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

Open Access Research

Solutions and Green’s function of the first order linear equation with reflection and initial conditions

Alberto Cabada* and Fernando Adrián Fernández Tojo

Author Affiliations

Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia, 15706, Spain

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Boundary Value Problems 2014, 2014:99  doi:10.1186/1687-2770-2014-99


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/99


Received:23 December 2013
Accepted:8 April 2014
Published:7 May 2014

© 2014 Cabada and Tojo; 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

This work is devoted to the study of the existence and sign of Green’s functions for first order linear problems with constant coefficients and initial (one point) conditions. We first prove a result on the existence of solutions of nth order linear equations with involutions via some auxiliary functions to later prove a uniqueness result in the first order case. We study then different situations for which a Green’s function can be obtained explicitly and derive several results in order to obtain information as regards the sign of the Green’s function. Once the sign is known, optimal maximum and anti-maximum principles follow.

Keywords:
equations with involutions; equations with reflection; Green’s functions; maximum principles; comparison principles; periodic conditions

1 Introduction

The study of functional differential equations with involutions (DEI) can be traced back to the solution of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M1">View MathML</a> by Silberstein (see [1]) in 1940. Briefly speaking, an involution is just a function f that satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M2">View MathML</a> for every x in its domain of definition. For most applications in analysis, the involution is defined on an interval of ℝ and in the majority of the cases, it is continuous, which implies it is decreasing and has a unique fixed point. Ever since that foundational paper of Silberstein, the study of problems with DEI has been mainly focused on those cases with initial conditions, with an extensive research in the case of the reflection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M3">View MathML</a>.

Wiener and Watkins study in [2] the solution of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M4">View MathML</a> with initial conditions. Equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M5">View MathML</a> has been treated by Piao in [3,4]. In [2,5-8] some results are introduced to transform this kind of problems with involutions and initial conditions into second order ordinary differential equations with initial conditions or first order two dimensional systems, granting that the solution of the last will be a solution to the first. Furthermore, asymptotic properties and boundedness of the solutions of initial first order problems are studied in [9] and [10], respectively. Second order boundary value problems have been considered in [8,11-13] for Dirichlet and Sturm-Liouville boundary value conditions, higher order equations has been studied in [14]. Other techniques applied to problems with reflection of the argument can be found in [15-17].

More recently, the papers of Cabada et al.[18,19] have further studied the case of the second order equation with two-point boundary conditions, adding a new element to the previous studies: the existence of a Green’s function. Once the study of the sign of the aforementioned function is done, maximum and anti-maximum principles follow. Other works in which Green’s functions are obtained for functional differential equations (but with a fairly different setting, like delay or normal equations) are, for instance, [20-25].

In this paper we try to answer to the following question: How is it possible find a solution of an initial problem with a differential equation with reflection? What is more, in which cases can a Green’s function be constructed and how can it be found?

Section 2 will have two parts. In the first one we construct the solutions of the nth order DEI with reflection, constant coefficients and initial conditions. In the second one we find the Green’s function for the order one case. In Section 3 we apply these findings in order to describe exhaustively the range of values for which suitable comparison results are fulfilled and we illustrate them with some examples.

2 Solutions of the problem

In order to prove an existence result for the nth order DEI with reflection, we consider the even and odd parts of a function f, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M7">View MathML</a> as done in [18].

2.1 The nth order problem

Consider the following nth order DEI with involution:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M8">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M10">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M11">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M12">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M13">View MathML</a>. A solution to this problem will be a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M14">View MathML</a>, that is, u is k times differentiable in the sense of distributions and each of the derivatives satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M15">View MathML</a> for every compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M16">View MathML</a>.

Theorem 2.1Assume that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18">View MathML</a>, functions such that satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M19">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M20">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M21">View MathML</a>

(2.4)

and also one of the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M22">View MathML</a>

Then problem (2.1) has a solution.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M23">View MathML</a>

Observe that φ is odd, ψ is even and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M24">View MathML</a>. So, in order to ensure the existence of solution of problem (2.1) it is enough to find y and z such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M26">View MathML</a> for, in that case, defining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M27">View MathML</a>, we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M28">View MathML</a>. We will deal with the initial condition later on.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M29">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M30">View MathML</a>

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M31">View MathML</a> is even if n is odd and vice versa. In particular, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M32">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M33">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M25">View MathML</a>.

All the same, by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M35">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M36">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M26">View MathML</a>.

Hence, defining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M38">View MathML</a> we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M41">View MathML</a>.

If we assume (h1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M42">View MathML</a> is clearly a solution of problem (2.1).

When (h2) is fulfilled a solution of problem (2.1) is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M43">View MathML</a>.

If (h3) holds, using the aforementioned construction we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M46">View MathML</a>. Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M47">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M48">View MathML</a>. Observe that the second part of condition (h2) is precisely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M49">View MathML</a>, and hence, defining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M50">View MathML</a> we see that w is a solution of problem (2.1). □

Remark 2.1 Having in mind condition (h1) in Theorem 2.1, it is immediate to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M51">View MathML</a> provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M52">View MathML</a>

In an analogous way for (h2), one can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M53">View MathML</a> when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M54">View MathML</a>

2.2 The first order problem

After proving the general result for the nth order case, we concentrate our work in the first order problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M55">View MathML</a>

(2.5)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M57">View MathML</a>. A solution of this problem will be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M58">View MathML</a>.

In order to do so, we first study the homogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M59">View MathML</a>

(2.6)

By differentiating and making the proper substitutions we arrive at the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M60">View MathML</a>

(2.7)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M61">View MathML</a>. Equation (2.7) presents three different cases:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62">View MathML</a>. In such a case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M63">View MathML</a> is a solution of (2.7) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M64">View MathML</a>. If we impose (2.6) onto this expression we arrive at the general solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M65">View MathML</a>

of (2.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66">View MathML</a>.

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67">View MathML</a>. Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M68">View MathML</a> is a solution of (2.7) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M64">View MathML</a>. To get (2.6) we arrive at the general solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M70">View MathML</a>

of (2.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66">View MathML</a>.

(C3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M72">View MathML</a>. In this a case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M73">View MathML</a> is a solution of (2.7) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M64">View MathML</a>. So, (2.6) holds provided that one of the two following cases is fulfilled:

(C3.1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M76">View MathML</a>

is the general solution of (2.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66">View MathML</a>, and

(C3.2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M79">View MathML</a>

is the general solution of (2.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M66">View MathML</a>.

Now, according to Theorem 2.1, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M83">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M84">View MathML</a>

(2.9)

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18">View MathML</a> can be obtained from the explicit expressions of the cases (C1)-(C3) by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M87">View MathML</a>.

Remark 2.2 Note that if u is in the case (C3.1), v is in the case (C3.2) and vice versa.

We have now the following properties of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18">View MathML</a>.

Lemma 2.2For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M90">View MathML</a>, the following properties hold.

(I) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M92">View MathML</a>for some real constantka.e.,

(II) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M94">View MathML</a>,

(III) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M95">View MathML</a>,

(IV) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M96">View MathML</a>.

Proof (I) and (III) can be checked by inspection of the different cases. (II) is a direct consequence of (I). (IV) is obtained from the definition of even and odd parts and (III). □

Now, Theorem 2.1 has the following corollary.

Corollary 2.3Problem (2.5) has a unique solution if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M97">View MathML</a>.

Proof Considering Lemma 2.2(III), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M18">View MathML</a>, defined as in (2.8) and (2.9), respectively, satisfy the hypothesis of Theorem 2.1, (h1), therefore a solution exists.

Now, assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M101">View MathML</a> are two solutions of (2.5). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M102">View MathML</a> is a solution of (2.6). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M102">View MathML</a> is of one of the forms covered in the cases (C1)-(C3) and, in any case, a multiple of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M17">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M105">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M106">View MathML</a>. Also, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M107">View MathML</a>, but we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M108">View MathML</a> as a hypothesis, therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M110">View MathML</a>. This is, problem (2.5) has a unique solution.

Assume now that w is a solution of (2.5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M111">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M112">View MathML</a> is also a solution of (2.5) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M106">View MathML</a>, which proves the result. □

This last theorem raises an obvious question: In which circumstances <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M108">View MathML</a>? In order to answer this question, it is enough to study the cases (C1)-(C3). We summarize this study in the following lemma, which can be checked easily.

Lemma 2.4<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M111">View MathML</a>only in the following cases:

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M117">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M118">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M120">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M121">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M123">View MathML</a>.

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M124">View MathML</a>. We define the oriented characteristic function of the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M125">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M126">View MathML</a>

Remark 2.3 The previous definition implies that, for any given integrable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M127">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M128">View MathML</a>

Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M129">View MathML</a>.

The following corollary gives the expression of the Green’s function for problem (2.5).

Corollary 2.5Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M97">View MathML</a>. Then the unique solution of problem (2.5) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M131">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M132">View MathML</a>

(2.10)

Proof First observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M133">View MathML</a> is bounded and of compact support for every fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M134">View MathML</a>, so the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M135">View MathML</a> is well defined. It is not difficult to verify, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M136">View MathML</a>, the following equalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M137">View MathML</a>

(2.11)

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M138">View MathML</a>

(2.12)

Thus, adding (2.11) and (2.12), it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M139">View MathML</a>.

We now check the initial condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M140">View MathML</a>

Using the construction of the solution provided in Theorem 2.1, it is an easy exercise to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M141">View MathML</a>

which proves the result. □

Denote now by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M142">View MathML</a> the Green’s function for problem (2.5) with coefficients a and b. The following lemma is analogous to [[18], Lemma 4.1].

Lemma 2.6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M143">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M144">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M145">View MathML</a> be a solution to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M139">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M147">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M148">View MathML</a>, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M149">View MathML</a>. On the other hand, by definition of v,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M150">View MathML</a>

therefore we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M152">View MathML</a>. □

As a consequence of the previous result, we arrive at the following immediate conclusion.

Corollary 2.7<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M142">View MathML</a>is positive if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M154">View MathML</a>is negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M155">View MathML</a>.

3 Sign of the Green’s function

In this section we use the above obtained expressions to obtain the explicit expression of the Green’s function, depending on the values of the constants a and b. Moreover, we study the sign of the function and deduce suitable comparison results.

We separate the study in three cases, taking into consideration the expression of the general solution of (2.6).

3.1 The case (C1)

Now, assume the case (C1), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M156">View MathML</a>. Using (2.10), we get the following expression of G for this situation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M157">View MathML</a>

which we can rewrite as

Studying the expression of G we can obtain maximum and anti-maximum principles. In order to do this, we will be interested in those maximal strips (in the sense of inclusion) of the kind <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M159">View MathML</a> where G does not change sign depending on the parameters.

So, we are in a position to study the sign of the Green’s function in the different triangles of definition. The result is the following.

Lemma 3.1Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62">View MathML</a>and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M161">View MathML</a>

Then the Green’s function of problem (2.5) is

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M163">View MathML</a>,

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M165">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166">View MathML</a>, the Green’s function of problem (2.5) is

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M168">View MathML</a>,

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M170">View MathML</a>,

and, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171">View MathML</a>, the Green’s function of problem (2.5) is

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M168">View MathML</a>,

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M170">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M176">View MathML</a>, the argument of the sin in (3.1c) is positive, so (3.1c) is positive for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M177">View MathML</a>. On the other hand, it is easy to check that (3.1a) is positive as long as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M178">View MathML</a>.

The rest of the proof continues similarly. □

As a corollary of the previous result we obtain the following one.

Lemma 3.2Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M62">View MathML</a>. Then we have the following:

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166">View MathML</a>, the Green’s function of problem (2.5) is non-negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M181">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171">View MathML</a>, the Green’s function of problem (2.5) is non-positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M183">View MathML</a>,

the Green’s function of problem (2.5) changes sign in any other strip not a subset of the aforementioned.

Proof The proof follows from the previous result together with the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M184">View MathML</a>

 □

Remark 3.1 Realize that the rectangles defined in the previous lemma are optimal in the sense that G changes sign in a bigger rectangle. The same observation applies to similar results we will prove for the other cases. This fact implies that we cannot have maximum or anti-maximum principles on bigger intervals for the solution, something that is widely known and which the following results, together with Example 3.4, illustrate.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M185">View MathML</a> changes sign at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M186">View MathML</a>, it is immediate to verify that, by defining function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M187">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M188">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M189">View MathML</a> otherwise, we have a solution of problem (2.5) that crosses the 0 line as close to the right of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M190">View MathML</a> as necessary. So the estimates are optimal for this case.

However, one can study problems with particular non-homogeneous part h for which the solution crosses 0 for a bigger interval. This is showed in the following example.

Example 3.1 Consider the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M192">View MathML</a>.

Clearly, we are in the case (C1). For this problem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M193">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M41">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a> is the solution of our problem.

Studying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a>, we can arrive at the conclusion that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a> is non-negative in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M198">View MathML</a>, being zero at both ends of the interval and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M199">View MathML</a>

Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M200">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M201">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M202">View MathML</a> sufficiently small. Furthermore, the solution is periodic of period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M203">View MathML</a> (see Figure 1).

thumbnailFigure 1. Graph of the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M204">View MathML</a>on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M205">View MathML</a>. Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a> is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M207">View MathML</a> and negative on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M208">View MathML</a>.

If we use Lemma 3.2, we find that, a priori, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M209">View MathML</a> is non-positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M210">View MathML</a> which we know is true by the study we have done of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a>, but this estimate is, as expected, far from the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M212">View MathML</a> in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a> is non-positive. This does not contradict the optimality of the a priori estimate, as we have showed before, some other examples could be found for which the interval where the solution has constant is arbitrarily close to the one given by the a priori estimate.

3.2 The case (C2)

We study here the case (C2). In this case, it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M214">View MathML</a>

which we can rewrite as

Studying the expression of G we can obtain maximum and anti-maximum principles. With this information, we can state the following lemma.

Lemma 3.3Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67">View MathML</a>and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M217">View MathML</a>

Then we have the following:

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166">View MathML</a>, the Green’s function of problem (2.5) is positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171">View MathML</a>, the Green’s function of problem (2.5) is negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M224">View MathML</a>, the Green’s function of problem (2.5) is negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M224">View MathML</a>, the Green’s function of problem (2.5) is positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M228">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M229">View MathML</a>, the Green’s function of problem (2.5) is positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M229">View MathML</a>, the Green’s function of problem (2.5) is negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M233">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M234">View MathML</a>, he argument of the sinh in (3.1d) is negative, so (3.2d) is positive. The argument of the sinh in (3.1c) is positive, so (3.2c) is positive. It is easy to check that (3.2a) is positive as long as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M235">View MathML</a>.

On the other hand, (3.2b) is always negative.

The rest of the proof continues similarly. □

As a corollary of the previous result we obtain the following one.

Lemma 3.4Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M67">View MathML</a>. Then we have the following.

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M234">View MathML</a>, the Green’s function of problem (2.5) is non-negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M238">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M239">View MathML</a>, the Green’s function of problem (2.5) is non-negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M240">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M241">View MathML</a>, the Green’s function of problem (2.5) is non-positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M242">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M243">View MathML</a>, the Green’s function of problem (2.5) is non-positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M244">View MathML</a>,

the Green’s function of problem (2.5) changes sign in any other strip not a subset of the aforementioned.

Example 3.2 Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M245">View MathML</a>

(3.3)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M246">View MathML</a>.

Clearly, we are in the case (C2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M247">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M248">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M249">View MathML</a>

With these equalities, it is straightforward to construct the unique solution w of problem (3.3). For instance, in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M250">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M251">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M252">View MathML</a>

Observe that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M255">View MathML</a>. Lemma 3.4 guarantees the non-negativity of w on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M256">View MathML</a>, but it is clear that the solution w is positive on the whole positive real line.

3.3 The case (C3)

We study here the case (C3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75">View MathML</a>. In this case, it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M258">View MathML</a>

which we can rewrite as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M259">View MathML</a>

Studying the expression of G we can obtain maximum and anti-maximum principles. With this information, we can prove the following lemma as we did with the analogous ones for cases (C1) and (C2).

Lemma 3.5Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75">View MathML</a>. Then, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166">View MathML</a>, the Green’s function of problem (2.5) is

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169">View MathML</a>,

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164">View MathML</a>,

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M266">View MathML</a>,

and, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171">View MathML</a>, the Green’s function of problem (2.5) is

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M169">View MathML</a>,

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162">View MathML</a>,

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M272">View MathML</a>.

As a corollary of the previous result we obtain the following one.

Lemma 3.6Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75">View MathML</a>. Then we have the following:

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M274">View MathML</a>, the Green’s function of problem (2.5) is non-negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M275">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171">View MathML</a>, the Green’s function of problem (2.5) is non-positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M277">View MathML</a>,

the Green’s function of problem (2.5) changes sign in any other strip not a subset of the aforementioned.

For this particular case we have another way of computing the solution to the problem.

Proposition 3.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M75">View MathML</a>and assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M279">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M280">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M281">View MathML</a>. Then problem (2.5) has a unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M282">View MathML</a>

Proof The equation is satisfied, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M283">View MathML</a>

The initial condition is also satisfied for, clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M284">View MathML</a>. □

Example 3.3 Consider the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M286">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M288">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M289">View MathML</a> we have a singularity at 0. We can apply the theory in order to get the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M290">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M291">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M292">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M209">View MathML</a> is positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M294">View MathML</a> and negative in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M295">View MathML</a> independently of λ, so the solution has better properties than the ones guaranteed by Lemma 3.6.

The next example shows that the estimate is sharp.

Example 3.4 Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M296">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M299">View MathML</a> is the characteristic function of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M300">View MathML</a>. Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M301">View MathML</a> is continuous. By means of the expression of the Green’s function for problem (3.4), we see that its unique solution is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M302">View MathML</a>

The a priori estimate on the solution tells us that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M303">View MathML</a> is non-negative at least in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M304">View MathML</a>. Studying the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M303">View MathML</a> (see Figure 2), it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M303">View MathML</a> is zero at 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M307">View MathML</a>, positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M308">View MathML</a> and negative in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M309">View MathML</a>.

thumbnailFigure 2. Graph of the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M310">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M311">View MathML</a>(dashed). Observe that u becomes zero at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M312">View MathML</a>.

The case (C3.2) is very similar,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M313">View MathML</a>

Lemma 3.8Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78">View MathML</a>. Then, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166">View MathML</a>, the Green’s function of problem (2.5) is

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M318">View MathML</a>,

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M320">View MathML</a>,

and, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166">View MathML</a>, the Green’s function of problem (2.5) is

negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M164">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M318">View MathML</a>,

positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M162">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M326">View MathML</a>.

As a corollary of the previous result we obtain the following one.

Lemma 3.9Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78">View MathML</a>. Then we have the following:

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M166">View MathML</a>, the Green’s function of problem (2.5) is non-negative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M240">View MathML</a>,

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M171">View MathML</a>the Green’s function of problem (2.5) is non-positive on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M244">View MathML</a>,

the Green’s function of problem (2.5) changes sign in any other strip not a subset of the aforementioned.

Again, for this particular case we have another way of computing the solution to the problem.

Proposition 3.10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M333">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M334">View MathML</a>. Then problem (2.5) has a unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M335">View MathML</a>

Proof The equation is satisfied, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M336">View MathML</a>

The initial condition is also satisfied for, clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M284">View MathML</a>. □

Example 3.5 Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M338">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M106">View MathML</a>. We can apply the theory in order to get the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M340">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M341">View MathML</a>.

Observe that the real function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M342">View MathML</a>

is positive on ℝ if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M343">View MathML</a> and negative on ℝ for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M344">View MathML</a>. Therefore, Lemma 3.9 guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M39">View MathML</a> will be positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M346">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M347">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M295">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M349">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors are thankful to the anonymous referees for the careful reading of the manuscript and suggestions. This work was supported in part by FEDER and Ministerio de Educación y Ciencia, Spain, project MTM2010-15314. The second author was supported by FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain.

References

  1. Silberstein, L: Solution of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M350">View MathML</a>. Philos. Mag.. 7(30), 185–186 (1940)

  2. Wiener, J, Watkins, W: A glimpse into the wonderland of involutions. Mo. J. Math. Sci.. 14(3), 175–185 (2002)

  3. Piao, D: Pseudo almost periodic solutions for differential equations involving reflection of the argument. J. Korean Math. Soc.. 41(4), 747–754 (2004)

  4. Piao, D: Periodic and almost periodic solutions for differential equations with reflection of the argument. Nonlinear Anal.. 57(4), 633–637 (2004). Publisher Full Text OpenURL

  5. Kuller, RG: On the differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M352">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/99/mathml/M353">View MathML</a>. Math. Mag.. 42, 195–200 (1969). Publisher Full Text OpenURL

  6. Shah, SM, Wiener, J: Reducible functional-differential equations. Int. J. Math. Math. Sci.. 8, 1–27 (1985). Publisher Full Text OpenURL

  7. Watkins, W: Modified Wiener equations. Int. J. Math. Math. Sci.. 27(6), 347–356 (2001). Publisher Full Text OpenURL

  8. Wiener, J: Generalized Solutions of Functional-Differential Equations, World Scientific, River Edge, NJ (1993)

  9. Watkins, W: Asymptotic properties of differential equations with involutions. Int. J. Pure Appl. Math.. 44(4), 485–492 (2008)

  10. Aftabizadeh, AR, Huang, YK, Wiener, J: Bounded solutions for differential equations with reflection of the argument. J. Math. Anal. Appl.. 135, 31–37 (1988). Publisher Full Text OpenURL

  11. Gupta, CP: Existence and uniqueness theorems for boundary value problems involving reflection of the argument. Nonlinear Anal.. 11(9), 1075–1083 (1987). Publisher Full Text OpenURL

  12. Gupta, CP: Two-point boundary value problems involving reflection of the argument. Int. J. Math. Math. Sci.. 10(2), 361–371 (1987). Publisher Full Text OpenURL

  13. O’Regan, D, Zima, M: Leggett-Williams norm-type fixed point theorems for multivalued mappings. Appl. Math. Comput.. 187(2), 1238–1249 (2007). Publisher Full Text OpenURL

  14. O’Regan, D: Existence results for differential equations with reflection of the argument. J. Aust. Math. Soc. A. 57(2), 237–260 (1994). Publisher Full Text OpenURL

  15. Andrade, D, Ma, TF: Numerical solutions for a nonlocal equation with reflection of the argument. Neural Parallel Sci. Comput.. 10, 227–233 (2002)

  16. Ma, TF, Miranda, ES, de Souza Cortes, MB: A nonlinear differential equation involving reflection of the argument. Arch. Math.. 40(1), 63–68 (2004)

  17. Wiener, J, Aftabizadeh, AR: Boundary value problems for differential equations with reflection of the argument. Int. J. Math. Math. Sci.. 8(1), 151–163 (1985). Publisher Full Text OpenURL

  18. Cabada, A, Tojo, FAF: Comparison results for first order linear operators with reflection and periodic boundary value conditions. Nonlinear Anal., Theory Methods Appl.. 78, 32–46 (2013)

  19. Cabada, A, Infante, G, Tojo, FAF: Nontrivial solutions of perturbed Hammerstein integral equations with reflections. Bound. Value Probl.. 2013, Article ID 86 (2013)

  20. Azbelev, NV, Domoshnitsky, A: A question concerning linear differential inequalities - I. Differ. Uravn. (Minsk). 27, 257–263 (1991)

  21. Azbelev, NV, Domoshnitsky, A: A question concerning linear differential inequalities - II. Differ. Uravn. (Minsk). 27, 641–647 (1991)

  22. Agarwal, RP, Berezansky, L, Braverman, E, Domoshnitsky, A: Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York (2012)

  23. Domoshnitsky, A: Maximum principles and nonoscillation intervals for first order Volterra functional differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal.. 15, 769–814 (2008)

  24. Domoshnitsky, A: Nonoscillation interval for n-th order functional differential equations. Nonlinear Anal., Theory Methods Appl.. 71, e2449–e32456 (2009). Publisher Full Text OpenURL

  25. Domoshnitsky, A, Maghakyan, A, Shklyar, R: Maximum principles and boundary value problems for first-order neutral functional differential equations. J. Inequal. Appl.. 2009, Article ID 141959 (2009)