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

Open Access Research

Higher order functional boundary value problems without monotone assumptions

João F Fialho13* and Feliz Minhós23

Author Affiliations

1 School of Mathematics, Physics and Technology, College of the Bahamas, Nassau, Bahamas

2 Department of Mathematics, School of Sciences and Technology, University of Évora, Évora, Portugal

3 Research Centre on Mathematics and Applications, University of Évora (CIMA-UE), Rua Romão Ramalho, 59, Évora, 7000-671, Portugal

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Boundary Value Problems 2013, 2013:81  doi:10.1186/1687-2770-2013-81

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/81


Received:15 December 2012
Accepted:22 March 2013
Published:10 April 2013

© 2013 Fialho and Minhós; licensee Springer

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

Abstract

In this paper, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M1">View MathML</a> a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2">View MathML</a>-Carathéodory function, it is considered the functional higher order equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M3">View MathML</a>

together with the nonlinear functional boundary conditions, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7">View MathML</a>, are continuous functions. It will be proved an existence and location result in presence of not necessarily ordered lower and upper solutions, without assuming any monotone properties on the boundary conditions and on the nonlinearity f.

1 Introduction

In this paper, it is considered the functional higher order boundary value problem, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M8">View MathML</a> composed by the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M9">View MathML</a>

(1)

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M10">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M11">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2">View MathML</a>-Carathéodory function, and the function boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M13">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7">View MathML</a>, are continuous functions without assuming monotone conditions or another type of variation.

The functional differential equation (1) can be seen as a generalization of several types of full differential and integro-differential equations and allow to consider delays, maxima or minima arguments, or another kind of global variation on the unknown function or its derivatives until order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M16">View MathML</a>. On the other hand, the functional dependence in (2) makes possible its application to a huge variety of boundary conditions, such as Lidstone, separated, multipoint, nonlocal and impulsive conditions, among others. As example, we mention the problems contained in [1-15]. A detailed list about the potentialities of functional problems and some applications can be found in [16].

Recently, functional boundary value problems have been studied by several authors following several approaches, as it can be seen, for example, in [17-24]. In this work, the lower and upper solutions method is applied together with topological degree theory, according some arguments suggested in [25-27].

The novelty of this paper consists in the following items:

• There is no monotone assumptions on the boundary functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7">View MathML</a>, by using adequate auxiliary functions and global arguments. This fact with the functional dependence on the unknown function and its derivatives till order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M19">View MathML</a> will allow that problem (1)-(2) can include the periodic and antiperiodic cases, which were not covered by the existent literature on functional boundary value problems. In this sense, the results in this area, as for instance [28-32], are improved, even for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M20">View MathML</a>, where equation (1) loses its functional part.

• No extra condition on the nonlinear part of (1) is considered, besides a Nagumo-type growth assumption. In fact, as far as we know, it is the first time where lower and upper solutions technique is used without such hypothesis on function f, by the use of stronger definitions for lower and upper solutions.

• No order between lower and upper solutions is assumed. Putting the ‘well ordered’ case on adequate auxiliary functions, it allows that lower and upper solutions could be well ordered, by reversed order or without a defined order.

The last section contains an example where the potentialities of the functional dependence on the equation and on the boundary conditions are explored.

2 Definitions and auxiliary functions

In this section, it will be introduced the notations and definitions needed forward together with some auxiliary functions useful to construct some ordered functions on the basis of the not necessarily ordered lower and upper solutions of the referred problem.

A Nagumo-type growth condition, assumed on the nonlinear part, will be an important tool to set an a priori bound for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M21">View MathML</a>th derivative of the corresponding solutions.

In the following, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M22">View MathML</a> denotes the usual Sobolev Spaces in I, that is, the subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M23">View MathML</a> functions, whose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M24">View MathML</a>th derivative is absolutely continuous in I and the mth derivative belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M25">View MathML</a> and the usual norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M26">View MathML</a>

for spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M28">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M29">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2">View MathML</a>-Carathéodory function, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M31">View MathML</a> is a continuous function for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M33">View MathML</a> is measurable for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M34">View MathML</a>; and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M35">View MathML</a> there is a real-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M36">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M37">View MathML</a>

and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M38">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M39">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M7">View MathML</a>.

The main tool to obtain the location part is the upper and lower solutions method. However, in this case, they must be defined as a pair, which means that it is not possible to define them independently from each other. Moreover, it is pointed out that lower and upper functions, and the correspondent first derivatives, are not necessarily ordered.

To introduce ‘some order’, some auxiliary functions must be defined.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M41">View MathML</a> define functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M43">View MathML</a>, as it follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M44">View MathML</a>

(3)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M45">View MathML</a>.

The Nagumo-type condition is given by next definition.

Definition 1 Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M49">View MathML</a>, and the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M50">View MathML</a>

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M51">View MathML</a> is said to verify a Nagumo-type condition in E if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M52">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M53">View MathML</a>

(4)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M54">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M55">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M56">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M57">View MathML</a>

The next result gives an a priori estimate for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M21">View MathML</a>th derivative of all possible solutions of (1).

Lemma 2There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M59">View MathML</a>such that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2">View MathML</a>-Carathéodory function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M61">View MathML</a>satisfying (4) and (5) and every solutionuof (1) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M62">View MathML</a>

(6)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M64">View MathML</a>

(7)

Moreover, the constantRdepends only on the functionsφand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M66">View MathML</a>) and not on the boundary conditions.

Proof The proof is similar to [[19], Lemma 2.1]. □

The upper and lower solution definition is then given by the following.

Definition 3 The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M67">View MathML</a> are a pair of lower and upper solutions for problem (1)-(2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M68">View MathML</a>, on I, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M69">View MathML</a>, and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M70">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M59">View MathML</a>, the following inequalities hold for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M72">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M73">View MathML</a>

(8)

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M74">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M75">View MathML</a>

(9)

3 Existence and location result

In this section, it is provided an existence and location theorem for the problem (1)-(2). More precisely, sufficient conditions are given for, not only the existence of a solution u, but also to have information about the location of u, and all its derivatives up to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M19">View MathML</a> order.

The arguments of the proof require the following lemma, given on [29].

Lemma 4For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M77">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M78">View MathML</a>, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M80">View MathML</a>

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M81">View MathML</a>the next two properties hold:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M82">View MathML</a>exists for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>.

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M84">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M85">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M86">View MathML</a>then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M87">View MathML</a>

Now, we are in a position to prove the main result of this paper.

Theorem 5Assume that there exists a pair of lower and upper solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M88">View MathML</a>of problem (1)-(2).

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M89">View MathML</a>is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2">View MathML</a>-Carathéodory function, satisfying a Nagumo-type condition in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M91">View MathML</a>

then problem (1)-(2) has at least one solutionusuch that

for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M49">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M96">View MathML</a>

(10)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M97">View MathML</a>is given by (7).

Proof Define the continuous functions, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M43">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M99">View MathML</a>

(11)

and the truncation, not necessarily continuous,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M100">View MathML</a>

with K given by (10).

Consider the modified problem composed by the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M101">View MathML</a>

(12)

and the boundary conditions, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M103">View MathML</a>

(13)

The proof will follow the next steps:

Step 1. Every solution u of problem (12)-(13), satisfies

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M105">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M59">View MathML</a> given in (10).

Let u be a solution of the modified problem (12)-(13). Assume, by contradiction, that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M109">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M110">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M111">View MathML</a>

As, by (13), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M113">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M114">View MathML</a>. So, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M115">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M116">View MathML</a>

(14)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M117">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M118">View MathML</a>

Now, since for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M119">View MathML</a> it is satisfied that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M120">View MathML</a>, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M121">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M123">View MathML</a> is nonincreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M124">View MathML</a>, this contradicts the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M126">View MathML</a>.

The inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M127">View MathML</a>, in I, can be proved in same way and so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M128">View MathML</a>

(15)

By (13) and (3), the following inequalities hold for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M130">View MathML</a>

Analogously, it can be obtained <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M131">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>.

The remaining inequalities are obtained by the same integration process.

Applying previous bounds in Lemma 2, and remarking that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M133">View MathML</a>

for K given by (10), it is obtained, by Lemma 2, the a priori bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M134">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M32">View MathML</a>. For details, see [[33], Lemma 2].

Step 2. Problem (12)-(13) has at least one solution.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M136">View MathML</a> let us consider the homotopic problem given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M137">View MathML</a>

(16)

and the boundary conditions, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M139">View MathML</a>

(17)

Let us consider the norms in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M140">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M141">View MathML</a>, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M142">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M143">View MathML</a>

Define the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M144">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M145">View MathML</a> and, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M148">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M149">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M150">View MathML</a> are continuous and f is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M2">View MathML</a>-Carathéodory function, then, from Lemma 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M152">View MathML</a> is continuous. Moreover, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M153">View MathML</a> is compact, it can be defined the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M154">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M155">View MathML</a>.

It is obvious that the fixed points of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M156">View MathML</a> coincide with the solutions of problem (16)-(17).

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M157">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M158">View MathML</a> and uniformly bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M140">View MathML</a>, we have that any solution of the problem (16)-(17), verifies the following a priori bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M160">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M161">View MathML</a> independent of λ.

In the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M162">View MathML</a>, the degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M163">View MathML</a> is well defined for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M164">View MathML</a> and, by the invariance under homotopy, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M165">View MathML</a>.

As the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M166">View MathML</a> is equivalent to the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M167">View MathML</a>

which has only the trivial solution, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M168">View MathML</a>. So, by degree theory, the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M169">View MathML</a> has at least one solution, that is, the problem (12)-(13) has at least a solution in Ω.

Step 3. Every solution u of problem (12)-(13) is a solution of (1)-(2).

Let u be a solution of the modified problem (12)-(13). By previous steps, function u fulfills equation (1). So, it will be enough to prove the following inequalities, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M43">View MathML</a>:

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M172">View MathML</a>

Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M173">View MathML</a>

(18)

Then, by (13), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M174">View MathML</a>. By previous steps, it is obtained the following contradiction with (18):

Applying similar arguments, it can be proved that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M176">View MathML</a>

and analogously, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M177">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M178">View MathML</a>

Also, using the same arguments and the same techniques, it can be proved that

 □

4 Example

This section contains a problem composed by an integro-differential equation with some functional boundary conditions, whose solvability is proved in presence of nonordered lower and upper solutions. We remark that such fact was not possible with the results in the current literature. This example does not model any particular problem arising in real phenomena. Our purpose consists on emphasizing the powerful of the developed theory in this paper by showing what kind of problems we can deal with.

Consider, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M180">View MathML</a>, the fourth-order equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M181">View MathML</a>

(19)

coupled with the boundary value conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M182">View MathML</a>

(20)

One can verify that functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M183">View MathML</a>

are, respectively, lower and upper solutions for the problem (19)-(20). Moreover, we deduce that

and

As the continuous function f verifies (4) and (5) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M186">View MathML</a> in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M187">View MathML</a>

then, by Theorem 5, there is a nontrivial solution u for problem (19)-(20) such that

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/81/mathml/M180">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally, read and approved the final version of the manuscript.

Acknowledgements

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

References

  1. Cabada, A, Grossinho, MR, Minhós, F: On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions. J. Math. Anal. Appl.. 285, 174–190 (2003). Publisher Full Text OpenURL

  2. Cuia, Y, Sun, J: Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces. Bound. Value Probl.. 2012, Article ID 107 (2012)

  3. Feng, H, Ji, D, Ge, W: Existence and uniqueness of solutions for a fourth-order boundary value problem. Nonlinear Anal.. 70, 3561–3566 (2009). Publisher Full Text OpenURL

  4. Franco, D, O’Regan, D, Perán, J: Fourth-order problems with nonlinear boundary conditions. J. Comput. Appl. Math.. 174, 315–327 (2005). Publisher Full Text OpenURL

  5. Graef, JR, Kong, L, Yang, B: Existence of solutions for a higher order multi-point boundary value problems. Results Math.. 53, 77–101 (2009). Publisher Full Text OpenURL

  6. Han, J, Liu, Y, Zhao, J: Integral boundary value problems for first order nonlinear impulsive functional integro-differential differential equations. Appl. Math. Comput.. 218, 5002–5009 (2012). Publisher Full Text OpenURL

  7. Kong, L, Wong, J: Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions. J. Math. Anal. Appl.. 367, 588–611 (2010). Publisher Full Text OpenURL

  8. Lu, H, Sun, L, Sun, J: Existence of positive solutions to a non-positive elastic beam equation with both ends fixed. Bound. Value Probl.. 2012, Article ID 56 (2012)

  9. Ma, D, Yang, X: Upper and lower solution method for fourth-order four-point boundary value problems. J. Comput. Appl. Math.. 223, 543–551 (2009). Publisher Full Text OpenURL

  10. Minhós, F, Gyulov, T, Santos, AI: Lower and upper solutions for a fully nonlinear beam equations. Nonlinear Anal.. 71, 281–292 (2009). Publisher Full Text OpenURL

  11. Pang, H, Ge, W: Existence results for some fourth order multi-point boundary value problem. Math. Comput. Model.. 49, 1319–1325 (2009). Publisher Full Text OpenURL

  12. Pao, CV, Wang, YM: Fourth-order boundary value problems with multi-point boundary conditions. Commun. Appl. Nonlinear Anal.. 16, 1–22 (2009)

  13. Pei, M, Chang, S, Oh, YS: Solvability of right focal boundary value problems with superlinear growth conditions. Bound. Value Probl.. 2012, Article ID 60 (2012)

  14. Zhang, X, Liu, L: Positive solutions of fourth-order multi-point boundary value problems with bending term. Appl. Math. Comput.. 194, 321–332 (2007). Publisher Full Text OpenURL

  15. Zhao, H: A note on upper and lower solutions method for fourth-order boundary value problems. Ann. Differ. Equ.. 24, 117–120 (2008)

  16. Cabada, A, Pouso, R, Minhós, F: Extremal solutions to fourth-order functional boundary value problems including multipoint condition. Nonlinear Anal., Real World Appl.. 10, 2157–2170 (2009). Publisher Full Text OpenURL

  17. Cabada, A, Fialho, J, Minhós, F: Non ordered lower and upper solutions to fourth order functional BVP. Discrete Contin. Dyn. Syst.. 2011, 209–218 suppl. (2011)

  18. Cabada, A, Minhós, F: Fully nonlinear fourth order equations with functional boundary conditions. J. Math. Anal. Appl.. 340(1), 239–251 (2008). Publisher Full Text OpenURL

  19. Cabada, A, Minhós, F, Santos, AI: Solvability for a third order discontinuous fully equation with functional boundary conditions. J. Math. Anal. Appl.. 322, 735–748 (2006). Publisher Full Text OpenURL

  20. Graef, J, Kong, L, Minhós, F: Higher order functional boundary value problems: existence and location results. Acta Sci. Math.. 77, 87–100 (2011)

  21. Graef, J, Kong, L, Minhós, F: Higher order boundary value problems with ϕ-Laplacian and functional boundary conditions. Comput. Math. Appl.. 61, 236–249 (2011). Publisher Full Text OpenURL

  22. Sun, Y, Han, Z: On forced oscillation of higher order functional differential equations. Appl. Math. Comput.. 218, 6966–6971 (2012). Publisher Full Text OpenURL

  23. Wang, W, Shen, J, Luo, Z: Multi-point boundary value problems for second-order functional differential equations. Comput. Math. Appl.. 56, 2065–2072 (2008). Publisher Full Text OpenURL

  24. Zhao, Z, Liang, J: Existence of solutions to functional boundary value problem of second-order nonlinear differential equation. J. Math. Anal. Appl.. 373, 614–634 (2011). Publisher Full Text OpenURL

  25. Minhós, F, Fialho, J: On the solvability of some fourth-order equations with functional boundary conditions. Discrete Contin. Dyn. Syst.. 2009, 564–573 suppl. (2009)

  26. Graef, J, Kong, L, Minhós, F, Fialho, J: On lower and upper solutions method for higher order functional boundary value problems. Appl. Anal. Discrete Math.. 5(1), 133–146 (2011). Publisher Full Text OpenURL

  27. Mawhin, J: Topological Degree Methods in Nonlinear Boundary Value Problems, Am. Math. Soc., Providence (1979)

  28. Minhós, F: Periodic solutions for some fully nonlinear fourth order differential equations. Discrete Contin. Dyn. Syst.. 2011, 1068 suppl. (2011)

  29. Wang, MX, Cabada, A, Nieto, JJ: Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions. Ann. Pol. Math.. 58, 221–235 (1993)

  30. Weng, S, Gao, H, Jiang, D, Hou, X: Upper and lower solutions method for fourth-order periodic boundary value problems. J. Appl. Anal.. 14, 53–61 (2008)

  31. Yao, M, Zhao, A, Yan, J: Anti-periodic boundary value problems of second order impulsive differential equations. Comput. Math. Appl.. 59, 3617–3629 (2010). Publisher Full Text OpenURL

  32. Zhang, Y: The existence of solutions to nonlinear second order periodic boundary value problems. Nonlinear Anal.. 76, 140–152 (2013)

  33. Grossinho, MR, Minhós, F, Santos, AI: A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition. Nonlinear Anal.. 70, 4027–4038 (2009). Publisher Full Text OpenURL