This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

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Three-point boundary value problems of fractional functional differential equations with delay

Yanan Li, Shurong Sun*, Dianwu Yang and Zhenlai Han

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

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Boundary Value Problems 2013, 2013:38  doi:10.1186/1687-2770-2013-38


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/38


Received:6 December 2012
Accepted:7 February 2013
Published:22 February 2013

© 2013 Li et al.; licensee Springer

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

Abstract

In this paper, we study three-point boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3">View MathML</a> denote Caputo fractional derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7">View MathML</a>. We use the Green function to reformulate boundary value problems into an abstract operator equation. By means of the Schauder fixed point theorem and the Banach contraction principle, some existence results of solutions are obtained, respectively. As an application, some examples are presented to illustrate the main results.

MSC: 34A08, 34K37.

Keywords:
fractional functional differential equation; delay; three-point boundary value problems; fixed point theorem; existence of solutions

1 Introduction

Fractional calculus is a branch of mathematics, it is an emerging field in the area of the applied mathematics that deals with derivatives and integrals of arbitrary orders as well as with their applications. The origins can be traced back to the end of the seventeenth century. During the history of fractional calculus, it was reported that the pure mathematical formulations of the investigated problems started to be addressed with more applications in various fields. With the help of fractional calculus, we can describe natural phenomena and mathematical models more accurately. Therefore, fractional differential equations have received much attention and the theory and its application have been greatly developed; see [1-6].

Recently, there have been many papers focused on boundary value problems of fractional ordinary differential equations [7-15] and an initial value problem of fractional functional differential equations [16-28]. But the results dealing with the boundary value problems of fractional functional differential equations with delay are relatively scarce [29-35]. It is well known that in practical problems, the behavior of systems not only depends on the status just at the present, but also on the status in the past. Thus, in many cases, we must consider fractional functional differential equations with delay in order to solve practical problems. Consequently, our aim in this paper is to study the existence of solutions for boundary value problems of fractional functional differential equations.

In 2011, Rehman [12] studied the existence and uniqueness of solutions to nonlinear three-point boundary value problems for the following fractional differential equation:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M14">View MathML</a> denote Caputo fractional derivatives. By the Banach contraction principle and the Schauder fixed point theorem, they obtained some new existence and uniqueness results.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M15">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M16">View MathML</a> the Banach space of all continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M17">View MathML</a> endowed with the sup-norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M18">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M19">View MathML</a>, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M20">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M21">View MathML</a> the element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M16">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M23">View MathML</a>

Enlightened by literature [12], in this paper we study the following three-point boundary value problem for the fractional functional differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M24">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3">View MathML</a> denote Caputo fractional derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M29">View MathML</a> is a continuous function associated with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M30">View MathML</a>

(1.2)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M31">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7">View MathML</a> and φ is an element of the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M34">View MathML</a>

To the best of our knowledge, no one has studied the existence of positive solutions for problem (1.1)-(1.2). The aim of this paper is to fill the gap in the relevant literatures. In this paper, we firstly give the fractional Green function and some properties of the Green function. Consequently, boundary value problem (1.1) and (1.2) is reduced to an equivalent Fredholm integral equation. Then we extend the existence results for boundary value problems of an ordinary fractional differential equation of δ-order (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M9">View MathML</a>) in [12] to a fractional functional differential equation of α-order (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>). As an application, some examples are presented to illustrate the main results.

2 Preliminaries

For the convenience of the reader, we give the following background material from fractional calculus theory to facilitate the analysis of boundary value problem (1.1) and (1.2). This material can be found in the recent literature; see [1,2,36].

Definition 2.1 ([1])

The fractional integral of order α (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M37">View MathML</a>) of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M38">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M39">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M40">View MathML</a> is the gamma function, provided that the right-hand side is point-wise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M41">View MathML</a>.

Definition 2.2 ([1])

The Caputo fractional derivative of order α (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M42">View MathML</a>) of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M38">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M44">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M40">View MathML</a> is the gamma function, provided that the right-hand side is point-wise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M41">View MathML</a>.

Obviously, the Caputo derivative for every constant function is equal to zero.

From the definition of the Caputo derivative, we can acquire the following statement.

Lemma 2.1 ([2])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M47">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M48">View MathML</a>

Lemma 2.2 ([2])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M37">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M50">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M52">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M54">View MathML</a>denotes the integer part ofα.

Next, we introduce the Green function of fractional functional differential equations boundary value problems.

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M58">View MathML</a>be continuous. Then the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M59">View MathML</a>

(2.1)

has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M60">View MathML</a>

(2.2)

where

(2.3)

Proof From equation (2.1), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M62">View MathML</a>

From Lemma 2.2, we have

(2.4)

According to (2.1), we know that

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M65">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M66">View MathML</a>

Therefore,

Now, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M68">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M69">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M70">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M71">View MathML</a>

Hence, we can conclude (2.2) holds, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M72">View MathML</a>

The proof is completed. □

Lemma 2.4 ([36] Schauder fixed point theorem)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M73">View MathML</a>be a complete metric space, Ube a closed convex subset ofD, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M74">View MathML</a>be the map such that the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M75">View MathML</a>is relatively compact inD. Then the operatorThas at least one fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M76">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M77">View MathML</a>

3 Main results

In this section, we discuss the existence and uniqueness of solutions for boundary value problem (1.1) and (1.2) by the Schauder fixed point theorem and the Banach contraction principle.

For convenience, we define the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M78">View MathML</a>. Also, if I is an interval of the real line ℝ, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M80">View MathML</a> we denote the set of continuous and continuously differentiable functions on I, respectively. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M81">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M82">View MathML</a>

(3.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M31">View MathML</a>, in view of the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M21">View MathML</a> and φ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M85">View MathML</a>

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M86">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M87">View MathML</a> is a continuous function, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M88">View MathML</a> in Lemma 2.3. We have by Lemma 2.3 that a function u is a solution of boundary value problem (1.1) and (1.2) if and only if it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M89">View MathML</a>

We define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M90">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M91">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M92">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M93">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M94">View MathML</a>

Theorem 3.1Assume the following:

(H1) There exists a nonnegative function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M95">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M96">View MathML</a>

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M98">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M99">View MathML</a>are nonnegative constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M100">View MathML</a>; or

(H2) There exists a nonnegative function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M95">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M102">View MathML</a>

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M98">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M99">View MathML</a>are nonnegative constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M106">View MathML</a>.

Then boundary value problem (1.1) and (1.2) has a solution.

Proof Suppose (H1) holds. Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M107">View MathML</a>

(3.3)

and define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M108">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M109">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M110">View MathML</a>

(3.4)

Also,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M111">View MathML</a>

Hence,

In view of (3.1) and (3.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M113">View MathML</a>

(3.5)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M114">View MathML</a>. The continuity of the operator T follows from the continuity of f and G.

Now, if (H2) holds, we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M115">View MathML</a>

(3.6)

and by the same process as above, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M116">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M114">View MathML</a>.

Now, we show that T is a completely continuous operator.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M118">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M120">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M121">View MathML</a>, in view of Lemma 2.3, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M122">View MathML</a>, then

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M124">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M125">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M126">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M127">View MathML</a>

Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M122">View MathML</a>, we have

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M124">View MathML</a>, in view of the definition of φ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M131">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M126">View MathML</a>, then

Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M122">View MathML</a>, we have

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M124">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M137">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M126">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M139">View MathML</a>

In any case, it implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M140">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M141">View MathML</a>, i.e., for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M142">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M143">View MathML</a>, independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M145">View MathML</a> and u, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M146">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M147">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M90">View MathML</a> is completely continuous. The proof is completed. □

For convenience, we denote

Theorem 3.2Assume that

(H3) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M150">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M151">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M153">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M154">View MathML</a>

then boundary value problem (1.1) and (1.2) has a unique solution.

Proof Consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M90">View MathML</a> defined by (3.2). Clearly, the fixed point of the operator T is the solution of boundary value problem (1.1) and (1.2). We will use the Banach contraction principle to prove that T has a fixed point. We first show that T is a contraction. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M20">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M157">View MathML</a>

(3.7)

By a similar method, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M158">View MathML</a>

(3.8)

In view of the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M159">View MathML</a>, we obtain

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M161">View MathML</a>

(3.9)

Clearly, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M162">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M163">View MathML</a>. Therefore, by (3.7) and (3.9), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M164">View MathML</a>

and T is a contraction. As a consequence of the Banach contraction principle, we get that T has a fixed point which is a solution of boundary value problem (1.1) and (1.2). □

4 Example

In this section, we will present some examples to illustrate our main results.

Example 4.1

Consider boundary value problems of the following fractional functional differential equations:

(4.1)

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3">View MathML</a> denote Caputo fractional derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M171">View MathML</a>.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M176">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M177">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M171">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M179">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M100">View MathML</a>, (H1) is satisfied and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M106">View MathML</a>, (H2) is satisfied. Therefore, by Theorem 3.1, boundary value problem (4.1) and (4.2) has a solution.

Example 4.2

Consider boundary value problems of the following fractional functional differential equations:

(4.3)

(4.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3">View MathML</a> denote Caputo fractional derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M171">View MathML</a>.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M190">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M191">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M192">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M153">View MathML</a>. Then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M20">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M196">View MathML</a>

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M197">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M198">View MathML</a>

Thus the condition (H3) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M199">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M190">View MathML</a>, we have

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M205">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M206">View MathML</a>

It implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M207">View MathML</a>

Then by Theorem 3.2, boundary value problem (4.3) and (4.4) has a unique solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors 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, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

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