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

Open Access Research

A one-dimensional prescribed curvature equation modeling the corneal shape

Isabel Coelho12, Chiara Corsato3 and Pierpaolo Omari3*

Author Affiliations

1 Area Departamental de Matemática, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro 1, Lisboa, 1950-062, Portugal

2 Département de Mathématique, Université Libre de Bruxelles, CP 214 Boulevard du Triomphe, Bruxelles, 1052, Belgium

3 Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, Trieste, 34127, Italy

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Boundary Value Problems 2014, 2014:127  doi:10.1186/1687-2770-2014-127

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/127


Received:17 December 2013
Accepted:2 May 2014
Published:20 May 2014

© 2014 Coelho et al.; licensee Springer.

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

Abstract

We prove existence, uniqueness, and stability of solutions of the prescribed curvature problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M1">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M3">View MathML</a>, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5">View MathML</a>. We also develop a linear monotone iterative scheme for approximating the solution. This equation has been proposed as a model of the corneal shape in the recent paper (Okrasiński and Płociniczak in Nonlinear Anal., Real World Appl. 13:1498-1505, 2012), where a simplified version obtained by partial linearization has been investigated.

Keywords:
mean curvature equation; mixed boundary condition; positive solution; existence; uniqueness; linear stability; order stability; Lyapunov stability; lower and upper solutions; monotone approximation; topological degree

1 Introduction

In this paper we study existence, uniqueness, stability, and approximation of classical solutions of the one-dimensional prescribed curvature problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M6">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5">View MathML</a> are given constants. This problem, together with its N-dimensional counterpart

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M9">View MathML</a>

(2)

has been proposed in [1-4] as a mathematical model for the geometry of the human cornea. However, in these papers a simplified version of (2) has been investigated, where the mean curvature operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M10">View MathML</a> has been replaced by its linearization <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M11">View MathML</a> around 0. In particular, it has been proved in [1] that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M12">View MathML</a>, then the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M13">View MathML</a>

(3)

has a unique solution which is the limit of a sequence of successive approximations. The above limitations on the parameters have recently been removed in [4].

Unlike all these works we tackle here the fully nonlinear problem (1) and we prove the existence of a unique solution for the whole range of positive parameters a, b. The study of problem (1) requires some care because, even if pairs of constant lower and upper solutions can easily be exhibited, the presence of the curvature term rules out in general the possibility of applying the standard existence results, due to the possible occurrence of derivative blow-up phenomena (see, e.g., [5]). On the other hand, the non-variational structure of (1) puts the problem, as it stands, out of the scope of the methods developed in [6-8] for the curvature equation. Nevertheless, we show that an a priori bound in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M14">View MathML</a> for all possible solutions can be obtained by an elementary, but delicate, argument which exploits the qualitative properties - positivity, monotonicity, and concavity - of the solutions themselves. These estimates eventually enable us to use a degree argument in order to prove the existence of solutions. The proof of the uniqueness is then based on suitable fixed point index calculations, which are performed via linearization. A similar approach, applied to an associated evolutionary problem, is exploited for detecting the linear stability of the solution.

Next, taking inspiration from [9,10], we develop a linear iterative scheme for approximating the solution by two monotone sequences of strict lower and upper solutions, starting from an explicit pair of constant lower and upper solutions. These two sequences, besides providing accurate two-sided bounds on the solution, yield the strict order stability and hence, according to [11], the (Lyapunov) asymptotic stability of the solution itself, yielding as well an explicitly computable estimate of its basin of attractivity. We finally illustrate the use of this approximation scheme in order to compute numerically the solution u of (1) for the same choice of the parameters a and b as the one considered in [1].

We finally mention that part of our results extends to the general N-dimensional problem (2); this topic will be discussed elsewhere.

2 Existence, qualitative properties and approximation

In this section we are concerned with the study of the existence, the qualitative properties and the approximation of classical solutions, i.e., belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a>, of problem (1), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5">View MathML</a> are fixed constants. Clearly, problem (1) can be written in the equivalent form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M18">View MathML</a>

(4)

Let us set for convenience, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M20">View MathML</a>

(5)

It is obvious that, due to the symmetry properties of the function f, the mixed problem (4) is equivalent to the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M21">View MathML</a>

Notations As usual, for functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M22">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M23">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M24">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M25">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M27">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M24">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M30">View MathML</a>. We also write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M31">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M24">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M33">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M34">View MathML</a> and, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M36">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M38">View MathML</a> denote the left Dini derivatives; this is equivalent to requiring that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M40">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M41">View MathML</a>. Whenever no confusion occurs, we omit the indication of the interval.

Existence, uniqueness, and linear stability

We start with a preliminary result, where some properties of the possible solutions of problem (1) are highlighted.

Lemma 2.1The following assertions hold.

(i) Any solutionuof (1) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M42">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M44">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>.

(ii) Any solutionuof (1) is such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M47">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M48">View MathML</a>.

(iii) Any solutionuof (1) is such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M49">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M50">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M51">View MathML</a>.

Proof In the following steps u denotes a solution of (1), or equivalently of (4). From the equation in (4) it follows that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M52">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M54">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M55">View MathML</a>

(6)

Step 1. Proof of (i). Let us first show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Assume by contradiction that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M58">View MathML</a>. The boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M59">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M60">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M61">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63">View MathML</a>. Condition (6) yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M64">View MathML</a>. Hence there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M66">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M67">View MathML</a> and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M68">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M67">View MathML</a>, which is a contradiction. Now suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M70">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M72">View MathML</a> and condition (6) yields again a contradiction. Hence we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. In a completely similar way we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M75">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>.

Next, in order to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M77">View MathML</a>

(7)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M43">View MathML</a>, it is sufficient to note that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M79">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M80">View MathML</a>, then (6) would yield <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M81">View MathML</a>, which is impossible. Moreover, as the constant function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M82">View MathML</a> is a solution of the equation in (4), the uniqueness of solutions for any Cauchy problem associated with this equation implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M83">View MathML</a>

(8)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>.

Step 2. Proof of (ii). As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63">View MathML</a>, assertion (i) implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M87">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M66">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M90">View MathML</a>. Let us show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M91">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M48">View MathML</a>. If this is not the case, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M93">View MathML</a>. Then, by (6), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M94">View MathML</a> and hence there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M95">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M96">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M97">View MathML</a>, which contradicts the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M98">View MathML</a>.

Let us now prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M99">View MathML</a>

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. By contradiction, assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M101">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M102">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M103">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M106">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M107">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M108">View MathML</a>. Since we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M109">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M110">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M111">View MathML</a> is decreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M112">View MathML</a>. We also know that the function au is decreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Hence the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M114">View MathML</a> is decreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M115">View MathML</a>. On the other hand, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M106">View MathML</a>, from the equation in (4) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M117">View MathML</a>

and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M118">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M110">View MathML</a>. Then the equation in (4) yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M120">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M121">View MathML</a>, which is a contradiction.

Step 3. Proof of (iii). Since by assertion (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M124">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, and then, from the equation in (4), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M126">View MathML</a>

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Multiplying by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M128">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M129">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> by assertion (ii), and integrating between 0 and 1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M131">View MathML</a>

(9)

On the one hand, using the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M133">View MathML</a>

On the other hand, the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M134">View MathML</a> and assertion (i) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M135">View MathML</a>

In conclusion, setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M136">View MathML</a>

(10)

we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M137">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M128">View MathML</a> is non-increasing, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M139">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>. □

We are now in position to prove the existence of a unique solution of problem (1), which is linearly stable.

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5">View MathML</a>be given. Then there exists a unique solutionuof (1) and it satisfies the conditions (i), (ii), and (iii). Further, uis linearly stable as an equilibrium of the parabolic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M143">View MathML</a>

(11)

Proof The proof is divided into three steps.

Step 1. Existence. Let us prove that there exists at least one solution of (1), or equivalenty of (4). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M144">View MathML</a> be the operator which associates with any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M145">View MathML</a> the unique solution w of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M146">View MathML</a>

Clearly, is completely continuous. Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M148">View MathML</a> be the Nemitski operator associated with f, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M149">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M150">View MathML</a>. The operator is continuous and maps bounded sets into bounded sets. Introduce the open bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M153">View MathML</a>

(12)

Finally, define a completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M154">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M155">View MathML</a>. The fixed points of are precisely the solutions of (4).

An inspection of the assertions of Lemma 2.1 shows that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M157">View MathML</a> satisfies, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M159">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M160">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M161">View MathML</a>. The invariance property of the degree under homotopy implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M162">View MathML</a>

(13)

where ℑ stands for the identity operator. Therefore there exists a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M163">View MathML</a> of , which is a solution of (4).

By Lemma 2.1, u satisfies the conditions (i), (ii), and (iii).

Step 2. Uniqueness. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M165">View MathML</a>. As the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M166">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M14">View MathML</a>, the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M168">View MathML</a> and, hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M169">View MathML</a> are of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M14">View MathML</a>, with Fréchet differentials

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M171">View MathML</a>

for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M150">View MathML</a>.

Observe that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M175">View MathML</a> is invertible. Indeed, let us fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172">View MathML</a> and assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M177">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M150">View MathML</a>. This means that w is the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M179">View MathML</a>

(14)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M180">View MathML</a>

(15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M19">View MathML</a>, the maximum principle [[5], Appendix, Theorem 5.2] implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M182">View MathML</a>. Hence the local inversion theorem applies to Φ at every point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172">View MathML</a> and thus any fixed point of is isolated. The compactness in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152">View MathML</a> of the set of all fixed points of then implies that is finite, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M189">View MathML</a> for some positive integer N.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M190">View MathML</a> the open ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152">View MathML</a> centered at u and having radius r. Pick <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M192">View MathML</a> so small that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M193">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M194">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M195">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M196">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M197">View MathML</a>. The excision and the additivity properties of the degree yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M198">View MathML</a>

(16)

where, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M199">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M200">View MathML</a>

denotes the fixed point index of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M201">View MathML</a>. Using again (15), we see as above that, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M203">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M204">View MathML</a>

has no non-trivial solution w. Accordingly, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M172">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M206">View MathML</a> does not have any eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M203">View MathML</a>. Therefore, we infer from [[12], Theorem 3.20] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M208">View MathML</a>

(17)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M199">View MathML</a>. Finally, by (13), (16), and (17) we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M210">View MathML</a>, i.e., there is a unique solution u of problem (4).

Step 3. Linear stability. The solution u of (4) is an equilibrium of the parabolic problem (11), in particular, it is a 1-periodic solution of (11). In order to show that u is linearly stable, and hence locally exponentially asymptotically stable, it is enough, after a standard cut-off argument, to show that the eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M211">View MathML</a>

(18)

does not have any eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M212">View MathML</a> (see, e.g., [[13], Chapter III.23]), or [[14], Chapter V.22]). Indeed, if w is a solution of (18) for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M212">View MathML</a>, then using again condition (15), together with the interior form of the parabolic maximum principle and the Hopf boundary point lemma (see, e.g., [[14], Chapter III.13]), we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M182">View MathML</a>. □

Monotone approximation and order stability

In this section we discuss approximation and stability of the solution of (1), or equivalently of (4). To this end, we define a linear iterative scheme that allows one to construct an increasing sequence of strict lower solutions and a decreasing sequence of strict upper solutions of (4) which converge in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a> to the unique solution u of (4), that is, of (1). Then, according to [11,13], we see that u is strictly order stable from above and from below and hence it is (Lyapunov) asymptotically stable as an equilibrium of the parabolic problem (11). In addition, the converging sequences of lower and upper solutions provide explicitly computable estimates of the basin of attractivity of the solution.

Lower and upper solutions Let us consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M216">View MathML</a>

(19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M217">View MathML</a> is locally Lipschitz continuous. A lower solution of (19) is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M218">View MathML</a> which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M219">View MathML</a>

Similarly an upper solution of (19) is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M220">View MathML</a> which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M221">View MathML</a>

Remark 2.1 The Lipschitz character of g implies (see [[5], Chapter 3, Proposition 1.7, Proposition 2.7]) that a lower solution α of (19), which is not a solution, is a strict lower solution, that is, any solution u of (19), such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M222">View MathML</a>, satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M223">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Similarly, an upper solution β of (19), which is not a solution, is a strict upper solution, that is, any solution u of (19), such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M225">View MathML</a>, satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M226">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>.

Remark 2.2 Any constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M228">View MathML</a> is a strict lower solution of (4) and any constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M229">View MathML</a> is a strict upper solution of (4). In particular, one can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M230">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M231">View MathML</a>. We wish to point out that, with this choice of lower and upper solutions, the existence of at least one solution u of problem (4) between α and β can be alternatively achieved by applying [[5], Chapter 2, Theorem 3.1]; the relevant observations being here the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M233">View MathML</a> and f satisfies the one-sided Nagumo condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M234">View MathML</a>

(20)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M19">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M236">View MathML</a>. We point out that one-sided Nagumo conditions were introduced for the first time by Kiguradze in [15].

Let us consider the following modified problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M237">View MathML</a>

(21)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M238">View MathML</a> is defined as follows. We first introduce an auxiliary function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M239">View MathML</a> by setting, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M240">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M241">View MathML</a>

where R is defined in (10). Then we set, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M240">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M243">View MathML</a>

(22)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244">View MathML</a> is locally Lipschitz continuous and satisfies the following conditions:

(h1) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M245">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M246">View MathML</a>

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M247">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M248">View MathML</a>;

(h2) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M249">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M250">View MathML</a>

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M251">View MathML</a>.

We can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M252">View MathML</a> in (h1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M253">View MathML</a> in (h2).

Remark 2.3 Any constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M228">View MathML</a> and any constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M229">View MathML</a> are, respectively, a strict lower and a strict upper solution of (21) too.

Lemma 2.3A function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M256">View MathML</a>is a solution of (21) if and only if it is a solution of (1).

Proof Let u be a solution of (21). In order to prove that u is also a solution of (1), or equivalently of (4), it is sufficient to show that u satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M258">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244">View MathML</a> satisfies the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M261">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M262">View MathML</a>

It can be verified, proceeding as in the proof of Lemma 2.1 and using the maximum principle, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M263">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M264">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>.

Next we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M258">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. The proof of assertion (ii) in Lemma 2.1 can be repeated verbatim in order to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M66">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M270">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Assume now that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M272">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M273">View MathML</a>. In particular, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M258">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M275">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M276">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277">View MathML</a>. By definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244">View MathML</a>, u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M279">View MathML</a>

(23)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M275">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M281">View MathML</a>

(24)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M129">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, we easily get from (23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M286">View MathML</a>

(25)

From (24), using again <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M129">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M290">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M291">View MathML</a>

(26)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M293">View MathML</a>

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M295">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, we finally get from (26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M297">View MathML</a>

(27)

Combining (25) and (27) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M298">View MathML</a>

which is precisely (9) in Step 3 of the proof of Lemma 2.1. As there we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M299">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Accordingly, u is a solution of (4).

Conversely, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244">View MathML</a> implies that any solution of (1), or equivalently of (4), is a solution of (21) as well. □

Let us consider the following auxiliary linear problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M302">View MathML</a>

(28)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M303">View MathML</a> is a continuous function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M305">View MathML</a> are given constants. Notice that problem (28) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M306">View MathML</a>. The following result is inspired from [9] and [[10], Chapter 5].

Lemma 2.4There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M307">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M145">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M310">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M312">View MathML</a>, the solutionwof (28) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M313">View MathML</a>

(29)

In addition, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M314">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M316">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M317">View MathML</a>

(30)

Proof Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319">View MathML</a> the respective solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M320">View MathML</a>

(31)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M321">View MathML</a>

(32)

Step 1. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M322">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M324">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M325">View MathML</a>

(34)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M326">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M327">View MathML</a>

(36)

A simple computation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M328">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M329">View MathML</a>

The conclusion then easily follows by direct calculations.

Step 2. There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M307">View MathML</a> such that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M332">View MathML</a>

(37)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M333">View MathML</a>

(38)

Let us first show that (37) holds. We have, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M334">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M335">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M337">View MathML</a>, we can conclude that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M338">View MathML</a> sufficiently large, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M339">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>. Namely, if we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M341">View MathML</a>

(39)

and we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M343">View MathML</a>

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M344">View MathML</a>, the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M345">View MathML</a>

holds as well. This yields the validity of (37).

As for (38), by the sign properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M347">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M348">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M349">View MathML</a> provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M350">View MathML</a>: this condition holds as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M351">View MathML</a>.

Fix now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M145">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M310">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M312">View MathML</a>. Let w be the solution of problem (28). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M357">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M358">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M359">View MathML</a>, then (29) trivially follows. Therefore suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M314">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M316">View MathML</a>. We can express w as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M363">View MathML</a>

(40)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>. Inequality (29) now reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M365">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>. The sign properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M369">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M370">View MathML</a> and the assumptions on h and m immediately yield (30). □

We introduce now a linear monotone iterative scheme for approximating the solution of (1); namely, we define by recurrence two sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372">View MathML</a> as follows:

let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M373">View MathML</a>be any constant, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M374">View MathML</a>, and, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M376">View MathML</a>be the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M377">View MathML</a>

(41)

let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M378">View MathML</a>be any constant, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M379">View MathML</a>, and, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M381">View MathML</a>be the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M382">View MathML</a>

(42)

Theorem 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5">View MathML</a>be given. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M307">View MathML</a>, given by (39), such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M386">View MathML</a>the sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372">View MathML</a>recursively defined in (41) and (42), respectively, converge in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a>to the unique solutionuof (21) and hence of (1). In addition, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>the following conditions hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a>is a strict lower solution and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392">View MathML</a>is a strict upper solution of (21), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M393">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>.

Proof Let us fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M396">View MathML</a> is given by (39).

Step 1. The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> is such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M399">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a> is a strict lower solution of (21).

The proof is done by induction. Define, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M403">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M404">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M405">View MathML</a>

(43)

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M406">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Hence the maximum principle implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M408">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M409">View MathML</a>, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Now, let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M411">View MathML</a> is a strict lower solution of (21). Using the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M412">View MathML</a>, together with conditions (h1) and (h2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M413">View MathML</a>

(44)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M404">View MathML</a> is a solution of (43), which is of the form of (28), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M416">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M312">View MathML</a>, Lemma 2.4 applies and yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M419">View MathML</a>

(45)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>. From (44), (45) and from the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M421">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M411">View MathML</a> is a strict lower solution of (21).

Assume now that, for some integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M423">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a> is strict lower solution of (21) satisfying the boundary conditions. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M425">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M426">View MathML</a>

(46)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a> is strict and satisfies the boundary conditions, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M428">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Hence the maximum principle yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M430">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M431">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Finally, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M376">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M434">View MathML</a>

(47)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M425">View MathML</a> is the solution of problem (46), which is of the form of (28), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M437">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M359">View MathML</a>, Lemma 2.4 applies and yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M440">View MathML</a>

(48)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>. From (47), (48), and from the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M442">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M376">View MathML</a> is a strict lower solution of (21), such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M444">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>.

In a similar way, one can prove the following conclusion.

Step 2. The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372">View MathML</a> is such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M448">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392">View MathML</a> is a strict upper solution of (21).

Step 3. We have, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M452">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M455">View MathML</a>

where, clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M457">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M458">View MathML</a>

(49)

By construction, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M459">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M461">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M462">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M463">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>.

Take now any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a> and suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M466">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M467">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. From (49) we infer, using the maximum principle, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M469">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Let us prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M471">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. We easily see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M473">View MathML</a>

(50)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M456">View MathML</a> is the solution of problem (49), which is of the form of (28), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M476">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M359">View MathML</a>, Lemma 2.4 applies and yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M479">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M480">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M45">View MathML</a>. The conclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M466">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M483">View MathML</a>, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>, then follows by induction.

Step 4. There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M486">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M488">View MathML</a>

(51)

We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M489">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M491">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M492">View MathML</a>. Hence we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M493">View MathML</a>

(52)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M423">View MathML</a>. Let us set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M495">View MathML</a>.

Suppose, by contradiction, that (51) does not hold, i.e., for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M496">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M497">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M498">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M499">View MathML</a>. Assume that the former eventuality occurs. By Step 1, using conditions (h1) and (h2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M500">View MathML</a>

(53)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M502">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M503">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M504">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M505">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M506">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M507">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M508">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M509">View MathML</a>. From (53) we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M510">View MathML</a>

(54)

The right-hand side of (54) diverges as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M511">View MathML</a>, then a contradiction follows.

In a completely similar way, we achieve the conclusion if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M512">View MathML</a>, or if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M499">View MathML</a>.

Step 5. The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a> to the solution u of (1).

It follows from the previous steps that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> is increasing and bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a>. Therefore there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M518">View MathML</a> which is the pointwise limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M519">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>; in particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M521">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M375">View MathML</a>. Moreover, by the Arzelà-Ascoli theorem, any subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M524">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> admits a subsequence which is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152">View MathML</a> to u. Then the whole sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M519">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152">View MathML</a> to u. From the equation in (41) we see that the convergence takes place in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a>. Hence u is a solution of problem (21) and, by Lemma 2.3 and Theorem 2.2, it is in fact the unique solution of problem (1).

In a similar way, one can prove the following conclusion.

Step 6. The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a> to the solution u of (1).

Thus the proof is completed. □

Corollary 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5">View MathML</a>be given. Then the unique solutionuof (21) is (Lyapunov) globally asymptotically stable as an equilibrium of the parabolic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M534">View MathML</a>

(55)

Proof Let us note that any lower, respectively upper, solution of (21) is a lower, respectively upper, solution of the parabolic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M535">View MathML</a>

(56)

Arguing as in the proof of Theorem 2.2 we see that u is the unique solution of (56). Then Theorem 2.5 implies that u is strictly order stable from below and from above. Actually, since any constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M536">View MathML</a> is a strict lower solution and any constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M537">View MathML</a> is a strict upper solution of (56), the results in [[11], Section 2.6] imply that u is (Lyapunov) globally asymptotically stable as a solution of (56) and hence as an equilibrium of (55). □

Remark 2.4 The definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M244">View MathML</a> implies that the solution u of (1) is strictly order stable from below and from above and (Lyapunov) asymptotically stable as an equilibrium of the parabolic problem (11).

Numerical experiments

We present here some experiments concerning the numerical approximation of the solution of problem (1), for the same choice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M539">View MathML</a> of the parameters as in [1].

The iterative scheme in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308">View MathML</a> We have computed various approximations, at different precision levels, of the unique solution u of problem (1) by implementing in MatLab the linear iterative scheme defined by (41) and (42); at each step of the iteration the resulting linear equations have been solved using the bvp4c routine with a 100-point grid. We have chosen <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M541">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M542">View MathML</a> given by (39), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M544">View MathML</a>. Theorem 2.5 guarantees that the approximating sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M519">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372">View MathML</a> are constituted by lower and upper solutions and monotonically converge to u, in an increasing or decreasing fashion, respectively; thus, for each n, the couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M548">View MathML</a> brackets the solution u, thus providing lower and upper estimates. In what follows the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M549">View MathML</a>-norm of a given function is intended to have been computed as the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M549">View MathML</a>-norm of its discretization on the given grid. We have denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M551">View MathML</a> the minimum number of iterations needed in order that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M552">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M553">View MathML</a>; the corresponding values are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M554">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M555">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M556">View MathML</a>. In Table 1 we have tabulated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M557">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M558">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M559">View MathML</a>, at the mesh points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M560">View MathML</a>; the graphs of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M557">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M558">View MathML</a> are displayed in Figure 1; whereas Figure 2 describes the rate of decay of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M563">View MathML</a>, as well as of the errors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M564">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M565">View MathML</a>, plotted against the number n of iterations. Here u denotes a reference approximation of the solution of (1), calculated using the same scheme up to a precision of 10−5. Although the lower solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a> converge slightly faster than the upper solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392">View MathML</a>, it is evident that the monotone iterative scheme defined by (41) and (42) turns out to be extremely slow.

thumbnailFigure 1. Graphs of the approximations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M568">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M569">View MathML</a>(in violet),<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M570">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M571">View MathML</a>(in green) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M572">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M573">View MathML</a>(in blue), with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M574">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M575">View MathML</a>, defined by (41), (42) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M576">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M577">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M578">View MathML</a>.

thumbnailFigure 2. Graphs of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M579">View MathML</a>(in blue),<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M580">View MathML</a>(in green) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M581">View MathML</a>(in violet), for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M576">View MathML</a>, plotted against the numbernof iterations.

Table 1. Values of the approximations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M557">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M558">View MathML</a>, defined by (41), (42) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M541">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M552">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M553">View MathML</a>

The iterative scheme in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M594">View MathML</a> We start from the obvious observation that the iterative scheme given by (41) and (42) is well defined for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M595">View MathML</a>; hence it is clear that, if the resulting sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372">View MathML</a> are Cauchy sequences in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M152">View MathML</a>, then, by the uniqueness of the solution of (1), they converge in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M15">View MathML</a> to u. Of course, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M600">View MathML</a> we cannot anymore guarantee that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a> is a lower solution, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M392">View MathML</a> is an upper solution, or the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M372">View MathML</a> enjoy any monotonicity property. Let us take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543">View MathML</a> in (41) and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> be the sequence of iterates obtained for some given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M595">View MathML</a>. The numerical experiments, we have performed for several different choices of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M594">View MathML</a>, show that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M371">View MathML</a> converges to u, but the magnitude of L strongly affects the speed of convergence; namely, as L decreases, the required number of iterations n in order that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M610">View MathML</a> goes beneath a prescribed threshold, decreases. In particular, the speed of convergence significantly increases as L approximates 1 and, for this choice of L, it becomes comparable even with the speed of Newton’s method. Indeed, if we fix an error tolerance of 10−3, the iterative scheme defined by (41), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M611">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543">View MathML</a>, converges in 4 iterations, whereas Newton’s method, starting from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M543">View MathML</a> too, converges in 2 iterations: these results are displayed in Tables 2 and 3. This computational remark suggests the possibility of using the iterative scheme also in case the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M308">View MathML</a> fails; however, its convergence properties should be theoretically analyzed.

Table 2. Values of the approximations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M391">View MathML</a>, defined by (41) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M611">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M617">View MathML</a>

Table 3. Values of the Newton approximations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M622">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M623">View MathML</a>

A comparison between the solutions of (1) and (3) Here we present a numerical comparison between the solution u of the fully nonlinear problem (1) and the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626">View MathML</a> of the partially linearized problem (3) investigated in [1]. We have approximated u by the lower solution obtained by implementing the monotone iterative scheme given by (41), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M627">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M541">View MathML</a> and stopping criterion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M629">View MathML</a>. An approximation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626">View MathML</a>, matching the one obtained in [1], has been calculated using the bvp4c routine of MatLab with a 100-point grid. Table 4 reports the values of u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626">View MathML</a> at the mesh points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M560">View MathML</a> and Figure 3 displays the graphs of u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626">View MathML</a>.

thumbnailFigure 3. Graphs ofu(in blue) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M634">View MathML</a>(in green).

Table 4. Values ofuand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M626">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

This paper was written under the auspices of INdAM-GNAMPA. The first named author has been supported by Fundação para a Ciência e a Tecnologia (SFRH/BD/61484/2009). The last two named authors have been supported by Università di Trieste, in the frame of the FRA projects ‘Equazioni differenziali ordinarie: aspetti qualitativi e numerici’ and ‘Nonlinear Ordinary Differential Equations: Qualitative Theory, Numerics and Applications’. They also wish to thank Igor Moret for some useful discussions.

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