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The local well-posedness for nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative

Cuihua Guo1*, Shulin Sun2 and Hongping Ren3

Author Affiliations

1 School of Mathematical Science, Shanxi University, Taiyuan, Shanxi, 030006, China

2 School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi, 041004, China

3 School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China

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Boundary Value Problems 2014, 2014:90  doi:10.1186/1687-2770-2014-90


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/90


Received:9 January 2014
Accepted:4 April 2014
Published:6 May 2014

© 2014 Guo 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 credited.

Abstract

We study the Cauchy problem of the nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and derivative: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M3">View MathML</a>, where a, b, and c are real numbers. We obtain the local well-posedness for the Cauchy problem with low regularity initial value data by the Fourier restriction norm method.

Keywords:
nonlinear fourth-order Schrödinger equation with derivative; Fourier restriction norm method; Cauchy problem

1 Introduction

In [1], Fibich et al. discussed the following fourth-order Schrödinger equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M4">View MathML</a>

(1.1)

They gave the sufficient conditions of the existence of solutions in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M5">View MathML</a>. For the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M6">View MathML</a>, the mass is still invariant under the scaling <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M7">View MathML</a>. We call this case mass-critical.

Meanwhile, in [1], Fibich et al. mentioned the physical motivation of (1.1). With nonparaxial effects, one obtained the perturbed nonlinear Schrödinger equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M8">View MathML</a>

(1.2)

Furthermore, with vectorial effects, one obtained the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M9">View MathML</a>

(1.3)

Evidently, the nonlinearities with derivatives appear. It is well known that nonlinearities with derivatives bring about more difficulties to solve the problem for us. Especially, there are so many nonlinearities with derivatives in (1.2) and (1.3).

So, in this paper we will study the following nonlinear fourth-order Schrödinger equation with mass-critical nonlinearity and two-order derivative in one dimension:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M10">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M11">View MathML</a> are complex-valued function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M12">View MathML</a> is the complex conjugate quantity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M11">View MathML</a>. a, b, and c are real numbers. We are interested in obtaining the well-posedness for the Cauchy problem of (1.1) with initial value data under low regularity (which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M15">View MathML</a>). Tao et al. obtained the global well-posedness for the Schrödinger equations with derivative (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M16">View MathML</a>) by the I-method (see [2,3]). Bourgain obtained the well-posedness for the nonlinear Schrödinger equation (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M17">View MathML</a>) by the Fourier restriction norm method (see [4]). The character of (1.4) lies in the coexistence of the mass-critical nonlinearity and the two-order derivative. We will discuss the local well-posedness for the fourth-order Schrödinger equation by the Fourier restriction norm method.

For the complicated case, we will discuss it in another paper.

First, we introduce the following notations. We define the Sobolev norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M18">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M19">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M20">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M21">View MathML</a> denotes the Fourier transformation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M22">View MathML</a>.

We also define the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M23">View MathML</a> (see [5]) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M24">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M25">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M26">View MathML</a> denotes the Fourier transformation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M11">View MathML</a>.

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M28">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M3">View MathML</a>) the fundamental solution operator of the fourth-order Schrödinger equation, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M31">View MathML</a> denotes the Fourier transformation of φ, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M32">View MathML</a> represents the inverse Fourier transformation.

We use C to denote various constants which may be different from in particular cases of use throughout.

The main result of this paper is the following theorem.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M34">View MathML</a>. Then the system (1.4) is locally solvable in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M35">View MathML</a>, i.e., for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M36">View MathML</a>, there exists a corresponding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M37">View MathML</a>such that the system (1.4) has a unique solution in the class

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M38">View MathML</a>

Moreover, the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M39">View MathML</a>is Lipschitz continuous from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M35">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M41">View MathML</a>.

In [6], Cui et al. obtained the local well-posedness with the initial condition satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M43">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M44">View MathML</a> in (1.4).

Thus from the above theorem, we can see the following result.

Remark 1.1 When mass-critical nonlinearity and nonlinearity with derivative appear at the same time, nonlinearity with second-order derivative plays more important role. This property is consistent with the classical Schrödinger equation which has both mass-critical nonlinearity and first-order derivative nonlinearity.

2 The preliminary estimates

Definition 2.1 For two integers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M46">View MathML</a>, we say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M47">View MathML</a> is an admissible pair if the following condition is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M48">View MathML</a>

We have the following Strichartz estimate (see [6]): For any admissible pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M47">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M50">View MathML</a>

(2.1)

Lemma 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M47">View MathML</a>is an admissible pair. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M53">View MathML</a>

(2.2)

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M54">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M55">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M56">View MathML</a>

(2.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M57">View MathML</a>.

Proof Firstly, we prove the inequality (2.2).

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M58">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M59">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M60">View MathML</a>.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52">View MathML</a>, using the Strichartz estimate, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M62">View MathML</a>

Next we prove (2.3), we only need to prove that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M64">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M65">View MathML</a>

(2.6)

Changing the variable to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M66">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M67">View MathML</a>

(2.7)

From (2.6) and (2.7), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M68">View MathML</a>

so that (2.5) holds.

Similarly, (2.4) follows from the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M69">View MathML</a>

This inequality has been proved in Theorem 4.1 of [7]. We omit its detailed proof here. □

Lemma 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M70">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M71">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M72">View MathML</a>

(2.8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M73">View MathML</a>is as same as in Lemma 2.1.

Proof Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M74">View MathML</a> is an admissible pair, so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M75">View MathML</a>

(2.9)

Therefore, using (2.7), Minkowski’s inequality, (2.9), and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M52">View MathML</a>, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M77">View MathML</a>

(2.10)

By interpolation [8] between (2.10) and the following relation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M78">View MathML</a>

we immediately obtain for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M70">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M80">View MathML</a>

 □

We take a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M81">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M82">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M84">View MathML</a>. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M85">View MathML</a>.

Similar to the proof of Lemma 3.2 in [9] (or see [[10], Lemmas 3.1-3.3], [[11], Lemma 2.3], [[12], Lemma 2.7]), we have the following estimates.

Lemma 2.3For any reals, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M87">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M88">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M89">View MathML</a>

Lemma 2.4[12]

Iff, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M91">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M92">View MathML</a>belong to a Schwartz space on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M93">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M94">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M95">View MathML</a>

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M98">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M99">View MathML</a>

(2.11)

Proof By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M100">View MathML</a> and duality, the inequality (2.11) is reduced to the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M101">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M102">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M103">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M106">View MathML</a>.

Without loss of generality, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M108">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M109">View MathML</a>.

We split the domain of integration into two cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M111">View MathML</a>.

Case I. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M111">View MathML</a>.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33">View MathML</a>, by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M114">View MathML</a>

Case II. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M110">View MathML</a>.

Subcase 1. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M116">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M117">View MathML</a>.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M118">View MathML</a>, by Lemma 2.4, the Hölder inequality, and Lemma 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M119">View MathML</a>

Subcase 2. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M120">View MathML</a>. We split this domain of integration in several pieces.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M121">View MathML</a> Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M122">View MathML</a>.

Similar to the proof of subcase 1, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M118">View MathML</a>, by Lemma 2.4, the Hölder inequality, and Lemma 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M125">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M126">View MathML</a> Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M127">View MathML</a>.

In this case, the proof is similar to that of the above case, so here we omit the detailed proof.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M128">View MathML</a> Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M129">View MathML</a>.

In this case, we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M130">View MathML</a>

(2.12)

By a straightforward calculation, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M131">View MathML</a>

(2.13)

3.0: Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M132">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M133">View MathML</a> holds.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M134">View MathML</a>, by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M135">View MathML</a>

3.1: Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M136">View MathML</a>. By (2.10) and (2.11), we immediately obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M137">View MathML</a>.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M98">View MathML</a>, by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M140">View MathML</a>

3.2: Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M141">View MathML</a>.

The proof is similar to that of the case 3.1, so here omit the detailed proof.

3.3: Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M142">View MathML</a>. By (2.10) and (2.11), we immediately obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M143">View MathML</a>.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M98">View MathML</a>, by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M146">View MathML</a>

 □

Lemma 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M34">View MathML</a>. Then we have the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M149">View MathML</a>

(2.14)

Proof Firstly, we prove (2.14) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M150">View MathML</a>.

By the definition, the Hölder inequality, and the Sobolev inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M151">View MathML</a>

(2.15)

That means

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M152">View MathML</a>

(2.16)

Again using the Hölder inequality, the Sobolev inequality in the variable x, and Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M153">View MathML</a>

(2.17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M158">View MathML</a> is admissible pair.

Combining (2.16) and (2.17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M159">View MathML</a>

(2.18)

Secondly, using the Leibniz rule for fractional power, in a similar way, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M160">View MathML</a>

(2.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M165">View MathML</a> is admissible pair for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M166">View MathML</a>.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M169">View MathML</a>; and for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M173">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M176">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M177">View MathML</a> in (2.19), noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M179">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M180">View MathML</a>

(2.20)

which completes the proof. □

3 Proof of main result

Proof of Theorem 1.1 We will prove Theorem 1.1 by using the Banach fixed point theorem. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M181">View MathML</a>. We rewrite (1.4) in integral form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M182">View MathML</a>

(3.1)

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M183">View MathML</a>

and define a mapping S as follows: For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M184">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M185">View MathML</a>

In the sequel we will prove that S is well defined and it is a contraction map on Z.

By Lemma 2.3 and Lemmas 2.5-2.6, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M186">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M187">View MathML</a>

If we take δ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M188">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M189">View MathML</a>.

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M190">View MathML</a>, in an analogous way to the above, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M191">View MathML</a>

Furthermore, if we take δ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M192">View MathML</a> then S is a contraction mapping of Z into itself. The desired result immediately follows from Banach’s fixed point theorem. That means that there is a unique solution which solves the Cauchy problem (1.4) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M181">View MathML</a>. The Lipschitz continuousness from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M35">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/90/mathml/M195">View MathML</a> is easily obtained from the above proof process. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This work is supported by Natural Science of Shanxi province (No. 2013011003-2, No. 2013011002-2 and No. 2010011001-1), Natural Science Foundation of China (No. 11071149), Research Project Supported by Shanxi Scholarship Council of China (No. 2011-011).

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