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

Open Access Research Article

Positive blow-up solutions of nonlinear models from real world dynamics

Jürgen Gschwindl1, Irena Rachůnková2*, Svatoslav Staněk2 and Ewa B Weinmüller1

Author Affiliations

1 Department for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, Wien, A-1040, Austria

2 Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 77146, Czech Republic

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Boundary Value Problems 2014, 2014:121  doi:10.1186/1687-2770-2014-121


Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciences


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/121


Received:13 December 2013
Accepted:5 May 2014
Published:16 May 2014

© 2014 Gschwindl et al.; licensee Springer.

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

Abstract

In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M2">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3">View MathML</a>. The differential equation is studied together with the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M5">View MathML</a>. We specify conditions for the data function h which guarantee that the set of all positive solutions to the above boundary value problem is nonempty. Further properties of the solutions are discussed and results of numerical simulations are presented.

MSC: 34B18, 34B16, 34A12.

Keywords:
singular ordinary differential equation of the second order; time singularities; blow-up, positive solutions; existence of solutions; polynomial collocation

1 Introduction

In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M6">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3">View MathML</a>.

Models in the form of (1) arise in many applications. Among others, they occur in the study of phase transitions of Van der Waals fluids [1-3], in population genetics, where they characterize the spatial distribution of the genetic composition of a population [4,5], in the homogeneous nucleation theory [6], in relativistic cosmology for particles which can be treated as domains in the universe [7], and in the nonlinear field theory, in particular, in context of bubbles generated by scalar fields of the Higgs type in Minkowski spaces [8].

Here, we assume that h is positive and satisfies the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M8">View MathML</a>. We define a positive solution of (1) as a function v which satisfies (1) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a>, is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10">View MathML</a>, and has absolutely continuous first derivative on each compact subinterval in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a>.

According to Lemma 4, if v is a positive solution of (1), then either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M12">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M13">View MathML</a>

In the literature, bounded solutions of (1) have been widely investigated; see e.g.[9-13]. Such solutions are characterized by the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M14">View MathML</a>. In contrast to this, some real problems lead to the investigation of unbounded solutions which are characterized by the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M15">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M16">View MathML</a> and which are called blow-up solutions. We refer to [14-16]. Here, we are interested in blow-up solutions of (1), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M17">View MathML</a>. In particular, (1) will be considered together with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M18">View MathML</a>

(2)

In this case, we speak about a positive solution of problem (1), (2). Let us denote by the set of all positive solutions to (1), (2). Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M21">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M23">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M24">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M25">View MathML</a>.

Our main goal is to find conditions for the data function h in (1), which guarantee that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26">View MathML</a> is nonempty for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M27">View MathML</a> and then to investigate the properties of this set. For example, we prove that the difference of any two functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29">View MathML</a>, retains its sign on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M30">View MathML</a>, and that there exist minimal and maximal solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M31">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29">View MathML</a>, cf. Theorem 5. If the interior of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M33">View MathML</a> is nonempty, we show that this interior is fully covered by ordered graphs of other functions belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29">View MathML</a>; see Theorem 6. Finally, in Theorem 7, the existence of a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M37">View MathML</a> is shown and all functions v from ℛ are uniquely characterized by the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M38">View MathML</a>

If we denote such v by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M39">View MathML</a> and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M40">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M41">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M42">View MathML</a>, then we find that the graphs of these functions do not intersect, cf. Theorem 8, and that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M43">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M44">View MathML</a> is compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a>; see Theorem 9.

The study of a structure of positive solutions to other types of ordinary differential equations can be found for example in [17-19].

Notation

Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a> the Banach space of functions continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a> equipped with the maximum norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M48">View MathML</a>

Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a> means the Banach space of functions having a continuous first derivative on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a> with the corresponding maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M51">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M52">View MathML</a> we denote the set of functions which are Lebesgue integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M54">View MathML</a> is the set of functions whose first derivative is absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M56">View MathML</a> is the set of functions having absolutely continuous first derivative on each compact subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a>. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M58">View MathML</a> satisfies the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M59">View MathML</a>, if the following three conditions hold:

(i) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M60">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M61">View MathML</a>.

(ii) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M62">View MathML</a> is continuous for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a>.

(iii) For each compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M64">View MathML</a> there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M65">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M66">View MathML</a>

For functions satisfying above conditions, we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M67">View MathML</a>.

Structure of the paper

The paper is organized as follows. In Section 2 we discuss properties of the solutions of the auxiliary Dirichlet problem (3), (4). We recapitulate previous results from [20] and also present new results in Theorems 1, 2, and 3. The main results of the paper can be found in Section 3, where we describe a relation between solutions of problem (3), (4) and blow-up solutions of problem (1), (2); see Theorem 4. Using this relation and the results of Section 2, we obtain various interesting properties of blow-up solutions; see Theorems 5 to 9. Section 4 contains three examples illustrating the theoretical findings. Final remarks and open problems are formulated in Section 5.

2 Auxiliary results

In this section we consider the auxiliary singular differential equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M68">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M69">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M71">View MathML</a>, (3) becomes a special case of (1) and therefore, results obtained for (3) apply for (1). For the further analysis, we assume that f satisfies the following conditions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M72">View MathML</a>,

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M73">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M75">View MathML</a>,

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M76">View MathML</a> is increasing in x for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M77">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M78">View MathML</a>

We now study (3) subject to the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M79">View MathML</a>

(4)

and require that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M80">View MathML</a>

(5)

holds. We call a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M81">View MathML</a> a positive solution of the Dirichlet problem (3), (4) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M83">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10">View MathML</a>, u satisfies the boundary conditions (4), and (3) holds for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M77">View MathML</a>. Clearly, for each positive solution u of (3), (4), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a> such that (5) is satisfied.

We now denote by the set of all positive solutions of problem (3), (4), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M88">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>.

In the following lemma we cite those results from [20] which will be used in the analysis of problem (1), (2).

Lemma 1Let (H1)-(H3) hold. Then the following statements hold:

(a) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a>is nonempty and there exist functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M92">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M93">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95">View MathML</a>.

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M99">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100">View MathML</a>.

(c) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M104">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a>, then either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M106">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100">View MathML</a>or there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M108">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M99">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M110">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M111">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M112">View MathML</a>.

(d) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a>and each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M114">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M116">View MathML</a>.

(e) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a>is a one-point set for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M118">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M119">View MathML</a>is at most countable.

(f) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M120">View MathML</a>, the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M121">View MathML</a>is compact in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a>.

(g) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95">View MathML</a>if and only if it is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M125">View MathML</a>

(6)

in the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a>.

We now formulate new results which complete those from [20]. We first establish a relation between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M127">View MathML</a> and the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M128">View MathML</a> if its interior is nonempty. This question was a short time ago an open problem [[20], Remark 4.4]. We note that the relation between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a> and the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M130">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M131">View MathML</a> having nonempty interior is described in Lemma 1(d).

Theorem 1Let (H1)-(H3) hold. Let us assume that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M132">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M133">View MathML</a>. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M134">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M135">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M136">View MathML</a>.

ProofStep 1. Auxiliary Dirichlet problem.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M137">View MathML</a>. Consider (3) subject to the Dirichlet conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M138">View MathML</a>

(7)

We claim that there exists a solution v to problem (3), (7) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M139">View MathML</a>

(8)

We show this result utilizing the method of lower and upper functions. It follows from (H1) that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M140">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M141">View MathML</a>

(9)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M142">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M143">View MathML</a> be given as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M144">View MathML</a>

We now consider the auxiliary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M145">View MathML</a>

(10)

It is not difficult to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M146">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M147">View MathML</a>. Hence, cf. (9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M148">View MathML</a>

(11)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M149">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M147">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M153">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M156">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M158">View MathML</a> solve (3) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M158">View MathML</a> are lower and upper functions of problem (3), (7) (see e.g.[21] or [22]). This fact together with (11) implies the existence of a solution v to problem (3), (7) satisfying (8), cf. [[21], Lemma 3.7]. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M162">View MathML</a>

and taking the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M163">View MathML</a> we obtaina<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M165">View MathML</a>, which together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M166">View MathML</a> gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M167">View MathML</a>.

We now prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M168">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M169">View MathML</a>. The proof is indirect. Let us assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M172">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M173">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M174">View MathML</a> on this interval, and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M175">View MathML</a>

From (8) and (9) we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M176">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M177">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M178">View MathML</a>

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M179">View MathML</a>

By the Gronwall lemma,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M180">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M181">View MathML</a>

which contradicts

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M182">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M172">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M149">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M185">View MathML</a> = 1 on this interval. Thus, v is a solution of problem (3), (7).

Step 2. Continuation of the solutionv.

It follows from the arguments given in Step 1 that v is a solution of problem (3), (7) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M188">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M189">View MathML</a>. It is easy to verify that the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M190">View MathML</a>

(12)

is satisfied for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M149">View MathML</a>. We now integrate the last equality two times over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M192">View MathML</a> and have (note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M193">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M194">View MathML</a>

Let u be a solution of problem (3), (7) on an interval J which is a left-continuation of v. Let us assume that u is not continuable. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M195">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M196">View MathML</a> and (12) with v replaced by u holds for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M197">View MathML</a>. The integration now yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M198">View MathML</a>

(13)

and we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M199">View MathML</a>

(14)

We show inequality (14) indirectly. Let us assume that (14) does not hold. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M200">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M201">View MathML</a>

and either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M202">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M203">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M202">View MathML</a>.

Since, by (H3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M205">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M207">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M208">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M209">View MathML</a>

which is not possible. The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M210">View MathML</a> can be discussed analogously. Hence, (14) holds.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M211">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M212">View MathML</a> and since u is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M213">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M214">View MathML</a>. The integration of the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M215">View MathML</a>

over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M216">View MathML</a> gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M217">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M218">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M219">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M220">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M221">View MathML</a>

Using the Gronwall lemma, we deduce that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M222">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M223">View MathML</a>

holds. This is a contradiction.

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M224">View MathML</a>, and then (14) yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M225">View MathML</a>. Consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M226">View MathML</a>, and the assertion of the theorem follows from (13). □

In the next corollary we extend the statement (d) from Lemma 1 to a large set of A values.

Corollary 1For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a>and each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M228">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M116">View MathML</a>.

Proof The result follows immediately from Lemma 1(d) and Theorem 1. □

Remark 1 Corollary 1 says that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M231">View MathML</a> is covered by the graphs of functions from , that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M233">View MathML</a>

By Lemma 1(b), (c) we know that functions from are uniquely determined by the values −c of their derivatives at the right end point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M235">View MathML</a> only in the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a> is a singleton set for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29">View MathML</a>. Since we cannot uniquely determine all functions from via their derivatives at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M235">View MathML</a>, see Lemma 1(e), we discuss their derivatives at the singular point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240">View MathML</a>.

Lemma 2Let (H1)-(H3) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M243">View MathML</a>

(15)

Proof It follows from Lemma 1(g) that (6) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M244">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M245">View MathML</a>, we have (note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M246">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M247">View MathML</a>

and (15) follows. □

Corollary 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M249">View MathML</a>

Proof The result follows from Lemma 2, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M250">View MathML</a>. □

Corollary 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M252">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M253">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M252">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M256">View MathML</a>. We can assume that for instance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M257">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M258">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M260">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261">View MathML</a> by Lemma 1(b)(c). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M262">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M263">View MathML</a>

which together with Corollary 2 gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M264">View MathML</a>. □

Corollary 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M267">View MathML</a>

(16)

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M268">View MathML</a>.

Proof Inequality (16) follows from Lemma 2 and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M269">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a> by (H2). □

Corollary 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M272">View MathML</a>

Proof The integration of (3) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M273">View MathML</a> gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M274">View MathML</a>

Using integration by parts we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M275">View MathML</a>

and, therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M276">View MathML</a>

Taking the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M277">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M278">View MathML</a>

On the other hand, it follows from Corollary 2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M279">View MathML</a>

Combining the above two equalities yields the result. □

Lemma 3Let (H1)-(H3) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a>be not a singleton set. Then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M282">View MathML</a>there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M285">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M287">View MathML</a>. Then it follows from Lemma 1(a) and (H3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M288">View MathML</a>

Hence, by Lemma 2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M289">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a> is a compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a> by Lemma 1(f), the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M292">View MathML</a> is closed. In fact, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M294">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M295">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M296">View MathML</a>. Then there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M297">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M299">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M300">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a>. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M302">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M303">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M304">View MathML</a>.

It remains to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M305">View MathML</a>. Assume that the equality does not hold. Then, from the structure of bounded and closed subsets of ℝ the existence of an open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M306">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M307">View MathML</a>, follows. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M308">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M309">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M310">View MathML</a>. Due to Lemma 1(c), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M312">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M314">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M316">View MathML</a>. By Corollary 1, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M317">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M318">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M319">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M321">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M323">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M261">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M325">View MathML</a>, which is not possible. □

Since functions from belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M54">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M328">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M329">View MathML</a>. Corollary 4 yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M268">View MathML</a>. Let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M331">View MathML</a>

(17)

Lemma 3 implies that functions from can be uniquely determined by the values of their derivatives at the singular point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240">View MathML</a>; see Theorem 2.

Theorem 2Let (H1)-(H3) hold. Then there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M336">View MathML</a>.

Proof We first show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M337">View MathML</a>

(18)

It follows from Lemma 3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M338">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M339">View MathML</a>, where we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M340">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a> is a singleton set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M342">View MathML</a>. In view of Corollary 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M343">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M95">View MathML</a>. Consequently, (18) follows.

Let us now choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M345">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M346">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284">View MathML</a>. The uniqueness of u follows from Corollary 3. □

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M345">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M349">View MathML</a> the unique element of satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M351">View MathML</a>.

Theorem 3Let (H1)-(H3) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M352">View MathML</a>is a convergent sequence and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M353">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M354">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M356">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M357">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M358">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M359">View MathML</a> by Corollary 4. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M360">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M361">View MathML</a> is compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a> by Lemma 1(f), the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M365">View MathML</a> be a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363">View MathML</a> which is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M368">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369">View MathML</a> for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M370">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M371">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M372">View MathML</a> and hence any subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M365">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363">View MathML</a> converging in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a> has the same limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M349">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M363">View MathML</a> is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M349">View MathML</a> is its limit. □

3 Blow-up solutions and their properties

This section contains the main results of the paper. First, we present a lemma which describes how positive solutions of (1) may behave at the singular point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240">View MathML</a>.

Lemma 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3">View MathML</a>and let the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M382">View MathML</a>be positive. Letvbe a positive solution of (1). Then either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M383">View MathML</a>

(19)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M384">View MathML</a>

(20)

Proof By Corollary 3.5 in [12], if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M385">View MathML</a>, then v satisfies (19). Now, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M386">View MathML</a>

(21)

Then v has to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M387">View MathML</a>

(22)

because otherwise, if (22) was not true, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M388">View MathML</a>, contradicting (21). Since h is positive, (1) indicates that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M389">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a> and hence there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M391">View MathML</a>. The next part of the proof is divided into three cases.

(i) Let us assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M392">View MathML</a>

(23)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M389">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a>, we deduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M395">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M396">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M41">View MathML</a>. Consequently, v is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a>. This together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M399">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M400">View MathML</a> yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M401">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M402">View MathML</a>. Integrating (1) and using (23), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M403">View MathML</a>

which is a contradiction to (22).

(ii) Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M404">View MathML</a>

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M405">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M406">View MathML</a>

(24)

Hence, v is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M408">View MathML</a>.

By integration, we obtain from (24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M409">View MathML</a>

and, by virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M410">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M411">View MathML</a>, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M412">View MathML</a>

which again is a contradiction.

(iii) Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M413">View MathML</a>

(25)

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M405">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M415">View MathML</a>

(26)

Hence, v is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M259">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M417">View MathML</a>. By integration, we obtain from (26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M418">View MathML</a>

and, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M410">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M420">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M421">View MathML</a>

In addition (25) yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M422">View MathML</a> and (20) follows. □

Now, we investigate the existence and properties of blow-up solutions of (1), for the case that the function h has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M423">View MathML</a>

In particular, we study the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M424">View MathML</a>

(27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M425">View MathML</a> and ψ, g satisfy the conditions

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M427">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M428">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M244">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M430">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M431">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432">View MathML</a> is increasing in x for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M434">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M435">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M437">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M438">View MathML</a>, ϕ is increasing and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M439">View MathML</a>

Together with (27) we consider the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M440">View MathML</a>

(28)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M441">View MathML</a>

(29)

We define a positive solution to problem (27), (28) as a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M442">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M399">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10">View MathML</a>, v satisfies the boundary conditions (28), and (27) holds for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a>.

Define a set by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M447">View MathML</a>

(30)

Clearly, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M448">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a> such that (29) holds.

Lemma 5Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M448">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M453">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M10">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M455">View MathML</a>, the integration of (27) over the subinterval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M456">View MathML</a> gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M457">View MathML</a>

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M458">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M460">View MathML</a>

which is not possible, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M461">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M462">View MathML</a>. □

Moreover, let us define sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M463">View MathML</a>

(31)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M464">View MathML</a>

(32)

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M25">View MathML</a>. Using the substitution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M466">View MathML</a>

(33)

we can rewrite (27) and obtain the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M467">View MathML</a>

(34)

We now introduce a function f,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M468">View MathML</a>

Under conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>), the function f satisfies (H1)-(H3); see the proof of [[20], Theorem 5.1]. Therefore, the results of Section 2 hold for problem (34), (4). As in Section 2, we define the sets and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M91">View MathML</a> for solutions of (34).

The following result is the key-stone to the analysis of the structure of the set ℛ and describes the relation between the sets and ℛ.

Theorem 4Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>. Let us assume that (33) holds. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M479">View MathML</a>.

Proof (⇒) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477">View MathML</a>. Then (28), (32), and (33) provide the following properties: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M481">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M482">View MathML</a>, and u is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a>. It follows from the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M484">View MathML</a>

that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M485">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M486">View MathML</a>

We now argue as in the proof of [[20], Lemma 3.3] and arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M487">View MathML</a>

(35)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M488">View MathML</a>. Since u is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M490">View MathML</a>. Then, by [[20], Lemma 2.1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M491">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M41">View MathML</a>, and hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M493">View MathML</a>. Now, [[23], Corollary 1] guarantees that u can be extended on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M82">View MathML</a> such that the equality (35) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369">View MathML</a>.

(⇐) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M479">View MathML</a>. Then it follows from (4), (5), and (33) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M500">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M501">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M502">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M503">View MathML</a> is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M11">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M505">View MathML</a>

yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M506">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M507">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M369">View MathML</a>, the equality (35) holds on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a>, and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M510">View MathML</a>

Hence, note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M512">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M513">View MathML</a>. As a result, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477">View MathML</a>. □

Now, we are in the position to provide results on the solvability of problem (27), (28) and formulate the properties of its solutions.

Theorem 5Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold. Then the following statements hold:

(a) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26">View MathML</a>is nonempty and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M519">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M520">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M488">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477">View MathML</a>.

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M524">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M526">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100">View MathML</a>.

(c) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M529">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M98">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M531">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a>, then either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M533">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M100">View MathML</a>or there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M108">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M526">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M110">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M538">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M112">View MathML</a>.

(d) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M26">View MathML</a>is a singleton set for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M541">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M119">View MathML</a>is at most countable.

Proof The result follows by combining results from Lemma 1(a), (b), (c), and (e) with those from Theorem 4. □

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M479">View MathML</a>, and if (33) holds, then Theorems 4 and 5 yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M545">View MathML</a>

(36)

The next theorem shows that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M546">View MathML</a>

is covered by graphs of the functions from ℛ.

Theorem 6Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold. Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a>and each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M550">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M552">View MathML</a>. In particular, if for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M554">View MathML</a>the inequality<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M555">View MathML</a>holds, then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M556">View MathML</a>there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M477">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M558">View MathML</a>.

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M550">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M561">View MathML</a>. Then, using (33) and (36), we deduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M562">View MathML</a>. Therefore, by Corollary 1, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M116">View MathML</a>. Consequently, (33) and Theorem 4 give <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M552">View MathML</a>. The last statement follows from Theorem 5(c). □

By Corollary 4, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M567">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M568">View MathML</a>

(37)

and hence, by (33) and (36) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M569">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M570">View MathML</a>

(38)

Using constants from the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M571">View MathML</a>, we can uniquely determine all functions in ℛ.

Theorem 7Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36">View MathML</a>be as in (38). Then the following statements hold:

(a) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M576">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M577">View MathML</a>

(39)

(b) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M578">View MathML</a>there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551">View MathML</a>satisfying (39).

Proof (a) Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551">View MathML</a>. Using (33), (37), and Theorem 4, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M581">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M582">View MathML</a>

(40)

By Lemma 1(c), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M583">View MathML</a>. Hence, if we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284">View MathML</a>, the first condition in (39) follows. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M585">View MathML</a>

(41)

and therefore the second condition in (39) holds.

(b) Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M578">View MathML</a>. Theorem 2 guarantees that there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M115">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M284">View MathML</a>. Using (33) and Theorem 4, we conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M589">View MathML</a>. Then v satisfies (40) and (41) which results in (39). It remains to prove that v is unique. Assume that there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M590">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M591">View MathML</a>, such that w satisfies (39). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M592">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M593">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M594">View MathML</a>. Consequently, by Theorem 2, we arrive at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M595">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M596">View MathML</a>, which is a contradiction. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36">View MathML</a> be given in (38) and choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M598">View MathML</a>. Keeping Theorem 7(b) in mind, there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M551">View MathML</a> satisfying (39). We denote such v by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M39">View MathML</a> and define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M601">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M602">View MathML</a>

(42)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M603">View MathML</a>. We now specify further properties of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M604">View MathML</a>.

Theorem 8Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36">View MathML</a>be from (38). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M608">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M609">View MathML</a>. Then either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M610">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M611">View MathML</a>or there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M612">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M610">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M614">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M615">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M112">View MathML</a>.

Proof According to (42), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M617">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M618">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M619">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M620">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M621">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M488">View MathML</a>. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M623">View MathML</a> and, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M624">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M625">View MathML</a> in a right neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M240">View MathML</a>. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M405">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M628">View MathML</a> and hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M629">View MathML</a>. (Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M630">View MathML</a>, then Theorem 5(b) yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M631">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M632">View MathML</a>, which is not possible.) The result now follows from Theorem 5(b) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M633">View MathML</a> and from Theorem 5(c) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M634">View MathML</a>. □

Corollary 6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M635">View MathML</a>be a convergent sequence and let us denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M353">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M637">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a>.

Proof The proof is indirect. Assume that the statement of the corollary does not hold. Then there exist ε, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M639">View MathML</a> and a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M640">View MathML</a>, we denote it again by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M640">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M642">View MathML</a>

(43)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M643">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M644">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M645">View MathML</a> be countable. Then there exists a decreasing subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M646">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M647">View MathML</a>. By Theorem 8, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M648">View MathML</a> is not increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M650">View MathML</a> on this interval. Using (33) and (42), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M651">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M652">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M653">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a> it follows from Theorem 3, note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M655">View MathML</a>, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M42">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M657">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M9">View MathML</a>. This fact together with the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M648">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M603">View MathML</a> gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M661">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a>, which contradicts (43). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M663">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a>. □

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M43">View MathML</a>, we now define a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M667">View MathML</a>

(44)

Theorem 9Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M36">View MathML</a>be from (38). For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M43">View MathML</a>the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666">View MathML</a>defined by (44) is compact in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a>.

Proof Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M674">View MathML</a>. By Theorem 8 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M675">View MathML</a>

According to (33), (42), and Theorem 4 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M676">View MathML</a>

(45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M677">View MathML</a>

(46)

Let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M678">View MathML</a>

Conditions (4) and (46) imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M679">View MathML</a>. Consequently, using Theorem 4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M680">View MathML</a>

(47)

Consider a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M681">View MathML</a>. Then there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M682">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M683">View MathML</a>. By Lemma 1(f), the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M684">View MathML</a> is compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a>. Therefore, there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M686">View MathML</a> which converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M49">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M688">View MathML</a>. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M689">View MathML</a>. Let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M690">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M691">View MathML</a>. This yields, by Corollary 6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M692">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M694">View MathML</a>. Thus we showed that for any sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666">View MathML</a>, there exists a uniformly converging subsequence on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M47">View MathML</a> with a limit in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M666">View MathML</a>. □

4 Examples

Using the open domain MATLAB Code bvpsuite, we numerically simulate three model problems in order to illustrate theoretical statements made above.

4.1 MATLAB Code bvpsuite

The MATLABTM software package bvpsuite is designed to solve BVPs in ODEs and differential algebraic equations. The solver routine is based on a class of collocation methods whose orders may vary from 2 to 8. Collocation has been investigated in context of singular differential equations of first and second order in [24,25], respectively. This method could be shown to be robust with respect to singularities in time and retains its high convergence order in case that the analytical solution is appropriately smooth. The code also provides an asymptotically correct estimate for the global error of the numerical approximation. To enhance the efficiency of the method, a mesh adaptation strategy is implemented, which attempts to choose grids related to the solution behavior, in such a way that the tolerance is satisfied with the least possible effort. The error estimate procedure and the mesh adaptation work dependably provided that the solution of the problem and its global error are appropriately smooth.b The code and the manual can be downloaded from http://www.math.tuwien.ac.at/~ewa webcite. For further information see [26]. This software proved useful for the approximation of numerous singular BVPs important for applications; see e.g.[9,27-29].

Example 1 The first example is used to comment on Theorem 5 and to show that if a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432">View MathML</a> in (27) is increasing in x, the graphs of positive solutions of problem (27), (28) cannot intersect. Here, we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M699">View MathML</a>

and thus problem (27), (28) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M700">View MathML</a>

(48)

Using substitution (33), which has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M701">View MathML</a>

(49)

we transform (48) onto the form, cf. (34),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M702">View MathML</a>

and we solve the terminal value problem given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M703">View MathML</a>

(50)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M704">View MathML</a>, and 4. Then, using (49), we recalculate these solution to their blow-up form and obtain different positive solutions v of problem (48). Note that their graphs are ordered after c, cf. Figure 1.

thumbnailFigure 1. Example 1. Solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705">View MathML</a> of (50) (first) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706">View MathML</a> of (48) (second) for different values of c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M707">View MathML</a> orange, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M708">View MathML</a> green, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M709">View MathML</a> red, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M710">View MathML</a> dark blue, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711">View MathML</a> magenta. We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy to satisfy the tolerances <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M712','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M712">View MathML</a>.

Example 2 We designed this example to see if the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M713">View MathML</a>) is essential for statement (b) of Theorem 5. To this aim, we choose a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432">View MathML</a> which is decreasing in x in (27). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M715">View MathML</a>

Then problem (27), (28) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M716">View MathML</a>

(51)

Using substitution (49), we rewrite (51) and have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M717">View MathML</a>

The resulting terminal value problem reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M718">View MathML</a>

(52)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M704">View MathML</a>, and 4. Then, using (49), we obtain different positive blow-up solutions of problem (51). However, their graphs are again ordered; see Figure 2. Hence, condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M713">View MathML</a>) seems not to be necessary for statement (b) of Theorem 5 to hold. On the other hand, the question, if another model with a decreasing nonlinearity g exists, where solutions cross each other and are not ordered after c, remains open.

thumbnailFigure 2. Example 2. Solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705">View MathML</a> of (52) (first) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706">View MathML</a> of (51) (second) for different values of c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M707">View MathML</a> orange, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M708">View MathML</a> green, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M709">View MathML</a> red, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M710">View MathML</a> dark blue, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711">View MathML</a> magenta. We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy to satisfy the tolerances <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M728','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M728">View MathML</a>.

Example 3 Since, Example 2 did not provide the solution crossings, we constructed another model problem, where in (27) the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M432">View MathML</a> oscillates in x. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M730">View MathML</a>

then problem (27), (28) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M731','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M731">View MathML</a>

(53)

Using (49), we again transform (53) and solve the terminal value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M732">View MathML</a>

(54)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M704">View MathML</a>, and 4. Then, using (49), we calculate the related positive blow-up solutions of problem (53). As can be seen in Figure 3, their graphs are again ordered after c and the question of solution crossings remains open even if we consider oscillating nonlinearities.

thumbnailFigure 3. Example 3. Solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705">View MathML</a> of (54) (first) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706">View MathML</a> of (53) (second) for different values of c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M707">View MathML</a> orange, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M708">View MathML</a> green, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M709">View MathML</a> red, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M710">View MathML</a> dark blue, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711">View MathML</a> magenta. We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy to satisfy the tolerances <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M741">View MathML</a>.

To show the advantage of the grid adaptation, we carried out the following additional test. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M711">View MathML</a>, we evaluated the solution u at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M743">View MathML</a> and solved the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M744">View MathML</a>

(55)

In Figure 4, solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705">View MathML</a> of (54) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706">View MathML</a> of (55), together with the final grids are shown on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M747">View MathML</a> (left) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M748">View MathML</a> (right). Due to the different smoothness of u and v, their accuracy strongly varies. While the absolute and relative errors of u (or rather their estimates) on the grid with 52 grid points is 10−13, the absolute and relative errors of v on the grid with exactly the same number of points is only 10−9. Note that the code has automatically adapted the location of the grid points to correctly reflect the solution behavior.

thumbnailFigure 4. Example 3,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M749','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M749">View MathML</a>. Solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M705">View MathML</a> of (54) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M706">View MathML</a> of (55), shown on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M747">View MathML</a> (first) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M753','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M753">View MathML</a> (second). We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy.

5 Conclusions

We have described the set ℛ of all positive solutions of problem (1), (2), where h has the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M754','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M754">View MathML</a> and assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M426">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M451">View MathML</a>) hold. By Theorem 5 we know that ℛ is nonempty and that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M29">View MathML</a> there exists at least one function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M758">View MathML</a> fulfilling <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M501">View MathML</a>. In addition, graphs of functions from ℛ do not intersect and ℛ has a minimal element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M760','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M760">View MathML</a>.

If we choose an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M761','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M761">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M762">View MathML</a>, then there exists a maximal element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M763','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M763">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M764">View MathML</a>. Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M765','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M765">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M766','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M766">View MathML</a>. According to Theorem 6, the interior of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M767">View MathML</a> is fully covered by graphs of functions from ℛ. Finally, we deduce from Theorems 7-9 that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M768">View MathML</a> is compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M45">View MathML</a>.

Example 1 illustrates the results of Section 3 and hence, according to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M713">View MathML</a>), we have chosen the increasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M771','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M771">View MathML</a> in (48). Figure 1 shows ordered graphs of solutions.

In contrast to this, in Example 2 and Example 3, we have chosen the decreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M772','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M772">View MathML</a> in (51) and the non-monotonous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M773','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M773">View MathML</a> in (53), respectively. We can see on Figure 2 and Figure 3 that the graphs of solutions are ordered in both cases. But to prove such order of solutions for non-increasing g is an open problem. On the other hand, the question of the construction of problem (27), (28) whose positive solutions cross each other remains open, as well.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final draft. IR and SS contributed to the analytical part of the paper and JG and EW contributed to the numerical part of the paper.

Acknowledgements

This research was supported by the grants IGA PrF_2013_013 and IGA-PrF_2014028. The authors are grateful to the referees for useful comments and suggestions which improved the paper.

End notes

  1. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M164">View MathML</a>.

  2. The required smoothness of higher derivatives is related to the order of the used collocation method.

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