Open Access

Accurate Asymptotic Formulas for Eigenvalues and Eigenfunctions of a Boundary-Value Problem of Fourth Order

Boundary Value Problems20102010:720235

DOI: 10.1155/2010/720235

Received: 7 July 2010

Accepted: 9 November 2010

Published: 28 November 2010

Abstract

In the present paper, we consider a nonself-adjoint fourth-order differential operator with the periodic boundary conditions. We compute new accurate asymptotic expression of the fundamental solutions of the given equation. Then, we obtain new accurate asymptotic formulas for eigenvalues and eigenfunctions.

1. Introduction

In the present work, we consider a nonself-adjoint fourth-order operator which is generated by the periodic boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq1_HTML.gif is a complex-valued function. Without lose of generality, we can assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq2_HTML.gif .

Spectral properties of Sturm-Liouville operator which is generated by the periodic and antiperiodic boundary conditions have been investigated by many authors, the results on this direct and references are given details in the monographs [15].

In this paper we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the fourth-order boundary-value problem (1.1), (1.2). For second-order differential equations, similar asymptotic formulas were obtained in [69]. We note that in [6, 10, 11], using the obtained asymptotic formulas for eigenvalues and eigenfunctions, the basis properties of the root functions of the operators were investigated.

The paper is organized as follows. In Section 2, we compute new asymptotic expression of the fundamental solutions of (1.1). In Section 3, we obtain new accurate asymptotic estimates for the eigenvalues. In Section 4, we have asymptotic formulas for eigenfunctions under the distinct conditions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq3_HTML.gif .

2. The Expression of the Fundamental Solutions

It is well known that (see [2, page 92]) if the complex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq4_HTML.gif -plane https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq5_HTML.gif is divided into eight sectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq6_HTML.gif , defined by the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ3_HTML.gif
(2.1)
then in each of these sectors (1.1) has four linear independent solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq7_HTML.gif , which are regular with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq8_HTML.gif in the sector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq9_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq10_HTML.gif sufficiently large and which satisfy the relation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ4_HTML.gif
(2.2)
where the numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq11_HTML.gif are the fourth roots of unity, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq13_HTML.gif . In general, the term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq14_HTML.gif at the formula (2.2) depends upon the smoothness of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq15_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq16_HTML.gif has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq17_HTML.gif continuous derivatives, then one can assert the existence of a representation (2.2) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq18_HTML.gif . Here, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq19_HTML.gif . The functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq20_HTML.gif satisfy the following recursion relations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ5_HTML.gif
(2.3)
Let us put, moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq22_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq23_HTML.gif . Thus, the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq24_HTML.gif are uniquely determined. Thus, we can find from (2.3) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq25_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ6_HTML.gif
(2.4)

3. The Asymptotic Formulas of Eigenvalues

It follows from the classical investigations (see [4, page 65]) that the eigenvalues of the problem (1.1), (1.2) (in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq26_HTML.gif ) consist of the pairs of the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq27_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq28_HTML.gif satisfying the following asymptotic formula:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ7_HTML.gif
(3.1)

for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq29_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq30_HTML.gif is a constant.

Theorem 3.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq31_HTML.gif . Then, the eigenvalues of the boundary-value problem (1.1), (1.2) form two infinite sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq33_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq34_HTML.gif is a big positive integer and have the following asymptotic formulas:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ8_HTML.gif
(3.2)

Proof.

By derivation of (2.2) up to third order with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq35_HTML.gif , the following relations are obtained:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ9_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq37_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ10_HTML.gif
(3.4)
Now let us substitute all these expressions into the characteristic determinant
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ11_HTML.gif
(3.5)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ12_HTML.gif
(3.6)
By long computations, for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq38_HTML.gif , we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ13_HTML.gif
(3.7)
Multiplying the last equation by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ14_HTML.gif
(3.8)
it becomes
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ15_HTML.gif
(3.9)
Hence, by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq39_HTML.gif , for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq40_HTML.gif , the following equations hold:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ16_HTML.gif
(3.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ17_HTML.gif
(3.11)
By Rouche's theorem, we have asymptotic estimates for the roots https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq43_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq44_HTML.gif , of (3.10) and (3.11), respectively, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq45_HTML.gif is a big positive integer
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ18_HTML.gif
(3.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ19_HTML.gif
(3.13)

From the relations (3.12), (3.13) and the relations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq46_HTML.gif , the asymptotic formulas (3.2) are valid for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq47_HTML.gif .

4. The Asymptotic Formulas for the Eigenfunctions

Now, we obtain asymptotic formulas for eigenfunctions under the distinct conditions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq48_HTML.gif .

Case 1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq49_HTML.gif and the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq50_HTML.gif holds. Based on the asymptotic expressions of the fundamental solutions of (1.1) and the asymptotic formulas for eigenvalues of the boundary-value problem (1.1), (1.2) up to order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq51_HTML.gif , the following result is valid.

Theorem 4.1.

If the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq52_HTML.gif holds, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq54_HTML.gif are of the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ20_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ21_HTML.gif
(4.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq55_HTML.gif is sufficiently large integer.

Proof.

Let us calculate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq56_HTML.gif up to order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq57_HTML.gif . Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ22_HTML.gif
(4.3)
we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ23_HTML.gif
(4.4)
Follows from the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq58_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq59_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq60_HTML.gif . Thus, we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq61_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq62_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ24_HTML.gif
(4.5)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ25_HTML.gif
(4.6)
By simple computations, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ26_HTML.gif
(4.7)
Hence, using the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ27_HTML.gif
(4.8)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ28_HTML.gif
(4.9)
Using the relations (3.3) and (3.12), for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq63_HTML.gif , we obtain (4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ29_HTML.gif
(4.10)
Similarly, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq64_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq65_HTML.gif , we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq66_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq67_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ30_HTML.gif
(4.11)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ31_HTML.gif
(4.12)
By similar computations we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ32_HTML.gif
(4.13)
Hence, using the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ33_HTML.gif
(4.14)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ34_HTML.gif
(4.15)
Hence, for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq68_HTML.gif , we obtain (4.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ35_HTML.gif
(4.16)

Case 2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq69_HTML.gif and the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq71_HTML.gif hold. Based on the asymptotic expressions of the fundamental solutions of (1.1) and the asymptotic formulas for eigenvalues of the boundary-value problem (1.1), (1.2) up to order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq72_HTML.gif , the following result is valid.

Theorem 4.2.

If the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq74_HTML.gif hold, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq75_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq76_HTML.gif are of the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ36_HTML.gif
(4.17)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ37_HTML.gif
(4.18)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq77_HTML.gif is sufficiently large integer.

Proof.

It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ38_HTML.gif
(4.19)
It follows from the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq78_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq79_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq80_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq81_HTML.gif . Thus, we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq82_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq83_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ39_HTML.gif
(4.20)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ40_HTML.gif
(4.21)
By simple computations, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ41_HTML.gif
(4.22)
Hence, using the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ42_HTML.gif
(4.23)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ43_HTML.gif
(4.24)
Using the relations (3.3) and (3.12), for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq84_HTML.gif , we obtain (4.17):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ44_HTML.gif
(4.25)
In similar way, we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq85_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq86_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ45_HTML.gif
(4.26)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ46_HTML.gif
(4.27)
By simple computations, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ47_HTML.gif
(4.28)
Hence, using the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ48_HTML.gif
(4.29)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ49_HTML.gif
(4.30)
Hence, for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq87_HTML.gif , we obtain (4.18):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ50_HTML.gif
(4.31)

Case 3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq88_HTML.gif and the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq89_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq90_HTML.gif hold. Based on the asymptotic expressions of the fundamental solutions of (1.1) and the asymptotic formulas for eigenvalues of the boundary-value problem (1.1), (1.2) up to order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq91_HTML.gif , the following result is valid.

Theorem 4.3.

If the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq92_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq93_HTML.gif hold, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq95_HTML.gif are of the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ51_HTML.gif
(4.32)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ52_HTML.gif
(4.33)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq96_HTML.gif is sufficiently large integer.

Proof.

It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ53_HTML.gif
(4.34)
From the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq98_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq99_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq100_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq101_HTML.gif . Thus, we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq102_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq103_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ54_HTML.gif
(4.35)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ55_HTML.gif
(4.36)
By simple calculations, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ56_HTML.gif
(4.37)
Hence, using the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ57_HTML.gif
(4.38)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ58_HTML.gif
(4.39)
Using the relations (3.3) and (3.12), for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq104_HTML.gif , we obtain (4.32)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ59_HTML.gif
(4.40)
In similar way, we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq105_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq106_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ60_HTML.gif
(4.41)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ61_HTML.gif
(4.42)
By simple computations, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ62_HTML.gif
(4.43)
By the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ63_HTML.gif
(4.44)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ64_HTML.gif
(4.45)
Hence, for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq107_HTML.gif , we obtain the relation (4.33)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ65_HTML.gif
(4.46)

Case 4.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq108_HTML.gif and the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq111_HTML.gif hold. Based on the asymptotic expressions of the fundamental solutions of (1.1) and the asymptotic formulas for eigenvalues of the boundary-value problem (1.1), (1.2) up to order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq112_HTML.gif , the following result is valid.

Theorem 4.4.

If the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq114_HTML.gif hold, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq116_HTML.gif are of the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ66_HTML.gif
(4.47)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ67_HTML.gif
(4.48)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq117_HTML.gif is sufficiently large integer.

Proof.

It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ68_HTML.gif
(4.49)
From the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq119_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq120_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq121_HTML.gif . Thus, we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq122_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq123_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ69_HTML.gif
(4.50)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ70_HTML.gif
(4.51)
By simple computations, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ71_HTML.gif
(4.52)
By the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ72_HTML.gif
(4.53)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ73_HTML.gif
(4.54)
Using the relations (3.3) and (3.12), for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq124_HTML.gif , we obtain (4.47):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ74_HTML.gif
(4.55)
In similar way, we can seek eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq125_HTML.gif corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq126_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ75_HTML.gif
(4.56)
By simple computations, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ76_HTML.gif
(4.57)
By the formula (2.2), we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ77_HTML.gif
(4.58)
Therefore, for the normalized eigenfunction, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ78_HTML.gif
(4.59)
Hence, for sufficiently large integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_IEq127_HTML.gif , we obtain the relation (4.48)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F720235/MediaObjects/13661_2010_Article_950_Equ79_HTML.gif
(4.60)

Declarations

Acknowledgments

This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK). The author would like to thank the referee and the editor for their helpful comments and suggestions. The author also would like to thank prof. Kh. R. Mamedov for useful discussions.

Authors’ Affiliations

(1)
Mathematics Department, Science and Arts Faculty, Mersin University

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Copyright

© The Author(s) Hamza Menken. 2010

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