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

In the case when the boundary conditions (2) are strongly regular, the results ofMikhailov [1] andKesellman [2] provide the Riesz basis property in L 2 (0, 1) 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):

We introduce the matrix of coefficients of the boundary conditions ( By () we denote the matrix composed of the th and th columns of matrix ,   = det().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: For example, boundary conditions with periodical or antiperiodical conditions form the (I)-type and can be determined in the following form: That is,  11 = − 12 ,  13 =  14 =  21 =  22 = 0, and  23 = − 24 .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  14 ̸ =  23 ,  34 = 0 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 () ≡ 0, 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  2 (0, 1), 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][9][10].

Statement of the Problem and Main Results
The spectral problem (8)-( 2) with boundary conditions of the (III)-type when () ≡ 0 is a non-self-adjoint problem in  2 (0, 1).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 (0) ̸ = (1).Moreover, using these asymptotic formulas, we prove that the root functions of these operators form a Riesz basis in the space  2 (0, 1) [13,14].
From [18] it follows that the E&AF system of problem ( 9)-( 11) is complete and minimal in  2 (0, 1).Moreover, the E&AF system at any () forms Riesz basis with brackets.Our aim is to show that the basis property in  2 (0, 1) 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   (−1, 1) 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  2 (, ).

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  ̸ = 0, 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): We represent function () as the biorthogonal expansion to the Fourier series by system {V 0 0 , V 0 1 }: Using ( 16), we find more convenient representation of determinant Δ 1 ().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: Δ 0 () = 0 implies that  (1)   =  0  = (2) 2 .Quantities  (2)   = [2 +  0 ( √ 2 + (1/ √ ))] 2 are roots of equation () = 0.
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 Δ 0 () is the characteristic determinant of the unperturbed Samarskii-Ionkin spectral problem,  0 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  =  0  , and function Δ 0 () has zeroes of the second order at these points.Hence, function Δ 1 () 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].
International Journal of Differential Equations

Partial Cases of the Characteristic Determinant
If coefficients  0 = 0 of ( 16) for all indexes , then  1  =  0  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  0 = 0 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 λ and positive integer m, there always exists function () such that λ will be eigenvalue of problem ( 9)-( 11) of m multiplicity.
From analysis of (19) it is also easy to see that Δ 1 ( 0  ) = 0 for all  > .That is, all eigenvalues  0  ,  > , 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  0  ,  > , is also preserved.Furthermore, from the orthogonal condition () ⊥  0 0 , () ⊥  0 1 at all  > , it follows that, in this case, Therefore, eigenfunctions  0 0 () and associated functions  0 1 () 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  2 (0, 1).
Proof.It is obvious that the set of the functions () ∈  2 (0, 1) represented as (16), coefficients of which asymptotically (i.e., beginning with some number) have the property  0 ̸ = 0,  1 = 0, is dense in  2 (0, 1).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  0 ̸ = 0,  1 = 0. Then from (18) it is not hard to see that  0  = (2) 2 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 We find an eigenfunction of problem ( 9)- (11).For large enough  =  0  = (2) 2 the first equation of the system from Section 3 becomes an identity, and the second equation is transformed into the following form: Since  0 ̸ = 0, then we write  2 by  1 .Therefore, eigenfunction of problem ( 9)-( 11) has the following form: Choose constant  1 from the biorthogonal condition It is easy to see that  1 = √ 2. Finally, we find the eigenfunction of problem ( 9)-( 11): By the direct calculation we find its norm in  2 (0, 1): Consequently, lim  → ∞ ‖ 1  ‖ ⋅ ‖ 1  ‖ = ∞.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  2 (0, 1).
Theorem 4 is proved.
Since adjoint operators at the same time have the Riesz basis property of root functions, therefore, we obtain the following.
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.