A new eigenvalue problem for the difference operator with nonlocal conditions

Mifodijus Sapagovas, Regimantas Čiupaila, Kristina Jakubėlienė, Stasys Rutkauskas Fakulty of Mathematics and Informatics, Vilnius University, Akademijos str. 4, LT-08412 Vilnius, Lithuania mifodijus.sapagovas@mii.vu.lt Vilnius Gediminas Technical University, Saulėtekio ave. 11, LT-10223 Vilnius Lithuania regimantas.ciupaila@vgtu.lt Department of Applied Mathematics, Kaunas University of Technology, Studentų str. 50, LT-51368 Kaunas, Lithuania kristina.jakubeliene@ktu.lt Institute of Data Science and Digital Technologies, Vilnius University, Akademijos str. 4, LT-08412 Vilnius, Lithuania


Introduction and problem statement
During last several decades of the development of differential equations theory and numerical analysis, there is an increased interest in problems with various types of nonlocal conditions. A separate class of these problems is eigenvalue problems of differential and difference operators. Eigenvalue problems of differential operator with nonlocal conditions can be interpreted as a separate case of the non-self-adjoint operators theory [12].
Eigenvalue problems of difference operators with nonlocal conditions usually arise when solving boundary problems by the finite difference method. The spectrum properties of difference operators with various nonlocal boundary conditions were explored for investigation of the stability of difference schemes [1,2,4,7,8,10,11]. Another sphere of such a spectrum analysis application is convergence of iterative methods for the systems of difference equations [17,19,22], in particular, for nonlinear elliptic equations with integral boundary conditions [22,23].
Many articles on the investigation of the partial differential equations with various types of nonlocal conditions were published presenting new mathematical models in heat conduction, thermoelasticity, underground water flow, biochemistry and so on. References to the original papers can be found in [2,19,21]. Solving these problems by the finite difference method, we meet unavoidably the problem of the structure of the spectrum of difference operator. Therefore, the eigenvalue problems could be interpreted as one of the methods of modeling.
In [5,6], the eigenvalue problem was investigated in connection with the existence, uniqueness and multiplicity of the solution of differential problems with nonlocal conditions.
The spectrum of differential and difference operators with nonlocal conditions is much more diverse and complicated as compared to the spectrum in the case of the classical boundary conditions (Dirichlet or Neumann). Let us take such an eigenvalue problem with the Bitsadze-Samarskii nonlocal condition [20]: d 2 u dx 2 + λu = 0, x ∈ (0, 1), u(0) = 0, u(1) = γu(ξ), ξ ∈ (0, 1), where γ, ξ are given real numbers. It has been proved that, depending on the values of these parameters, in the spectrum of both differential and difference operators, there can be zero, positive, negative or complex values. Besides, though the matrix of a difference problem is non-symmetrical (except the case γ = 0), we can determine intervals of γ and ξ in which all the eigenvalues are real and positive.
Next, let us take the corresponding difference problem where h = 1/N , ξ = Sh, which is equivalent to (N − 1)-order matrix A eigenvalue problem Au = λu, u = {u i }, i = i, . . . , N − 1. It has been proved that, under certain values of γ and ξ, matrix A may have a parasitic eigenvalue without any correspondence as h → 0. For example, if there exists a matrix eigenvalue λ = 4/h 2 with the corresponding eigenvector where c = 0 is any real number. As h → 0, the limit of the eigenvector does not exist. In paper [18], one more singularity of the spectrum of a difference operator with nonlocal conditions is indicated. Let us take a differential eigenvalue problem with integral conditions d 2 u dx 2 + λu = 0, x ∈ (0, 1), The following difference problem corresponds to it with the approximation error O(h 2 ): This difference eigenvalue problem for all the values γ 1 , γ 2 and h, except one case as h = 2/(γ 1 + γ 2 ), is equivalent to the eigenvalue problem Au = λu, where A is the (N − 1)-order matrix. If γ 1 + γ 2 > 2 and h < 2/(γ 1 + γ 2 ), then all eigenvalues of difference operator are positive except one that is negative. This negative eigenvalue tends to infinity (−∞) as h → 2/(γ 1 +γ 2 ). This fact is well illustrated by numerical experiment when γ 1 + γ 2 is quite large positive number. If h > 2/(γ 1 + γ 2 ), then all eigenvalues are positive, and one of them tends to infinity (+∞) as h → 2/(γ 1 + γ 2 ). In the case h = 2/(γ 1 + γ 2 ), difference eigenvalue problem cannot be written in the matrix form Au = λu.
In this paper, we consider a differential eigenvalue problem and a difference eigenvalue problem, corresponding to it. Such a difference eigenvalue problem was investigated in paper [3] in which some necessary conditions for the parameters γ 1 , γ 2 and ξ were obtained in order that zero, positive, negative or complex eigenvalues might exist. In this paper, we have investigated in detail the spectrum of differential and difference operators and drew a new qualitative conclusions. Particularly, we have proved that, depending on the parameters γ 1 , γ 2 , ξ and h, the spectrum structure of a difference operator can be essentially different from the spectrum structure both of differential operator and that of matrix. https://www.mii.vu.lt/NA 2 Eigenvalue problem of a differential operator We investigate the spectrum structure of a differential operator, defined by formulas (1)-(3). First, we analyze when there exist real eigenvalues by separate three cases: λ = 0, λ > 0 and λ < 0. Theorem 1. The number λ = 0 is an eigenvalue of differential problem (1)- (3) if and only if the following condition is true: Proof. As λ = 0, the general solution of equation (1) is where c 1 and c 2 are arbitrary constants. By substituting this expression into nonlocal conditions (2) and (3), we obtain a system of equations with two unknowns c 1 and c 2 This system has a nontrivial solution if and only if After elementary rearrangement, it follows (4) from this equality.
where α k are roots of the equation Proof. As λ > 0, the general solution of (1) is After substituting this solution into nonlocal conditions (2) and (3), we have a system For nontrivial solution (c 1 , c 2 ) of this system, the necessary and sufficient condition is After elementary rearrangements, hence it follows (7).
Proof. Let us consider three qualitatively different cases of the parameters γ 1 and γ 2 .
The phenomenon of continuous spectrum also takes a place in the theory of boundary value problems for degenerate elliptic equations [14,15].
where β 0 > 0 is the root of the equation Proof. As λ < 0, the general solution of equation (1) is After substituting this expression into nonlocal condition (2) and (3), we get The necessary and sufficient condition for the existence of a nontrivial solution (c 1 , c 2 ) of this system is as follows: As earlier, it follows (10) from this equality.
Next, we explore when equation (10) has at least one root.

Eigenvalue problem of a difference operator
Let us write a difference problem of eigenvalues that approximates differential problem (1)-(3) with the approximation error O(h 2 ): https://www.mii.vu.lt/NA We denote here h = 1/N , ξ = Sh, 1 − ξ = (N − S)h; N and S are integer numbers. In other words, the grid is uniform, and ξ is a mesh point (1 − ξ is also a mesh point). Note that 1 S N/2 − 1 and h ξ 1/2 − h. We call the number λ an eigenvalue of the difference problem, if with this number there exists a nontrivial solution (eigenvector) of problem (14)- (15).
Let us analyze the spectrum (the set of all eigenvalues) of difference problem (14)- (15). The proofs of lemmas and theorems presented below, according to methodology are analogous to that used in Section 2. Therefore, in our proofs, we emphasize only that what is different.
Proof. First, let us pay attention that condition (16) is coincident with condition (4) of Theorem 1.
As λ = 0, the general solution of difference equation (14) is where c 1 and c 2 are arbitrary constants. By substituting this expression of u i into nonlocal conditions (15), we obtain a system just like system (5) in the proof of Theorem 1.
are defined by the formula where α k are roots of the equation in the interval (0, π/h).
Proof. First of all, note that equation (19) is coincident with equation (7) in Lemma 1. However, expressions and numbers of eigenvalues are different than in Lemma 1.
Since the inequality follows from condition (17), we can introduce into equation (14) a new unknown α instead of λ: Hence, formula (18) follows. Now, equation (14) becomes as follows: and its general solution is By substituting this expression of u i into nonlocal conditions (15), we obtain a system coincident with system (8) in the demonstration of Lemma 1. So, the rest of this proof now is coincident with that one of Lemma 1.
Proof. According to assumption of the theorem, all possible values of γ 1 and γ 2 can be separated into three qualitatively different cases.
We shall indicate one interesting fact.
https://www.mii.vu.lt/NA Remark 2. If ξ = h, i.e., S = 1 and γ 1 = γ 2 = ±1, then it follows from formula (22) that difference problem (14)- (15) has no positive eigenvalue λ that satisfies the inequality 0 < λ < 4/h 2 . Thereby we admit that from the equality ξ = h it not follows (1)-(3) is illpossessed problem. The matter is that h → 0, but ξ = const. Hence, the spectrum of the difference problem (14)-(15) is empty set with only one concrete value h, i.e., h = ξ (see Section 6, case 2)). With increasing or decreasing value h this phenomenon disappears. The conclusion follows directly from equation (19). Let us find an eigenvector as γ 1 = γ 2 = 1, i.e., in the case of continuous spectrum. Consider any fixed number λ 0 ∈ (0, 4/h 2 ) as an eigenvalue. In accordance with (20), we calculate α 0 ∈ (0, π/h) from the equality The eigenvector is of form (21). Choosing c 1 = 1, from system (8) we calculate Thus, the eigenvector corresponding to the eigenvalue λ 0 is as follows: where α 0 satisfies equality (24). Note that, differently than in the case of the differential problem (Theorem 2), we have found not all the positive eigenvalues of the difference problem, but only the eigenvalues from the interval (0, 4/h 2 ). Under certain additional conditions, there may exist one more eigenvalue λ 4/h 2 of the difference problem (see below Theorems 7 and 8). As far as the authors are acquainted, for the first time, the existence of such an eigenvalue in the case of nonlocal conditions was noticed most likely in paper [20].
where β > 0 is the root of the equation Proof. If λ < 0, then 1 − λh 2 /2 > 1, therefore we can introduce a new unknown β by the relation Hence, formula (25) follows. Equation (14) becomes as follows: The general solution of this equation is By substituting this expression into nonlocal conditions (15), we get system (11). A further proof of the lemma is coincident with that of Lemma 2.
Like as Theorem 3, the next theorem is proved.
In Fig. 1, in the presence of the fixed value ξ = 0.4, in the grey areas of the coordinate plane (γ 1 , γ 2 ), there exists a negative eigenvalue of difference problem (14)- (15).
According to Theorems 3 and 6, there exists a negative eigenvalue of both differential and difference operators under the same conditions (12).
is satisfied.

The general solution of this equation is
After substituting the expression of this solution into conditions (15), we obtain https://www.mii.vu.lt/NA Hence, the necessary and sufficient condition for the existence of nontrivial solution After elementary rearrangement, hence we derive (27).
Assume that N is an even number. Then condition (27) is coincident with condition (16). Thus, we obtained the following conclusion.
If this condition is satisfied, then where β is a unique root of the equation in the interval (0, ∞).

The general solution of this equation is
After substituting this expression into (15), we obtain a system, analogous to system (11): By equating a determinant of this system to zero, after elementary rearrangement, we obtain (31). The further proof of the theorem is analogous to that of Theorem 3.
The inferences analogous to Corollaries 3 and 5 are true.

Corollary 7.
As N is an even number, the existence condition of the eigenvalues λ < 0 and λ > 4/h 2 is the same.
Remark 3. The result of Theorems 7 and 8 on the existence conditions for the eigenvalues λ = 4/h 2 and λ > 4/h 2 is proper on for a difference but not a differential problem. The matter is that eigenvectors (28) and (32), corresponding to these eigenvalues, have no limit as h → 0.

Complex eigenvalues
Differential operator of (1)- (3) is not self-adjoint. Therefore, there may exist complex eigenvalues. Such a statement is also right for difference operator of (14)- (15). We investigate when there exist complex eigenvalues of a difference operator, since for a differential operator, both the investigation methods and their results are analogous.
In this section, we denote an imaginary unit by the letter i, i.e., i = √ −1. Therefore, in equation (14), we shall use the index j instead of index i.

Lemma 5.
If there exist complex eigenvalues λ k of difference problem (14)-(15), they are defined by the following formula: where q k = α k ± iβ are complex roots of the equation Proof. If λ is a complex number in equation (14), we can introduce a new complex quantity q by the formula where q = α ± iβ. Hence, it follows that Note that, in the case where λ is a complex number, there must be α = 0 and β = 0. The condition β = 0 is coincident with the condition that q is a real number, so λ is positive.
https://www.mii.vu.lt/NA After replacement (35), equation (14) becomes as follows: Its general solution is where c 1 and c 2 are arbitrary complex constants. By substituting this expression in nonlocal conditions (15) we obtain Hence just like in the proof of Lemma 3, we get that the nontrivial solution (c 1 , c 2 ) exists if and only if condition (34) is fulfilled. In case this equation has complex roots q k = α k ± iβ k , α k = 0, β k = 0, then the corresponding eigenvalue λ k is defined by (33).
If ξ is a real numbers, then all the roots q k are real.
In this case, we can rewrite equation (34) as follows: the roots q k of which are only real. An analogous proposition is right in the case We analyze one more complicated case where all eigenvalues of the difference operator are real.
5 The difference eigenvalue problem as a generalized matrix eigenvalue problem In paper [3], it is stated difference eigenvalue problem (14)- (15) can be written as a matrix eigenvalue problem where A is (N −1)-order matrix, u = (u 1 , u 2 , . . . , u N −1 ) T . The expression of matrix A is written as well. Though the expression of matrix A is written correctly, problem (14)- (15) is not equivalent to problem (40). We shall repeat these reasoning, presented in paper [3], and will correct one inaccuracy in it. At the same time, we note that the authors of paper, after writing problem (14)- (15) in incorrect form (40), nowhere use such a form. We will write a slightly different matrix from of problem (14)- (15), and we will comment it in Section 6.
Let us write equation (14) more in detail: We substitute the expression u 0 = γ 1 u N from (15) into the first equation of system (41). Analogously, we substitute the expression u S = γ 2 u N −S from (15) into three equations of system (41) as i = S − 1, S, S + 1. After substitution, we obtain a new form of problem (14)- (15) System (42), together with (15), is equivalent to system (14)- (15). Now system (42) can be written in the matrix form The expression of the matrix A is written correct in [3]. Since u (1) = u (2) , the problem Au (1) = λu (2) is not an eigenvalue problem. Note that, neither by the method proposed in [3], not by any other way, difference eigenvalue problem (14)-(15) cannot be written in form (40). However, this problem (14)- (15) can be written as generalized matrix eigenvalue problem [9,13] Au = λBu, where A and B are the N -order matrices. A specific feature of such a problem is that B is a singular matrix (B −1 does not exists). Let us write system (14)- (15) in the following way: In this system, we take two steps of equivalent rearrangement. The first step: write the expression u 0 from (44) into equation (45) in which i = 1. So we obtain − 2u 1 + u 2 + γ 1 u N + λh 2 u 1 = 0.
The second step: subtract equation (46) from equation (45) in which i = S + 1. Instead of equation (46), we obtain the new equation In this way, we get a new system −2u 1 + u 2 + γ 1 u N + λh 2 u 1 = 0, By adding equation (44) this system is equivalent to initial system (44)-(46). Now system (47) can be written as generalised matrix eigenvalue problem (43), where B is https://www.mii.vu.lt/NA singular matrix [16]. Next, we write expressions of N -order matrices A and B: Since the order of matrix A and B is N , det(A − λB) is not higher than the N -order polynomial. It would be not right assert that the characteristic polynomial is of N -order because that is possible only in the case when exists B −1 . Equation (48) yields the following assertion.

Illustrative example
Let us take a concrete example: ξ = 0.25, N = 4 and γ 1 , γ 2 are varying parameters. By means of this rather elementary example we illustrate the substance and variety of the spectrum of difference operator with nonlocal conditions. So we get the following eigenvalue problem: We transform this eigenvalue problem into a matrix form Au = λBu in accordance with the methodology described in Section 5. To this end, we substitute u 0 from (49) into (50), and we change equation (53) by a difference of (51) and (53). Thus, we derive −2u 1 + u 2 + γ 1 u 4 + λh 2 u 1 = 0, Next, we calculate the fourth-order determinant: In the general case, we obtain the second-order characteristic polynomials. To be precise, depending on the values of γ 1 and γ 2 , we obtain not higher than the second-order polynomial (we remind, N = 4). Besides, note that, in the case γ 1 = γ 2 , we can write equation (54) as follows: Let us take several concrete values of γ 1 and γ 2 . Case 1. γ 1 = γ 2 = ±1. Characteristic equation (54) becomes an identity 0 = 0 for all the values λ. Thus, the spectrum of difference operator (49)-(53) is continuous (see Corollaries 3,4,6).
Case 2. γ 1 = γ 2 = ±1. Characteristic equation (54) becomes as follows: what it is not true. So the spectrum of difference equation is an empty set (see Remark 2).
https://www.mii.vu.lt/NA value under nonlocal conditions. Besides, the dependence is more clearly defined not by the values of separate parameters γ 1 , γ 2 and ξ, but using generalized parameter values as well as interdependence of these generalized values. The conditions (equalities and inequalities) that include these generalized parameters in the coordinates plane (γ 1 , γ 2 ) are related with a hyperbola. By comparing the published results of the considered subject, we have to note that, perhaps for the first time, it has been proved that the spectrum of difference operators with nonlocal conditions can be continuous or coincident with an empty set. Namely, if the parameters γ 1 , γ 2 satisfy the condition γ 1 = γ 2 = 1 or γ 1 = γ 2 = −1, any real or complex number is an eigenvalue of both the difference and differential operator. Meanwhile, the spectrum of the difference operator as an empty set can be in case ξ = h. In addition, the number of eigenvalues of difference operator (14)-(15) depends on the parameters γ 1 , γ 2 and ξ rather in a complicated way.
The eigenvalue problem of a difference operator with nonlocal conditions investigated in our paper differs by many properties from the eigenvalue problems of a differential operator and that of matrix.
The fact that the spectrum of a difference operator with concrete values of parameters γ 1 , γ 2 has some unusual properties is conditioned by form of nonlocal conditions. Without an exhaustive examination of this issue, we refer to one typical property of nonlocal conditions analyzed in this paper.
Boundary interval points x = 0 and x = 1 are included only in one nonlocal condition. Another nonlocal condition is defined only at interior points of the interval under consideration. In essence, that is the main reason why the eigenvalue problem of difference operator cannot be expressed in the matrix form Au = λu with the (N − 1)-order matrix. Note that this property is typical not only of nonlocal conditions (15). Nonlocal conditions of different type can have such property, for example, integral conditions 1 0 u(x) dx = 0, ξ2 ξ1 u(x) dx = 0, 0 < ξ 1 < ξ 2 < 1.
If we exchange nonlocal condition (3) by a more general one u(ξ) = γ 2 u(η), 1 2 < η < 1, many propositions, proved in this paper, will be correct. Taking apart, conclusions about a continuous spectrum will be true also. Without doubt, in each concrete case of new nonlocal condition, more exhaustive researches are necessary. The spectrum structure of difference operators is rather important in the investigation of stability and convergence of difference schemes as well as iterative methods for systems of difference equations. These issues are not the research object of this article as well as the spectrum structure of the corresponding two-dimensional differential operators. We left these problems for future research.
The main aim of our paper was to pay attention to some unusual properties of the spectrum structure of difference operators with nonlocal conditions. The research we performed allows us to make the following conclusion. The spectrum of the difference operator with nonlocal conditions may essentially differ from the spectrum of the differential operators or the one of the matrices. Therefore, the eigenvalue problem of the difference operator with nonlocal conditions is worth to be self-contained object of the investigation.