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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Positive solutions of nonlinear Dirichlet BVPs in ODEs with time and space singularities

Irena Rachůnková1*, Alexander Spielauer2, Svatoslav Staněk1 and Ewa B Weinmüller2

Author Affiliations

1 Department of Mathematical Analysis, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, CZ-771 46, Czech Republic

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

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Boundary Value Problems 2013, 2013:6  doi:10.1186/1687-2770-2013-6

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/6


Received:17 October 2012
Accepted:21 December 2012
Published:16 January 2013

© 2013 Rachůnková et al.; licensee Springer

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

Abstract

In this paper, we discuss the existence of positive solutions to the singular Dirichlet boundary value problems (BVPs) for ordinary differential equations (ODEs) of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2">View MathML</a>. The nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3">View MathML</a> may be singular for the space variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M6">View MathML</a>, the differential operator on the left-hand side of the differential equation is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M7">View MathML</a>. Sufficient conditions for the existence of positive solutions of the above BVPs are formulated and asymptotic properties of solutions are specified. The theory is illustrated by numerical experiments computed using the open domain MATLAB code bvpsuite, based on polynomial collocation.

MSC: 34B18, 34B16, 34A12.

Keywords:
singular ordinary differential equation of the second order; time singularities; space singularities; positive solutions; existence of solutions; polynomial collocation

1 Introduction

In the present work, we analyze the existence of positive solutions to the singular Dirichlet BVP,

(1a)

(1b)

Here, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2">View MathML</a> and f satisfies the local Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M12">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M15">View MathML</a>. Let us recall that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M17">View MathML</a>, satisfies the local Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M18">View MathML</a> if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M19">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M20">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M21">View MathML</a> is continuous for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M22">View MathML</a>,

(iii) for each compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M23">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M24">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M25">View MathML</a>

For such functions, we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M26">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3">View MathML</a> may become singular when the space variables x and/or y vanish, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3">View MathML</a> may become unbounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M30">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M31">View MathML</a>, and/or it may be unbounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M30">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M34">View MathML</a>. Finally, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M6">View MathML</a>, Eq. (1a) has a singularity of the first kind at the time variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M7">View MathML</a> because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M37">View MathML</a>

The differential operator on the left-hand side of Eq. (1a) can be equivalently written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M38">View MathML</a> and, after the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M39">View MathML</a>, it takes the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M40">View MathML</a>, which arises in numerous important applications. Operators of such type were studied in phase transitions of Van der Waals fluids [1-4], in population genetics, especially in models for the spatial distribution of the genetic composition of a population [5,6], in the homogeneous nucleation theory [7], in relativistic cosmology for description of particles which can be treated as domains in the universe [8], and in the nonlinear field theory [9], in particular, when describing bubbles generated by scalar fields of Higgs type in the Minkowski spaces [10].

The aim of this paper is to study the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M41">View MathML</a> which is fundamentally different from the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M42">View MathML</a>. The latter setting was studied in [11,12], where the structure and properties of the set of all positive solutions to (1a) and (1b) were investigated (the cardinality of this set is a continuum).

In the sequel, we introduce the basic notation and state the preliminary results required in the analysis of problem (1a) and (1b). Here, we focus our attention on the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M41">View MathML</a> and prove the existence of at least one positive solution of (1a) and (1b). In contrast to [11,12], we consider the more general situation in which f may be also singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5">View MathML</a>. This means that we have to deal with the following additional difficulties.

Let u be a positive solution of problem (1a) and (1b), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3">View MathML</a> has a singularity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M47">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M49">View MathML</a> and hence f is unbounded in a neighborhood of the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M50">View MathML</a>. Unfortunately, we do not know the exact position of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M51">View MathML</a> and therefore, it is not possible to construct a universal Lebesgue integrable majorant for all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M52">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53">View MathML</a> are positive solutions of a sequence of auxiliary regular problems. Consequently, the Lebesgue dominated convergence theorem is not applicable and we have to use arguments based on the Vitali convergence theorem instead; see Lemma 2. Another tool used in the proofs is a combination of regularization and sequential techniques with the Leray-Schauder nonlinear alternative.

The investigation of singular Dirichlet BVPs has a long history and a lot of methods for their analysis are available. One of the most important ones is the topological degree method providing various fixed point theorems and existence alternative theorems; see, e.g., Lemma 1. For more information on the topological degree method and its application to numerous BVPs, including Dirichlet problems, we refer the reader to the monographs by Mawhin [13-15].

Throughout this paper, we work with the following conditions on the function f in (1a):

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M54">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M55">View MathML</a>.

(H2) There exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M57">View MathML</a>

(H3) For a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M59">View MathML</a>, the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M60">View MathML</a>

holds, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M63">View MathML</a> are positive, h is nondecreasing in both its arguments, g and r are nonincreasing, and

By

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M65">View MathML</a>

we denote the norms in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M67">View MathML</a>, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M68">View MathML</a> denotes the set of functions whose first derivative is absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M70">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/2013/1/6/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M71">View MathML</a>. We use the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M72">View MathML</a> to denote the Lebesgue measure of ℳ.

Definition 1 We say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73">View MathML</a> is a positive solution of problem (1a) and (1b) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M74">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M75">View MathML</a>, u satisfies the boundary conditions (1b) and (1a) holds for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M22">View MathML</a>.

Remark 1 Let a function g have the properties specified in (H3). Then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M78">View MathML</a>, and it follows from the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M79">View MathML</a>

that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M80">View MathML</a>

(2)

In order to prove that the singular problem (1a) and (1b) has a positive solution, we use regularization and sequential techniques. To this end, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81">View MathML</a> we define functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M83">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M84">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M85">View MathML</a>

respectively. Then it follows from (H1) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M86">View MathML</a> and (H2) and (H3) yield

(3)

(4)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M89">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M90">View MathML</a>

(6)

As a first step in the analysis, we investigate auxiliary regular BVPs of the form

(7a)

(7b)

To show the solvability of problem (7a) and (7b), we use the following alternative of Leray-Schauder type which follows from [[16], Theorem 5.1].

Lemma 1LetEbe a Banach space, Ube an open subset ofEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M93">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M94">View MathML</a>is a compact operator. Then either

A1: ℱ has a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M95">View MathML</a>, or

A2: there exists an element<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M96">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M97">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M98">View MathML</a>.

In limit processes, we apply the following Vitali convergence theorem; cf.[17-19].

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M99">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M100">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>. Then the following statements are equivalent:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M103">View MathML</a>,

(ii) the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M104">View MathML</a>is uniformly integrable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>.

We recall that a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M106">View MathML</a> is called uniformly integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M109">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M111">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M112">View MathML</a>

The paper is organized as follows. In Section 2, we collect auxiliary results used in the subsequent analysis. Section 3 is devoted to the study of limit properties of solutions to Eq. (7a). In Section 4, we investigate auxiliary regular problems associated with the singular problem (1a) and (1b). We show their solvability and describe properties of their solutions. An existence result for the singular problem (1a) and (1b) is given in Section 5. Finally, in Section 6, we illustrate the theoretical findings by means of numerical experiments.

Throughout the paper <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2">View MathML</a>.

2 Preliminaries

In this section, auxiliary statements necessary for the subsequent analysis are formulated.

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102">View MathML</a>and

Then

(i) rcan be extended on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M117">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M118">View MathML</a>,

(ii) Hcan be extended on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M120">View MathML</a>, and the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M121">View MathML</a>

(8)

holds for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>.

Proof (i) It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M123">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M124">View MathML</a>

(9)

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M125">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M127">View MathML</a> follows.

(ii) Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M128">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M130">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131">View MathML</a>. We now show that p can be extended on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> in such a way that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M133">View MathML</a>. Integrating by parts yields

(10)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M136">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M137">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M138">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M133">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M140">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131">View MathML</a>, we see that H can be extended on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M143">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M144">View MathML</a>

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M145">View MathML</a>

(11)

Hence, cf. (10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M146">View MathML</a>

and therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M147">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M120">View MathML</a>. Finally, it follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M150">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M151">View MathML</a>. Since, by (11), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M152">View MathML</a>, we see that equality (8) is satisfied for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a> which completes the proof. □

Lemma 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M99">View MathML</a>be a uniformly integrable sequence on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M156">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>. Then the sequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M158">View MathML</a>

(12)

Proof It follows from Lemma 2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M159">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81">View MathML</a>, where L is a positive constant. Recall that by Lemma 3(i), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M161">View MathML</a>. Let us assume that (12) does not hold. Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M164">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M166">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M167">View MathML</a>

(13)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M169">View MathML</a> are bounded sequences, we may assume that they are convergent, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M170">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M171">View MathML</a>, then (cf. (9))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M172">View MathML</a>

which contradicts (13). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M173">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M174">View MathML</a> and since the uniform integrability of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M104">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M177">View MathML</a>

we conclude from the relation

that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M179">View MathML</a>

The last equality contradicts (13). Consequently, (12) holds and the result follows. □

Lemma 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M181">View MathML</a>

(14)

Proof Since (cf. (10))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M182">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>, estimate (14) holds. □

3 Limit properties of solutions to Eq. (7a)

Here, we investigate asymptotic properties of solutions of (7a). We also provide a related integral equation this solution satisfies.

Lemma 6Let (H1) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M184">View MathML</a>satisfy Eq. (7a) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M186">View MathML</a>. Thenucan be extended on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73">View MathML</a>, and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M189">View MathML</a>such that the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M190">View MathML</a>

(15)

holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>.

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192">View MathML</a> and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M193">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M22">View MathML</a>. In order to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102">View MathML</a>, define for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M197">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M198">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M199">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M201">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M203">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M205">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M206">View MathML</a>. Consequently, by the Lebesgue dominated convergence theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M207">View MathML</a>.

We now discuss the linear Euler differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M208">View MathML</a>

(16)

Let H be the function given in Lemma 3. By Lemma 3(ii), H can be extended on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M120">View MathML</a> and −H satisfies (16) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>. Therefore, each function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M212">View MathML</a> which satisfies Eq. (16) a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> has the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M214">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131">View MathML</a>, with some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M216">View MathML</a>. By assumption we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M184">View MathML</a> satisfies (16) a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>, and therefore there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M219">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M131">View MathML</a>. Since by assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M222">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M71">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M224">View MathML</a>. Consequently, the function u can be extended on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> in the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M68">View MathML</a> and (15) holds on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. □

Corollary 1Let (H1) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73">View MathML</a>be a solution of Eq. (7a). Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M229">View MathML</a>such that equality (15) is satisfied for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>.

Proof The result holds by Lemma 6 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M231">View MathML</a>. □

Remark 2 Corollary 1 says that the set of all solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73">View MathML</a> of Eq. (7a) depends on one parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M229">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a>.

4 Auxiliary regular problems

In order to prove the solvability of problem (7a) and (7b), we first have to investigate the problem

(17a)

(17b)

depending on the parameter λ. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M56">View MathML</a> is from (H2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192">View MathML</a>.

The following result shows that the solvability of problem (17a) and (17b) is equivalent to the solvability of an integral equation in the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>.

Lemma 7Let (H1) hold. Thenuis a solution of problem (17a) and (17b) if and only ifuis a solution of the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M240">View MathML</a>

(18)

in the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>.

Proof Let u be a solution of Eq. (17a). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73">View MathML</a>, and by Corollary 1 (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M243">View MathML</a> replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M244">View MathML</a>), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M189">View MathML</a> such that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M246">View MathML</a>

holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M249">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M250">View MathML</a>. Consequently, if u is a solution of problem (17a) and (17b), then u is a solution of Eq. (18) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>.

Let u be a solution of Eq. (18) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M253">View MathML</a>. Hence, Lemma 3(ii) (with ρ replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M254">View MathML</a>) guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73">View MathML</a> and u is a solution of Eq. (17a). Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M256">View MathML</a>. Consequently, u is a solution of problem (17a) and (17b) which completes the proof. □

The following results provide bounds for solutions of problem (17a) and (17b).

Lemma 8Let (H1)-(H3) hold. Then there exists a positive constantS (independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M258">View MathML</a>) such that for all solutionsuof problem (17a) and (17b), the estimates

(19)

(20)

hold. Moreover, for any solutionuof problem (17a) and (17b), there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M261">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M262">View MathML</a>

(21)

Proof Let u be a solution of problem (17a) and (17b). Then by Lemma 7, equality (18) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>. Since by (5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M264">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>, the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M266">View MathML</a>

follows from (18). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M267">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M268">View MathML</a> because g is nonincreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M269">View MathML</a>. Due to Remark 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M270">View MathML</a>, which means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M271">View MathML</a>

(22)

It is clear that L is independent of the choice of solution u to problem (17a) and (17b) and independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M258">View MathML</a>.

We now show that inequality (21) holds for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M274">View MathML</a>. Differentiation of (18) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M275">View MathML</a>

(23)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M276">View MathML</a>, it follows from (5) and (23) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M277">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M278">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M278">View MathML</a> vanishes at a unique point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M261">View MathML</a> due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M256">View MathML</a>. The inequality (21) now follows from the relations

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M284">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M285">View MathML</a>, and

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

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M287">View MathML</a>

(24)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M288">View MathML</a>. Taking into account (6), (9), (18), (22), (24), and Lemma 5, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M289">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M290">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M292">View MathML</a>

(25)

and therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M293">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M295">View MathML</a>. By (H3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M296">View MathML</a>

Consequently, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M297">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M298">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M299">View MathML</a>. Now, due to (26), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M300">View MathML</a>, and therefore, by (25), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M301">View MathML</a>. □

We are now in the position to prove the existence result for problem (7a) and (7b).

Lemma 9Let (H1)-(H3) hold. Then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192">View MathML</a>, problem (7a) and (7b) has a solutionusatisfying inequalities (19)-(21), whereSis a positive constant independent ofn.

Proof Let S be a positive constant in Lemma 8 and let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M303">View MathML</a>

Then Ω is an open and bounded subset of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>. Keeping in mind Lemma 3, define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M305">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M306">View MathML</a>

(27)

By Lemma 7, any fixed point of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307">View MathML</a> is a solution of problem (7a) and (7b). In order to show the existence of a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307">View MathML</a>, we apply Lemma 1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M310">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M311">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M312">View MathML</a>. Especially, we show that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M313">View MathML</a> is a compact operator, and

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M314">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M315">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M316">View MathML</a>.

We first verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307">View MathML</a> is a continuous operator. To this end, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M318">View MathML</a> be a convergent sequence, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M319">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M321">View MathML</a>

It follows from Lemma 5 and (9) that

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M324">View MathML</a>. In particular, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M196">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M326">View MathML</a>

(28)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M327">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M102">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M330">View MathML</a>

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M331">View MathML</a> by the Lebesgue dominated convergence theorem. Hence, by (28), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307">View MathML</a> is a continuous operator. We now show that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M333">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M335">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M336">View MathML</a> bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a> that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M338">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M339">View MathML</a>

Then by Lemma 5 and (9), the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M340">View MathML</a>

are satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M342">View MathML</a>, and therefore, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M343">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a>. Moreover, the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M345">View MathML</a>

holds for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M342">View MathML</a> (cf. (9)). Consequently, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M348">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. Hence, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M333">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a> by the Arzelà-Ascoli theorem. As a result, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307">View MathML</a> is a compact operator and the condition (i) follows.

Due to the fact that by Lemma 7 any fixed point u of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M353">View MathML</a> is a solution of problem (17a) and (17b), Lemma 8 guarantees that u satisfies inequality (20). Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M354">View MathML</a> has property (ii). Consequently, by Lemmas 1 and 8, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192">View MathML</a>, problem (7a) and (7b) has a solution u satisfying estimates (19)-(21). □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53">View MathML</a> be a solution of problem (7a) and (7b) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192">View MathML</a>. The following property of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358">View MathML</a> is an important prerequisite for solving problem (1a) and (1b).

Lemma 10Let (H1)-(H3) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53">View MathML</a>be a solution of problem (7a) and (7b) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M192">View MathML</a>. Then the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358">View MathML</a>is uniformly integrable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>.

Proof

By Lemma 9, the inequalities

(29)

(30)

(31)

hold, where S is a positive constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M366">View MathML</a>. Hence, by (3) and (4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M367">View MathML</a>

(32)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M369">View MathML</a>, see Remark 1. Since the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M370">View MathML</a> is uniformly integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> (cf. [[20], criterion A.4], [21,22]), it follows from (32) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358">View MathML</a> is uniformly integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a> and the result follows. □

5 The existence result for BVP (1a) and (1b)

This section is devoted to the main result on the existence of positive solutions to the original BVP (1a) and (1b).

Theorem 1Let (H1)-(H3) hold. Then problem (1a) and (1b) has at least one positive solution.

Proof By Lemma 9, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81">View MathML</a>, problem (7a) and (7b) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53">View MathML</a> satisfying inequalities (29)-(31), where S is a positive constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M366">View MathML</a>. Moreover, by Lemma 10, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M377">View MathML</a> is uniformly integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. We now prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M379">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M53">View MathML</a> is a fixed point of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M307">View MathML</a> given in (27), the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M383">View MathML</a>

holds for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M58">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M81">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M386">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M387">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M388">View MathML</a>. By Lemma 3(i), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M389">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M390">View MathML</a>. Integrating by parts yields

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M392">View MathML</a>

(33)

follows. By Lemma 4 (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M393">View MathML</a>), the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M394">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. Since the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M358">View MathML</a> is uniformly integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M398">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. Hence, it follows from (33), that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M400">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. We summarize: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M402">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M404">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M69">View MathML</a>. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M406">View MathML</a>. Using appropriate subsequences, if necessary, we can assume, by the Arzelà-Ascoli theorem and the Bolzano-Weierstrass theorem, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M402">View MathML</a> is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M168">View MathML</a> is convergent in ℝ. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M410">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M411">View MathML</a>. With <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M412">View MathML</a> in (29)-(31), we conclude

In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M249">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M416">View MathML</a>

(34)

it follows from Lemma 2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M417">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M418">View MathML</a>. We now deduce from the inequality (cf. Lemma 5)

that

Taking the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M412">View MathML</a> in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M422">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M423">View MathML</a>

(35)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M424">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M73">View MathML</a> by Lemma 3(ii). This means that u is a positive solution of problem (1a) and (1b) and the result follows. □

6 Numerical simulations

For the numerical simulation, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M426">View MathML</a> and use an alternative formulation of problem (1a) and (1b),

(36a)

(36b)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M429">View MathML</a> is a parameter. We can use the above formulation because problem (1a) and (1b) is solvable for f satisfying the assumptions of Theorem 1 and, therefore, solutions of problem (1a) and (1b) can be computed as solutions of problem (36a) and (36b) using the proper value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M430">View MathML</a> depending on f. The values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M431">View MathML</a> are provided for given f in Examples 1 and 2, below.

The reason for changing the boundary conditions from (1b) to (36b) is that the differential equation (36a) subject to (1b) is not well posed; see [23]. However, to enable successful numerical treatment, well-posedness of the model is crucial. This property means that Eq. (36a) subject to proper boundary conditions has at least a locally unique solution,a and this solution depends continuously on the problem data. The well-posedness of the problem is important for two reasons. First of all, it allows to express errors in the solution of the analytical problem in terms of modeling errors and data errors (all measured via appropriate norms). Therefore, when the errors in the data become smaller due to more precise modeling or smaller measurement inaccuracies, the errors in the solution will decrease. The second reason is that the well-posedness decides if the numerical simulation will be at all successful. If the analytical problem is ill-posed, then the inevitable round-off errors can become extremely magnified and fully spoil the accuracy of the approximation.

In what follows, we work with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M432">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M433">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M435">View MathML</a> and, according to the next numerical approach (see Section 6.2), we consider Eq. (36a), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M436">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M437">View MathML</a>

(37)

By [23], problem (37), (36b) is well posed and therefore it is suitable for the numerical treatment. To see this, we need to look at a general solution of the homogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M438">View MathML</a>

(38)

If we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M439">View MathML</a>, we arrive at the characteristic polynomial of (38),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M440">View MathML</a>

whose roots <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M441">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M442">View MathML</a> are positive. Therefore, conditions for u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M278">View MathML</a> can be prescribed at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M444">View MathML</a> as it is done in (36b).

6.1 MATLAB Code bvpsuite

To illustrate the analytical results discussed in the previous section, we solved numerically examples of the form (36a) and (36b) using a MATLAB™ software package bvpsuite 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 two to eight. Collocation has been investigated in the 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 the 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. 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., [3,9,27,28].

6.2 Preliminaries

Before dealing with two nonlinear models specified in Sections 6.3 and 6.4, we have to compute numerical solutions for a simpler linearc model of the form

(39a)

(39b)

where a was chosen as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M447">View MathML</a>. Since in this case the exact solution is given, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M448">View MathML</a>, the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M449">View MathML</a> is available, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M450">View MathML</a>, respectively. In Figure 1, the numerical solutions of BVPs (39a) and (39b) are shown. They will be used as starting values for the numerical solution of Examples 1 and 2; see Sections 6.3 and 6.4, respectively. All numerical results have been obtained using collocation with five Gaussian collocation points on an equidistant grid (justified by a very simple solution structure) with the step size 0.01.

thumbnailFigure 1. Problem (39a) and (39b): Numerical solutions for different values ofa.

6.3 Example 1

We first investigate the following problem:

(40a)

(40b)

The nonlinearity f in (40a) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M453">View MathML</a>

(41)

and it satisfies (H1)-(H3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M454">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M455">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M456">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M457">View MathML</a>

It follows from Theorem 1 that there exists at least one value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M458">View MathML</a> such that the related solution u of problem (40a) and (40b) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M459">View MathML</a> is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M460">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a>. Using formula (35), we now determine an interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M462">View MathML</a> containing all admissible values of c.

Let u be a solution of problem (1a) and (1b) with f from (41). Then by (35), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M463">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M464">View MathML</a>

(42)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M465">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M466">View MathML</a>

(43)

Then (42) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M467">View MathML</a> and due to (35) and (41),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M468">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M469">View MathML</a>

(44)

In order to solve the nonlinear problem (40a) and (40b), we first have to solve a series of auxiliary problems for parameter-dependent differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M470">View MathML</a>

(45)

We begin the calculations with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M471">View MathML</a> and increase its value gradually until we arrive at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M472">View MathML</a>; cf. (40a). In each step we use the solution of the previous problem to solve the next one. The aim is to find a good starting value for both the solution u and the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M449">View MathML</a> before solving the BVP (40a) and (40b), i.e., find the final value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M474">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a>.

In the case of Example 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M476">View MathML</a>, this chain has the following structure:

1. Numerical approximation of BVP (39a) and (39b) is used as an initial guess for ODE (45) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M477">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M479">View MathML</a>.

2. Use the above approximation as an initial guess for ODE (45) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M480">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482">View MathML</a>.

3. Use the above approximation as an initial guess for ODE (45) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M483">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482">View MathML</a>.

4. Use the above approximation as an initial guess for ODE (45) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M486">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482">View MathML</a>.

After the last step, we have solved problem (40a) and (40b) subject to boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M482">View MathML</a>. In this case, the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M491">View MathML</a> was not small enough to consider it a reasonable approximation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a>. Therefore, we use a shooting idea combined with a bisection strategy to find a better value for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M493">View MathML</a>. The complete numerical results for Example 1 can be found in Table 1 and Figure 2.

thumbnailFigure 2. Problem (40a) and (40b): Numerical solutions for different values ofa. Values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M494">View MathML</a>.

Table 1. Problem (40a) and (40b): Complete data of the numerical simulation for different values ofa

6.4 Example 2

The above approach has been also accordingly applied for Example 2. Here, we consider the problem

(46a)

(46b)

The right-hand side f in Eq. (46a) now reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M499">View MathML</a>

(47)

and has a singularity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4">View MathML</a>. The function f satisfies conditions (H1)-(H3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M454">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M502">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M456">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M504">View MathML</a>

Theorem 1 guarantees the existence of at least one <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M458">View MathML</a> such that a solution u of problem (46a) and (46b) is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M460">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a> holds. We now again determine an interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M508">View MathML</a> containing all such values of c. Let u be a solution of problem (1a) and (1b) with f given in (47). Inequality (19) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M509">View MathML</a>

and hence by (35),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M510">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M511">View MathML</a>

(48)

For Example 2, the auxiliary ODE is constructed using ODE (39a),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M512">View MathML</a>

(49)

For all values of a, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M513">View MathML</a> and analogously carry out the path-following in δ first. The related chain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M476">View MathML</a> is as follows.

1. Numerical approximation of BVP (39a) and (39b) is used as an initial guess for ODE (49) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M477">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M517">View MathML</a>.

2. Use the above approximation as an initial guess for ODE (49) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M480">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M517">View MathML</a>.

3. Use the above approximation as an initial guess for ODE (49) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M483">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M523">View MathML</a>.

4. Use the above approximation as an initial guess for ODE (49) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M524">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M526">View MathML</a>.

5. Use the above approximation as an initial guess for ODE (49) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M486">View MathML</a> subject to terminal conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M529">View MathML</a>.

After the last step, we have solved BVP (46a) and (46b) subject to boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M529">View MathML</a>, but also, in this case, the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M491">View MathML</a> is too large and we have to find a better value for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M493">View MathML</a>. The complete numerical results for Example 2 can be found in Table 2 and Figure 3.

thumbnailFigure 3. Problem (46a) and (46b): Numerical solutions for different values ofa. Values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M534">View MathML</a>.

Table 2. Problem (46a) and (46b): Complete data of the numerical simulation for different values ofa

7 Conclusions

In the present article, we deal with the existence of positive solutions to the singular Dirichlet problem of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M537">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M2">View MathML</a>, and the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M3">View MathML</a> may be singular at the space variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M4">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M5">View MathML</a>. The main result for the existence of positive solutions of the above BVP is Theorem 1. It is illustrated by numerical simulations using the MATLAB code bvpsuite, based on polynomial collocation. For the successful numerical treatment, the above problem has to be reformulated to obtain its well-posed form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M542">View MathML</a>

Here, it is only known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M543">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M431">View MathML</a> can be specified depending on functions f arising in Examples 1 and 2. Now, a simple shooting method combined with the bisection idea is used to find c in such a way that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/6/mathml/M234">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

IR and SS contributed to the analytical part of the work and AS and EBW contributed to its numerical part. All authors read and approved the final version of the manuscript.

Acknowledgements

Dedicated to Jean Mawhin on the occasion of his 70th birthday.

This research was supported by the grant Matematické modely a struktury, PrF-2012-017. The authors thank the referees for suggestions which improved the paper.

End notes

  1. This BVP can have more than one solution, but they may not lay close together.

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

  3. The nonlinear term in f has been omitted; see (40a) and (46a).

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