Open Access Research

Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments

Rubén Figueroa* and Rodrigo López Pouso

Author Affiliations

Department of Mathematical Analysis, University of Santiago de Compostela, Spain

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Boundary Value Problems 2012, 2012:7  doi:10.1186/1687-2770-2012-7


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/7


Received:13 May 2011
Accepted:20 January 2012
Published:20 January 2012

© 2012 Figueroa and Pouso; 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

We prove some new results on existence of solutions to first-order ordinary differential equations with deviated arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the unknown solutions. Our existence results lean on new definitions of lower and upper solutions introduced in this article, and we show with an example that similar results with the classical definitions are false. We also introduce an example showing that the problems considered need not have the least (or the greatest) solution between given lower and upper solutions, but we can prove that they do have minimal and maximal solutions in the usual set-theoretic sense. Sufficient conditions for the existence of lower and upper solutions, with some examples of application, are provided too.

1 Introduction

Let I0 = [t0, t0 + L] be a closed interval, r ≥ 0, and put I- = [t0 - r, t0] and I = I- I0. In this article, we are concerned with the existence of solutions for the following problem with deviated arguments:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M1">View MathML</a>

(1)

where f : I × ℝ2 → ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M2">View MathML</a> are Carathéodory functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M3">View MathML</a> is a continuous nonlinear operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M4">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M5">View MathML</a> denotes the set of real functions which are continuous on the interval J.

For example, our framework admits deviated arguments of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M6">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M7">View MathML</a>

We define a solution of problem (1) to be a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M9">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M10">View MathML</a> is absolutely continuous on I0) and x fulfills (1).

In the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11">View MathML</a> we consider the usual pointwise partial ordering, i.e., for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M12">View MathML</a> we define γ1 γ2 if and only if γ1(t) ≤ γ2(t) for all t I. A solution of (1), x*, is a minimal (respectively, maximal) solution of (1) in a certain subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M13">View MathML</a> if x* Y and the inequality x x*, (respectively, x x*) implies x = x*, whenever x is a solution to (1) and x Y. We say that x* is the least (respectively the greatest) solution of (1) in Y if x* x (respectively x* x) for any other solution x Y. Notice that the least solution in a subset Y is a minimal solution in Y, but the converse is false in general, and an analgous remark is true for maximal and greatest solutions.

Interestingly, we will show that problem (1) may have minimal (maximal) solutions between given lower and upper solutions and not have the least (greatest) solution. This seems to be a peculiar feature of equations with deviated arguments, see [1] for an example with a second-order equation. Therefore, we are obliged to distinguish between the concepts of minimal solution and least solution (or maximal and greatest solutions), unfortunately often identified in the literature on lower and upper solutions.

First-order differential equations with state-dependent deviated arguments have received a lot of attention in the last years. We can cite the recent articles [2-7] which deal with existence results for this kind of problems. For the qualitative study of this type of problems we can cite the survey of Hartung et al. [8] and references therein.

As main improvements in this article with regard to previous works in the literature we can cite the following:

(1) The deviating argument τ depends at each moment t on the global behavior of the solution, and not only on the values that it takes at the instant t.

(2) Delay problems, which correspond to differential equations of the form x'(t) = f(t, x(t), x(t - r)) along with a functional start condition, are included in the framework of problem (1). This is not allowed in articles [3-6].

(3) No monotonicity conditions are required for the functions f and τ, and they need not be continuous with respect to their first variable.

This article is organized as follows. In Section 2, we state and prove the main results in this article, which are two existence results for problem (1) between given lower and upper solutions. The first result ensures the existence of maximal and minimal solutions, and the second one establishes the existence of the greatest and the least solutions in a particular case. The concepts of lower and upper solutions introduced in Section 2 are new, and we show with an example that our existence results are false if we consider lower and upper solutions in the usual sense. We also show with an example that our problems need not have the least or the greatest solution between given lower and upper solutions. In Section 3, we prove some results on the existence of lower and upper solutions with some examples of application.

2 Main results

We begin this section by introducing adequate new definitions of lower and upper solutions for problem (1).

Notice first that τ(t, γ) ∈ I = I- I0 for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M14">View MathML</a>, so for each t I0we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M15">View MathML</a>

Definition 1 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M16">View MathML</a>, with α β on I, are a lower and an upper solution for problem (1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M17">View MathML</a>and the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M18">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M19">View MathML</a>

(3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M20">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M21">View MathML</a>.

Remark 1 Definition 1 requires implicitly that Λ be bounded in [α, β].

On the other hand, the values

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M22">View MathML</a>

are really attained for almost every fixed t I0 thanks to the continuity of f(t, α(t), ·) and f(t, β(t), ·) on the compact set E(t).

Now we introduce the main result of this article.

Theorem 1 Assume that the following conditions hold:

(H1) (Lower and upper solutions) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M16">View MathML</a>, with α β on I, which are a lower and an upper solution for problem (1).

(H2) (Carathéodory conditions)

(H2) - (a) For all x, y ∈ [mintI α(t), maxtI β(t)] the function f(·,x,y) is measurable and for a.a. t I0, all x ∈ [α(t), β(t)] and all y E(t) (as defined in Definition 1) the functions f(t, ·, y) and f(t, x, •) are continuous.

(H2) - (b) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M23">View MathML</a>the function τ(·, γ) is measurable and for a.a. t I0 the operator τ(t, ·) is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11">View MathML</a>(equipped with it usual topology of uniform convergence).

(H2) - (c) The nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M24">View MathML</a>is continuous.

(H3) (L1-bound) There exists ψ L1( I0) such that for a.a. t I0, all x ∈ [α(t), β(t)] and all y E(t) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M25">View MathML</a>

Then problem (1) has maximal and minimal solutions in [α, β].

Proof. As usual, we consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M26">View MathML</a>

and the modified problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M27">View MathML</a>

(4)

Claim 1: Problem (4) has a nonempty and compact set of solutions. Consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M28">View MathML</a> which maps each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M29">View MathML</a> to a continuous function defined for each t I- as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M30">View MathML</a>

and for each t I0 as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M31">View MathML</a>

It is an elementary matter to check that T is a completely continuous opera-tor from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11">View MathML</a> into itself (one has to take Remark 1 into account). Therefore, Schauder's Theorem ensures that T has a nonempty and compact set of fixed points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11">View MathML</a>, which are exactly the solutions of problem (4).

Claim 2: Every solution x of (4) satisfies α x β on I and, therefore, it is a solution of (1) in [α, β]. First, notice that if x is a solution of (4) then p(·,x(·)) ∈ [α, β]. Hence the definition of lower solution implies that for all t I- we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M32">View MathML</a>

Assume now, reasoning by contradiction, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M33">View MathML</a> on I0. Then we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M34">View MathML</a> and ε > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M35">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M36">View MathML</a>

(5)

Therefore, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M37">View MathML</a> we have p(t, x(t)) = α(t) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M38">View MathML</a>

so for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M39">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M40">View MathML</a>

Hence for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M37">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M41">View MathML</a>

a contradiction with (5).

Similar arguments prove that all solutions x of (4) obey x β on I. Claim 3: The set of solutions of problem (1) in [α, β] has maximal and minimal elements. The set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M42">View MathML</a>

is nonempty and compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M11">View MathML</a>, beacuse it coincides with the set of fixed points of the operator T. Then, the real-valued continuous mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M43">View MathML</a>

attains its maximum and its minimum, that is, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M45">View MathML</a>

(6)

Now, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M46">View MathML</a> is such that x x* on I then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M47">View MathML</a> and, by (6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M48">View MathML</a>. So we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M49">View MathML</a> which, along with x x*, implies that x = x* on I. Hence x* is a maximal element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M50">View MathML</a>. In the same way, we can prove that x* is a minimal element.

One might be tempted to follow the standard ideas with lower and upper solutions to define a lower solution of (1) as some function α such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M51">View MathML</a>

(7)

and an upper solution as some function β such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M52">View MathML</a>

(8)

These definitions are not adequate to ensure the existence of solutions of (1) between given lower and upper solutions, as we show in the following example.

Example 1 Consider the problem with delay

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M53">View MathML</a>

(9)

Notice that functions α(t) = 0 and β(t) = 1, t ∈ [-1, 1], are lower and upper solutions in the usual sense for problem (9). However, if x is a solution for problem (9) then for a.a. t ∈ [0, 1] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M54">View MathML</a>

so for all t ∈ [0,1] we compute

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M55">View MathML</a>

and then x(t) < α(t) for all t ∈ (0,1]. Hence (9) has no solution at all between α and β.

Remark 2 Notice that inequalities (2) and (3) imply (7) and (8), so lower and upper solutions in the sense of Definition 1 are lower and upper solutions in the usual sense, but the converse is false in general.

Definition 1 is probably the best possible for (1) because it reduces to some definitions that one can find in the literature in connection with particular cases of (1). Indeed, when the function τ does not depend on the second variable then for all t I0 we have E(t) = [α(τ(t)), β(τ(t))] in Definition 1. Therefore, if f is nondecreasing with respect to its third variable, then Definition 1 and the usual definition of lower and upper solutions are the same (we will use this fact in the proof of Theorem 2). If, in turn, f is nonincreasing with respect to its third variable, then Definition 1 coincides with the usual definition of coupled lower and upper solutions (see for example [5]).

In general, in the conditions of Theorem 1 we cannot expect problem (1) to have the extremal solutions in [α, β] (that is, the greatest and the least solutions in [α, β ]). This is justified by the following example.

Example 2 Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M56">View MathML</a>

(10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M57">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M58">View MathML</a>.

First we check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M59">View MathML</a>, t I0, are lower and upper solutions for problem (10). The definition of f implies that for all (t, x, y) ∈ I0 × ℝ2 we have |f(t, x, y)| ≤ 1, so for all t I0 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M60">View MathML</a>

where, according to Definition 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M61">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M62">View MathML</a>, so α and β are, respectively, a lower and an upper solution for (10), and then condition (H1) of Theorem 1 is fulfilled. As conditions (H2) and (H3) are also satisfied (take, for example, ψ ≡ 1) we deduce that problem (1) has maximal and minimal solutions in [α, β]. However we will show that this problem does not have the extremal solutions in [α, β].

The family xx(t) = λ cos t, t I0, with λ ∈ [-1,1], defines a set of solutions of problem (10) such that α xλ β for each λ ∈ [-1,1]. Notice that the zero solution is neither the least nor the greatest solution of (10) in [α, β]. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M63">View MathML</a> be an arbitrary solution of problem (10) and let us prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64">View MathML</a> is neither the least nor the greatest solution of (10) in [α, β]. First, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64">View MathML</a> changes sign in I0 then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64">View MathML</a> cannot be an extremal solution of problem (10) because it cannot be compared with the solution x ≡ 0. If, on the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M65">View MathML</a> in I0 then the differential equation yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M66">View MathML</a> a.e. on I0, which implies, along with the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M67">View MathML</a>, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M68">View MathML</a> for all t I0. Reasoning in the same way, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M69">View MathML</a> in I0 implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M70">View MathML</a>. Hence, problem (10) does not have extremal solutions in [α, β].

The previous example notwithstanding, existence of extremal solutions for problem (1) between given lower and upper solutions can be proven under a few more assumptions. Specifically, we have the following extremality result.

Theorem 2 Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M71">View MathML</a>

(11)

If (11) satisfies all the conditions in Theorem 1 and, moreover, f is nondecreasing with respect to its third variable and Λ is nondecreasing in [α, β], then problem (11) has the extremal solutions in [α, β].

Proof. Theorem 1 guarantees that problem (11) has a nonempty set of solutions between α and β. We will show that this set of solutions is, in fact, a directed set, and then we can conclude that it has the extremal elements by virtue of [9, Theorem 1.2].

According to Remark 2, the lower solution α and the upper solution β satisfy, respectively, inequalities (7) and (8) and, conversely, if α and β satisfy (7) and (8) then they are lower and upper solutions in the sense of Definition 1.

Let x1, x2 ∈ [α, β] be two solutions of problem (11). We are going to prove that there is a solution x3 ∈ [α, β] such that xi x3 (i = 1, 2), thus showing that the set of solutions in [α, β] is upwards directed. To do so, we consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M72">View MathML</a>, t I0, which is absolutely continuous on I0. For a.a. t I0 we have either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M73">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M74">View MathML</a>

and, since f is nondecreasing with respect to its third variable, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M75">View MathML</a>

We also have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M76">View MathML</a> in I- because Λ is nondecreasing, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M64">View MathML</a> is a lower solution for problem (11). Theorem 1 ensures now that (11) has at least one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M77">View MathML</a>.

Analogous arguments show that the set of solutions of (11) in [α, β] is downwards directed and, therefore, it is a directed set.

Next we show the applicability of Theorem 2.

Example 3 Let L > 0 and consider the following differential equation with reflection of argument and a singularity at x = 0:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M78">View MathML</a>

(12)

In this case, the function defining the equation is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M79">View MathML</a>, which is nondecreasing with respect to y. On the other hand, functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M80">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M81">View MathML</a>

are lower and upper solutions for problem (12). Indeed, for t ∈ [-L,0] we have -2t k(t) ≤ -4t and for a.a. t I0 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M82">View MathML</a>

Hence α and β are lower and upper solutions for problem (12) by virtue of Remark 2.

Finally, for a.a. t I0 and all y ∈ [α(-t), β(-t)] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M83">View MathML</a>

so problem (12) has the extremal solutions in [α, β ]. Notice that f admits a Carathéodory extension to I0 × ℝ outside the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M84">View MathML</a>

so Theorem 2 can be applied.

In fact, we can explicitly solve problem (12) because the differential equation and the initial condition yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M85">View MathML</a>

hence problem (12) has a unique solution (see Figure 1) which is given by

thumbnailFigure 1. Solution of (12) bracketed by the lower and the upper solution.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M86">View MathML</a>

3 Construction of lower and upper solutions

In general, condition (H1) is the most difficult to check among all the hypotheses in Theorem 1. Because of this, we include in this section some sufficient conditions on the existence of linear lower and upper solutions for problem (1) in particular cases We begin by considering a problem of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M87">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M89">View MathML</a>.

Proposition 1 Assume that f is a continuous function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M90">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M91">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M92">View MathML</a>

(16)

Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M93">View MathML</a>such that the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M94">View MathML</a>

(17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M95">View MathML</a>

(18)

are, respectively, a lower and an upper solution for problem (13), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M96">View MathML</a>

In particular, problem (13) has maximal and minimal solutions between α and β, and this does not depend on the choice of τ.

Proof. Conditions (15) and (16) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M97">View MathML</a>

so there exists y1 < min{0, φt} such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M98">View MathML</a>

(19)

On the other hand, condition (14) implies that there exists y2 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M99">View MathML</a>

(20)

Let λ = min{f(y): y1 y y2}. By condition (15) and continuity of f, there exists y3 y1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M100">View MathML</a>

(21)

and this choice of y3 also provides that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M101">View MathML</a>

(22)

and, by virtue of (19),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M102">View MathML</a>

(23)

Now, define α as in (17), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M103">View MathML</a>. Notice that α(t) ≤ k(t) for all t I-, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M104">View MathML</a> for all t I0 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M105">View MathML</a>

so we deduce from (22) and (23) that for all t I0 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M106">View MathML</a>

(24)

In the same way, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M107">View MathML</a> such that β defined as in (18) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M108">View MathML</a> satisfies that β(t) ≥ k(t) for all t I- and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M109">View MathML</a>

(25)

So we deduce from (24) and (25) that α and β are lower and upper solutions for problem (13).

Example 4 The function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M110">View MathML</a>

satisfies all the conditions in Proposition 1 for every compact interval I0. So the corresponding problem (13) has at least one solution for any choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M112">View MathML</a>.

We use now the ideas of Proposition 1 to construct lower and upper solutions for the general problem (1).

Proposition 2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M113">View MathML</a>and let f : I0 × ℝ2 → ℝ be a Carathéodory function. Assume that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M114">View MathML</a>such that for a.a. t I0 and all y ∈ ℝ we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M115">View MathML</a>

(26)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M116">View MathML</a>

(27)

Moreover, assume that the next conditions involving Fα and Fβ hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M117">View MathML</a>

(28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M118">View MathML</a>

(29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M119">View MathML</a>

(30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M120">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M121">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M122">View MathML</a>

(33)

Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M123">View MathML</a>such that α and β defined as in (17), (18) are lower and upper solutions for problem (1) with Λ = 0, and this does not depend on the choice of τ.

Proof. Reasoning in the same way as in the proof of Proposition 1, we obtain that there exists m ≥ 0 such that α(t) ≤ φ* for all t I- and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M124">View MathML</a>

As α(t) ≤ φ* for all t I, we obtain by virtue of (26) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M125">View MathML</a>

In the same way there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M126">View MathML</a> such that β(t) ≥ φ* for all t I- and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M127">View MathML</a>

Therefore, α and β are lower and upper solutions for problem (1).

Example 5 Let F be the function defined in Example 4 and consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M128">View MathML</a>

(34)

where γ ≥ 0, L > 0, and g is a nonnegative Carathéodory function.

In this case, we have φ* = -π, φ* ≈ 0.5611, and the function f(t,x,y) which defines the equation satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M129">View MathML</a>

so in particular conditions (26) and (27) hold. As conditions (28)-(33) also hold (see Example 4) we obtain that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/7/mathml/M130">View MathML</a> such that α and β defined as in (17), (18) are lower and upper solutions for problem (34) for any choice of τ. In particular, if there exists ψ L1(I 0) such that for a.a. t I0 and all x ∈ [α(t), β(t)] we have g(t, x) ≤ ψ(t), then problem (34) has maximal and minimal solutions between α and β.

Remark 3 Notice that the lower and upper solutions obtained both in Propositions 1 and 2 satisfy a slightly stronger condition than the one required in Definition 1.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Both authors' contributions to this paper are similar and it is impossible to say which part corresponds to each author's work. All authors read and approved the final manuscript.

Acknowledgements

This study was partially supported by the FEDER and Ministerio de Edu-cación y Ciencia, Spain, project MTM2010-15314.

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