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On positive solutions for a class of singular nonlinear fractional differential equations

Mohamed Jleli and Bessem Samet*

Author Affiliations

Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

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Boundary Value Problems 2012, 2012:73  doi:10.1186/1687-2770-2012-73

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/73


Received:29 March 2012
Accepted:18 May 2012
Published:12 July 2012

© 2012 Jleli and Samet; 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 study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2">View MathML</a> is a real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3">View MathML</a> is the Riemann-Liouville fractional derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M4">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M5">View MathML</a> (f is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6">View MathML</a>). Our approach is based on a coupled fixed point theorem on ordered metric spaces. An example is given to illustrate our main result.

MSC: 34A08, 34B16, 47H10.

Keywords:
singular fractional differential equation; positive solution; coupled fixed point; coupled lower and upper solution; ordered metric space

1 Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, fluid flows, electrical networks, viscoelasticity, aerodynamics, and many other branches of science. For details, see [1-11].

In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, etc.); see [2,4,5,11].

Recently, there have been many exciting developments in the field of fixed point theory on partially ordered metric spaces. The first result in this direction was given by Turinici [12]. In [13], Ran and Reurings extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result by Ran and Reurings was further extended and refined by many authors; see [14-19].

Very recently, Shurong Sun et al.[20] discussed the existence and uniqueness of a positive solution to the singular nonlinear fractional differential equation boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2">View MathML</a> is a real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3">View MathML</a> is the Riemann-Liouville fractional derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M10">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M11">View MathML</a> (f is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M13">View MathML</a> is nondecreasing for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M14">View MathML</a>.

Motivated by the above mentioned work, in this paper we investigate the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M15">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M16">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2">View MathML</a> is a real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3">View MathML</a> is the Riemann-Liouville fractional derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M4">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M20">View MathML</a> (f is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6">View MathML</a>), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M14">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M23">View MathML</a> is nondecreasing with respect to the first component, and it is decreasing with respect to the second component. Our approach is based on a recent coupled fixed point theorem on ordered metric spaces established by Harjani et al.[17]. We end the paper with an example that illustrates our main result.

2 Preliminaries

In this section, we recall some basic definitions and properties from fractional calculus theory. For more details about fractional calculus, we refer the readers to [1,3,10].

Definition 2.1 The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24">View MathML</a> of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M25">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M26">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M28">View MathML</a> denotes the integer part of number α, provided that the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M29">View MathML</a>.

Definition 2.2 The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M25">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M32">View MathML</a>

provided that the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M29">View MathML</a>.

From the definition of the Riemann-Liouville derivative, we can obtain the following statement.

Lemma 2.1 (see [10])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24">View MathML</a>. If we assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M35">View MathML</a>, then the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M36">View MathML</a>

has<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M39">View MathML</a>as unique solutions, whereNis the smallest integer greater than or equal toα.

Lemma 2.2 (see [10])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M35">View MathML</a>with a fractional derivative of order<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M24">View MathML</a>that belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M42">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M43">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M39">View MathML</a>, whereNis the smallest integer greater than or equal toα.

The Green function of fractional differential equation boundary value problem is given by

Lemma 2.3 (see [10])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2">View MathML</a>. The unique solution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M48">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M49">View MathML</a>

(4)

is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M50">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M51">View MathML</a>

Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M52">View MathML</a>is called the Green function of boundary value problem (3)-(4).

The following properties of the Green function will be used later.

Lemma 2.4 (see [10])

The following properties hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M53">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M57">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M54">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M61">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62">View MathML</a> be a partially ordered set endowed with a metric d such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M63">View MathML</a> is complete metric space. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M64">View MathML</a> be a given mapping.

Definition 2.3 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62">View MathML</a> is directed if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M66">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M67">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M69">View MathML</a>.

Definition 2.4 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M70">View MathML</a> is regular if the following conditions hold: (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M71">View MathML</a>) = if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M72">View MathML</a> is a nondecreasing sequence in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M73">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M74">View MathML</a> for all n;; (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M75">View MathML</a>) = if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M76">View MathML</a> is a decreasing sequence in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M77">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M78">View MathML</a> for all n..

Example 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M80">View MathML</a>, be the set of real continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M81">View MathML</a>. We endow X with the standard metric d given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M82">View MathML</a>

We define the partial order ⪯ on X by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M83">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M84">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M85">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M86">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M87">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M89">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62">View MathML</a> is directed. Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M72">View MathML</a> be a nondecreasing sequence in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M92">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M93">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M94">View MathML</a>. Then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M96">View MathML</a> is a nondecreasing sequence of real numbers converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M97">View MathML</a>. Thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M98">View MathML</a> for all n, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M74">View MathML</a> for all n. Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M76">View MathML</a> is a decreasing sequence in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M101">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M93">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M103">View MathML</a>, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M78">View MathML</a> for all n. Then we proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M105">View MathML</a> is regular.

Definition 2.5 (see [15])

An element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M66">View MathML</a> is called a coupled fixed point of F if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M108">View MathML</a>.

Definition 2.6 (see [15])

We say that F has the mixed monotone property if for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M110">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M111">View MathML</a>

Denote by Φ the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M112">View MathML</a> satisfying: (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M113">View MathML</a>) = φ is continuous;; (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M114">View MathML</a>) = φ is nondecreasing;; (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M115">View MathML</a>) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M116">View MathML</a>..

The following two lemmas are fundamental in the proofs of our main results.

Lemma 2.5 (see [17])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62">View MathML</a>be a partially ordered set and suppose that there exists a metricdonXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M63">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M64">View MathML</a>be a mapping having the mixed monotone property onXsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M120">View MathML</a>

(5)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M121">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M122">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M123">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M124">View MathML</a>. Suppose also that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M70">View MathML</a>is regular and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M126">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M127">View MathML</a>

ThenFhas a coupled fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M128">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M129">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M76">View MathML</a>are the sequences inXdefined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M131">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M132">View MathML</a>

(6)

Lemma 2.6 (see [17])

Adding to the hypotheses of Lemma 2.5 the condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M62">View MathML</a>is regular, we obtain the uniqueness of the coupled fixed point. Moreover, we have the equality<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M134">View MathML</a>.

3 Main result

Let Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M135">View MathML</a> be endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M136">View MathML</a>. We define the partial order ⪯ on E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M137">View MathML</a>

(7)

In Example 2.1, we proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M138">View MathML</a> with the classic metric given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M139">View MathML</a>

satisfies the following properties: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M138">View MathML</a> is directed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M141">View MathML</a> is regular.

Define the closed cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M142">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M143">View MathML</a>

where 0 denotes the zero function.

Definition 3.1 (see [15])

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M144">View MathML</a> is a coupled lower and upper solution to (1)-(2) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M146">View MathML</a>

Our main result is the following.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M149">View MathML</a>is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M150">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M151">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M152">View MathML</a>. Assume that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M153">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M154">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M156">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M158">View MathML</a>

(8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M160">View MathML</a>. Suppose also that (1)-(2) has a coupled lower and upper solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M161">View MathML</a>. Then the boundary value problem (1)-(2) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M162">View MathML</a>. The sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M164">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M165">View MathML</a>

converge uniformly to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M166">View MathML</a>.

Proof Suppose that u is a solution of boundary value problem (1)-(2). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M167">View MathML</a>

(9)

We define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M168">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M169">View MathML</a>

• Step 1. We shall prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M170">View MathML</a>

(10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M171">View MathML</a>. Let us prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M172">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M173">View MathML</a>

By the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M174">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M175">View MathML</a>, it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M176">View MathML</a>. Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M177">View MathML</a>. We have to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M178">View MathML</a>

We distinguish three cases:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M179">View MathML</a>

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M180">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M181">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M175">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M183">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M184">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M185">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M186">View MathML</a>

Using Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M187">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M188">View MathML</a> denotes the beta function.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M190">View MathML</a>. In this case,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M191">View MathML</a>

Now, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M192">View MathML</a>

Case 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M194">View MathML</a>. The proof is similar to that of Case 2, so we omit it.

Thus we proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M195">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M175">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M197">View MathML</a>. Moreover, taking into account Lemma 2.4 and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M198">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M200">View MathML</a>, our claim (10) is proved. Now the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M201">View MathML</a>

is well defined.

• Step 2. We shall prove that F has the mixed monotone property with respect to the partial order ⪯ given by (7).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M202">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M203">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M204">View MathML</a>. From (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M205">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M185">View MathML</a>. This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M207">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M208">View MathML</a>

which gives us that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M209">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157">View MathML</a>, and then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M211">View MathML</a>

Then F has the mixed monotone property.

• Step 3. We shall prove that F satisfies the contractive condition (5) for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M212">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M215">View MathML</a>. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157">View MathML</a>, using (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M217">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M218">View MathML</a>

(11)

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M219">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M220">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M221">View MathML</a>

Now, using the above inequality, (11) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M222">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M223">View MathML</a>

Thus we proved that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M215">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M227">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M229">View MathML</a>.

• Step 4. Existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M230">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M232">View MathML</a>.

We take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M233">View MathML</a>, the coupled lower and upper solution to (1)-(2).

Now, from Lemmas 2.5 and 2.6, there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M234">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M235">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M166">View MathML</a> is the unique positive solution to (1)-(2). The convergence of the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M238">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M166">View MathML</a> follows immediately from (6). □

Now, we end this paper with the following example.

Example 3.1 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M240">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M241">View MathML</a>

(13)

In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M242">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M243">View MathML</a>. Note that f is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M245">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M246">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M247">View MathML</a>. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M154">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M157">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M252">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M253">View MathML</a>

Consider now, the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M161">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M256">View MathML</a>. Using Lemma 2.4(iv), one can show easily that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M257">View MathML</a> is a coupled lower and upper solution to (12)-(13).

Finally, applying Theorem 3.1, we deduce that (12)-(13) has one and only one positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M162">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgement

This work was supported by the Research Center, College of Science, King Saud University.

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