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An inverse problem related to a half-linear eigenvalue problem

Wei-Chuan Wang1* and Yan-Hsiou Cheng2

Author Affiliations

1 Center for General Education, National Quemoy University, Kinmen, 892, Taiwan, ROC

2 Department of Mathematics and Information Education, National Taipei University of Education, Taipei, 106, Taiwan, ROC

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Boundary Value Problems 2014, 2014:65  doi:10.1186/1687-2770-2014-65


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/65


Received:7 December 2013
Accepted:14 March 2014
Published:24 March 2014

© 2014 Wang and Cheng; 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 an inverse problem on the half-linear Dirichlet eigenvalue problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3">View MathML</a> and r is a positive function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>. Using eigenvalues and nodal data (the lengths of two consecutive zeros of solutions), we reconstruct <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M5">View MathML</a> and its derivatives. Our method is based on (Law and Yang in Inverse Probl. 14:299-312, 779-780, 1998; Shen and Tsai in Inverse Probl. 11:1113-1123, 1995), and our result extends the result in (Shen and Tsai in Inverse Probl. 11:1113-1123, 1995) for the linear case to the half-linear case.

MSC: 34A55, 34B24, 47A75.

1 Introduction

The subject under investigation is the half-linear eigenvalue problem consisting of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M6">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3">View MathML</a>, and r is a positive function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>. By [1-4], it is well known that the problem (1) has countably many eigenpairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M10">View MathML</a>, and the eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M11">View MathML</a> has exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M12">View MathML</a> nodal points in (0,1), say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M13">View MathML</a>. In this paper, we intend to give the representation of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14">View MathML</a> and its derivatives in (1) by using eigenvalues and nodal points. This formation is treated as the reconstruction formula. Such a problem is called an inverse nodal problem and has attracted researchers’ attention. Readers can refer to [5-7] for the linear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M15">View MathML</a>), and to [8,9] for the general case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M2">View MathML</a>).

In [8,9], inverse nodal problems on

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M17">View MathML</a>

(2)

are considered. The authors in [8] studied (2) with Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M18">View MathML</a>

while the authors in [9] studied (2) with eigenparameter dependent boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M19">View MathML</a>

Both of them first used the modified Prüfer substitution to derive the asymptotic expansion of eigenvalues and nodal points and then gave the reconstruction formula of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M20">View MathML</a> by using the nodal data. Note that the authors did not deal with the derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M20">View MathML</a> in [8,9]. Besides, the author in [10] considered the same issue for the p-Laplacian energy-dependent Sturm-Liouville problem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M22">View MathML</a>

coupling with the Dirichlet boundary conditions.

In [6], Shen and Tsai studied the inverse nodal problem on the string equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M23">View MathML</a> with Dirichlet boundary conditions. They employed the standard difference operator △ to give a reconstruction formulas for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M24">View MathML</a> and its derivatives. On the other hand, Law and Yang [7] not only studied the inverse nodal problems for the linear problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M25">View MathML</a>

with separated boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M26">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M27">View MathML</a>, and gave a reconstruction formulas for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M20">View MathML</a> and its derivatives, but they also mentioned that the formulas for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14">View MathML</a> and its derivatives in the string equation with (3) are still valid. They applied the difference quotient operator δ in the formulas.

The main aim and methods of this study are basically the same as the ones in [6,7]. Here we employ a modified Prüfer substitution on (1) derived by the generalized sine function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M30">View MathML</a>. The well-known properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M30">View MathML</a> can be referred to [1,2,4], etc. It shall be mentioned that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M32">View MathML</a> is not <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M33">View MathML</a> at odd multiples of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M34">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M35">View MathML</a>, and not <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M36">View MathML</a> at even multiples of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M34">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M38">View MathML</a>. These lead to that the reconstruction formulas for Nth derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M39">View MathML</a>, in [6,7] cannot be extended to the half-linear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3">View MathML</a>) in this article.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M41">View MathML</a> the first zero of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M30">View MathML</a> in the positive axis. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M44">View MathML</a>, and the nodal length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M45">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M46">View MathML</a>. The following is our first result.

Theorem 1Consider (1) and supposeris continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>. For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M48">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M49">View MathML</a>for the sake of simplicity. Then the following asymptotic formula is valid:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M50">View MathML</a>

(4)

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M51">View MathML</a>, the error term can be replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M52">View MathML</a>.

Now, define the difference operator △ and the difference quotient operator δ as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M53">View MathML</a>

This δ-operator discretizes the differential operator in a nice way. It resembles the difference quotient operator in finite difference. For the derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M54">View MathML</a>, we have the following result.

Theorem 2Consider (1) and supposeris<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M36">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>. For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M48">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M49">View MathML</a>for the sake of simplicity. Then as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M59">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M60">View MathML</a>

(5)

This paper is organized as follows. In Section 2, we give the asymptotic estimates for eigenvalues. This step makes us know such quantities well. It is necessary to specify the orders of the expansion terms in the proofs of the main results. In Section 3, the proofs of the main theorems are given.

2 Asymptotic estimates for eigenvalues

Before we prove the main results, we derive the eigenvalue expansion. Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M61">View MathML</a>, the last term in (6) can be replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M62">View MathML</a> (cf. [[2], p.171]). In [[3], Theorem 2.5], they proved the error term is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M63">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M51">View MathML</a>. The smoothness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14">View MathML</a> increases, and the smaller error can be derived.

Theorem 3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14">View MathML</a>is a positive<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M33">View MathML</a>-function defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>. Then the asymptotic estimate yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M69">View MathML</a>

(6)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M70">View MathML</a> be a solution of (1) and define a Prüfer-type substitution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M71">View MathML</a>

(7)

Then a direct calculation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M72">View MathML</a>

(8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M73">View MathML</a>

(9)

With each eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M74">View MathML</a> of (1), one can associate a Prüfer angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M75">View MathML</a> via (7) if one also specifies the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M76">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M77">View MathML</a> . In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M78">View MathML</a>. Integrating both sides of (8) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M80">View MathML</a>

(10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M81">View MathML</a>. Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M82">View MathML</a> is valid in some subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M83">View MathML</a>, the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M84">View MathML</a> will be constant in this subinterval. This implies that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M14">View MathML</a> depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M74">View MathML</a> in this subinterval from (8). This will contradict our original problem. Hence, the points satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M82">View MathML</a> shall be isolated. Then, in (8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M88">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M89">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M90">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M91">View MathML</a>. Dropping the function variable t, one has the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M92">View MathML</a>

(11)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M93">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M94">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M95">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M41">View MathML</a>-periodic function. Moreover, by integration by parts, (11) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M97">View MathML</a>

(12)

for sufficiently large n. Substituting (11)-(12) into (10), the eigenvalue estimates (6) can be derived. □

3 Proofs of Theorems 1-2

Proof of Theorem 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M49">View MathML</a>. By the Sturm comparison theorem for the p-Laplacian (cf.[4,11,12]etc.), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M99">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M101">View MathML</a> are the maximal and minimal values of r on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M102">View MathML</a>, respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M103">View MathML</a>

(13)

In particular, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M104">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M105">View MathML</a>

(14)

Moreover, by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M54">View MathML</a> and (13), there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M107">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M108">View MathML</a>

(15)

On the other hand, by the mean value theorem and (14), one also has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M109">View MathML</a>

(16)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M110">View MathML</a> between x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M111">View MathML</a>. Therefore, (15) and (16) complete the proof. □

Before we prove Theorem 2, one has to derive the asymptotic behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M54">View MathML</a> at the nodal points. Note that the series in (11) is uniformly convergent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M4">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M114">View MathML</a>. Integrating (8) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M102">View MathML</a> and applying (11), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M116">View MathML</a>

(17)

By the Taylor expansion theorem and integration by parts, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M117">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M118">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M119">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M120">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M121">View MathML</a>

(18)

Multiplying (18) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M122">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M123">View MathML</a>

For convenience, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M124">View MathML</a>

(19)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M125">View MathML</a>

Remark 1 Note that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M126">View MathML</a> is defined by the integral of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M127">View MathML</a>. By the unlike property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M126">View MathML</a> is not a smooth function. This is the main reason that the result in [[7], Theorem 5.1] does not hold in our case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M3">View MathML</a>.

Then the following lemmas are necessary to the proof of Theorem 2 and the superscript will be dropped for the sake of convenience, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M132">View MathML</a>.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M133">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M134">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M135">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M136">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M137">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M138">View MathML</a>is replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M139">View MathML</a>, the above result is still valid. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M140">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M135">View MathML</a>.

Proof The first part of the proof is followed by the mean value theorem and the asymptotic estimates for nodal length (14), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M142">View MathML</a>

It is also valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M143">View MathML</a> by similar arguments. On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M144">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M145">View MathML</a>

and it is also valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M146">View MathML</a>. Finally, by (15) and (16), we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M147">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M148">View MathML</a>

 □

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M149">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M151">View MathML</a>

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M152">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M154">View MathML</a>

Proof First applying the identity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M155">View MathML</a>, (14) and Lemma 1, one can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M156">View MathML</a>

The second part is similar to the one of the first part. So it is omitted here. □

The following corollary is similar to [[7], Lemma 2.3]. We give the proof for the convenience of the readers.

Corollary 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M157">View MathML</a>. Define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M158">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M160">View MathML</a>

Proof By the mean value theorem for integrals, for every j there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M162">View MathML</a>. Then applying Lemmas 1-2, we complete the proof. □

Proof of Theorem 2 Recall (19). By Theorem 3 and Lemma 2, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M164">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>. By Theorem 3 and Corollary 1, one can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M166">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>. By Theorem 3, Lemma 1 and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M126">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M170">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M171">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>. Hence, one can find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M173">View MathML</a>

(20)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>. To complete the proof, it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M175">View MathML</a>

(21)

for sufficiently large n. Obviously, (21) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M176">View MathML</a> by (14) and (16). When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>, by the Taylor theorem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M178">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M179">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M180">View MathML</a>

(22)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M150">View MathML</a>; i.e., by the mean value theorem and (22),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M182">View MathML</a>

(23)

Successively, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/65/mathml/M183">View MathML</a>

(24)

Therefore, substituting (20) into (23)-(24), this completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The author WCW was a major contributor in writing the manuscript. The author YHC performed the literature review. Both of them performed the final editing of the manuscript. They also gave their final approval of the version to be submitted and any revised version.

Acknowledgements

The authors express their thanks to the referees for helpful comments. The first author is supported in part by the National Science Council, Taiwan under contract number NSC 102-2115-M-507-001. The author YH Cheng is supported by the National Science Council, Taiwan under contract numbers NSC 102-2115-M-152 -002.

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