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

Open Access Research

On a certain way of proving the solvability for boundary value problems

Arnold Y Lepin

Author Affiliations

Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia

Boundary Value Problems 2014, 2014:111  doi:10.1186/1687-2770-2014-111

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/111


Received:13 December 2013
Accepted:28 April 2014
Published:13 May 2014

© 2014 Lepin; 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

A certain way of replacing a given boundary value problem by another one, a solution of which solves also the original problem, is considered.

MSC: 34B15.

Keywords:
boundary value problems; upper and lower functions

Research

Consider the solvability of the boundary value problem (BVP)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M1">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M2">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M3">View MathML</a> is strictly increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4">View MathML</a> for fixed t and x, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M5">View MathML</a> satisfies the Caratheodory conditions, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M6">View MathML</a> is measurable in I for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M8">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M9">View MathML</a> for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M10">View MathML</a>, and for any compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M11">View MathML</a> there exists function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M12">View MathML</a> such, that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M13">View MathML</a>, the estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M14">View MathML</a> holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M16">View MathML</a>, α is the lower function, β the upper function.

This boundary value problem is replaced by another one, which is dependent on the parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M18">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M19">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M20">View MathML</a> is strictly increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4">View MathML</a> for fixed t and x, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M5">View MathML</a> satisfies the Caratheodory conditions.

Definition 1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M23">View MathML</a> is a solution of (1), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M24">View MathML</a> is absolutely continuous on I and (1) is satisfied almost everywhere on I.

We provide below definitions of generalized upper and lower functions and the generalized solution along with Theorem 1 from [1-3]. This is needed to prove the main result.

Definition 2 The class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M25">View MathML</a> consists of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M26">View MathML</a>, which possess the property: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M27">View MathML</a> there exist the left derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M28">View MathML</a> and the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M29">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M30">View MathML</a>; for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M31">View MathML</a> there exist the right derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M32">View MathML</a> and the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M33">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M34">View MathML</a>, and, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M36">View MathML</a>.

The class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M37">View MathML</a> consists of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M38">View MathML</a>, which possess the following property: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M39">View MathML</a> there exist the left derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M40">View MathML</a> and the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M41">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M42">View MathML</a>; for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M31">View MathML</a> there exist the right derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M44">View MathML</a> and the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M45">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M46">View MathML</a>, and, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M48">View MathML</a>.

Definition 3 We call a bounded function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M49">View MathML</a> a generalized lower function and write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50">View MathML</a>, if in any interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M51">View MathML</a>, where this function satisfies the Lipschitz condition, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M53">View MathML</a> where the derivative exists, the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M54">View MathML</a>

holds. We will call a bounded function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M55">View MathML</a> a generalized upper function and write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56">View MathML</a>, if in any interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M51">View MathML</a>, where this function satisfies the Lipschitz condition, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M53">View MathML</a> where the derivative exists, the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M60">View MathML</a>

holds.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M61">View MathML</a> will be called a generalized solution, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M62">View MathML</a>.

A generalized solution has a derivative at any point, possibly infinite, either −∞ or +∞, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M64">View MathML</a>; if in some interval the derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M4">View MathML</a> does not attain the values −∞ or +∞, then x is a solution of (1) in this interval.

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M68">View MathML</a>. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M69">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M70">View MathML</a>there exists a generalized solution of the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M71">View MathML</a>

(4)

In addition to conditions on α and β the compactness conditions are needed for solvability of the boundary value problem (1)-(2). The Nagumo condition [4] for φ-Laplacian and the Schrader condition [5] are sufficient conditions for compactness of a set of solutions. We accept the following compactness conditions.

Definition 4 We say that the compactness condition is fulfilled, if for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M73">View MathML</a> any generalized solution of the Dirichlet problem (4) is a solution.

It is clear that this condition is weaker than the Schrader condition.

A set of solutions of the Dirichlet problem (4) will be denoted by S.

Remark 1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M68">View MathML</a> and the compactness condition is fulfilled, then the Dirichlet problem (4) has a solution.

Theorem 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M56">View MathML</a>and the compactness condition be fulfilled. If the boundary value problem (3) has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M79">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M17">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M81">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M82">View MathML</a>

then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M83">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M84">View MathML</a>solves the boundary value problem (1)-(2).

Proof Notice that the results in [6] imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M85">View MathML</a>. Suppose the contrary. Let the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M86">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M88">View MathML</a> tend to infinity. Consider the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M89">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M91">View MathML</a> . We can assume, without loss of generality, that it converges in any rational points of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M92">View MathML</a> to the function u, located between α and β. Notice that without loss of generality for any interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M93">View MathML</a> it follows from the boundedness of u and the Mean Value Formula that there exists an interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M94">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M95">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M97">View MathML</a>, and u satisfy the Lipschitz condition with constant L in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M98">View MathML</a>. The u can be extended by continuity to the entire interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M98">View MathML</a>, and thus we obtain a function u that satisfies the Lipschitz condition. It follows from the Lipschitz condition that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M100">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M102">View MathML</a>. It is clear that the derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M103">View MathML</a> converge to the derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M104">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M102">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> is a solution of (1) in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M98">View MathML</a>. Continuing the construction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> on both sides, one gets a solution of (1) on the maximal interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M109">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M110">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M111">View MathML</a> is either −∞ or +∞. Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M112">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M113">View MathML</a> is either −∞ or +∞. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M115">View MathML</a> is not −∞ or +∞, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> can be continued to a. Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M118">View MathML</a> is not −∞ nor +∞, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> can be continued to b. By repeating this construction, find an open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M120">View MathML</a> in I, where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> is defined and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> is a solution of (1) on intervals from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M120">View MathML</a>. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M124">View MathML</a> is closed and nowhere dense. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M125">View MathML</a> the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M126">View MathML</a> is equal to −∞ or +∞. Indeed, assuming the contrary and acting as above, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M127">View MathML</a>. Extend <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> to irrational points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M129">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M130">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M131">View MathML</a>, and in the remaining cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M132">View MathML</a>. The above limits exist since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> is monotone in neighborhood of any point from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M129">View MathML</a>. Similarly we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M137">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> is a generalized solution of (1). It follows from the compactness condition that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M101">View MathML</a> is a solution of (1). Let us show that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M103">View MathML</a> uniformly converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M104">View MathML</a>. Suppose the contrary is true. We assume, without loss of generality, that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M142">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M143">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M88">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M148">View MathML</a>. Consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M91">View MathML</a> . We can assume, without loss of generality, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M91">View MathML</a> , and this contradicts the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M153">View MathML</a>. The uniform convergence is proved. We can conclude now that all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M154">View MathML</a> are the solutions of the boundary value problem (1)-(2). □

Remark 2 Theorem 2 gives the possibility to prove the solvability of boundary value problems if the solvability of more simple boundary value problems is known.

Remark 3 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M155">View MathML</a> and the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M156">View MathML</a> hold for a solution x of the boundary value problem (1)-(2), then the compactness condition (Definition 4) can be weakened.

Definition 5 We will say that the compactness condition holds if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M73">View MathML</a> all generalized solutions of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M159">View MathML</a>

are classical solutions.

Example One way to use Theorem 2 is to verify that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M18">View MathML</a>, the following conditions are satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M164">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M165">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M167">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M169">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M170">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

The author participated in drafting, revising and commenting on the manuscript. The author read and approved the final manuscript.

Acknowledgements

The author sincerely thanks the reviewers for their valuable suggestions and useful comments. This research was supported by the Institute of Mathematics and Computer Science, University of Latvia.

References

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  2. Lepin, AY, Lepin, LA: Boundary Value Problems for Ordinary Differential Equations, Zinatne, Riga (1988) (Russian)

  3. Lepin, AY, Lepin, LA: Generalized lower and upper functions for φ-Laplacian equations. Differ. Equ. (2014, in press)

  4. Nagumo, M: Über die Differentialgleichung <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/111/mathml/M171">View MathML</a>. Proc. Phys. Math. Soc. Jpn.. 19(3), 861–866 (1937)

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  6. Lepin, AY: Compactness of generalized solutions between the generalized lower and upper functions. Proc. LUMII Math. Differ. Equ.. 11, 22–24 (2011)