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

Open Access Research

Positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions

Xingqiu Zhang

Author Affiliations

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China

School of Mathematics, Liaocheng University, Liaocheng, Shandong, 252059, P.R. China

Boundary Value Problems 2012, 2012:123  doi:10.1186/1687-2770-2012-123

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/123


Received:20 July 2012
Accepted:10 October 2012
Published:24 October 2012

© 2012 Zhang; 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 article, by employing a fixed point theorem in cones, we investigate the existence of a positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions. We also obtain some relations between the solution and Green’s function.

MSC: 26A33, 34B15, 34B16, 34G20.

Keywords:
fractional differential equations; integral boundary value problem; positive solution; semipositone; cone

1 Introduction

In this article, we consider the existence of a positive solution for the following singular semipositone fractional differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M6">View MathML</a> is the standard Riemann-Liouville derivative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M7">View MathML</a> may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M8">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M9">View MathML</a>. Since the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M10">View MathML</a> may change sign, the problem studied in this paper is called the semipositone problem in the literature which arises naturally in chemical reactor theory. Up to now, much attention has been attached to the existence of positive solutions for semipositone differential equations and the system of differential equations; see [1-11] and references therein to name a few.

Boundary value problems with integral boundary conditions for ordinary differential equations arise in different fields of applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics. Moreover, boundary value problems with integral conditions constitute a very interesting and important class of problems. They include two-point, three-point, multi-point, and nonlocal boundary value problems as special cases, which have received much attention from many authors. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [12], Karakostas and Tsamatos [13], Lomtatidze and Malaguti [14], and the references therein.

On the other hand, fractional differential equations have been of great interest for many researchers recently. This is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various fields of science and engineering such as control, porous media, electromagnetic, and other fields. For an extensive collection of such results, we refer the readers to the monographs by Samko et al.[15], Podlubny [16] and Kilbas et al.[17]. For the case where α is an integer, a lot of work has been done dealing with local and nonlocal boundary value problems. For example, in [18] Webb studied the nth-order nonlocal BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M11">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M12">View MathML</a> can have singularities, and the nonlinearity f satisfies Carathéodory conditions. Under weak assumptions, Webb obtained sharp results on the existence of positive solutions under a suitable condition on f. In [19] Hao et al. consider the nth-order singular nonlocal BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M14">View MathML</a> is a parameter, a may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M15">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M17">View MathML</a> may also have singularity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M18">View MathML</a>.

In two recent papers [20] and [21], by means of the fixed point theory and fixed point index theory, the authors investigated the existence and multiplicity of positive solutions for the following two kinds of fractional differential equations with integral boundary value problems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M19">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M23">View MathML</a> are the standard Riemann-Liouville derivative and the Caputo fractional derivative, respectively.

To the author’s knowledge, there are few papers in the literature to consider fractional differential equations with integral boundary value conditions. Motivated by above papers, the purpose of this article is to investigate the existence of positive solutions for the more general fractional differential equations BVP (1). Firstly, we derive corresponding Green’s function known as fractional Green’s function and argue its positivity. Then a fixed point theorem is used to obtain the existence of positive solutions for BVP (1). We also obtain some relations between the solution and Green’s function. From the example given in Section 4, we know that λ in this article may be greater than 2 and η may take the value 1. Therefore, compared with that in [21], BVP (1) considered in this article has a more general form.

The rest of this article is organized as follows. In Section 2, we give some preliminaries and lemmas. The main result is formulated in Section 3, and an example is worked out in Section 4 to illustrate how to use our main result.

2 Preliminaries and several lemmas

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M25">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M26">View MathML</a> is a Banach space. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M29">View MathML</a>.

For the reader’s convenience, we present some necessary definitions from fractional calculus theory and lemmas. They can be found in the recent literature; see [14-17].

Definition 2.1 The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M31">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M32">View MathML</a>

provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M33">View MathML</a>.

Definition 2.2 The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30">View MathML</a> of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M31">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M36">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M38">View MathML</a> denotes the integer part of the number α, provided that the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M33">View MathML</a>.

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

Lemma 2.1 ([17])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30">View MathML</a>. If we assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M41">View MathML</a>, then the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M42">View MathML</a>

has<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M45">View MathML</a>, as unique solutions, whereNis the smallest integer greater than or equal toα.

Lemma 2.2 ([17])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M41">View MathML</a>with a fractional derivative of order<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M30">View MathML</a>that belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M48">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M49">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M45">View MathML</a>, whereNis the smallest integer greater than or equal toα.

In the following, we present Green’s function of the fractional differential equation boundary value problem.

Lemma 2.3 Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M52">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M53">View MathML</a>

(2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M5">View MathML</a>, is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M58">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M59">View MathML</a>

(3)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61">View MathML</a>is called the Green function of BVP (2). Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M63">View MathML</a>.

Proof We may apply Lemma 2.2 to reduce (2) to an equivalent integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M64">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M65">View MathML</a>. Consequently, the general solution of (2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M66">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M67">View MathML</a>, one gets that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M68">View MathML</a>. On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M69">View MathML</a> combining with

yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M71">View MathML</a>

Therefore, the unique solution of the problem (2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M72">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M73">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M74">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M75">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M76">View MathML</a>

The proof is complete. □

Lemma 2.4 The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61">View MathML</a>defined by (3) satisfies

(a1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M79">View MathML</a>;

(a2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M79">View MathML</a>;

(a3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M79">View MathML</a>;

(a4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M84">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M85">View MathML</a>is not decreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M86">View MathML</a>;

(a5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M88">View MathML</a>,

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M91">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M93">View MathML</a>,

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M95">View MathML</a>,

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M97">View MathML</a>,

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M100">View MathML</a>,

From above, (a1), (a2), (a3), (a5) are complete. Clearly, (a4) is true. The proof is complete. □

Throughout this article, we adopt the following conditions.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M2">View MathML</a> and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M105">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M106">View MathML</a>

(H2) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M107">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M108">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M109">View MathML</a>;

(H3) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M110">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M111">View MathML</a>

Let

(4)

Obviously, Q is a cone in a Banach space E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M113">View MathML</a> is an ordering Banach space.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M114">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M115">View MathML</a> is defined as that in (H1). It follows from Lemma 2.4 and (H3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M116">View MathML</a>

(6)

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M117">View MathML</a> and it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M118">View MathML</a>

(7)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M121">View MathML</a>. Consequently, by (6) and Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M122">View MathML</a>

(8)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M123">View MathML</a>, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M124">View MathML</a>

We define an operator A as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M125">View MathML</a>

(9)

Lemma 2.5Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M126">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M127">View MathML</a>) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M128">View MathML</a>is completely continuous.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M129">View MathML</a>, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M130">View MathML</a>. By (H1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M131">View MathML</a>

(10)

By (10) and Lemma 2.4, we have

(11)

which together with (H3) means that operator A defined by (9) is well defined.

Now, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M128">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M134">View MathML</a>, by (H1) we have by (9) and Lemma 2.4 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M135">View MathML</a>

which means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M136">View MathML</a>

(12)

It follows from (12) and Lemma 2.4 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M137">View MathML</a>

Thus, A maps Q into Q.

Finally, we prove that A maps Q into Q is completely continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M138">View MathML</a> be any bounded set. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M140">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M141">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M142">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M144">View MathML</a>, by (H3) and (11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M145">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M146">View MathML</a> is uniformly bounded.

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M61">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M148">View MathML</a>, it is uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M63">View MathML</a> as well. Thus, for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M144">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M151">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M152">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M154">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M155">View MathML</a>

(13)

Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M156">View MathML</a>, we get by (10) and (13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M157">View MathML</a>

which implies that the operator A is equicontinuous. Thus, the Ascoli-Arzela theorem guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M146">View MathML</a> is a relatively compact set.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M160">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M161">View MathML</a>). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M162">View MathML</a> is bounded. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M163">View MathML</a>, by (10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M164">View MathML</a>

(14)

By (9), we have

(15)

It follows from (14), (15), (H1), (H3), and the Lebesgue dominated convergence theorem that A is continuous. Thus, we have proved the continuity of the operator A. This completes the complete continuity of A. □

To prove the main result, we need the following well-known fixed point theorem.

Lemma 2.6 (Fixed point theorem of cone expansion and compression of norm type [22])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M166">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M167">View MathML</a>be two bounded open sets in a Banach spaceEsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M168">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M169">View MathML</a>be a completely continuous operator, whereθdenotes the zero element ofEandPa cone ofE. Suppose that one of the two conditions holds:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M171">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M173">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M171">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M177">View MathML</a>.

ThenAhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M178">View MathML</a>.

3 Main result

Theorem 3.1Assume that conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M126">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M127">View MathML</a>) are satisfied. Then the singular semipositone BVP (1) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M181">View MathML</a>. Furthermore, there exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M182">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M183">View MathML</a>

(16)

Proof Firstly, we show that the operator A has a fixed point in Q. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M184">View MathML</a>

where r is the same as that defined in (H3). For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M185">View MathML</a>, by (10) and (12), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M186">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M187">View MathML</a>

which together with (H3) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M188">View MathML</a>

(17)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M189">View MathML</a> in (H2), it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M190">View MathML</a>

(18)

By (H3), we know that there exists a natural number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M191">View MathML</a> big enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M192">View MathML</a>

(19)

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M193">View MathML</a>

(20)

By (H2), we know there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M194">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M195">View MathML</a>

(21)

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M196">View MathML</a>

(22)

In the following, we are in a position to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M197">View MathML</a>

(23)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M198">View MathML</a>, by (8) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M199">View MathML</a>

which together with (18), (19), (22), and (H3) implies that

(24)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M109">View MathML</a>, it follows from (H1), (20), (21), (22), and (24) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M203">View MathML</a>

(25)

By (25), we know that (23) holds. So, (17), (23), and Lemma 2.6 guarantee that A has at least one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M204">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M206">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M207">View MathML</a>

(26)

By simple computation, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M208">View MathML</a>

(27)

Secondly, we show BVP (1) has a positive solution. It follows from (8) and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M209">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M210">View MathML</a>

which combined with (19) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M211">View MathML</a>

(28)

By (27) and (28), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M212">View MathML</a>

(29)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M121">View MathML</a>. It follows from (28) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M215">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M216">View MathML</a>

(30)

By (7), (29), and (30), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M217">View MathML</a>

Thus, we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M181">View MathML</a> is a positive solution for BVP (1).

Finally, we show that (16) holds. From (26) and Lemma 2.4, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M219">View MathML</a>

(31)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M220">View MathML</a>, (30), and (31) mean that (16) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M222">View MathML</a> holds. This completes the proof of Theorem 3.1. □

4 Example

Consider the following singular semipositone fractional differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M223">View MathML</a>

(32)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M224">View MathML</a>. It is clear (32) has the form of (1), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M227">View MathML</a>. By simple computation, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M229">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M230">View MathML</a>

Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M231">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M232">View MathML</a>

(33)

It follows from the left side of (33) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M233">View MathML</a>

(34)

Considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M234">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M235">View MathML</a>

(35)

By (34) and (35) we know (H1) holds. Obviously, (H2) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M236">View MathML</a>.

Now, we check (H3). By simple computation, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M241">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M242">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/123/mathml/M244">View MathML</a>. Thus, (H3) is valid. It follows from Theorem 3.1 that BVP (32) has at least one positive solution.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author thanks the referee for his/her careful reading of the manuscript and useful suggestions. The project is supported financially by the Foundation for Outstanding Middle-Aged and Young Scientists of Shandong Province (Grant No. BS2010SF004), a Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J10LA53, No. J11LA02), the China Postdoctoral Science Foundation (Grant No. 20110491154) and the National Natural Science Foundation of China (Grant No. 10971179).

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