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The equiconvergence of the eigenfunction expansion for a singular version of one-dimensional Schrodinger operator with explosive factor

Zaki FA El-Raheem1* and AH Nasser2

Author Affiliations

1 Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

2 Faculty of Industrial Education, Helwan University, Cairo, Egypt

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Boundary Value Problems 2011, 2011:45  doi:10.1186/1687-2770-2011-45

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/45


Received:7 June 2011
Accepted:23 November 2011
Published:23 November 2011

© 2011 El-Raheem and Nasser; 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

This paper is devoted to prove the equiconvergence formula of the eigenfunction expansion for some version of Schrodinger operator with explosive factor. The analysis relies on asymptotic calculation and complex integration. The paper is of great interest for the community working in the area.

(2000) Mathematics Subject Classification

34B05; 43B24; 43L10; 47E05

Keywords:
Eigenfunctions; Asymptotic formula; Contour integration; Equiconvergence

1 Introduction

Consider the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M2">View MathML</a>

(1.2)

where q(x) is a non-negative real function belonging to L1[0, π], λ is a spectral parameter, and ρ(x) is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M3">View MathML</a>

(1.3)

In [1], the author studied the asymptotic formulas of the eigenvalues, and eigenfunctions of problem (1.1)-(1.2) and proved that the eigenfunctions are orthogonal with weight function ρ(x). In [2], the author also studied the eigenfunction expansion of the problem(1.1)-(1.2). The calculation of the trace formula for the eigenvalues of the problem(1.1)-(1.2) is to appear. We mention here the basic definition and results from [1] that are needed in the progress of this work. Let φ(x, λ), ψ(x, λ) be the solutions of the problem (1.1)-(1.2) with the boundary conditions φ (0, λ) = 0, φ'(0, λ) = 1, ψ(π, λ) = 0, ψ'(π, λ) = 1 and let W(λ) = φ(x, λ)ψ'(x, λ) - ψ(x, λ)φ'(x, λ) be the Wronskian of the two linearly independent solutions φ(x, λ), ψ(x, λ). It is known that W is independent of x so that for x = a let W(λ) = Ψ(λ), the eigenvalues of (1.1)-(1.2) coincide with the roots of the equation Ψ(λ) = 0, which are simple. It is easy to see that the roots of Ψ(λ) = 0 are simple. The function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M4">View MathML</a>

(1.4)

is called the Green's function of the Dirichlet problem (1.1)-(1.2). This function satisfies for λ = λk the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M5">View MathML</a>

(1.5)

where λk are the eigenvalues of the Dirichlet problem (1.1)-(1.2) and ak ≠ 0, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M6">View MathML</a> are the normalization numbers of the eigenfunctions of the same problem (1.1)-(1.2). We consider now the Dirichlet problem (1.1)-(1.2) in the simple form of q(x) ≡ 0. For q(x) = 0, the Dirichlet problem (1.1)-(1.2) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M7">View MathML</a>

(1.6)

Let the eigenfunctions of the problem (1.6) be characterized by the index "o," i.e., φo(x, λ) and ψo(x, λ) are the solutions of the problem (1.6) in cases of ρ(x) = 1 and ρ(x) = -1, respectively, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M8">View MathML</a>

(1.7)

From (1.7), we notice that φo(x, λ) ψo(x, λ) are defined on parts of the interval [0, π], and these formulas must be extended to all intervals [0, π] to enable us to study the Green's function R(x, ξ, λ) in case of q(x) ≡ 0. The following lemma study this extension

Lemma 1.1 The solutions φo(x, λ) and ψo(x, λ) have the following asymptotic formulas

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M9">View MathML</a>

(1.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M10">View MathML</a>

(1.9)

Proof: The fundamental system of solutions of the equation -y″ = s2y, (0 ≤ x a) is y1(x, s) = sin sx, y2(x, s) = cos sx. Similarly, the fundamental system of the equation y″ = s2y, (a < x π) is z1(x, s) = sinh s(π - x), z2(x, s) = cosh s(π - x). So that the solutions φo(x, λ) and ψo(x, λ), over [0, π], can be written in the forms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M11">View MathML</a>

(1.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M12">View MathML</a>

(1.11)

The constants ci,i = 1, 2, 3, 4 are calculated from the continuity of φo(x, λ) and ψo(x, λ) together with their first derivatives at the point x = a, from which it can be easily seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M13">View MathML</a>

(1.12)

Substituting (1.12) into (1.10), we get (1.8). In a similar way, we calculate the constants c3, c4 where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M14">View MathML</a>

(1.13)

Substituting (1.12) and (1.13) into (1.10) and (1.11), respectively, we get the required relations (1.8) and (1.9)

2 The function R(x, ξ, λ) and the equiconvergence

The Green's function plays an important role in studying the equiconvergence theorem, so that, in addition to R(x, ξ, λ), we must study the corresponding Green's function for q(x) ≡ 0. Let Ro(x, ξ, λ) be the Green's function of problem (1.6), which is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M15">View MathML</a>

(2.1)

where the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M16">View MathML</a>

(2.2)

satisfies the following inequality on Γn, which is defined by (2.21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M17">View MathML</a>

(2.3)

Following [2], we state some basic asymptotic relations that are useful in the discussion. The solutions φ(x, λ) and ψ(x, λ) of the Dirichlet problem (1.1)-(1.2) have the following asymptotic formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M18">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M19">View MathML</a>

(2.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M20">View MathML</a>

(2.6)

As we introduce in (1.4), the function R(x, ξ, λ) is the Green's function of the problem (1.1)-(1.2), and Ro(x, ξ, λ) is the corresponding Green's function of the problem (1.6). In the following lemma, we prove an important asymptotic relation for the Green's function

Lemma 2.2 For q(x) ∈ L1(0, π) and by the help of the asymptotic formulas (2.4), (2.5) for φ(x, λ) and ψ(x, λ), respectively, the Green's function R(x, ξ, λ) satisfies the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M21">View MathML</a>

(2.7)

where r(x, ξ, λ), λ ∈ Γn, n → ∞, satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M22">View MathML</a>

(2.8)

Proof: From (2.4) and (2.5), the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M23">View MathML</a>

takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M24">View MathML</a>

(2.9)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M25">View MathML</a>

(2.10)

The function Ψo(λ) is given by (2.2). for x ξ, we discuss three possible cases:

(i) 0 ≤ x ≤ ξ ≤ a (ii) a x ≤ ξ ≤ π (iii) 0 ≤ x a ≤ ξ ≤ π.

The case (i) 0 ≤ x ξ a

From (1.4) and using (2.4) and (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M27">View MathML</a>

Using (2.9), (2.10), and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M28">View MathML</a>

So that from (2.1), for 0 ≤ x ξ a, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M29">View MathML</a>

(2.11)

The case (ii) a x ξ π.

Again, from (1.4) and using (2.4) and (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M30">View MathML</a>

Using (2.9), (2.10), and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M31">View MathML</a>

So that from (2.1), for a x ξ π, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M32">View MathML</a>

(2.12)

The case (iii) 0 ≤ x a ξ π.

From (1.4) and using (2.4) and (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M33">View MathML</a>

Using (2.9), (2.10), and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M34">View MathML</a>

So that from (2.1), for a x ξ π, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M35">View MathML</a>

(2.13)

The asymptotic formulas of R(x, ξ, λ) in case of ξ x remains to be evaluated and this, in turn, consists of three cases

(i*) 0 ≤ ξ ≤ x a (ii*) a ≤ ξ ≤ x ≤ π (iii*) 0 ≤ ξ ≤ a x ≤ π.

The case (i*) 0 ≤ ξ x a from (1.4) and using (2.4) and (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M37">View MathML</a>

Using (2.9), (2.10), and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M38">View MathML</a>

So that from (2.1), for a ξ x a, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M39">View MathML</a>

(2.14)

The case (ii*) a ξ x π from (1.4) and using (2.4) and (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M40">View MathML</a>

Using (2.9), (2.10), and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M41">View MathML</a>

So that from (2.1), for a ξ x π, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M42">View MathML</a>

(2.15)

The case (iii*) 0 ≤ ξ x a x π from (1.4) and using (2.4) and (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M43">View MathML</a>

Using (2.9), (2.10), and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M44">View MathML</a>

So that from (2.1), for a ξ x a, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M45">View MathML</a>

(2.16)

Now from (2.11) and (2.14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M46">View MathML</a>

(2.17)

also, from (2.12) and (2.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M47">View MathML</a>

(2.18)

As a result of the last discussion from (2.13), (2.16), (2.17), and (2.18), we deduce that R(x, ξ, λ) obeys the asymptotic relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M48">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M22">View MathML</a>

(2.19)

We remind here that the main purpose of this paper is to prove the equiconvergence of the eigenfunction expansion of the Dirichlet problem (1.1)-(1.2). We introduce the following notations, let Δn,f(x) denotes the nth partial sum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M49">View MathML</a>

(2.20)

where, from [1], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M50">View MathML</a>. It should be noted here, from [2], that as n → ∞, the series (2.20) converges uniformly to a function f(x) ∈ L2(0, π, ρ(x)). Let also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M51">View MathML</a> be the corresponding nth partial sum as (2.20), for the Dirichlet problem (1.1)-(1.2) in case of q(x) ≡ 0. The equiconvergence of the eigenfunction expansion means that the difference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M52">View MathML</a> uniformly converges to zero as n → ∞, x ∈ [0, π]. In the following theorem, we prove the equiconvergence theorem of the expansions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M53">View MathML</a>. This means that the two expansions have the same condition of convergence. Following [1], the contour Γn is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M54">View MathML</a>

(2.21)

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M55">View MathML</a> the upper half of the contour Γn, Ims ≥ 0, and let Ln be the contour, in λ-domain, formed from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M55">View MathML</a> by the mapping λ = s2. From (1.4), it is obvious that the poles of R(x, ξ, λ) are the roots of the function Ψ(s), which is the spectrum of the problem (1.1)-(1.2).

Theorem 2.1 Under the validity of lemma 1.1 and lemma 2.2, the following relation of equiconvergence holds true

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M56">View MathML</a>

(2.22)

Proof: Multiply both sides of (2.7) by ρ(ξ) f (ξ) and then integrating from 0 to π, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M57">View MathML</a>

where f(x) ∈ L2[0, π, ρ(x)]. We multiply the last equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M58">View MathML</a> and then integrating over the contour Ln in the λ-domain, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M59">View MathML</a>

(2.23)

From equation (1.5), we have the following

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M60">View MathML</a>

(2.24)

Applying Cauchy residues formula to the first integral of (2.23) and using (2.24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M61">View MathML</a>

(2.25)

Similarly, we carry out the same procedure to the second integral of (2.23) and we get an expression analogous to (2.25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M62">View MathML</a>

(2.26)

So that from (2.25), (2.26), and (2.23), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M63">View MathML</a>

from which it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M64">View MathML</a>

(2.27)

The last Equation (2.27) is an essential relation in the proof of the theorem, because the theorem is established if we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M65">View MathML</a> tends to zero uniformly, x ∈ [0, π]. We use the same technique as in [3] We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M66">View MathML</a>

(2.28)

where M1 and M2 are constants.

We treat now the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M67">View MathML</a> in (2.30). Let δ > 0 be a sufficiently small number and let λ = s2, so that, for x, ξ ∈ [0, a], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M68">View MathML</a>

(2.29)

This means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M69">View MathML</a>

(2.30)

where C1, C2, and C3 are independent of x, n and δ. In a similar way, we estimate the second integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M70">View MathML</a> in (2.30) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M71">View MathML</a>

(2.31)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M72">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M73">View MathML</a> are independent of x, n, and δ. Substituting (2.30) and (2.31) into (2.28) and using (2.29), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M74">View MathML</a>

(2.32)

where A,B, and C are constants independent of x, n, and δ. We apply now the property of absolute continuity of Lesbuge integral to the function f(x) ∈ L1[0, π].

∀ ϵ > 0, ∃ δ > 0 is sufficiently small such that ∫|x-ξ|≤δ |f(ξ)|dξ ≤ ϵ, where ϵ is independent of x (the set {ξ : |x - ξ| ≤ δ} is measurable). Fixing δ in (2.32), there exists N such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M75">View MathML</a> and e-δn < ϵ, so that (2.32) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M76">View MathML</a>

(2.33)

Since ϵ is sufficiently small as we please, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M77">View MathML</a> as n → ∞, uniformly with respect to x ∈ [0, π], which completes the proof.

3 The conclusion and comments

It should be noted here that, the theorem of equiconvergence of the eigenfunction expansion is one of interesting analytical problem that arising in the field of spectral analysis of differential operators, see [4-6]. In [3], the author studied the equiconvergence theorem of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M78">View MathML</a>

(3.34)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M79">View MathML</a>

(3.35)

There are many differences between problems (3.34)-(3.35) and the present one (1.1)-(1.2), and the differences are as follows:

1- The boundary conditions of (3.35) is separated boundary conditions, whereas (1.2) is the Dirichlet-Dirichlet condition

2- The eigenfunctions of (3.34)-(3.35) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M80">View MathML</a>

(3.36)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M81">View MathML</a>

(3.37)

3- The contour of integration is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M82">View MathML</a>

(3.38)

4- The remainder function r(x, ξ, λ) admits the following inequality for λ ∈ Γn, n → ∞.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M83">View MathML</a>

(3.39)

Although there are four differences between the two problems, we find that the proof of the equiconvergence formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/45/mathml/M84">View MathML</a> as n → ∞ is similar. So as long as the proof of the equiconvergence relation is carried out by means of the contour integration, we obtain the uniform convergence of the series (2.20)

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The two authors typed read and approved the final manuscript also they contributed to each part of this work equally.

Acknowledgements

We are indebted to an anonymous referee for a detailed reading of the manuscript and useful comments and suggestions, which helped us improve this work. This work was supported by the research center of Alexandria University.

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