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On calculation of eigenvalues and eigenfunctions of a Sturm-Liouville type problem with retarded argument which contains a spectral parameter in the boundary condition

Erdoğan Şen* and Azad Bayramov

Author Affiliations

Department of Mathematics, Faculty of Arts and Science, Namık Kemal University, 59030 Tekirdağ, Turkey

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Journal of Inequalities and Applications 2011, 2011:113  doi:10.1186/1029-242X-2011-113

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2011/1/113


Received:7 June 2011
Accepted:17 November 2011
Published:17 November 2011

© 2011 Şen and Bayramov; 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 study, a discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition and with transmission conditions at the point of discontinuity is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions.

MSC (2010): 34L20; 35R10.

Keywords:
differential equation with retarded argument; transmission conditions; asymptotics of eigenvalues and eigenfunctions

1 Introduction

Boundary-value problems for differential equations of the second order with retarded argument were studied in [1-5], and various physical applications of such problems can be found in [2].

The asymptotic formulas for the eigenvalues and eigenfunctions of boundary problem of Sturm-Liouville type for second order differential equation with retarded argument were obtained in [5].

The asymptotic formulas for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the spectral parameter in the boundary condition were obtained in [6].

In the articles [7-9], the asymptotic formulas for the eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions and with the boundary conditions which include spectral parameter were obtained.

In this article, we study the eigenvalues and eigenfunctions of discontinuous boundary-value problem with retarded argument and a spectral parameter in the boundary condition. Namely, we consider the boundary-value problem for the differential equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M1">View MathML</a>

(1)

on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a>, with boundary conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M3">View MathML</a>

(2)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M4">View MathML</a>

(3)

and transmission conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M5">View MathML</a>

(4)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M6">View MathML</a>

(5)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M7">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M8">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M9">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M10">View MathML</a>, the real-valued function q(x) is continuous in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a> and has a finite limit <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M11">View MathML</a>, the real-valued function Δ(x) ≥ 0 continuous in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a> and has a finite limit <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M12">View MathML</a>, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M8">View MathML</a>; <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M13">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M10">View MathML</a>; λ is a real spectral parameter; p1, p2, γ1, γ2, δ1, δ2 are arbitrary real numbers and |γi| + |δi| ≠ 0 for i = 1, 2. Also, γ1δ2p1 = γ2δ1p2 holds.

It must be noted that some problems with transmission conditions which arise in mechanics (thermal condition problem for a thin laminated plate) were studied in [10].

Let w1(x, λ) be a solution of Equation 1 on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M14">View MathML</a>, satisfying the initial conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M15">View MathML</a>

(6)

The conditions (6) define a unique solution of Equation 1 on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M14">View MathML</a> [2, p. 12].

After defining above solution, we shall define the solution w2 (x, λ) of Equation 1 on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16">View MathML</a> by means of the solution w1(x, λ) by the initial conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M17">View MathML</a>

(7)

The conditions (7) are defined as a unique solution of Equation 1 on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16">View MathML</a>.

Consequently, the function w (x, λ) is defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a> by the equality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M18">View MathML</a>

is a such solution of Equation 1 on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a>; which satisfies one of the boundary conditions and both transmission conditions.

Lemma 1. Let w (x, λ) be a solution of Equation 1 and λ > 0. Then, the following integral equations hold:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M19">View MathML</a>

(8)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M20">View MathML</a>

(9)

Proof. To prove this, it is enough to substitute <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M21">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M22">View MathML</a> instead of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M23">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M24">View MathML</a> in the integrals in (8) and (9), respectively, and integrate by parts twice.

Theorem 1. The problem (1)-(5) can have only simple eigenvalues.

Proof. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M25">View MathML</a> be an eigenvalue of the problem (1)-(5) and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M26">View MathML</a>

be a corresponding eigenfunction. Then, from (2) and (6), it follows that the determinant

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M27">View MathML</a>

and by Theorem 2.2.2 in [2], the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M28">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M29">View MathML</a> are linearly dependent on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M14">View MathML</a>. We can also prove that the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M30">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M31">View MathML</a> are linearly dependent on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16">View MathML</a>. Hence,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M32">View MathML</a>

(10)

for some K1 ≠ 0 and K2 ≠ 0. We must show that K1 = K2. Suppose that K1 K2. From the equalities (4) and (10), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M33">View MathML</a>

Since δ1 (K1 - K2) ≠ 0, it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M34">View MathML</a>

(11)

By the same procedure from equality (5), we can derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M35">View MathML</a>

(12)

From the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M31">View MathML</a> is a solution of the differential equation (1) on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16">View MathML</a> and satisfies the initial conditions (11) and (12) it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M36">View MathML</a> identically on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M16">View MathML</a> (cf. [2, p. 12, Theorem 1.2.1]).

By using we may also find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M37">View MathML</a>

From the latter discussions of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M31">View MathML</a>, it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M36">View MathML</a> identically on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a>. But this contradicts (6), thus completing the proof.

2 An existance theorem

The function ω(x, λ) defined in Section 1 is a nontrivial solution of Equation 1 satisfying conditions (2), (4) and (5). Putting ω(x, λ) into (3), we get the characteristic equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M38">View MathML</a>

(13)

By Theorem 1.1, the set of eigenvalues of boundary-value problem (1)-(5) coincides with the set of real roots of Equation 13. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M39">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M40">View MathML</a>.

Lemma 2. (1) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M41">View MathML</a>. Then, for the solution w1(x, λ) of Equation 8, the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M42">View MathML</a>

(14)

(2) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M43">View MathML</a>. Then, for the solution w2 (x, λ) of Equation 9, the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M44">View MathML</a>

(15)

Proof. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M45">View MathML</a>. Then, from (8), it follows that, for every λ > 0, the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M46">View MathML</a>

If s ≥ 2q1, we get (14). Differentiating (8) with respect to x, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M47">View MathML</a>

(16)

From (16) and (14), it follows that, for s ≥ 2q1, the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M48">View MathML</a>

Hence,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M49">View MathML</a>

(17)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M50">View MathML</a>. Then, from (9), (14) and (17), it follows that, for s ≥ 2q1, the following inequalities holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M51">View MathML</a>

Hence, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M43">View MathML</a>, we get (15).

Theorem 2. The problem (1)-(5) has an infinite set of positive eigenvalues.

Proof. Differentiating (9) with respect to x, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M52">View MathML</a>

(18)

From (8), (9), (13), (16) and (18), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M53">View MathML</a>

(19)

Let λ be sufficiently large. Then, by (14) and (15), Equation 19 may be rewritten in the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M54">View MathML</a>

(20)

Obviously, for large s, Equation 20 has an infinite set of roots. Thus, the theorem is proved.

3 Asymptotic formulas for eigenvalues and eigenfunctions

Now, we begin to study asymptotic properties of eigenvalues and eigenfunctions. In the following, we shall assume that s is sufficiently large. From (8) and (14), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M55">View MathML</a>

(21)

From (9) and (15), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M56">View MathML</a>

(22)

The existence and continuity of the derivatives <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M57">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M58">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M59">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M60">View MathML</a>, follows from Theorem 1.4.1 in [?].

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M61">View MathML</a>

(23)

Theorem 3. Let n be a natural number. For each sufficiently large n, there is exactly one eigenvalue of the problem (1)-(5) near <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M62">View MathML</a>.

Proof. We consider the expression which is denoted by O(1) in Equation 20. If formulas (21)-(23) are taken into consideration, it can be shown by differentiation with respect to s that for large s this expression has bounded derivative. It is obvious that for large s the roots of Equation 20 are situated close to entire numbers. We shall show that, for large n, only one root (20) lies near to each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M63">View MathML</a>. We consider the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M64">View MathML</a>. Its derivative, which has the form <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M65">View MathML</a>, does not vanish for s close to n for sufficiently large n. Thus, our assertion follows by Rolle's Theorem.

Let n be sufficiently large. In what follows, we shall denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M66">View MathML</a> the eigenvalue of the problem (1)-(5) situated near <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M63">View MathML</a>. We set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M67">View MathML</a>. From (20), it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M68">View MathML</a>. Consequently

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M69">View MathML</a>

(24)

The formula (24) makes it possible to obtain asymptotic expressions for eigenfunction of the problem (1)-(5). From (8), (16) and (21), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M70">View MathML</a>

(25)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M71">View MathML</a>

(26)

From (9), (22), (25) and (26), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M72">View MathML</a>

(27)

By putting (24) in (25) and (27), we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M73">View MathML</a>

Hence, the eigenfunctions un(x) have the following asymptotic representation:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M74">View MathML</a>

Under some additional conditions, the more exact asymptotic formulas which depend upon the retardation may be obtained. Let us assume that the following conditions are fulfilled:

(a) The derivatives q'(x) and Δ(x) exist and are bounded in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a> and have finite limits <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M75">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M76">View MathML</a>, respectively.

(b) Δ'(x) ≤ 1 in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M2">View MathML</a>, Δ(0) = 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M77">View MathML</a>.

Using (b), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M78">View MathML</a>

(28)

From (25), (27) and (28), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M79">View MathML</a>

(29)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M80">View MathML</a>

(30)

Under the conditions (a) and (b), the following formulas

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M81">View MathML</a>

(31)

can be proved by the same technique in Lemma 3.3.3 in [?]. Putting these expressions into (19), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M82">View MathML</a>

and using γ1δ2p1 = γ2δ1p2 we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M83">View MathML</a>

Dividing by s and using <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M67">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M84">View MathML</a>

Hence,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M85">View MathML</a>

and finally

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M86">View MathML</a>

(32)

Thus, we have proven the following theorem.

Theorem 4. If conditions (a) and (b) are satisfied, then the positive eigenvalues <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M66">View MathML</a> of the problem (1)-(5) have the (32) asymptotic representation for n → ∞.

We now may obtain a sharper asymptotic formula for the eigenfunctions. From (8) and (29),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M87">View MathML</a>

(33)

Replacing s by sn and using (32), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M88">View MathML</a>

(34)

From (16) and (29), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M89">View MathML</a>

(35)

From (9), (30), (31), (33) and (35), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M90">View MathML</a>

Now, replacing s by sn and using (32), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M91">View MathML</a>

(36)

Thus, we have proven the following theorem.

Theorem 5. If conditions (a) and (b) are satisfied, then the eigenfunctions un(x) of the problem (1)-(5) have the following asymptotic representation for n → ∞:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/113/mathml/M92">View MathML</a>

where u1n(x) and u2n(x) defined as in (34) and (36), respectively.

4 Conclusion

In this study, first, we obtain asymptotic formulas for eigenvalues and eigenfunctions for discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition. Then, under additional conditions (a) and (b) the more exact asymptotic formulas, which depend upon the retardation obtained.

5 Competing interests

The authors declare that they have no completing interests.

6 Authors' contributions

Establishment of the problem belongs to AB (advisor). ES obtained the asymptotic formulas for eigenvalues and eigenfunctions. All authors read and approved the final manuscript.

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