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

- Hamza Menken
^{1}Email author

**2010**:720235

**DOI: **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.

## 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

## References

- Dunford N, Schwartz JT:
*Linear Operators, Part 3 Spectral Operators, Wiley Classics Library*. John Wiley & Sons, New York, NY, USA; 1970.Google Scholar - Levitan BM, Sargsjan IS:
*Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators*. American Mathematical Society, Providence, RI, USA; 1975:xi+525.MATHGoogle Scholar - Marchenko VA:
*Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications*.*Volume 22*. Birkhäuser, Basel, Switzerland; 1986:xii+367.View ArticleGoogle Scholar - Naimark MA:
*Linear Differential Operators. Part 1*. Frederick Ungar, New York, NY, USA; 1967:xiii+144.Google Scholar - Rofe-Beketov FS, Kholkin AM:
*Spectral Analysis of Differential Operators, World Scientific Monograph Series in Mathematics*.*Volume 7*. World Scientific Publishing, Singapore; 2005:xxiv+438.MATHGoogle Scholar - Kerimov NB, Mamedov KR:
**On the Riesz basis property of root functions of some regular boundary value problems.***Mathematical Notes*1998,**64**(4):483-487. 10.1007/BF02314629MathSciNetView ArticleMATHGoogle Scholar - Chernyatin VA:
**Higher-order spectral asymptotics for the Sturm-Liouville operator.***Ordinary Differential Equations*2002,**38**(2):217-227.MathSciNetView ArticleMATHGoogle Scholar - Mamedov KR, Menken H:
**On the basisness in****of the root functions in not strongly regular boundary value problems.***European Journal of Pure and Applied Mathematics*2008,**1**(2):51-60.MathSciNetMATHGoogle Scholar - Mamedov KR, Menken H:
**Asymptotic formulas for eigenvalues and eigenfunctions of a nonselfadjoint Sturm-Liouville operator.**In*Further Progress in Analysis*. Edited by: Begehr HGW, Çelebi AO, Gilbert RP. World Scientific Publishing; 2009:798-805.View ArticleGoogle Scholar - Kurbanov VM:
**A theorem on equivalent bases for a differential operator.***Doklady Akademii Nauk*2006,**406**(1):17-20.MathSciNetMATHGoogle Scholar - Menken H, Mamedov KR:
**Basis property in****of the root functions corresponding to a boundary-value problem.***Journal of Applied Functional Analysis*2010,**5**(4):351-356.MathSciNetMATHGoogle Scholar

## Copyright

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