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

Open Access Research

Existence of positive solutions for a kind of periodic boundary value problem at resonance

Mirosława Zima* and Piotr Drygaś

Author Affiliations

Institute of Mathematics, University of Rzeszów, Rejtana 16 A, Rzeszów, 35-959, Poland

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Boundary Value Problems 2013, 2013:19  doi:10.1186/1687-2770-2013-19


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/19


Received:20 December 2012
Accepted:21 January 2013
Published:11 February 2013

© 2013 Zima and Drygaś; 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 the paper we provide sufficient conditions for the existence of positive solutions for some second-order differential equation subject to periodic boundary conditions. Our method employs a Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. Two examples are given to illustrate the main result of the paper.

Keywords:
periodic boundary value problem; positive solution; coincidence equation

1 Introduction

In the paper we are interested in the existence of positive solutions for the periodic boundary value problem (PBVP)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M3">View MathML</a> are continuous functions. Our study is motivated by current activity of many researchers in the area of theory and applications of PVBPs; see, for example, [1-4] and references therein. In particular, in a recent paper [1], Chu, Fan and Torres have studied the existence of positive periodic solutions for the singular damped differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M4">View MathML</a>

by combining the properties of the Green’s function of the PBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M5">View MathML</a>

(2)

with a nonlinear alternative of Leray-Schauder type (see, for example, [5]). It should be noted that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M6">View MathML</a> was the key assumption used in [1]. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M7">View MathML</a>, then PBVP (2) has nontrivial solutions, which means that the problem we are concerned with here, that is, PBVP (1), is at resonance. There are several methods to deal with the resonant PBVPs. For example, in [6], Torres studied the existence of a positive solution for the PBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M8">View MathML</a>

by considering the equivalent problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M9">View MathML</a>

via Krasnoselskii’s theorem on cone expansion and compression. Further results in this direction can be found in [7] and [8]. In [9] Rachůnková, Tvrdý and Vrkoč applied the method of upper and lower solutions and topological degree arguments to establish the existence of nonnegative and nonpositive solutions for the PBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M10">View MathML</a>

(3)

The same PBVP was studied by Wang, Zhang and Wang in [10]. Their existence and multiplicity results on positive solutions are based on the theory of a fixed point index for A-proper semilinear operators on cones developed by Cremins [11].

The goal of our paper is to provide sufficient conditions that ensure the existence of positive solutions of (1) with the function h positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11">View MathML</a>. Our general result is illustrated by two examples. The method we use in the paper is to rewrite BVP (1) as a coincidence equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M12">View MathML</a>, where L is a Fredholm operator of index zero and N is a nonlinear operator, and to apply the Leggett-Williams norm-type theorem for coincidences obtained by O’Regan and Zima [12]. We would like to emphasize that the idea of results of [11] and [12], as well as these of [13-15], goes back to the celebrated Mawhin’s coincidence degree theory [16]. For more details on this significant tool, its modifications and wide applications, we refer the reader to [17-22] and references therein.

In this paper, for the first time, the existence theorem from [12] is used for studying the boundary value problem with the nonlinearity f depending also on the derivative. In general, the presence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M13">View MathML</a> in f makes the problem much harder to handle. We point out that, to the best of our knowledge, there are only a few papers on PBVPs that discuss such a nonlinearity; we refer the reader to [15,23-25] for some results of that type. We also complement several results in the literature, for example, in [1,26] and [27]. It is evident that the existence theorems for PBVP (1) can be established by the shift method used in [6], that is, one can employ the results of [1] to the periodic problem we study here. However, the conditions imposed on f in [1] are not comparable with ours.

2 Coincidence equation

For the convenience of the reader, we begin this section by providing some background on cone theory and Fredholm operators in Banach spaces.

Definition 1 A nonempty subset C, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M14">View MathML</a>, of a real Banach space X is called a cone if C is closed, convex and

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M15">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M17">View MathML</a>,

(ii) x, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M18">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M19">View MathML</a>.

Every cone induces a partial ordering in X as follows: for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M20">View MathML</a>, we say that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M21">View MathML</a>

The following property holds for every cone in a Banach space.

Lemma 1[28]For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M22">View MathML</a>, there exists a positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M23">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M24">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16">View MathML</a>.

Consider a linear mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M26">View MathML</a> and a nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M27">View MathML</a>, where X and Y are Banach spaces. If L is a Fredholm operator of index zero, that is, ImL is closed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M28">View MathML</a>, then there exist continuous projections <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M32">View MathML</a> (see, for example, [14,16]). Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M33">View MathML</a>, there exists an isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M34">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M35">View MathML</a> the restriction of L to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M36">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M35">View MathML</a> is an isomorphism from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M38">View MathML</a> to ImL and its inverse

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M39">View MathML</a>

is defined.

As a result, the coincidence equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M12">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M41">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M42">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M43">View MathML</a> be a retraction, that is, a continuous mapping such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M44">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16">View MathML</a>. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M46">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M48">View MathML</a> be open bounded subsets of X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M50">View MathML</a>. Assume that

1L is a Fredholm operator of index zero,

2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M51">View MathML</a> is continuous and bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M52">View MathML</a> is compact on every bounded subset of X,

3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M53">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M55">View MathML</a>,

4ρ maps subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M56">View MathML</a> into bounded subsets of C,

5<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M57">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M58">View MathML</a> stands for the Brouwer degree,

6 there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M59">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M60">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M61">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M62">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M63">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M64">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M16">View MathML</a>,

7<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M67">View MathML</a>.

Theorem 1[12]

Under the assumptions 1-7the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M12">View MathML</a>has a solution in the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M69">View MathML</a>.

In the next section, we use Theorem 1 to prove the existence of a positive solution for PBVP (1). For applications of Theorem 1 to nonlocal boundary value problems at resonance, we refer the reader to [22], [29] and [30].

3 Periodic boundary value problem

We now provide sufficient conditions for the existence of positive solutions for PBVP (1). For convenience and ease of exposition, we make use of the following notation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M70">View MathML</a>

(4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M71">View MathML</a>

(5)

We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M72">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11">View MathML</a>. Moreover, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M74">View MathML</a>

(6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M75">View MathML</a>

(7)

where M is a positive constant.

We assume that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M3">View MathML</a> are continuous functions.

We also assume that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M83">View MathML</a> and a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M84">View MathML</a> such that

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M85">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M86">View MathML</a>,

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M87">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88">View MathML</a>,

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M90">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M91">View MathML</a>,

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M92">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M94">View MathML</a>,

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M96">View MathML</a>,

(H7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M97">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M99">View MathML</a>.

Theorem 2Under the assumptions (H1)-(H7), PBVP (1) has a positive solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M101">View MathML</a> denote the supremum norm in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M102">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M103">View MathML</a>. Consider the Banach spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M104">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M105">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M106">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M101">View MathML</a>.

We write problem (1) as a coincidence equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M108">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M109">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M110">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M111">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M112">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M113">View MathML</a>

where ψ is given by (5).

Clearly, ImL is closed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M114">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M115">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M116">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M117">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M118">View MathML</a>, which gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M119">View MathML</a>. Consequently, L is Fredholm of index zero, and the assumption 1 is satisfied.

Define the projections <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M29">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M121">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M30">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M123">View MathML</a>

It is a routine matter to show that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M124">View MathML</a>, the inverse <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M125">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M35">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M127">View MathML</a>

with the kernel k defined by (6). Clearly, the assumption 2 is satisfied. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M128">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M129">View MathML</a>

Then J is an isomorphism from ImQ to KerL. Next, consider a cone

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M130">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M131">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M132">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M133">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M134">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M135">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M48">View MathML</a> are open bounded subsets of X, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M49">View MathML</a>.

To verify 3, suppose that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M139">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M140">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M141">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M142">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M144">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M145">View MathML</a>

(8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M146">View MathML</a>

(9)

There are two cases to consider.

1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M147">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M148">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M149">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M150">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M151">View MathML</a>, contrary to the assumption (H3). Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M152">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M153">View MathML</a>, BCs (9) imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M154">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M155">View MathML</a> which contradicts (H3) again.

2. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M156">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M148">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M158">View MathML</a>. Observe that (H2) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M159">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M91">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M150">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M163">View MathML</a>, we get from (8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M164">View MathML</a>

(10)

a contradiction. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M165">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M166">View MathML</a>

(11)

contrary to (H5). By similar arguments, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M152">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M153">View MathML</a>, BCs (9) and (H4) imply either (10) or (11). Thus, 3 is fulfilled.

Next, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M169">View MathML</a>, define (see [15])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M170">View MathML</a>

Clearly, ρ is a retraction and maps subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M56">View MathML</a> into bounded subsets of C, so 4 holds.

To verify 5, it is enough to consider, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M173">View MathML</a>, the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M174">View MathML</a>

Observe that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M175">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M176">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M178">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M179">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M180">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M181">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M182">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M183">View MathML</a> and in view of (H3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M184">View MathML</a>

which is a contradiction. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M185">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M186">View MathML</a>, hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M187">View MathML</a>

which contradicts (H2). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M190">View MathML</a>. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M191">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M192">View MathML</a>

We next show that 6 holds. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M193">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M88">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M196">View MathML</a>, and by (H6) and (H7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M197">View MathML</a>

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M198">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M61">View MathML</a>, so 6 is satisfied.

Finally, we must check if 7 holds. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M200">View MathML</a>, then in view of (H2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M201">View MathML</a>

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M202">View MathML</a>, we have from (H2) and (H7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M203">View MathML</a>

Thus, 7 is fulfilled and the assertion follows. □

We now give two examples illustrating Theorem 2. Some calculations have been made with Mathematica. In the first example, the function h is constant, while in the second <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M204">View MathML</a> and f is independent of t.

Example 1

Consider the following PBVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M205">View MathML</a>

(12)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M209">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M210">View MathML</a>

Moreover, (7) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M211">View MathML</a> reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M212">View MathML</a>

and the assumptions (H2)-(H7) are met with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M219">View MathML</a>. By Theorem 2, problem (12) has a positive solution.

Example 2

Consider the PBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M220">View MathML</a>

(13)

In this case, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M223">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M224">View MathML</a>

The assumptions of Theorem 2 are fulfilled with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/19/mathml/M232">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MZ and PD contributed equally to the manuscript and read and approved its final version.

Acknowledgements

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

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