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Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential

Etibar S Panakhov1 and Murat Sat2*

Author affiliations

1 Department of Mathematics, Firat University, Elazig, 23119, Turkey

2 Department of Mathematics, Erzincan University, Erzincan, 24100, Turkey

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Citation and License

Boundary Value Problems 2013, 2013:49  doi:10.1186/1687-2770-2013-49

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/49


Received:12 December 2012
Accepted:21 February 2013
Published:8 March 2013

© 2013 Panakhov and Sat; licensee Springer

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

Abstract

In this paper, we are concerned with an inverse problem for the Sturm-Liouville operator with Coulomb potential using a new kind of spectral data that is known as nodal points. We give a reconstruction of q as a limit of a sequence of functions whose nth term is dependent only on eigenvalue and its associated nodal data. It is mentioned that this method is based on the works of Law and Yang, but we have applied the method to the singular Sturm-Liouville problem.

MSC: 34L05, 45C05.

Keywords:
Coulomb potential; nodal point; reconstruction formula

1 Introduction

Inverse problems of spectral analysis imply the reconstruction of a linear operator from some or other of its spectral characteristics. Such characteristics are spectra (for different boundary conditions), normalizing constants, spectral functions, scattering data, etc. An early important result in this direction, which gave vital impetus for further development of inverse problem theory, was obtained in [1]. At present, inverse problems are studied for certain special classes of ordinary differential operators. Inverse problems from two spectra are the most simple in their formulation and well studied in [2,3]. An effective method of constructing a regular and singular Sturm-Liouville operator from a spectral function or from two spectra is given in [4-7].

We note that the details of the inverse problem for singular equations are given in the monographs [8-11] and references therein.

In some recent interesting works [12,13], Hald and McLaughlin and Browne and Sleeman have taken a new approach to inverse spectral theory for the Sturm-Liouville problem. The novelty of these works lies in the use of nodal points as the given spectral data. In recent years, inverse nodal problems have been studied by several authors [14-21]etc.

In this paper, we deal with an inverse nodal problem for the Sturm-Liouville operator with Coulomb potential. We have reconstructed the potential function q from the nodal points of eigenfunctions, provided q is smooth enough. The method is based on a series of works by Law and Yang [14,17].

Before giving the main results, we mention some physical properties of the Sturm-Liouville operator with Coulomb potential. Learning about the motion of electrons moving under the Coulomb potential is of significance in quantum theory. Solving these types of problems allows us to find energy levels not only for a hydrogen atom but also for single valence electron atoms such as sodium. For hydrogen atom, the Coulomb potential is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M1">View MathML</a>, where r is the radius of the nucleus, e is electronic charge. According to this, we use the time-dependent Schrödinger equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M2">View MathML</a>

where Ψ is the wave function, ħ is Planck’s constant and m is the mass of electron. In this equation, if the Fourier transform is applied

it will convert to energy equation dependent on the situation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M4">View MathML</a>

Therefore, energy equation in the field with the Coulomb potential becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M5">View MathML</a>

If this hydrogen atom is substituted to other potential area, then the energy equation becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M6">View MathML</a>

If we make the necessary transformation, then we can get a Sturm-Liouville equation with Coulomb potential

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M7">View MathML</a>

where λ is a parameter which corresponds to the energy [22].

We consider the singular Sturm-Liouville problem

(1.1)

(1.2)

(1.3)

in which the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M11">View MathML</a>, A, H are finite numbers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M12">View MathML</a>. Next, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M13">View MathML</a> the solution of (1.1) satisfying the initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M14">View MathML</a>

(1.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M15">View MathML</a> be the nth eigenvalue and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M17">View MathML</a> be nodal points of the nth eigenfunction. Also, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M18">View MathML</a> be the ith nodal domain of the nth eigenfunction and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M19">View MathML</a> be the associated nodal length. We also define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M20">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M21">View MathML</a>.

2 Main results

In this section, we try to obtain some asymptotic results and a reconstruction formula for the potential q, which has been obtained as a solution of an inverse nodal problem.

Lemma 2.1The solution of problem (1.1)-(1.3) has the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M22">View MathML</a>

(2.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M23">View MathML</a>.

Proof Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M13">View MathML</a> satisfies equation (1.1), we get

By integrating the first term twice on the right-hand side by parts and taking the conditions into account (1.2), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M26">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M23">View MathML</a>. □

Lemma 2.2The eigenvalues of problem (1.1)-1.3) are the roots of (1.3). This spectral characteristic satisfies the following asymptotic expression[23]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M28">View MathML</a>

(2.2)

where

Lemma 2.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M30">View MathML</a>. Then, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31">View MathML</a>,

(2.3)

(2.4)

Proof By using some iterations and trigonometric calculations in (2.1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M34">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M13">View MathML</a> is equal to zero and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M36">View MathML</a> is not close to zero, then

Now, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M39">View MathML</a>. Because Taylor’s expansion for the arctangent function is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M40">View MathML</a>

for some integer i, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M41">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M42">View MathML</a>

The nodal length is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M43">View MathML</a>

This completes the proof of Lemma 2.3. □

Lemma 2.4Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M44">View MathML</a>. Then, for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M45">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M47">View MathML</a>

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M49">View MathML</a> almost everywhere. Thus, given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M50">View MathML</a>, when n is sufficiently large and for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51">View MathML</a>,

This proves Lemma 2.4. □

Theorem 2.1The potential function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M53">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M54">View MathML</a>

for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46">View MathML</a>. We note that the asymptotic expression for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M57">View MathML</a>in Theorem 2.1 implies that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M58">View MathML</a>.

Proof When we consider (2.4) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M59">View MathML</a>

so that

By Lemma 2.4

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M61">View MathML</a>

for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51">View MathML</a>.

It remains to show that for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M64">View MathML</a>

tends to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31">View MathML</a>. Take a sequence of continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66">View MathML</a> which converges to q in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M67">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66">View MathML</a> has a subsequence converging to q almost everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M69">View MathML</a>. We call this subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66">View MathML</a>. Take any x such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M71">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M72">View MathML</a>. Then for a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M73">View MathML</a>, we can fix a large k such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M74">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M75">View MathML</a>

By Lemma 2.3,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M76">View MathML</a>

and so it tends to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31">View MathML</a>. By Lemma 2.4, the first term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M78">View MathML</a> satisfies, when n is sufficiently large,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M79">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M80">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M66">View MathML</a> is continuous, this term is arbitrarily every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M82">View MathML</a>. Hence we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M83">View MathML</a>. This proves Theorem 2.1. □

Lemma 2.5We take a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M84">View MathML</a>converges to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M85">View MathML</a>, then, for any large enoughn, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M87">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M88">View MathML</a>

Proof By (2.4) and observation that the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M89">View MathML</a> is constant on any nodal interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M90">View MathML</a>, we obtain

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M87">View MathML</a> this term converges to zero. □

Lemma 2.6Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M93">View MathML</a>, then as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M94">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M96">View MathML</a>

Proof Firstly, let us show that if q is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M97">View MathML</a>, the result is satisfied. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M98">View MathML</a>. By using the intermediate value theorem, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M99">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M100">View MathML</a>

If x is close enough to a, the difference can be arbitrarily small. Then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M73">View MathML</a>, when n is large enough, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M46">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M103">View MathML</a>

In the above process, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M104">View MathML</a>. The estimate also holds if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M105">View MathML</a>. Hence if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M107">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M72">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M109">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M110">View MathML</a> can be arbitrarily small. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M111">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M112">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M30">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M114">View MathML</a> convergent to q in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M115">View MathML</a>. Hence, fix n sufficiently large,

From the above process and Lemma 2.5, when k is large enough, the first two terms are arbitrarily small. Hence, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M117">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M118">View MathML</a>

 □

Theorem 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M119">View MathML</a>converges toqin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M120">View MathML</a>.

Proof When we consider the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M119">View MathML</a>, we obtain that

It suffices to show that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M124">View MathML</a>

By using (2.4) we have

Hence, we only need to prove that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M31">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M127">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M128">View MathML</a>

(2.5)

From Lemma 2.6, the first limit holds and the second limit also holds. On the other hand, the sequence of functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M129">View MathML</a>

converges to 0 for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M51">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M131">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/49/mathml/M132">View MathML</a>

Then, we may apply the Lebesque dominated convergence theorem to show that (2.5) is valid. The proof of Theorem 2.2 is completed. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MS wrote the first draft and ESP corrected and improved the final version. Both authors read and approved the final draft.

Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and remarks which led to a great improvement of the article.

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