Lemmas 3.2 and 3.3. Let, moreover, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M236">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M237">View MathML</a>

(52)

By virtue of Lemma 3.3, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M238">View MathML</a>

has no nontrivial solution. Hence, by virtue of Proposition 2.1, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M239">View MathML</a>

(53)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M62">View MathML</a>. Moreover, by virtue of Lemma 3.3, the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M241">View MathML</a>

(54)

holds, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M242">View MathML</a>

On the other hand, on account of Lemma 3.1 and (55), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M243">View MathML</a>

(55)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M244">View MathML</a>

In view of (53), (55), and (56), the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M245">View MathML</a> is uniformly bounded and equicontinuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M201">View MathML</a>. Hence, by virtue of the Arzelá-Ascoli lemma, we can suppose, without loss of generality, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M247">View MathML</a>

(56)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M203">View MathML</a> and, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M249">View MathML</a>

(57)

Taking into account (54), one can easily verify, by a direct calculation, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M250">View MathML</a>

Hence, in view of (57) and (58), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M251">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208">View MathML</a> is a solution of equation (1).

Further, by virtue of Lemma 3.2 and (55), the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M254">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M255">View MathML</a>

are fulfilled. Hence, on account of (57), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M256">View MathML</a>

and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M258">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M208">View MathML</a> is a solution of problem (1), (2). □

Proof of Theorem 1.2 According to Theorem 1.1, problem (1), (2) has a unique solution u. By virtue of Lemma 3.3, the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M260">View MathML</a>

holds. On the other hand, it follows from Lemma 3.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M261">View MathML</a>

The latter two inequalities imply (9) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M262">View MathML</a>

 □

Proof of Corollary 1.1 By virtue of Theorem 1.1, problems (1), (2) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M61">View MathML</a>), (2) have unique solutions u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M62">View MathML</a>, respectively. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M265">View MathML</a>

(58)

Then it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M266">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M267">View MathML</a>

(59)

Hence, by virtue of Theorem 1.2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M268">View MathML</a>

Taking now into account (12), (59), and (60), we get (13) and (14). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Published results were supported by the project ‘Popularization of BUT R&D results and support systematic collaboration with Czech students’ CZ.1.07/2.3.00/35.0004 and by Grant No. FSI-S-11-3 ‘Modern methods of mathematical problem modelling in engineering’. Research was also supported by RVO: 67985840.

References

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  2. Kiguradze, IT, Shekhter, BL: Singular boundary value problems for second order ordinary differential equations. J. Sov. Math.. 43(2), 2340–2417 (1988). Publisher Full Text OpenURL