- Research Article
- Open Access

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

- Hamza Menken
^{1}Email author

**2010**:720235

https://doi.org/10.1155/2010/720235

© The Author(s) Hamza Menken. 2010

**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.

## Keywords

- Periodic Boundary Condition
- Simple Computation
- Fundamental Solution
- Asymptotic Formula
- Recursion Relation

## 1. Introduction

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 [1–5].

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 [6–9]. 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

## 3. The Asymptotic Formulas of Eigenvalues

for sufficiently large integer , where is a constant.

Theorem 3.1.

Proof.

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.

where is sufficiently large integer.

Proof.

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.

where is sufficiently large integer.

Proof.

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.

where is sufficiently large integer.

Proof.

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.

where is sufficiently large integer.

Proof.

## 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

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## Copyright

This article is published under license to BioMed Central Ltd. 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.