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Existence and uniqueness of solutions for periodic-integrable boundary value problem of second order differential equation

Hongtu Hua12*, Fuzhong Cong1 and Yi Cheng12

Author Affiliations

1 Fundamental Department, Aviation University of Air Force, Changchun, 130022, People’s Republic of China

2 Institute of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

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Boundary Value Problems 2012, 2012:89  doi:10.1186/1687-2770-2012-89

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/89


Received:26 March 2012
Accepted:16 July 2012
Published:7 August 2012

© 2012 Hua 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 deal with one kind of second order periodic-integrable boundary value problem. Using the lemma on bilinear form and Schauder’s fixed point theorem, we give the existence and uniqueness of solutions for the problem under Lazer type nonresonant condition.

MSC: 34B15, 34B16, 37J40.

Keywords:
lemma on bilinear forms; Schauder’s fixed point theorem; existence and uniqueness; periodic-integrable boundary value problems

1 Introduction and main results

In this paper, we consider the solutions to the following periodic-integrable boundary value problem (for short, PIBVP):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M2">View MathML</a> is a given T-periodic function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M4">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M5">View MathML</a> is T-periodic in t.

Throughout this paper, we assume

(A1) there exist two constants m and M such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M6">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M8">View MathML</a>;

(A2) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M9">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M10">View MathML</a>

Recently, boundary value problems with integral conditions have been studied extensively [6-10]. As we all know Lazer type conditions are essential for the existence and uniqueness of periodic solutions of equations [1-4]. In [5] the existence of periodic solutions has been considered for the following second order equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M11">View MathML</a>

Motivated by the above works, we will consider periodic-integrable boundary value problem (1.1). The main result obtained by us is the following theorem.

Theorem 1Assume that (A1) and (A2) are satisfied. Then PIBVP (1.1) has a unique solution.

This paper is organized as follows. Section 2 deals with a linear problem. There, using the bilinear lemma developed by Lazer, one proves the uniqueness of solutions for linear equations. In Section 3, applying the result in Section 2 and Schauder’s fixed point theorem, we complete the proof of Theorem 1.

2 Linear equation

Consider the following linear PIBVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M12">View MathML</a>

(2.1)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M2">View MathML</a> is a given T-periodic function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M4">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M16">View MathML</a> is a T-periodic function. Assume that

(L1) there exist two constants m and M such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M17">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3">View MathML</a>. Moreover, m and M suit (A2).

Theorem 2Assume that (L1) and (A2) are satisfied, then PIBVP (2.1) has only a trivial solution.

In order to prove Theorem 2, let us give some following concepts.

First, for any interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M19">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M20">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M21">View MathML</a> is a linear space with the norm as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M22">View MathML</a>

Define a bilinear form on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M21">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M24">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M26">View MathML</a>. Let

where N suits assumption (L1), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M32">View MathML</a> are some constants. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M33">View MathML</a>.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M4">View MathML</a>, we can obtain that there exist two constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M38">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M3">View MathML</a>. Then from assumptions (L1) and (A2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M40">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M41">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M42">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M43">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M44">View MathML</a> is positive definite on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M45">View MathML</a> and negative definite on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M46">View MathML</a>. By the lemma in [1], we assert that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M47">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M48">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M49">View MathML</a>.

For every x on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M50">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M51">View MathML</a>, we introduce an auxiliary function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M52">View MathML</a>

The following lemma is very useful in our proofs.

Lemma 1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M54">View MathML</a>are continuous and satisfy (L1) and (A2), then the following two points boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M55">View MathML</a>

(2.2)

has only a trivial solution.

Proof It is clear that 0 is a solution of two points boundary value problem (2.2). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M56">View MathML</a> is a solution of problem (2.2), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M57">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M58">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M59">View MathML</a>

by using (2.2). Integrating the first terms by parts, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M60">View MathML</a>

By assumption (L1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M44">View MathML</a> is positive definite on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M45">View MathML</a> and negative definite on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M46">View MathML</a>. These show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M64">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M65">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M66">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M67">View MathML</a>. The proof of Lemma 1 is ended. □

Proof of Theorem 2 It is clear that PIBVP (2.1) has at least one solution, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M68">View MathML</a>. Assume that PIBVP (2.1) possesses a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M69">View MathML</a>. The proof is divided into three parts.

Case 1: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M70">View MathML</a>. By Lemma 1 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M72">View MathML</a>), PIBVP (2.1) has only a trivial solution. This contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M69">View MathML</a>.

Case 2: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M74">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M75">View MathML</a>

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M76">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M77">View MathML</a>, there are at least two points in the set S, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M79">View MathML</a>. By Lemma 1 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M81">View MathML</a>) the two points boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M82">View MathML</a>

(2.3)

only has a trivial solution. Hence we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M84">View MathML</a>. By the definitions of a and b, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M85">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M86">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M87">View MathML</a>

This contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M89">View MathML</a>.

Case 3: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M90">View MathML</a>. This case is similar to Case 2.

Thus, we complete the proof of Theorem 2. □

Theorem 3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M54">View MathML</a>are continuous and satisfy (L1) and (A2), then the following PIBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M93">View MathML</a>

(2.4)

has a unique solution.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M95">View MathML</a> be two linear independent solutions of the following linear equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M96">View MathML</a>

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M97">View MathML</a> is a solution of PIBVP (2.1), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M99">View MathML</a> are constants. Then by the boundary value conditions of (2.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M100">View MathML</a>

By Theorem 3, PIBVP (2.1) has only a trivial solution, which shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M101">View MathML</a>

(2.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M102">View MathML</a> be a solution of PIBVP (2.4), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M103">View MathML</a> is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M104">View MathML</a>

From the boundary value conditions, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M105">View MathML</a>

From (2.5) constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M107">View MathML</a> are unique. Thus, PIBVP (2.4) has only one solution. □

3 Nonlinear equations

Let us prove Theorem 1. Rewrite (1.1) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M108">View MathML</a>

(3.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M109">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M110">View MathML</a>

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M111">View MathML</a>, introduce an auxiliary PIBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M112">View MathML</a>

(3.2)

To prove the main result, we need the following Lemma 2.

Lemma 2Iffsatisfies (A1) and (A2), then for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M111">View MathML</a>, PIBVP (3.2) has only one solution, denoted as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M114">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M115">View MathML</a>.

Proof From condition (A2), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M116">View MathML</a>

By Theorem 3, PIBVP (3.2) has only one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M114">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M118">View MathML</a> does not hold, there would exist a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M121">View MathML</a>. Choose a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M122">View MathML</a>, without loss of generality, express as itself, such that the sequences are weakly convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M123">View MathML</a>. Denote the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M124">View MathML</a>. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M125">View MathML</a>.

Because the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M126">View MathML</a>

is bounded convex in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M123">View MathML</a>, by the Mazur theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M128">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M129">View MathML</a>

By the Arzela-Ascoli theorem, passing to a subsequence, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M130">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M131">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M132">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M134">View MathML</a>.

By

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M135">View MathML</a>

one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M136">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M137">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M65">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M139">View MathML</a>.

From PIBVP (3.2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M140">View MathML</a>

(3.3)

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M141">View MathML</a> is a nontrivial solution of the following PIBVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M142">View MathML</a>

(3.4)

On the other hand, by Theorem 2, PIBVP (3.4) has only zero, which leads to a contradiction. The proof of Lemma 2 is completed. □

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M143">View MathML</a>

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M144">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M145">View MathML</a>. Applying Lemma 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M146">View MathML</a>.

Lemma 3OperatorFis completely continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M147">View MathML</a>.

Proof We first prove that F is continuous. Given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M148">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M149">View MathML</a>. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M150">View MathML</a>. From the definition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M151">View MathML</a>

(3.5)

We would prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M152">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M153">View MathML</a>. If not, then there would be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M154">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M155">View MathML</a>

Utilizing Lemma 2 and Arzela-Ascoli theorem, passing to a subsequence, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M156">View MathML</a>. Similar to the proof of Lemma 2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M157">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M158">View MathML</a>

(3.6)

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M159">View MathML</a>

Hence, from Theorem 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M160">View MathML</a>. This implies F is continuous. By Lemma 2, for any bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M162">View MathML</a> is also bounded. Hence, applying the continuity of F and Arzela-Ascoli theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M162">View MathML</a> is relatively compact. This shows F is completely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M147">View MathML</a>. The proof of Lemma 3 is completed. □

Proof of Theorem 1 By Lemma 2, Lemma 3 and Schauder’s fixed point theorem, F has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M165">View MathML</a>, that is, PIBVP (1.1) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M166">View MathML</a>.

The following is to prove uniqueness. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M168">View MathML</a> be any two solutions of equation (1.1). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M169">View MathML</a> is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M170">View MathML</a>

Employing (A2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M171">View MathML</a>

Hence by Theorem 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/89/mathml/M172">View MathML</a>. The uniqueness is proved. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, HH, FC and YC contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

The authors are grateful to the referees for their useful comments. The research of F. Cong was partially supported by NSFC Grant (11171350) and Natural Science Foundation of Jilin Province of China (201115133).

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