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

Open Access Research

Green’s function for Sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application

Jie Zhou and Meiqiang Feng*

Author Affiliations

School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, Republic of China

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Boundary Value Problems 2014, 2014:69  doi:10.1186/1687-2770-2014-69

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/69


Received:6 January 2014
Accepted:5 March 2014
Published:25 March 2014

© 2014 Zhou and Feng; 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 credited.

Abstract

In this paper, we discuss the expression and properties of Green’s function for boundary value problems of nonlinear Sturm-Liouville-type fractional order impulsive differential equations. Its applications are also given. Our results are compared with some recent results by Bai and Lü.

Keywords:
fractional differential equation; impulse; Green’s function; existence; fixed point theorem

1 Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. An excellent account in the study of fractional differential equations can be found in [1-5]. For the basic theory and recent development of the subject, we refer to a text by Lakshmikantham et al.[6]. For more details and examples, see [7-23] and the references therein.

Integer-order impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see, for instance, [24-26]. Recently, the boundary value problems of impulsive differential equations of integer order have been studied extensively in the literature (see [27-36]).

On the other hand, impulsive differential equations of fractional order play an important role in theory and applications, see [37-46] and the references therein. However, as pointed out in [38,39], the fractional impulsive differential equations have not been addressed so extensively and many aspects of these problems are yet to be explored. For example, the theory using Green’s function to express the solution of fractional impulsive differential equations has not been investigated till now. Now, in this paper, we shall study the expression of the solution of fractional impulsive differential equations by using Green’s function.

Consider the following nonlinear boundary value problem of fractional impulsive differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M2">View MathML</a> is the Caputo fractional derivative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M4">View MathML</a> is a continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M5">View MathML</a> is a continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M6">View MathML</a> are continuous functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M8">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M13">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M14">View MathML</a> has a similar meaning for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M15">View MathML</a>.

Some special cases of (1.1) have been investigated. For example, Bai and Lü [13] considered problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M17">View MathML</a>. By using the fixed point theorem in cones, they proved some existence and multiplicity results of positive solutions of problem (1.1).

At the end of this section, it is worth mentioning that it is an important method to express the solution of differential equations by Green’s function. According to the previous work, we find that the solution of impulsive differential equations with integer order can be expressed by Green’s function of the case without impulse. For example, Green’s function of the following boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M18">View MathML</a>

(1.2)

can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M19">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M20">View MathML</a>. The solution of problem (1.2) can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M21">View MathML</a>

If we consider impulsive differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M22">View MathML</a>

(1.3)

then the solution of problem (1.3) can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M23">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25">View MathML</a>.

Naturally, one wishes to know whether or not the same result holds for the fractional order case. We first study the fractional order differential equations with Caputo derivatives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M26">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M2">View MathML</a> is the Caputo fractional derivative. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M28">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M29">View MathML</a>

Then we study whether the solution of fractional order impulsive differential equations with Caputo derivatives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M30">View MathML</a>

(1.5)

can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M31">View MathML</a>

Thus, it is an interesting problem, and so it is worthwhile to study. We will give the answers in the following sections.

The organization of this paper is as follows. In Section 2, we present the expression and properties of Green’s function associated with problem (1.1). In Section 3, we give some preliminaries about the operator and the fixed point theorem. In Section 4, we get some existence results for problem (1.1) by means of some standard fixed point theorems. The final section of the paper contains two examples to illustrate our main results.

2 Expression and properties of Green’s function

Consider the following fractional impulsive boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M32">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M2">View MathML</a> is the Caputo fractional derivative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M6">View MathML</a> are continuous functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M8">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M42">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M14">View MathML</a> has a similar meaning for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M15">View MathML</a>.

Theorem 2.1The solution of problem (2.1) can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M45">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M46">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M47">View MathML</a>

(2.4)

Proof Suppose that x is a solution of (2.1). Then, for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M48">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M49">View MathML</a>

(2.5)

It follows from (2.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M50">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M51">View MathML</a>, then, for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M52">View MathML</a>, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M53">View MathML</a>

Using the impulse conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M55">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M56">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M57">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M58">View MathML</a>, repeating the above procedure, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M59">View MathML</a>

(2.6)

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M61">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M62">View MathML</a>

By the boundary conditions, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M63">View MathML</a>

(2.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M64">View MathML</a>

(2.8)

where η is defined in (2.4).

Substituting (2.7), (2.8) into (2.6), we obtain (2.2). This completes the proof. □

Remark 2.1 It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M65">View MathML</a> is Green’s function of the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M66">View MathML</a>

Remark 2.2 The expression of the solution of problem (2.1) is simpler than that of [37-46].

From (2.3), we can prove the following results.

Proposition 2.1For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M67">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M68">View MathML</a>

(2.9)

Proposition 2.2For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M67">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M70">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M71">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M72">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M73">View MathML</a>

For the sake of convenience, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M74">View MathML</a>

(2.10)

Then it follows from (2.9) and (2.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M75">View MathML</a>

(2.11)

From the proof of Theorem 2.1 we have the following results.

Proposition 2.3The solution of fractional impulsive differential equations can be expressed by Green’s function, and it is not Green’s function of the corresponding fractional differential equations, but Green’s function of the corresponding integer order differential equations.

3 Preliminaries

In this section, we give some preliminaries for discussing the solvability of problem (1.1) as follows.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M76">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M77">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M78">View MathML</a> is a Banach space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M79">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M80">View MathML</a> is a Banach space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M81">View MathML</a>

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M82">View MathML</a> with its Caputo derivative of order q existing on J is a solution of problem (1.1) if it satisfies (1.1).

We give the following hypotheses:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M83">View MathML</a> is a continuous function, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M84">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M85">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M5">View MathML</a> is a continuous function;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M6">View MathML</a> are continuous functions.

It follows from Theorem 2.1 that:

Lemma 3.1If (H1)-(H3) hold, then a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M82">View MathML</a>is a solution of problem (1.1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M89">View MathML</a>is a solution of the impulsive fractional integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M90">View MathML</a>

(3.1)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M92">View MathML</a>

(3.2)

Using Lemma 3.1, problem (1.1) reduces to a fixed point problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M93">View MathML</a>, where T is given by (3.2). Thus problem (1.1) has a solution if and only if the operator T has a fixed point.

From (3.2) and Lemma 3.1, it is easy to obtain the following result.

Lemma 3.2Assume that (H1)-(H3) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91">View MathML</a>is completely continuous.

Proof Note that the continuity of f, ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M96">View MathML</a> together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M98">View MathML</a> ensures the continuity of T.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M99">View MathML</a> be bounded. Then there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M102">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M106">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M107">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M108">View MathML</a>

(3.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M109">View MathML</a>

(3.4)

Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M111">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M112">View MathML</a>

(3.5)

On the other hand, from (3.5), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M113">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M114">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M115">View MathML</a>

(3.6)

It follows from (3.3), (3.5) and (3.6) that T is equicontinuous on all subintervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25">View MathML</a>. Thus, by the Arzela-Ascoli theorem, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91">View MathML</a> is completely continuous. □

To prove our main results, we also need the following two lemmas.

Lemma 3.3 (See [47,48]) (Schauder fixed point theorem)

LetDbe a nonempty, closed, bounded, convex subset of aB-spaceX, and suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M119">View MathML</a>is a completely continuous operator. ThenThas a fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M120">View MathML</a>.

Lemma 3.4 (See [47,48]) (Leray-Schauder fixed point theorem)

LetXbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M121">View MathML</a>be a completely continuous operator. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M122">View MathML</a>

is bounded, thenThas a fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M123">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M124">View MathML</a>

4 Existence of solutions

In this section, we apply Lemma 3.3, Lemma 3.4 and the contraction mapping principle to establish the existence of solutions of problem (1.1). Let us begin by introducing some notation. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M125">View MathML</a>

Theorem 4.1Assume that (H1)-(H3) hold. Suppose further that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M126">View MathML</a>

(4.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M127">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M128">View MathML</a>

hereγis defined in (3.4). Then problem (1.1) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M129">View MathML</a>.

Proof We shall use Schauder’s fixed point theorem to prove that T has a fixed point. First, recall that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91">View MathML</a> is completely continuous (see the proof of Lemma 3.2).

On account of (4.1), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M133">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M134">View MathML</a>

(4.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M135">View MathML</a>

(4.3)

By the definition of ξ, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M136">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M137">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M138">View MathML</a>

(4.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M139">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M140">View MathML</a>

(4.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M141">View MathML</a>

(4.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M143">View MathML</a> are positive constants.

It follows from (3.2) and (4.4)-(4.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M144">View MathML</a>

(4.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M145">View MathML</a> is defined by (4.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M146">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M147">View MathML</a>

Similarly, from (3.2) and (4.4)-(4.6), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M148">View MathML</a>

(4.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M145">View MathML</a> is defined by (4.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M150">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M151">View MathML</a>

It follows from (4.7) and (4.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M152">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M153">View MathML</a>

Hence, we can choose a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M154">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M155">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M156">View MathML</a>

Consequently, Lemma 3.3 implies that T has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M157">View MathML</a>, and the proof is complete. □

Remark 4.1 Condition (4.1) is certainly satisfied if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M158">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M161">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M163">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M164">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25">View MathML</a>).

Theorem 4.2Assume that (H1)-(H3) hold. In addition, letf, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M95">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M96">View MathML</a>satisfy the following conditions:

(H4) There exists a nonnegative function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M168">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M169">View MathML</a>on a subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M170">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M171">View MathML</a>

(H5) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M172">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M173">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25">View MathML</a>.

Then problem (1.1) has at least one solution.

Proof It follows from Lemma 3.2 that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M91">View MathML</a> is completely continuous.

Next, we show that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M177">View MathML</a>

is bounded.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M178">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M179">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M180">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M182">View MathML</a>

(4.9)

It follows from (H4), (H5), (3.2) and (4.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M183">View MathML</a>

(4.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M184">View MathML</a>

It follows from (4.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M185">View MathML</a>

(4.11)

Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M111">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M188">View MathML</a>

which, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159">View MathML</a>, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M190">View MathML</a>

(4.12)

It follows from (4.11) and (4.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M191">View MathML</a>

(4.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M192">View MathML</a>, here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M193">View MathML</a>

So it follows from (4.13) that the set V is bounded. Thus, as a consequence of Lemma 3.4, the operator T has at least one fixed point. Consequently, the problem (1.1) has at least one solution. This finishes the proof. □

Finally we consider the existence of a unique solution for problem (1.1) by applying the contraction mapping principle.

Theorem 4.3Assume that (H1)-(H3) hold. In addition, letf, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M95">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M96">View MathML</a>satisfy the following conditions:

(H6) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M196">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M197">View MathML</a>

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M199">View MathML</a>.

(H7) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M200">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M201">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M25">View MathML</a>.

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M204">View MathML</a>

(4.14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M205">View MathML</a>

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M206">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M207">View MathML</a>are defined in (2.10), then problem (1.1) has a unique solution.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M208">View MathML</a>. Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M159">View MathML</a>, it follows from (H6), (H7) and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M210">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M211">View MathML</a>

Consequently, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M212">View MathML</a>, where Λ is defined by (4.14). As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M213">View MathML</a>, therefore T is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. The proof is complete. □

5 Examples

To illustrate how our main results can be used in practice, we present two examples.

Example 5.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M215">View MathML</a>. We consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M216">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M220">View MathML</a> are positive real numbers.

Conclusion Problem (5.1) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M221">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M222">View MathML</a>.

Proof It follows from (5.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M223">View MathML</a>

(5.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M224">View MathML</a>

(5.3)

From the definition of ω, f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M225">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M226">View MathML</a>, it is easy to see that (H1)-(H3) hold.

On the other hand, it follows from (5.2) and (5.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M227">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M228">View MathML</a>

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M231">View MathML</a>. Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M233">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M234">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M235">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M237">View MathML</a>, and therefore (4.1) is satisfied because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M238">View MathML</a>

Thus, our conclusion follows from Theorem 4.1. □

Example 5.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M215">View MathML</a>. We consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M241">View MathML</a>

(5.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M248">View MathML</a> are positive real numbers with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M249">View MathML</a>.

Conclusion Problem (5.4) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M221">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M251">View MathML</a>.

Proof It follows from (5.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M252">View MathML</a>

(5.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M253">View MathML</a>

(5.6)

From the definition of ω, f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M225">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M226">View MathML</a>, we can obtain that (H1)-(H3) hold.

On the other hand, from (5.5) and (5.6) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M256">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/69/mathml/M257">View MathML</a>

Therefore, the conditions (H4) and (H5) of Theorem 4.2 are satisfied. Thus, Theorem 4.2 gives our conclusion. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All results belong to JZ and MF. All authors read and approved the final manuscript.

Acknowledgements

This work is sponsored by the project NSFC (11301178, 11171032). The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.

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