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

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.

In the present work, we consider a nonself-adjoint fourth-order operator which is generated by the periodic boundary conditions:

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 .

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

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

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:

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

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:

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:

Proof.

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

where , and

Now let us substitute all these expressions into the characteristic determinant

where

By long computations, for sufficiently large , we obtain that

Multiplying the last equation by

it becomes

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

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

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

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

where is sufficiently large integer.

Proof.

Let us calculate up to order . Since

we obtain that

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

Then,

By simple computations, we obtain

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

Therefore, for the normalized eigenfunction, we get

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

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

Then,

By similar computations we obtain

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

Therefore, for the normalized eigenfunction, we get

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

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

where is sufficiently large integer.

Proof.

It is clear that

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

Then,

By simple computations, we have

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

Therefore, for the normalized eigenfunction, we get

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

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

Then,

By simple computations, we get

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

Therefore, for the normalized eigenfunction, we get

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

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

where is sufficiently large integer.

Proof.

It is clear that

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

Then,

By simple calculations, we get

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

Therefore, for the normalized eigenfunction, we get

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

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

Then,

By simple computations, we get

By the formula (2.2), we can write

Therefore, for the normalized eigenfunction, we get

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

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

where is sufficiently large integer.

Proof.

It is clear that

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

Then

By simple computations, we get

By the formula (2.2), we can write

Therefore, for the normalized eigenfunction, we get

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

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

By simple computations, we get

By the formula (2.2), we can write

Therefore, for the normalized eigenfunction, we get

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

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.

1. Dunford, N, Schwartz, JT: Linear Operators, Part 3 Spectral Operators, Wiley Classics Library, John Wiley & Sons, New York, NY, USA (1970)

2. Levitan, BM, Sargsjan, IS: Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators,p. xi+525. American Mathematical Society, Providence, RI, USA (1975)

3. Marchenko, VA: Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications,p. xii+367. Birkhäuser, Basel, Switzerland (1986)

4. Naimark, MA: Linear Differential Operators. Part 1,p. xiii+144. Frederick Ungar, New York, NY, USA (1967)

5. Rofe-Beketov, FS, Kholkin, AM: Spectral Analysis of Differential Operators, World Scientific Monograph Series in Mathematics,p. xxiv+438. World Scientific Publishing, Singapore (2005)

6. Kerimov, NB, Mamedov, KR: On the Riesz basis property of root functions of some regular boundary value problems. Mathematical Notes. 64(4), 483–487 (1998). Publisher Full Text

7. Chernyatin, VA: Higher-order spectral asymptotics for the Sturm-Liouville operator. Ordinary Differential Equations. 38(2), 217–227 (2002)

8. 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. 1(2), 51–60 (2008)

9. Mamedov, KR, Menken, H: Asymptotic formulas for eigenvalues and eigenfunctions of a nonselfadjoint Sturm-Liouville operator. In: Begehr HGW, Çelebi AO, Gilbert RP (eds.) Further Progress in Analysis, pp. 798–805. World Scientific Publishing (2009)

10. Kurbanov, VM: A theorem on equivalent bases for a differential operator. Doklady Akademii Nauk. 406(1), 17–20 (2006)

11. Menken, H, Mamedov, KR: Basis property in of the root functions corresponding to a boundary-value problem. Journal of Applied Functional Analysis. 5(4), 351–356 (2010)