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Positive solutions of fractional differential equations at resonance on the half-line

Yi Chen and Xianhua Tang*

Author Affiliations

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China

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Boundary Value Problems 2012, 2012:64  doi:10.1186/1687-2770-2012-64


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/64


Received:19 January 2012
Accepted:30 April 2012
Published:22 June 2012

© 2012 Chen and Tang; 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

This article deals with the differential equations of fractional order on the half-line. By the recent Leggett-Williams norm-type theorem due to O’Regan and Zima, we present some new results on the existence of positive solutions for the fractional boundary value problems at resonance on unbounded domains.

MSC: 26A33, 34A08, 34A34.

Keywords:
fractional order; half-line; coincidence degree; at resonance

1 Introduction

In this article, we are concerned with the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M2">View MathML</a> is the Riemann-Liouville fractional derivative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M4">View MathML</a> satisfies the following condition: (H) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M4">View MathML</a> is continuous and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M6">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M7">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M10">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M11">View MathML</a>

.

The problem (1.1) happens to be at resonance in the sense that the kernel of the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M2">View MathML</a> is not less than one-dimensional under the boundary value conditions.

Fractional calculus is a generalization of the ordinary differentiation and integration. It has played a significant role in science, engineering, economy, and other fields. Some books on fractional calculus and fractional differential equations have appeared recently (see [1-3]); furthermore, today there is a large number of articles dealing with the fractional differential equations (see [4-15]) due to their various applications.

In [8], the researchers dealt with the existence of solutions for boundary value problems of fractional order of the form

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M4">View MathML</a> is continuous. The results are based on the fixed point theorem of Schauder combined with the diagonalization method.

In [9], Su and Zhang studied the following fractional differential equations on the half-line using Schauder’s fixed point theorem

Employing the Leray-Schauder alternative theorem, in [12], Zhao and Ge considered the fractional boundary value problem

However, the articles on the existence of solutions of fractional differential equations on the half-line are still few, and most of them deal with the problems under nonresonance conditions. And as far as we know, recent articles, such as [4,6,7], investigating resonant problems are on the finite interval.

Motivated by the articles [16-20], in this article we study the differential equations (1.1) under resonance conditions on the unbounded domains. Moreover, we have successfully established the existence theorem by the recent Leggett-Williams norm-type theorem due to O’Regan and Zima. To our best knowledge, there is no article dealing with the resonant problems of fractional order on unbounded domains by the theorem.

The rest of the article is organized as follows. In Section 2, we give the definitions of the fractional integral and fractional derivative, some results about fractional differential equations, and the abstract existence theorem. In Section 3, we obtain the existence result of the solution for the problem (1.1) by the recent Leggett-Williams norm-type theorem. Then, an example is given in Section 4 to demonstrate the application of our result.

2 Preliminaries

First of all, we present some fundamental facts on the fractional calculus theory which we will use in the next section.

Definition 2.1 ([1-3])

The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M18">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M19">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M20">View MathML</a>

(2.1)

provided that the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M21">View MathML</a>.

Definition 2.2 ([1-3])

The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M18">View MathML</a> of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M19">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M24">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M25">View MathML</a>, provided that the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M26">View MathML</a>.

Lemma 2.1 ([1,9])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M27">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M28">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M29">View MathML</a>

(2.3)

Lemma 2.2 ([9])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M18">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M32">View MathML</a>

(2.4)

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M34">View MathML</a>, whereNis the smallest integer greater than or equal toν.

Now, let us recall some standard facts and the fixed point theorem due to O’Regan and Zima, and these can be found in [16,17,21-23].

Let X, Z be real Banach spaces. Consider an operation equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M35">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M36">View MathML</a> is a linear operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M37">View MathML</a> is a nonlinear operator. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M38">View MathML</a> and ImL is closed in Z, then L is called a Fredholm mapping of index zero. And if L is a Fredholm mapping of index zero, there exist linear continuous projectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M39">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M44">View MathML</a>. Then it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M45">View MathML</a> is invertible. We denote the inverse of this map by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M46">View MathML</a>. For ImQ is isomorphic to KerL, there exists an isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M47">View MathML</a>.

It is known that the coincidence equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M48">View MathML</a> is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M49">View MathML</a>

A nonempty convex closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M50">View MathML</a> is called a cone if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M51">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M53">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M54">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M55">View MathML</a>.

Note that C induces a partial order ⪯ in X by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M56">View MathML</a>

The following lemma is valid for every cone in a Banach space.

Lemma 2.3 ([17,23])

LetCbe a cone in the Banach spaceX. Then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M57">View MathML</a>, there exists a positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M58">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M59">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M60">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M61">View MathML</a> be a retraction, i.e., a continuous mapping such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M62">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M60">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M64">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M65">View MathML</a>

Theorem 2.1 ([16,17])

LetCbe a cone inXand let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M67">View MathML</a>be open bounded subsets ofXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M69">View MathML</a>. Assume that: 1 = Lis a Fredholm operator of index zero;; 2 = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M70">View MathML</a>is continuous and bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M71">View MathML</a>is compact on every bounded subset ofX;; 3 = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M72">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M74">View MathML</a>;; 4 = γmaps subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M75">View MathML</a>into bounded subsets ofC;; 5 = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M76">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M77">View MathML</a>stands for the Brouwer degree;; 6 = there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M79">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M80">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M82">View MathML</a>is such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M83">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M84">View MathML</a>;; 7 = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M85">View MathML</a>;; 8 = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M86">View MathML</a>..Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M87">View MathML</a>has a solution in the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M88">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M89">View MathML</a>

with the norm

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M91">View MathML</a>

equipped with the norm

Remark 2.1 It is easy for us to prove that and are Banach spaces.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M95">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M96">View MathML</a>

(2.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M97">View MathML</a>

(2.6)

Then the multi-point boundary value problem (1.1) can be written by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M98">View MathML</a>

Definition 2.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M99">View MathML</a> is called a solution of the problem (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M100">View MathML</a> and u satisfied Equation (1.1).

Next, similar to the compactness criterion in [12,24], we establish the following criterion, and it can be proved in a similar way.

Lemma 2.4<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101">View MathML</a>is a relatively compact set inXif and only if the following conditions are satisfied:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101">View MathML</a>is uniformly bounded, that is, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M103">View MathML</a>such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M104">View MathML</a>, .

(b) The functions from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101">View MathML</a>are equicontinuous on any compact subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M107">View MathML</a>, that is, let J be a compact subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M107">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M109">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M110">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M113">View MathML</a>

(c) The functions from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M101">View MathML</a>are equiconvergent, that is, given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M115">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M116">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M117">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M119">View MathML</a>.

3 Main results

In this section, we will present the existence theorem for the fractional differential equation on the half-line. In order to prove our main result, we need the following lemmas.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M120">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M121">View MathML</a>is the solution of the following fractional differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M122">View MathML</a>

if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M123">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M124">View MathML</a>

Proof In view of Lemmas 2.1 and 2.2, we can certify the conclusion easily, so we omit the details here. □

Lemma 3.2The operatorLis a Fredholm mapping of index zero. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M125">View MathML</a>

(3.1)
and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M126">View MathML</a>

(3.2)

Proof It is obvious that Lemma 3.1 implies (3.1) and (3.2). Now, let us focus our minds on proving that L is a Fredholm mapping of index zero.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M128">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M129">View MathML</a>. Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M131">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40">View MathML</a> is a continuous linear projector. In fact, for an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M129">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M134">View MathML</a>

that is to say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M40">View MathML</a> is idempotent.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M136">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M129">View MathML</a> is an arbitrary element. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M139">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M140">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M141">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M142">View MathML</a> can be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M144">View MathML</a> , for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M145">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M146">View MathML</a>, by (3.2), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M147">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M148">View MathML</a>, and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M149">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M150">View MathML</a>, thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M151">View MathML</a>.

Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M152">View MathML</a>, and observing that ImL is closed in Z, so L is a Fredholm mapping of index zero. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M39">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M154">View MathML</a>

(3.4)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M39">View MathML</a> is a linear continuous projector and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M156">View MathML</a>

Also, proceeding with the proof of Lemma 3.2, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M157">View MathML</a>.

Consider the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M158">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M159">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M160">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M161">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M162">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M163">View MathML</a>.

Define the linear isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M164">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M165">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M166">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M167">View MathML</a>

(3.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M168">View MathML</a>

Then, it is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M169">View MathML</a>

(3.6)

Now, we state the main result on the existence of the positive solutions to the problem (1.1) in the following.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M170">View MathML</a>satisfy the condition (H). Assume that there exist six nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M171">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M172">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M173">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M174">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M175">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M176">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M177">View MathML</a>

(3.8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M179">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M180">View MathML</a>is defined by (3.12), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M181">View MathML</a>is bounded on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M185">View MathML</a>,

(3.9)

(3.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M188">View MathML</a>

(3.11)

Then the problem (1.1) has at least one positive solution in domL.

Proof For the simplicity of notation, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M189">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M190">View MathML</a>

(3.12)

Consider the cone

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M191">View MathML</a>

Set

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M194">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M67">View MathML</a> are an open bounded set of X.

Step 1: In view of Lemma 3.2, the condition 1 of Theorem 2.1 is fulfilled.

Step 2: By virtue of Lemma 2.4, we can get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M70">View MathML</a> is continuous and bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M71">View MathML</a> is compact on every bounded subset of X, which ensures that the assumption 2 of Theorem 2.1 holds.

Step 3: Suppose that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M200">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M201">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M202">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M203">View MathML</a>

(3.13)

From (3.7) and (3.8), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M204">View MathML</a>

(3.14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M205">View MathML</a>

(3.15)

On account of the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M206">View MathML</a>

and considering (3.14) and (3.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M207">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M208">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M209">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M210">View MathML</a>

By (3.9), (3.10) and (3.13), we obtain that

which is a contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M199">View MathML</a>. Therefore, 3 is satisfied.

Step 4: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M213">View MathML</a>, then we can verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M61">View MathML</a> is a retraction and 4 holds.

Step 5: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M215">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M216">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M184">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M218">View MathML</a>. Inspired by Aijun and Wang [5], we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M219">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M220">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M221">View MathML</a>.

Define homeomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M222">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M223">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M224">View MathML</a>

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M225">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M226">View MathML</a> by (3.8) and (3.11).

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M227">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M228">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M230">View MathML</a>, then we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M231">View MathML</a>. Also, in view of (3.8),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M232">View MathML</a>

It is a contradiction. Besides, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M233">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M234">View MathML</a>, which is impossible. Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M221">View MathML</a>.

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M238">View MathML</a>

which shows that 5 is true.

Step 6: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M239">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M240">View MathML</a>

And we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M241">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M242">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M243">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M80">View MathML</a>, we have that

Therefore, combining (3.6), (3.8) and (3.11), we get that

Thus, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M80">View MathML</a>. So, 6 holds.

Step 7: For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M249">View MathML</a>, from (3.8) and (3.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M250">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M85">View MathML</a>. Hence, 7 holds.

Step 8: For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M252">View MathML</a>, by (3.6), (3.8) and (3.11), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M253">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M86">View MathML</a>, that is, 8 is satisfied.

Hence, applying Theorem 2.1, the problem (1.1) has a positive solution in the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M88">View MathML</a>. □

4 Examples

To illustrate our main result, we will present an example.

Example 4.1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M256">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M257">View MathML</a>, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M258">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M259">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M260">View MathML</a>

It is easy for us to certify that f satisfies the condition (H).

Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M261">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M262">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M263">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M264">View MathML</a>

Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M175">View MathML</a> satisfies (3.11).

Meanwhile, by simple computation we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/64/mathml/M266">View MathML</a>

Thus, to sum up the points which we have just indicated, by Theorem 3.1, we can conclude that the problem (4.1) has at least one positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors typed, read, and approved the final manuscript.

Acknowledgement

This project is supported by the Hunan Provincial Innovation Foundation For Postgraduate (NO. CX2011B079) and the National Natural Science Foundation of China (NO. 11171351).

References

  1. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

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