On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems

Imanbaev, N. S.

doi:https://doi.org/10.1155/2015/641481

International Journal of Differential Equations

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Research Article | Open Access

Volume 2015 | Article ID 641481 | https://doi.org/10.1155/2015/641481

On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems

N. S. Imanbaev^1,2

Academic Editor: Patricia J. Y. Wong

Received19 Jul 2015

Revised19 Nov 2015

Accepted03 Dec 2015

Published14 Dec 2015

Abstract

We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of Samarskii-Ionkin type boundary conditions.

1. Introduction

In space we consider an operator , generated by the following ordinary differential expression:and the boundary value conditions of the general form:In the case when the boundary conditions (2) are strongly regular, the results of Mikhailov [1] and Kesellman [2] provide the Riesz basis property in of the eigenfunction and associated functions (E&AF) system of the problem. In the case when the boundary conditions are regular but not strongly regular, the question on basis property of E&AF system is not yet completely resolved.

We introduce the matrix of coefficients of the boundary conditions (2):

By we denote the matrix composed of the th and th columns of matrix , . Let the boundary conditions (2) be regular but not strongly regular. According to [3, page 73], if the following conditions hold:then the boundary conditions (2) are equivalent regular, but not strongly boundary, conditions.

In [4] Makin suggested dividing all regular, but not strongly regular, boundary conditions into four types:(I), ;(II), ;(III), ;(IV), .

For example, boundary conditions with periodical or antiperiodical conditions form the (I)-type and can be determined in the following form:That is, , , and . These conditions will be equivalent to boundary value conditions, given by matrix , where the following options are possible:and the boundary conditions with “the lowest coefficients” form the (II)-type. The boundary value conditions defined as , form the (III)-type. These conditions are always equivalent to boundary conditions, given by matrix :This case will be the aim of our research in this paper.

Moreover, Makin [4] allocated the one type of nonstrongly regular boundary value conditions, when E&AF system of the spectral problem with boundary conditions of the general form (2) forms Riesz basis at any potentials .

When , the problem about basis property of E&AF system of the problem with general regular boundary conditions has been completely resolved in [5].

In [6, 7] questions on convergence of eigenfunction expansion of the Dirac operator in vector-matrix form and the Hill operator, forming Reitz basis in , with regular, but not strongly regular, boundary value conditions have been considered.

Questions on basisness of eigenfunctions of the differential operators with involution have been studied in [8–10].

2. Statement of the Problem and Main Results

The spectral problem (8)-(2) with boundary conditions of the (III)-type when is a non-self-adjoint problem in . For the case of non-self-adjoint initial operator the question about preservation of the basis properties with some (weak in a certain sense) perturbation was studied in [11]. Riesz basis property of eigenfunctions and associated functions of periodic and antiperiodic Sturm-Liouville problems was considered in [12]. We obtain asymptotic formulas for eigenvalues and eigenfunctions of periodic and antiperiodic Sturm-Liouville problems with boundary conditions, which are not strongly regular, when is a complex-valued absolutely continuous function, and . Moreover, using these asymptotic formulas, we prove that the root functions of these operators form a Riesz basis in the space [13, 14].

In [15, 16] questions on stability of basis properties of the periodic problem for (8) were investigated with integral perturbation of the boundary conditions (2), when , of the (I)-type, that is, at the conditions , . Moreover, in [17], the similar questions have been studied when . In this paper we consider the spectral problem close to research of [17] when , with integral perturbation of the boundary conditions (2) when , belonging to the (III)-type: If , then problem (9)–(11) is called Samarskii-Ionkin problem [11].

From [18] it follows that the E&AF system of problem (9)–(11) is complete and minimal in . Moreover, the E&AF system at any forms Riesz basis with brackets. Our aim is to show that the basis property in of the E&AF system of problem (9)–(11) is not stable at small changes of kernel of integral perturbation.

In [19] the construction method of the characteristic determinant of the spectral problem with integral perturbation of the boundary conditions has been suggested. The spectral properties of nonlocal problems have been considered in [20].

The basis properties in of root functions of the nonlocal problem for the equations with involution have been studied in [21]. Instability of basis properties of root functions of the Schrodinger operator with nonlocal perturbation of the boundary condition has been investigated in [22]. In [23, 24] they extended some spectral properties of regular Sturm-Liouville problems to the special type discontinuous boundary value problem, consisting of the Sturm-Liouville equation together with eigenparameter that depended on boundary conditions and two supplementary transmission conditions; we construct the resolvent operator and prove theorems on expansions in terms of eigenfunctions in modified Hilbert space

3. Characteristic Determinant of the Problem

In this section we use the method of our paper [19] to construct the characteristic determinant of the problem with integral perturbation of the boundary condition.

One aspect of this problem is the fact that an adjoint problem to (9)–(11) is the spectral problem for the loaded differential equation [16]:Firstly, we construct the characteristic determinant of the spectral problem. Representing the general solution of (9) by the following formula when ,and satisfying it by the boundary conditions (10) and (11), we obtain the linear system concerning the coefficients :Its determinant is a characteristic determinant of problem (9)–(11):When , we get the characteristic determinant of the unperturbed Samarskii-Ionkin problem. Denote it by The number is simple eigenvalue of the unperturbed Samarskii-Ionkin problem, and is the corresponding eigenfunction. Other eigenvalues of the unperturbed problem (9)–(11) are double: , and is the eigenfunction corresponding to them; is the associated function. Due to the biorthogonal property , we have the eigenfunction and the associated function of the problem adjoint to the Samarskii-Ionkin problem.

We represent function as the biorthogonal expansion to the Fourier series by system :Using (16), we find more convenient representation of determinant . To do it, first, we evaluate integrals in (15). Simple calculations show that

Using the obtained results, determinant (15) is reduced to the following form by the standard conversions: implies that . Quantities are roots of equation

These roots are eigenvalues of the perturbed spectral problems (9)–(11).

Therefore, we prove the following.

Theorem 1. Characteristic determinant of the spectral problem with perturbed boundary value conditions (9)–(11) can be represented as (18), where is the characteristic determinant of the unperturbed Samarskii-Ionkin spectral problem, are Fourier coefficients of the biorthogonal expansion (16) of the function by the E&AF system of adjoint unperturbed Samarskii-Ionkin spectral problem.

Function from (18) has a pole of the first order at points , and function has zeroes of the second order at these points. Hence, function represented by formula (18) is an entire analytical function of variable .

The characteristic determinant, which is an entire analytical function, related to the problem on eigenvalues of differential operator of the third order with nonlocal boundary conditions has been studied in [25].

4. Partial Cases of the Characteristic Determinant

If coefficients of (16) for all indexes , then is a double eigenvalue of the perturbed problem (9)–(11).

More simple characteristic determinant (18) is in the case when is represented as (16) with the finite first sum. That is, when there exists a number such that for all In this case formula (18) takes the following form: From this partial case (18) it is easy to establish the following.

Corollary 2. For any numbers given in advance, that is, complex and positive integer , there always exists function such that will be eigenvalue of problem (9)–(11) of multiplicity.

From analysis of (19) it is also easy to see that for all . That is, all eigenvalues , , of the Samarskii-Ionkin problem are eigenvalues of the perturbed spectral problem (9)–(11). Moreover, it is not hard to see that multiplicity of the eigenvalues , , is also preserved.

Furthermore, from the orthogonal condition , at all , it follows that, in this case, Therefore, eigenfunctions and associated functions of the unperturbed Samarskii-Ionkin problem for all satisfy the boundary conditions (10), (11), and perturbed Samarskii-Ionkin spectral problem consequently. Hence, the functions are eigenfunctions and associated functions of the perturbed problem (9)–(11). Accordingly, in this case, E&AF system of the perturbed problem (9)–(11) and the E&AF system of the unperturbed Samarskii-Ionkin problem (forming Riesz basis) differ from each other only in a finite number of the first members. Hence, the E&AF system of the perturbed system (9)–(11) also forms Riesz basis in .

A set of the functions , represented as the finite series (16), is dense in . Therefore, we prove the following.

Theorem 3. Let , ; that is, the boundary conditions (10), (11) are equivalent to the type-(III) with the integral perturbation. Then the set of the functions such that the E&AF system of the perturbed Samarskii-Ionkin problem (9)–(11) forms Riesz basis in is dense in .

5. Instability of Basis Property

Now we prove that the basis property of E&AF system of the perturbed Samarskii-Ionkin problem (9)–(11) is instable at arbitrarily small integral perturbation of the boundary condition (11).

Theorem 4. If , , that is, the boundary conditions (10), (11) belong to the type-(III), then the set of the functions such that E&AF system of the perturbed Samarskii-Ionkin problem (9)–(11) does not form even a simple basis in is also dense in .

Proof. It is obvious that the set of the functions represented as (16), coefficients of which asymptotically (i.e., beginning with some number) have the property , , is dense in . Therefore, to prove the theorem, it is enough to show that for these functions the E&AF system of the problem does not form a simple basis.
Let be a large enough number such that , . Then from (18) it is not hard to see that is a simple eigenvalue of problem (9)–(11). By the direct calculation it is easy to get that the corresponding eigenfunction to this value of the adjoint problem (12) is and .
We find an eigenfunction of problem (9)–(11). For large enough the first equation of the system from Section 3 becomes an identity, and the second equation is transformed into the following form:Since , then we write by . Therefore, eigenfunction of problem (9)–(11) has the following form:Choose constant from the biorthogonal condition . It is easy to see that . Finally, we find the eigenfunction of problem (9)–(11):By the direct calculation we find its norm in :From the Young theorem [26, Theorem 276, page 240] it follows that Therefore, Consequently, . Thus, necessary condition of basis property does not hold (see [11] and references in it) and, therefore, it does not form even a simple basis in .
Theorem 4 is proved.

Since adjoint operators at the same time have the Riesz basis property of root functions, therefore, we obtain the following.

Corollary 5. Let ; that is, the boundary conditions (10), (11) belong to the type-(III). Then the set of the functions such that the system of eigenfunctions of the problem (12) for the loaded differential equation forms Riesz basis in is everywhere dense in . The set is also everywhere dense in .

The results of the paper, in contrast to [18], show instability of basis property of root functions of the problem with an integral perturbation of boundary conditions of the type-(III), which are regular, but not strongly regular.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research is financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant no. 0825/GF4).

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Copyright © 2015 N. S. Imanbaev. 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.

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