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Existence and uniqueness of positive solution to singular fractional differential equations

Yongqing Wang1*, Lishan Liu12 and Yonghong Wu2

Author Affiliations

1 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, People’s Republic of China

2 Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA, 6845, Australia

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Boundary Value Problems 2012, 2012:81  doi:10.1186/1687-2770-2012-81

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/81


Received:6 April 2012
Accepted:10 July 2012
Published:28 July 2012

© 2012 Wang et al.; licensee Springer

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

Abstract

In this paper, we discuss the existence and uniqueness of a positive solution to the following singular fractional differential equation with nonlocal boundary value conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M6">View MathML</a> is the standard Riemann-Liouville derivative, f may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M8">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M9">View MathML</a>.

MSC: 34B10, 34B15.

Keywords:
fractional differential equation; positive solution; iterative scheme; singular boundary value problem

1 Introduction

In this paper, we consider the following fractional differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M10">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M6">View MathML</a> is the standard Riemann-Liouville derivative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M16">View MathML</a> may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M8">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M9">View MathML</a>. In this paper, by a positive solution to (1.1), we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M20">View MathML</a> which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M21">View MathML</a>, positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M22">View MathML</a> and satisfies (1.1).

Recently, many results were obtained dealing with the existence of solutions for nonlinear fractional differential equations by using the techniques of nonlinear analysis; see [1-23] and references therein. The multi-point boundary value problems (BVP for short) have provoked a great deal of attention, for example [13-19]. In [10], the authors discussed some positive properties of the Green function for Direchlet-type BVP of nonlinear fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M23">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M6">View MathML</a> is the standard Riemann-Liouville derivative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M26">View MathML</a>. By using the Krasnosel’skii fixed point theorem, the existence of positive solutions were obtained under suitable conditions on f.

In [14], the authors investigated the existence and multiplicity of positive solutions by using some fixed point theorems for the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M27">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M31">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M34">View MathML</a> satisfied Carathéodory type conditions.

In [20,21], the authors considered the fractional differential equation given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M35">View MathML</a>

(1.4)

In order to obtain the existence of positive solutions of (1.4), they considered the following fractional differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M36">View MathML</a>

(1.5)

In [20], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M37">View MathML</a>, and g, h have different monotone properties. By using the fixed point theorem for the mixed monotone operator, Zhang obtained (1.4) and had a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M38">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M39">View MathML</a>. But the results are not true since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M40">View MathML</a> is a positive solution of (1.5), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M41">View MathML</a>. What causes it lies in the unsuitable using of properties of the Green function.

In [21], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M43">View MathML</a> is increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M45">View MathML</a>. By using the positive properties of the Green function obtained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M46">View MathML</a> and fixed point theory for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M47">View MathML</a> concave operator, the authors obtained the uniqueness of a positive solution for the BVP (1.4).

Motivated by the works mentioned above, in this paper we aim to establish the existence and uniqueness of a positive solution to the BVP (1.1). Our work presented in this paper has the following features. Firstly, the BVP (1.1) possesses singularity, that is, f may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M8">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M9">View MathML</a>. Secondly, we impose weaker positivity conditions on the nonlocal boundary term, that is, some of the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M51">View MathML</a> can be negative. Thirdly, the unique positive solution can be approximated by an iterative scheme.

The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. We also develop some new positive properties of the Green function. In Section 3, we discuss the existence and uniqueness of a positive solution of the BVP (1.1), we also give an example to demonstrate the application of our theoretical results.

2 Preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can be found in recent literature.

Definition 2.1 The fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M52">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M53">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M54">View MathML</a>

provided the right-hand side is defined pointwise on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M55">View MathML</a>.

Definition 2.2 The fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M52">View MathML</a> of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M53">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M58">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M60">View MathML</a> denotes the integral part of the number α, provided the right-hand side is pointwisely defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M55">View MathML</a>.

Definition 2.3 By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M62">View MathML</a>, we mean <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M63">View MathML</a>.

Lemma 2.1 ([3])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M52">View MathML</a>. Then the following equality holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M21">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M67">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M70">View MathML</a>.

Set

(2.1)

(2.2)

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M74">View MathML</a>

(2.4)

For the convenience in presentation, we here list the assumption to be used throughout the paper.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M77">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M78">View MathML</a>.

Remark 2.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M79">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M80">View MathML</a>), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M82">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M83">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M84">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M5">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M77">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M78">View MathML</a>.

Lemma 2.2 ([14])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M90">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M91">View MathML</a>

Lemma 2.3Assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75">View MathML</a>) holds, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M93">View MathML</a>. Then the unique solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M94">View MathML</a>

(2.5)

is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M95">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M96">View MathML</a>is called the Green function of BVP (2.5).

Proof From Lemma 2.1, we have the solution of (2.5) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M97">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M98">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M99">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M100">View MathML</a>.

By Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M101">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M102">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M103">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M104">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M105">View MathML</a>

Therefore, the solution of (2.5) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M106">View MathML</a>

 □

Lemma 2.4The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M107">View MathML</a>has the following properties:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M108">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M109">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M110">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M111">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M112">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M109">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M114">View MathML</a>

(2.6)

Proof It is obvious that (1), (2) hold. In the following, we will prove (3).

(i) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M115">View MathML</a>, noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M116">View MathML</a>, we have

(2.7)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M118">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M119">View MathML</a>

(2.8)

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M120">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M121">View MathML</a>, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M122">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M123">View MathML</a>

(2.9)

(ii) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M124">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M125">View MathML</a>

(2.10)

On the other hand, clearly we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M126">View MathML</a>

(2.11)

(2.8)-(2.11) implies (3) holds. □

By Lemma 2.4 we have the following results.

Lemma 2.5Assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75">View MathML</a>) holds, then the Green function defined by (2.3) satisfies:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M129">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M131">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M129">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M134">View MathML</a>

Lemma 2.6Assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75">View MathML</a>) holds, then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M136">View MathML</a>satisfies:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M129">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M131">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M142">View MathML</a>.

For convenience, we list here two more assumptions to be used later:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M144">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M146">View MathML</a> is nondecreasing on u, nonincreasing on v, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M147">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M148">View MathML</a>

(2.12)

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M150">View MathML</a>

(2.13)

Remark 2.2 Inequality (2.12) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M151">View MathML</a>

(2.14)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M152">View MathML</a> be endowed with the maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M153">View MathML</a>. Define a cone P by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M154">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M155">View MathML</a>

(2.15)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M156">View MathML</a>, where θ is the zero element of E. We have the following lemma.

Lemma 2.7Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149">View MathML</a>) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M159">View MathML</a>.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M160">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M161">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M162">View MathML</a>

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149">View MathML</a>) and (2) of Lemma 2.6, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M165">View MathML</a>

(2.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M166">View MathML</a>. This implies that A is well defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M167">View MathML</a>.

On the other hand, by (3) of Lemma 2.6, we have

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M169">View MathML</a>. Combining with (2.16), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M159">View MathML</a>. □

Remark 2.3 By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143">View MathML</a>) and (2.15), A is a mixed monotone operator.

3 Main results

Theorem 3.1Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149">View MathML</a>) hold. Then the BVP (1.1) has a unique positive solution.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M174">View MathML</a>, by Remark 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M175">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M176">View MathML</a>, noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M177">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M178">View MathML</a> small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M179">View MathML</a>

(3.1)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M180">View MathML</a>

(3.2)

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M181">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M182">View MathML</a>

(3.3)

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M183">View MathML</a>

(3.4)

Noticing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M184">View MathML</a>

therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M185">View MathML</a>

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M186">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M187">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M188">View MathML</a>

By induction, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M189">View MathML</a>

(3.5)

By (3.4), (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M190">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M191">View MathML</a> is a Cauchy sequence. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M192">View MathML</a> is a Cauchy sequence. Noticing (3.4), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M193">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M191">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M192">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M197">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M198">View MathML</a>

(3.6)

(3.5) and (3.6) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M199">View MathML</a>

(3.7)

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M200">View MathML</a>.

By the mixed monotone property of A and (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M201">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M202">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M203">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M200">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M195">View MathML</a> is a positive fixed point of A.

In the following, we will prove the positive fixed point of A is unique.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M206">View MathML</a> is a positive fixed point of A. By Lemma 2.6, we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M207">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M208">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M209">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M210">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M211">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M212">View MathML</a>, which contradicts the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M213">View MathML</a>. Consequently, the positive fixed point of A is unique.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M214">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M215">View MathML</a>

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M216">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M217">View MathML</a>. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M218">View MathML</a>. By Lemma 2.5 and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149">View MathML</a>), we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M221">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M222">View MathML</a>

Lemma 2.3 implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M214">View MathML</a> is a positive solution of (1.1).

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M224">View MathML</a> is a positive solution of (1.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M225">View MathML</a>

By Lemma 2.5, we have there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M161">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M227">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M228">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M229">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M230">View MathML</a>

which implies u is a positive fixed point of A.

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M214">View MathML</a> is the unique positive solution of the BVP (1.1). □

Remark 3.1 The unique positive solution y of (1.1) can be approximated by the iterative schemes: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M176">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M234">View MathML</a> be defined as (3.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M237">View MathML</a> , then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M238">View MathML</a>.

Example 3.1 (A 4-point BVP with coefficients of both signs)

Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M239">View MathML</a>

(3.8)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M240">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M241">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M242">View MathML</a>

By direct calculations, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M77">View MathML</a>, which implies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M75">View MathML</a>) holds.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M246">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M248">View MathML</a> is nondecreasing on x and nonincreasing on y. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M249">View MathML</a>

(3.9)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M250">View MathML</a>

Therefore (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M143">View MathML</a>) holds. It is easy to get that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/81/mathml/M149">View MathML</a>) holds. Therefore, the assumptions of Theorem 3.1 are satisfied. Thus Theorem 3.1 ensures that the BVP (3.8) has a unique positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to the anonymous referee for his/her valuable suggestions. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11126231) and Project of Shandong Province Higher Educational Science and Technology Program (J11LA06). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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