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This article is part of the series A Tribute to Professor Ivan Kiguradze.

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Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian

Hongling Lu, Zhenlai Han* and Shurong Sun

Author Affiliations

School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, P.R. China

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Boundary Value Problems 2014, 2014:26  doi:10.1186/1687-2770-2014-26

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/26


Received:17 August 2013
Accepted:10 January 2014
Published:30 January 2014

© 2014 Lu et al.; licensee Springer.

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

Abstract

In this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M5">View MathML</a> are the standard Caputo fractional derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M12">View MathML</a> is continuous. By means of the properties of the Green’s function, Leggett-Williams fixed-point theorems, and fixed-point index theory, several new sufficient conditions for the existence of at least two or at least three positive solutions are obtained. As an application, an example is given to demonstrate the main result.

MSC: 34A08, 34B18, 35J05.

Keywords:
Sturm-Liouville boundary value problem; positive solution of fractional differential equation; Leggett-Williams fixed-point theorem; fixed-point index theory; p-Laplacian operator

1 Introduction

During the past decades, much attention has been focused on the study of equations with p-Laplacian differential operator. The motivation for those works stems from the applications in the modeling of different physical and natural phenomena: non-Newtonian mechanics [1], system of Monge-Kantorovich partial differential equations [2], population biology [3], nonlinear flow laws [4], combustion theory [5]. There exist a very large number of papers devoted to the existence of solutions for the equation with p-Laplacian operator.

The ordinary differential equation with p-Laplacian operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M13">View MathML</a>

subject to various boundary conditions, has been studied by many authors, see [6,7] and the references therein.

The existence of positive solutions of the differential equation with p-Laplacian operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M14">View MathML</a>

satisfying different boundary conditions have been established by using fixed-point theorems and monotone iterative technique, see [8,9] and the references therein.

In [10], Hai considered the existence of positive solutions for the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M15">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M18">View MathML</a> and λ is a positive parameter, f is p-superlinear or p-sublinear at ∞ and maybe singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M19">View MathML</a>.

However, few papers can be found in the literature on the existence of multiple positive solutions for the third-order Sturm-Liouville boundary value problem with p-Laplacian.

In [11], Zhai and Guo studied the third-order Sturm-Liouville boundary value problem with p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M26">View MathML</a>. By means of the Leggett-Williams fixed-point theorems, some existence and multiplicity results of positive solutions are obtained. In later work, Yang and Yan [12] also studied the above problem by means of the fixed-point index method.

Recently, fractional differential equations have been of great interest. The motivation for those works stems from both the intensive development of the theory of fractional calculus itself and the applications such as economics, engineering and other fields [13-17]. Much attention has been focused on the study of the existence and multiplicity of solutions or positive solutions for boundary value problems of fractional differential equations by the use of techniques of nonlinear analysis (fixed-point theorems [18-25], upper and lower solutions method [26], fixed-point index theory [27,28], coincidence theory [29], etc.).

Although the boundary value problems of fractional differential equation with p-Laplacian have been studied in many literature, only few papers can be found in the literature on the existence of multiple positive solutions for the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian. As the extension and supplement of some results in [11,12], in this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian subject Robin boundary value conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M27">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M5">View MathML</a> are the standard Caputo fractional derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M12">View MathML</a> is continuous. By means of the properties of the Green’s function, Leggett-Williams fixed-point theorems and fixed-point index theory, we establish the existence of at least two or at least three positive solutions for the Sturm-Liouville boundary value problem (1.1). As an application, an example is given to demonstrate the main result.

The rest of this paper is organized as follows. In Section 2, we shall introduce some definitions and lemmas to prove our main results. In Section 3, we state our main results. We prove our main results by Leggett-Williams fixed-point theorems and fixed-point index theory in Section 4. As an application, an example is presented to illustrate our main result in Section 5.

2 Preliminaries and lemmas

For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate analysis of problem (1.1). These materials can be found in the recent literature, see [14,17,18,31-33].

Definition 2.1 ([17])

The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M40">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M41">View MathML</a>

provided the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M42">View MathML</a>.

Definition 2.2 ([17])

The Caputo fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39">View MathML</a> of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M40">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M45">View MathML</a>

where n is the smallest integer greater than or equal to α, provided that the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M42">View MathML</a>.

Remark 2.1 ([14])

By Definition 2.2, under natural conditions on the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M47">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M48">View MathML</a> the Caputo derivative becomes a conventional nth derivative of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M47">View MathML</a>.

Remark 2.2 ([18])

As a basic example, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M50">View MathML</a>

given in particular that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M52">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M53">View MathML</a> is the Caputo fractional derivative, and n is the smallest integer greater than or equal to α.

From the definition of the Caputo derivative and Remark 2.2, we can obtain the following statement.

Lemma 2.1 ([17])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39">View MathML</a>. Then the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M55">View MathML</a>

has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M56">View MathML</a>

as the unique solution, wherenis the smallest integer greater than or equal toα.

Lemma 2.2 ([17])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M39">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M58">View MathML</a>. Then the following equality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M59">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M61">View MathML</a>, wherenis the smallest integer greater than or equal to α.

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2">View MathML</a>. Then the boundary value problem of the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M64">View MathML</a>

(2.1)

has a unique solution,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M65">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M66">View MathML</a>

(2.2)

Proof By the Lemma 2.2, we can reduce the equation of problem (2.1) to an equivalent integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M67">View MathML</a>

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M68">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M69">View MathML</a>

From the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M71">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M72">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M73">View MathML</a>

Hence, the unique solution of (2.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M74">View MathML</a>

which completes the proof. □

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M3">View MathML</a>. Then the boundary value problem of the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M77">View MathML</a>

(2.3)

has a unique solution,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M78">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M79">View MathML</a>is defined as (2.2).

Proof From Lemma 2.2 and the boundary value problem (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M80">View MathML</a>

that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M81">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M82">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M83">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M84">View MathML</a>. Thus, the boundary value problem (2.3) is equivalent to the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M85">View MathML</a>

Lemma 2.3 implies that boundary value problem (2.3) has a unique solution,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M86">View MathML</a>

which completes the proof. □

Lemma 2.5The Green’s function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M87">View MathML</a>defined by (2.2) is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M88">View MathML</a>.

Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M89">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M87">View MathML</a>also has the following properties:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M91">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M92">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M93">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M94">View MathML</a>;

(3) there exists a positive numberλsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M95">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M97">View MathML</a>.

The method of proof is similar to Lemma 3.2 in [30], and we omit it here.

Definition 2.3 ([31])

Let E be a real Banach space and P be a nonempty, convex closed set in E. We say that P is a cone if it satisfies the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M98">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M100">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M101">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M102">View MathML</a>, where θ denotes the null element of E.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M103">View MathML</a> is a cone, we denote the order induced by P on E by ≤. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M104">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M105">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M106">View MathML</a>.

Definition 2.4 ([31])

The map φ is said to be a nonnegative continuous concave functional on P of a real Banach space E provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M107">View MathML</a> is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M108">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M110">View MathML</a>.

Definition 2.5 ([31])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M111">View MathML</a> be given and let φ be a nonnegative continuous concave functional on the cone P. Define the convex sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M114">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M117">View MathML</a>.

Lemma 2.6 (Leggett-Williams [32])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M118">View MathML</a>be a completely continuous operator and letφbe a nonnegative continuous concave functional onPsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M119">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M120">View MathML</a>. Suppose that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M121">View MathML</a>such that

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M122">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M123">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M124">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M125">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M126">View MathML</a>;

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M123">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M128">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M129">View MathML</a>.

ThenThas at least three fixed points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M131">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M132">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M133">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M136">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M137">View MathML</a>.

Lemma 2.7 ([32])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M138">View MathML</a>be a completely continuous operator and letφbe a nonnegative continuous concave functional onPsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M139">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M120">View MathML</a>. Suppose that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M141">View MathML</a>such that

(B1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M142">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M123">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M128">View MathML</a>;

(B2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M125">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M126">View MathML</a>;

(B3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M147">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M120">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M149">View MathML</a>.

ThenThas at least two fixed points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M130">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M131">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M152">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M154">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M155">View MathML</a>.

Lemma 2.8 ([31])

LetPbe a closed convex set in a Banach spaceEand let Ω be a bounded open set such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M156">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M157">View MathML</a>be a compact map. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M158">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M159">View MathML</a>.

(C1) (Existence) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M160">View MathML</a>, then T has a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M161">View MathML</a>.

(C2) (Normalization) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M163">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M164">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M165">View MathML</a>.

(C3) (Homotopy) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M166">View MathML</a>be a compact map such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M167">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M159">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M169">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M170">View MathML</a>.

(C4) (Additivity) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M172">View MathML</a>are disjoint relatively open subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M161">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M158">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M175">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M176">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M177">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M178">View MathML</a>).

Lemma 2.9 ([33])

LetPbe a cone in a Banach spaceE. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M179">View MathML</a>, define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M180">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M181">View MathML</a>is a compact map such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M158">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M183">View MathML</a>. Thus, one has the following conclusions:

(D1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M184">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M185">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M186">View MathML</a>;

(D2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M187">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M185">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M189">View MathML</a>.

3 Main theorems

In this section, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M190">View MathML</a> be the Banach space of continuous functions endowed with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M191">View MathML</a>, and the ordering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M192">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M193">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M194">View MathML</a>. Define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M195">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M196">View MathML</a>

where λ is given as in Lemma 2.5.

For convenience of the reader, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M197">View MathML</a>

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M198">View MathML</a>be the operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M199">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M200">View MathML</a>is completely continuous.

Proof By Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M201">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M202">View MathML</a>. In view of non-negativity and continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M205">View MathML</a>, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M206">View MathML</a> is continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M207">View MathML</a> be bounded, i.e., there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M208">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M209">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M211">View MathML</a>, then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M213">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M214">View MathML</a> is uniformly bounded. Further for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M194">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M217">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M218">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M210">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M221">View MathML</a>

That is to say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M214">View MathML</a> is equicontinuous. By the Arzela-Ascoli theorem, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M206">View MathML</a> is completely continuous. The proof is completed. □

We are now ready to prove our main results.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224">View MathML</a>be nonnegative continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225">View MathML</a>. Assume that there exist constantsa, bwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M226">View MathML</a>such that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M227">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M228">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M229">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M230">View MathML</a>.

Then the boundary value problem (1.1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M234">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M235">View MathML</a>, whereλis given as in Lemma 2.5.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M236">View MathML</a> be the nonnegative continuous concave functional defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M237">View MathML</a>

Evidently, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M239">View MathML</a>.

It’s easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M240">View MathML</a> is completely continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M241">View MathML</a>. We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M242">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M243">View MathML</a>

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M244">View MathML</a>. Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M245">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M246">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M247">View MathML</a>. Thus for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M247">View MathML</a>, from assumption (H1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M249">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M250">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M251">View MathML</a>

Therefore, condition (B1) of Lemma 2.7 is satisfied. Now if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M252">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M253">View MathML</a>. By assumption (H2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M254">View MathML</a>

which shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M255">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M256">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M252">View MathML</a>. This shows that condition (B2) of Lemma 2.7 is satisfied. Finally, we show that (B3) of Lemma 2.7 also holds. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M258">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M259">View MathML</a>, then by the definition of cone P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M260">View MathML</a>

So condition (B3) of Lemma 2.7 is satisfied. Thus using Lemma 2.7, T has at least two fixed points. Consequently, the boundary value problem (1.1) has at least two positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M263">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M235">View MathML</a>. The proof is completed. □

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224">View MathML</a>be nonnegative continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225">View MathML</a>. Assume that there exist constantsa, b, cwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M269">View MathML</a>such that

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M229">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M230">View MathML</a>;

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M227">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M228">View MathML</a>;

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M274">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M275">View MathML</a>.

Then the boundary value problem (1.1) has at least three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M278">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M281">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M282">View MathML</a>, whereλis given as in Lemma 2.5.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M283">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M284">View MathML</a>. By assumption (H5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M285">View MathML</a>

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M118">View MathML</a>. Using the same arguments as in the proof of Lemma 3.1, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M118">View MathML</a> is a completely continuous operator. It follows from the conditions (H3) and (H4) in Theorem 3.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M288">View MathML</a>. Similarly with the proof of Theorem 3.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M255">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M290">View MathML</a>

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M291">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M292">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M293">View MathML</a>

So all the conditions of Lemma 2.6 are satisfied. Thus using Lemma 2.6, T has at least three fixed points. So, the boundary value problem (1.1) has at least three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M278">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M282">View MathML</a>. The proof is completed. □

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224">View MathML</a>be nonnegative continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225">View MathML</a>. If the following assumptions are satisfied:

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M303">View MathML</a>;

(H7) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M304">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M305">View MathML</a>

then the boundary value problem (1.1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M308">View MathML</a>.

Proof From Lemma 3.1, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M309">View MathML</a> is completely continuous. In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M310">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M311">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M312">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M313">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M314">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M315">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M316">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M317">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M318">View MathML</a>. Hence, Lemma 2.9 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M319">View MathML</a>

(3.1)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M320">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M321">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M322">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M323">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M324">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M325">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M326">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M327">View MathML</a>. By using the method to get (3.1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M328">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M317">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M330">View MathML</a>. Thus, from Lemma 2.9, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M331">View MathML</a>

(3.2)

Finally, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M332">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M333">View MathML</a>, by (H7), we then get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M334">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M335">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M336">View MathML</a>. Using Lemma 2.9 again, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M337">View MathML</a>

(3.3)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M338">View MathML</a>, by the additivity of fixed-point index and (3.1)-(3.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M339">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M340">View MathML</a>

Hence, T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M342">View MathML</a>, and has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M344">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a> are positive solutions of the boundary value problem (1.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M347">View MathML</a>. The proof is completed. □

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M224">View MathML</a>be nonnegative continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M225">View MathML</a>. If the following assumptions are satisfied:

(H8) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M350">View MathML</a>;

(H9) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M351">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M352">View MathML</a>

Then the boundary value problem (1.1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M355">View MathML</a>.

Proof From Lemma 3.1, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M309">View MathML</a> is completely continuous. In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M357">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M358">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M359">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M360">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M361">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M362">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M363">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M335">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M365">View MathML</a>. Hence, Lemma 2.9 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M366">View MathML</a>

(3.4)

Next, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M367">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M368">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M369">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M370">View MathML</a>. We consider two cases.

Case 1: Suppose that f is bounded, which implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M371">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M372">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M374">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M375">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M377">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M378">View MathML</a>

Case 2: Suppose that f is unbounded. In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M379">View MathML</a> being continuous, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M380">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M381">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M382">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M99">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M384">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M385">View MathML</a>

So, in either case, if we always choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M386">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M387">View MathML</a>

Thus, from Lemma 2.9, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M388">View MathML</a>

(3.5)

Finally, Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M389">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M390">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M391">View MathML</a>, by (H9), and we then obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M392">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M317">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M394">View MathML</a>. An application of Lemma 2.9 again shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M395">View MathML</a>

(3.6)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M396">View MathML</a>; by the additivity of fixed-point index and (3.4)-(3.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M397">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M398">View MathML</a>

Hence, T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M400">View MathML</a>, and it has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M402">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a> are positive solutions of the boundary value problem (1.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M405">View MathML</a>. The proof is completed. □

4 Example

In this section, we present an example to illustrate the main result.

Example 4.1 We consider the boundary value problem of the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M406">View MathML</a>

(4.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M407">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M408">View MathML</a>. We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M409">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M410">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M411">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M412">View MathML</a>. By a simple calculation, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M413">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M414">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M415">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M417">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M418">View MathML</a>, evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M269">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M420">View MathML</a>

Consequently, all the conditions of Theorem 3.2 are satisfied. With the use of Theorem 3.2, the boundary value problem (4.1) has at least three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M232">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M278">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M424">View MathML</a>

5 Conclusion

In this paper, the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian are investigated, the existence of at least two or at least three positive solutions for the fractional differential equations with Robin boundary conditions are given by using Leggett-Williams fixed-point theorems and the fixed-point index theory, respectively.

It is worth emphasizing that our work presented in this article has the following features: Firstly, the boundary conditions in (1.1) are important Robin boundary conditions. Secondly, our results improve and extend the main results of [11,12] for the Sturm-Liouville boundary value problems of integer-order differential equations with p-Laplacian. For example, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M425">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M426">View MathML</a>, then the problem (1.1) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M427">View MathML</a>

(5.1)

which is studied in [11,12]. Furthermore, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M428">View MathML</a>, problem (1.1) is the usual form of third-order Sturm-Liouville boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M429">View MathML</a>

(5.2)

The method can be applied on the Sturm-Liouville boundary value problems of higher-order fractional differential equations with p-Laplacian and boundary conditions involving fractional derivatives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/26/mathml/M430">View MathML</a>

(5.3)

Based on this paper, one can consider boundary value problems of fractional differential equations with parameters, and also one can do further research on eigenvalue problems of fractional differential equations with p-Laplacian.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009).

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