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Open Access Research Article

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

Hamza Menken

Author Affiliations

Mathematics Department, Science and Arts Faculty, Mersin University, 33343 Mersin, Turkey

Boundary Value Problems 2010, 2010:720235  doi:10.1155/2010/720235


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2010/1/720235


Received:7 July 2010
Accepted:9 November 2010
Published:28 November 2010

© 2010 The Author(s) Hamza Menken.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:

(11)

(12)

where is a complex-valued function. Without lose of generality, we can assume that .

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 .

2. The Expression of the Fundamental Solutions

It is well known that (see [2, page 92]) if the complex -plane is divided into eight sectors , defined by the inequalities

(21)

then in each of these sectors (1.1) has four linear independent solutions , which are regular with respect to in the sector for sufficiently large and which satisfy the relation

(22)

where the numbers are the fourth roots of unity, that is, and . In general, the term at the formula (2.2) depends upon the smoothness of the function . If has continuous derivatives, then one can assert the existence of a representation (2.2) with . Here, we assume that . The functions satisfy the following recursion relations:

(23)

Let us put, moreover, , , for . Thus, the functions are uniquely determined. Thus, we can find from (2.3) that

(24)

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 ) consist of the pairs of the sequences , satisfying the following asymptotic formula:

(31)

for sufficiently large integer , where is a constant.

Theorem 3.1.

Assume that . Then, the eigenvalues of the boundary-value problem (1.1), (1.2) form two infinite sequences , , where is a big positive integer and have the following asymptotic formulas:

(32)

Proof.

By derivation of (2.2) up to third order with respect to , the following relations are obtained:

(33)

where , and

(34)

Now let us substitute all these expressions into the characteristic determinant

(35)

where

(36)

By long computations, for sufficiently large , we obtain that

(37)

Multiplying the last equation by

(38)

it becomes

(39)

Hence, by , for sufficiently large , the following equations hold:

(310)

(311)

By Rouche's theorem, we have asymptotic estimates for the roots and , , , of (3.10) and (3.11), respectively, where is a big positive integer

(312)

(313)

From the relations (3.12), (3.13) and the relations , the asymptotic formulas (3.2) are valid for .

4. The Asymptotic Formulas for the Eigenfunctions

Now, we obtain asymptotic formulas for eigenfunctions under the distinct conditions on .

Case 1.

Assume that and the condition 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 , the following result is valid.

Theorem 4.1.

If the condition holds, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues and are of the form

(41)

(42)

where is sufficiently large integer.

Proof.

Let us calculate up to order . Since

(43)

we obtain that

(44)

Follows from the condition that for . Thus, we can seek eigenfunction corresponding in the form

(45)

Then,

(46)

By simple computations, we obtain

(47)

Hence, using the formula (2.2), we can write

(48)

Therefore, for the normalized eigenfunction, we get

(49)

Using the relations (3.3) and (3.12), for sufficiently large integer , we obtain (4.1)

(410)

Similarly, since for , we can seek eigenfunction corresponding in the form

(411)

Then,

(412)

By similar computations we obtain

(413)

Hence, using the formula (2.2), we can write

(414)

Therefore, for the normalized eigenfunction, we get

(415)

Hence, for sufficiently large integer , we obtain (4.2)

(416)

Case 2.

Assume that and the conditions and 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 , the following result is valid.

Theorem 4.2.

If the conditions and hold, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues and are of the form

(417)

(418)

where is sufficiently large integer.

Proof.

It is clear that

(419)

It follows from the conditions , that for . Thus, we can seek eigenfunction corresponding in the form

(420)

Then,

(421)

By simple computations, we have

(422)

Hence, using the formula (2.2), we can write

(423)

Therefore, for the normalized eigenfunction, we get

(424)

Using the relations (3.3) and (3.12), for sufficiently large integer , we obtain (4.17):

(425)

In similar way, we can seek eigenfunction corresponding in the form

(426)

Then,

(427)

By simple computations, we get

(428)

Hence, using the formula (2.2), we can write

(429)

Therefore, for the normalized eigenfunction, we get

(430)

Hence, for sufficiently large integer , we obtain (4.18):

(431)

Case 3.

Assume that and the conditions and 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 , the following result is valid.

Theorem 4.3.

If the conditions and hold, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues and are of the form

(432)

(433)

where is sufficiently large integer.

Proof.

It is clear that

(434)

From the conditions and , we have for . Thus, we can seek eigenfunction corresponding in the form

(435)

Then,

(436)

By simple calculations, we get

(437)

Hence, using the formula (2.2), we can write

(438)

Therefore, for the normalized eigenfunction, we get

(439)

Using the relations (3.3) and (3.12), for sufficiently large integer , we obtain (4.32)

(440)

In similar way, we can seek eigenfunction corresponding in the form

(441)

Then,

(442)

By simple computations, we get

(443)

By the formula (2.2), we can write

(444)

Therefore, for the normalized eigenfunction, we get

(445)

Hence, for sufficiently large integer , we obtain the relation (4.33)

(446)

Case 4.

Assume that and the conditions , and 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 , the following result is valid.

Theorem 4.4.

If the conditions and hold, then eigenfunctions of the boundary-value problem (1.1), (1.2) corresponding the eigenvalues and are of the form

(447)

(448)

where is sufficiently large integer.

Proof.

It is clear that

(449)

From the conditions and , we have for . Thus, we can seek eigenfunction corresponding in the form

(450)

Then

(451)

By simple computations, we get

(452)

By the formula (2.2), we can write

(453)

Therefore, for the normalized eigenfunction, we get

(454)

Using the relations (3.3) and (3.12), for sufficiently large integer , we obtain (4.47):

(455)

In similar way, we can seek eigenfunction corresponding in the form

(456)

By simple computations, we get

(457)

By the formula (2.2), we can write

(458)

Therefore, for the normalized eigenfunction, we get

(459)

Hence, for sufficiently large integer , we obtain the relation (4.48)

(460)

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.

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