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Existence results of positive solutions for boundary value problems of fractional differential equations

Guoqing Chai

Author Affiliations

College of Mathematics and Statistics, Hubei Normal University, Hubei, 435002, P.R. China

Boundary Value Problems 2013, 2013:109  doi:10.1186/1687-2770-2013-109

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/109


Received:15 October 2012
Accepted:15 April 2013
Published:29 April 2013

© 2013 Chai; 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 are concerned with the following fractional equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M1">View MathML</a>

with the boundary value conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M2">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M3">View MathML</a> is the standard Caputo derivative with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a> and δ, γ are constants with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>. By applying a new fixed point theorem on cone and Krasnoselskii’s fixed point theorem, some existence results of positive solution are obtained.

MSC: 34A08, 34B15, 34B18.

Keywords:
fractional differential equations; existence results; fixed point theorem; positive solution

1 Introduction

In this paper, we are concerned with the existence of positive solutions for the fractional equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M7">View MathML</a>

(1.1)

with the boundary value conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M8">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M3">View MathML</a> is the standard Caputo derivative with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a> and δ, γ are constants with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>.

Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, etc. (see [1-5]). There has been a significant development in the study of fractional differential equations and inclusions in recent years, see the monographs of Podlubny [5], Kilbas et al.[6], Lakshmikantham et al.[7], Samko et al.[8], Diethelm [9], and the survey by Agarwal et al.[10]. For some recent contributions on fractional differential equations, see [9-25] and the references therein.

On the other hand, it is well known that the fourth-order boundary value problem describes the deformations of an elastic beam in equilibrium state. Owing to its importance in physics, the existence of solutions to this problem has been studied by many authors; see, for example, [26-30] and references therein. Recently, there have been a few papers dealing with the existence of solutions for fractional equations of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M13">View MathML</a>.

In [14], Xu et al. discussed the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M14">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M16">View MathML</a> is nonnegative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M17">View MathML</a> is the Riemann-Liouville fractional derivative of order α. The existence results of positive solutions are obtained by applying the Leray-Schauder nonlinear alternative theorem.

In [15], Liang and Zhang studied the following nonlinear fractional boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M18">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M20">View MathML</a> is nondecreasing relative to u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M17">View MathML</a> is the Riemann-Liouville fractional derivative of order α. By means of the lower and upper solution method and fixed point theorems, some results on the existence of positive solutions were obtained.

In [16], Agarwal and Ahmad studied the solvability of the following anti-periodic boundary value problem for a nonlinear fractional differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M22">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>. The existence results were obtained by the nonlinear alternative theorem.

Inspired by above work, the author will be concerned with the boundary value problem (BVP for short in the sequel) (1.1)-(1.2). To the best of our knowledge, no contribution exists concerning the existence of solutions for BVP (1.1)-(1.2). In the present paper, by applying a new fixed point theorem on cone and Krasnoselskii’s fixed point theorem, some existence results of positive solution for BVP (1.1)-(1.2) are obtained. It is worth to point out that the results in this paper are also new even for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M24">View MathML</a> relative to the corresponding literature with regard to the fourth-order boundary value problem. In addition, the conditions imposed in this paper are easily verified.

The organization of this paper is as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. In Section 3, we put forward and prove our main results. Finally, we give two examples to demonstrate our main results.

2 Preliminaries

In this section, we introduce some preliminary facts which are useful throughout this paper.

Let ℕ be the set of positive integers, ℝ be the set of real numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M25">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M26">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M27">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M28">View MathML</a> the Banach space endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M30">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M31">View MathML</a>.

Definition 2.1[6]

The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M33">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M34">View MathML</a>

Definition 2.2[6]

The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M33">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M37">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M39">View MathML</a> denotes the integer part of α.

Definition 2.3[6]

The Caputo fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32">View MathML</a> of a function y on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M41">View MathML</a> is defined via the above Riemann-Liouville derivatives by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M42">View MathML</a>

Lemma 2.1[6]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M44">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M45">View MathML</a>

holds on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M46">View MathML</a>.

Lemma 2.2[17]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M47">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M49">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M50">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M51">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M52">View MathML</a>

holds on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M53">View MathML</a>.

For convenience, we first list some hypotheses which will be used throughout this paper.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M55">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M60">View MathML</a>, consider the following BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M61">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M62">View MathML</a>

(2.2)

We have the following lemma, which is important in this paper.

Lemma 2.3Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M64">View MathML</a>is a solution of BVP (2.1)-(2.2) iff<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M65">View MathML</a>has the expression as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M66">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M67">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M68">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M69">View MathML</a>

(2.6)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M64">View MathML</a> be a solution of (2.1)-(2.2). Then by Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M71">View MathML</a>

(2.7)

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M72">View MathML</a>

Thus, by the boundary value condition (2.2), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M73">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M74">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M75">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M76">View MathML</a>

(2.11)

From (2.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M77">View MathML</a>

(2.12)

Substituting (2.12) into (2.10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M78">View MathML</a>

(2.13)

So, by (2.13), (2.12), and (2.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M79">View MathML</a>

(2.14)

Thus, from (2.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M80">View MathML</a>

(2.15)

Hence, from (2.7) together with (2.12)-(2.15), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M81">View MathML</a>

(2.16)

Noticing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M82">View MathML</a>

by Definition 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M83">View MathML</a>

Conversely, if u has the expression (2.3), then from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M84">View MathML</a>, we can easily verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M85">View MathML</a>

(2.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M86">View MathML</a>

(2.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M87">View MathML</a>

(2.19)

hold for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and u satisfies the boundary condition (2.2).

Again, from (2.16) and Lemma 2.1, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M90">View MathML</a>. In addition, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M84">View MathML</a>, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M92">View MathML</a> from (2.19). □

For the forthcoming analysis, we need to introduce some new notations.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M93">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M94">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M95">View MathML</a>

It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M96">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M97">View MathML</a> noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>.

We also need the following lemma, which will play an important role in obtaining our main results in Section 3.

Lemma 2.4Under the assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>), Green’s function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M102">View MathML</a>has the following properties:

(1) Gis continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M103">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108">View MathML</a>.

Proof (1) Observing the expression of Green’s function given by (2.4)-(2.6), the conclusion (1) of Lemma 2.4 is obvious.

(2) We first show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108">View MathML</a>.

In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124">View MathML</a>, then by (2.5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M125">View MathML</a>

(2.20)

Owing to the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M128">View MathML</a>. Thus, we immediately obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M129">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124">View MathML</a> from (2.20) together with the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>.

Similarly, we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M133">View MathML</a>

(2.21)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M135">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134">View MathML</a>.

To summarize, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M137">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M138">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M108">View MathML</a>.

Now, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M140">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M141">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M137">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M143">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M144">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M104">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105">View MathML</a>.

(3) The proof is divided into four steps.

Step 1. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M109">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M105">View MathML</a>.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M149">View MathML</a>, then by (2.5) and the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M153">View MathML</a>

(2.22)

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M154">View MathML</a>, then by an argument similar to (2.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M155">View MathML</a>

Summing up the above analysis (i)-(ii), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M156">View MathML</a>

Step 2. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M157">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113">View MathML</a>.

In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112">View MathML</a>, then by (2.5) combined with the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M165">View MathML</a>

(2.23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M166">View MathML</a>, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M167">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M169">View MathML</a>, then by an argument similar to (2.23), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M170">View MathML</a>

(2.24)

So, by (2.23)-(2.24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M111">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112">View MathML</a>.

Step 3. Now, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M174">View MathML</a>

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124">View MathML</a>, then by (2.20) and keeping in mind that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M179">View MathML</a>

(2.25)

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134">View MathML</a>, then by an argument similar to (2.25), from (2.21), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M181">View MathML</a>

Summing up the above analysis (i)-(ii), and noting Step 2 of the proof as before, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M182">View MathML</a>

Step 4. It remains to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M183">View MathML</a>

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124">View MathML</a>, then by (2.20) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M185">View MathML</a>, we know that the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M186">View MathML</a>

hold for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M113">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M189">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M190">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M191">View MathML</a>

Similarly, we can obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M192">View MathML</a>

The proof is complete. □

Now, we introduce a cone as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M193">View MathML</a>

It is easy to check that the above set P is a cone in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M194">View MathML</a>, which will be used in the sequel.

We define an operator T on P as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M195">View MathML</a>

(2.26)

Obviously, under the assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>), the operator T is well defined. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M198">View MathML</a>

(2.27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M134">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M204">View MathML</a> are given by (2.20)-(2.21), respectively.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M205">View MathML</a> is a positive solution of BVP (1.1)-(1.2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M208">View MathML</a>, and u satisfies BVP (1.1)-(1.2).

By Lemma 2.3, it is easy to know that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M205">View MathML</a> is a positive solution of BVP (1.1)-(1.2) iff <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a> is a nonzero fixed point of T. So, we can focus on seeking the existence of a nonzero fixed point of T in P.

Finally, for the remainder of this section, we give the following two theorems, which are fundamental in the proof of our main results.

Let X be a Banach space, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M211">View MathML</a> be a cone. Suppose that the functions α, β satisfy the following condition:

(D) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M212">View MathML</a> are continuous convex functionals satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M214">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M216">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M217">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M215">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M219">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M220">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M221">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M222">View MathML</a> is a constant.

Lemma 2.5[31]

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M224">View MathML</a>, Lare constants with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M226">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M227">View MathML</a>

Set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M228">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229">View MathML</a>is a completely continuous operator satisfying

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M230">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M232">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M234">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M235">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M238">View MathML</a>) there is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M239">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M240">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M241">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M243">View MathML</a>.

ThenThas at least one fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M244">View MathML</a>.

Lemma 2.6[32]

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M246">View MathML</a>are two open subsets ofXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M247">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229">View MathML</a>be a completely continuous operator such that either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M250">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M252">View MathML</a>, or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M250">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M252">View MathML</a>.

ThenThas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M257">View MathML</a>.

3 Main results

We first prove the following lemma to obtain our main results.

Lemma 3.1Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>) hold. Then the operatorTdefined by (2.26) mapsPintoP, andTis completely continuous.

Proof It is well known that the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M261">View MathML</a> are equivalent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M262">View MathML</a>. So, we can consider that the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M262">View MathML</a> is equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M264">View MathML</a> in the following proof.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>, in view of the conclusion (1)-(2) of Lemma 2.4 and the hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M266">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M271">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a> observing (2.26)-(2.27). Moreover, the conclusion (3) of Lemma 2.4 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M273">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M274">View MathML</a>

(3.2)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M187">View MathML</a>.

From (3.1)-(3.2), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M278">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M279">View MathML</a>

Similarly, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M280">View MathML</a>

In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M281">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229">View MathML</a>.

Now, we show that the operator T is compact on P.

In fact, let U be an arbitrary bounded set in P. Then there exists a positive number L such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M283">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M284">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M285">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M284">View MathML</a>.

In terms of Lemma 2.4, it follows from (2.26)-(2.27) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M289">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M290">View MathML</a>

(3.4)

Because the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M291">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M292">View MathML</a> are integrable on I, the formulae (3.3)-(3.4) yield that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M294">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M295">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M296">View MathML</a>. That is, TU is uniformly bounded.

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M297">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M298">View MathML</a>, by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M299">View MathML</a>, the formula (2.26) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M300">View MathML</a>

(3.5)

According to (2.4)-(2.5) and by applying the mean value theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M301">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M302">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M303">View MathML</a>.

Similarly, there is another constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M304">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M305">View MathML</a>

(3.7)

Again, because the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M291">View MathML</a> is integrable on I, the absolute continuity of integral of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M307">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M308">View MathML</a> ensures that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M309">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M310">View MathML</a>

(3.8)

So, (3.5) together with (3.6)-(3.8) implies that there exists a constant N such that the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M311">View MathML</a>

holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M284">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M297">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M298">View MathML</a>. That is, the set TU is equicontinuous.

Similarly, we can deduce that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M315">View MathML</a> is also equicontinuous in terms of (2.27).

So, as a consequence of the Arzelà-Ascoli theorem, we have that TU is a compact set.

Now, we come to prove the operator T is continuous on P.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M316">View MathML</a> be an arbitrary sequence in P with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M317">View MathML</a>. Then there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M226">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M319">View MathML</a>

According to the uniform continuity of f on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M320">View MathML</a>, for an arbitrary number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M321">View MathML</a>, there is a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M322">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M323">View MathML</a>

(3.9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M325">View MathML</a>.

Thus, in view of Lemma 2.4, from (2.26)-(2.27) and (3.9), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M326">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M327">View MathML</a>

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M325">View MathML</a>. That is, T is continuous on P. □

We are now in a position to state and prove the first theorem in the article. Let constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M330">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M331">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M332">View MathML</a>.

Theorem 3.1Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>) hold. In addition, there are two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M224">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M337">View MathML</a>such thatfsatisfies the following condition:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M338">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M339">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M340">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M341">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M342">View MathML</a>.

Then BVP (1.1)-(1.2) has at least one positive solutionusatisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M343">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M344">View MathML</a>.

Proof We already know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229">View MathML</a> is completely continuous by Lemma 3.1.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M346">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M347">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>. It is easy to verify that the functions α, β satisfy the condition (D).

Choose a constant L large enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M349">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M350">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M351">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M352">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M353">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M354">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M355">View MathML</a>. Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M356">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M357">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M359">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M360">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M361">View MathML</a>.

Consider the following ancillary BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M362">View MathML</a>

(3.10)

Obviously, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M356">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M364">View MathML</a> according to the continuity of f. Thus, by an argument similar to that in Lemma 3.1, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M365">View MathML</a> given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M366">View MathML</a> is also completely continuous on P and maps P into P.

We will prove that T has at least one nonzero fixed point in P by applying Lemma 2.5. The approach is divided into four steps.

Step 1. We first show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M367">View MathML</a>

(3.11)

In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>, owing to the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M369">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M372">View MathML</a>.

On the other hand, applying the mean value theorem, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M374">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M375">View MathML</a>. Therefore, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M376">View MathML</a> from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M378">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>. So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M380">View MathML</a>

keeping in mind that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M112">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>.

Step 2. Now, we come to verify that the conditions corresponding to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M384">View MathML</a>) in Lemma 2.5 hold.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M386">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M388">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M390">View MathML</a>. Thus, in view of (3.11), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M391">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M392">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>. So, according to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M338">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M395">View MathML</a>

Thus, from (2.27) and Lemma 2.4, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M396">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M397">View MathML</a>, noting that the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M398">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M231">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M400">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M401">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M405">View MathML</a>. Thus, from (3.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M406">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M407">View MathML</a>

from the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M338">View MathML</a>). Therefore, in view of Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M409">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M410">View MathML</a>, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M411">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M233">View MathML</a>.

Step 3. We verify that the conditions corresponding to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M413">View MathML</a>) in Lemma 2.5 hold.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>, owing to the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M415">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M416">View MathML</a>, from the meaning of M, we have immediately that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M417">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M418">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M419">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M420">View MathML</a> from the choice of L. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M421">View MathML</a>.

Step 4. Finally, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M422">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M423">View MathML</a>. Then, by an argument similar to that in Lemma 3.1, we can know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M424">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M425">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M427">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a> from (3.11). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M429">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M240">View MathML</a>. Again,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M431">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M243">View MathML</a>.

So, the conditions corresponding to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M238">View MathML</a>) in Lemma 2.5 hold.

Summing up the above steps 1-4 and applying Lemma 2.5, we obtain that BVP (3.10) has at least one positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M435">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M436">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M437">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M438">View MathML</a> from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M439">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M440">View MathML</a> by (3.13). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M441">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M443">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M445">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and so u is a positive solution of BVP (1.1)-(1.2). The proof is complete. □

Now, we state another theorem in this paper. Let us begin with introducing some notations.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M448">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M449">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M450">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M451">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M452">View MathML</a>. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M453">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M454">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M456">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M457">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M458">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M355">View MathML</a>) are given in Lemma 2.4.

Theorem 3.2Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M54">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M56">View MathML</a>) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M462">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M463">View MathML</a>, then BVP (1.1)-(1.2) has at least one positive solution.

Proof As described in the proof of Theorem 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M229">View MathML</a> is completely continuous. Again, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M462">View MathML</a>, it follows that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M466">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M467">View MathML</a>

(3.12)

holds when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M468">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M469">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M470">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M471">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M472">View MathML</a>. Now, we show that the following relation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M473">View MathML</a>

(3.13)

In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M474">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M476">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M477">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M478">View MathML</a>

(3.14)

Thus, from (3.12), (3.14), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M479">View MathML</a>

(3.15)

Hence, (2.26) together with (3.15) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M480">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M481">View MathML</a>

(3.16)

Similarly, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M482">View MathML</a>

(3.17)

Therefore, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M453">View MathML</a>, from (3.16)-(3.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M484">View MathML</a>

So, the relation (3.13) holds.

Now, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M463">View MathML</a>, it follows that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M486">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M487">View MathML</a>

(3.18)

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M488">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M469">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M470">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M491">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M492">View MathML</a>. We prove the following relation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M493">View MathML</a>

(3.19)

In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M494">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M496">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M388">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M500">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M88">View MathML</a>. So, by (3.18), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M502">View MathML</a>

Therefore, from (2.26) and in view of Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M503">View MathML</a>

(3.20)

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M504">View MathML</a>

(3.21)

Similarly, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M505">View MathML</a>

(3.22)

Consequently, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M454">View MathML</a>, from (3.21)-(3.22), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M507">View MathML</a>

So, the relation (3.19) holds.

Summing up (3.13) and (3.19), applying Lemma 2.6, the operator T has at least one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M508">View MathML</a>. Thus u is a positive solution of BVP (1.1)-(1.2). The proof is complete. □

Example 3.1 Consider the following BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M509">View MathML</a>

(3.23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, and f is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M513">View MathML</a>

where constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M514">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M515">View MathML</a> are two positive numbers. Then BVP (3.23) has at least one positive solution.

In fact, assume that the notations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M517">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M518">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M519">View MathML</a> are described in Theorem 3.1. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M520">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M521">View MathML</a>. Then the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M522">View MathML</a>

holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M340">View MathML</a>, and the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M524">View MathML</a>

holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M525">View MathML</a>.

So, by Theorem 3.1, BVP (3.23) has at least one positive solution.

Example 3.2 Consider the following BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M526">View MathML</a>

(3.24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M5">View MathML</a>, and f is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M530">View MathML</a>

where constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M514">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M515">View MathML</a> are two positive numbers and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M533">View MathML</a>. Then BVP (3.24) has at least one positive solution.

In fact, observing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M534">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/109/mathml/M535">View MathML</a>, the conclusion follows from Theorem 3.2.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author sincerely thanks the anonymous referees for their valuable suggestions and comments which have greatly helped improve this article. Article is supported by the Natural Science Foundation of Hubei Provincial Education Department (D20102502).

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