Abstract

This paper is concerned with the eigenvalues of perturbed higher-order discrete vector boundary value problems. A suitable admissible function space is first introduced, a new variational formula of eigenvalues is then established under certain nonsingularity conditions, and error estimates of eigenvalues of problems with small perturbation are finally given by using the variational formula. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the nonsingularity conditions. In addition, two special perturbed cases are discussed.

1. Introduction

Consider the following -order vector difference equation: with the boundary condition whereis the forward difference operator; that is, ; is a -dimensional column vector-valued function on interval of integer, and ; () and () are Hermitian matrices, (); and are matrices satisfying the following self-adjoint condition [1, Lemma 2.1]: , are -dimensional vectors; denotes the transpose of ; denotes the complex conjugate transpose of ; and is the spectral parameter.

Higher-order discrete linear problems also have been investigated by some scholars besides second-order discrete Sturm-Liouville problems and discrete linear Hamiltonian systems (cf. [2–14] and their references). Zhou [15] and Grzegorczyk and Werbowski [7] studied a higher-order linear difference equation in which the leading coefficient is equal to 1 and established some criteria for the oscillation of solutions. Shi and Chen [1] investigated higher-order discrete linear boundary value problems (1)-(2) and obtained some spectral results, including Rayleigh’s principle, the minimax theorem, the dual orthogonality, and the number of eigenvalues. These results establish the theoretical foundation for our further research. Ren and Shi [16] discussed the defect index of singular symmetric linear difference equations of order with real coefficients and one singular endpoint and showed that the defect index satisfies the inequalities and that all values of in this range are realized. However, because of the characteristics of higher-order difference equations, compared with the research of second-order difference equations and discrete Hamiltonian systems, it is more difficult to study higher-order difference equations. Thus, there are few references in higher-order difference equations. For more information about higher-order discrete linear problems, the reader is referred to [6, 12, 14].

Recently, we have studied second-order discrete Sturm-Liouville problems and obtained error estimates of eigenvalues of perturbed problem under some hypotheses in [17]. Motivated by the ideas and methods used in [17], we extend the results to -order discrete vector boundary value problems (1)-(2) by means of the results obtained in [1]. Although the method is similar, the problems we investigate in this paper are more complex, since they are not only of higher order but also of higher dimension.

If is nonsingular on , then the -order vector difference equation (1) can be converted into the discrete linear Hamiltonian system studied in [18]: where

However, hypothesis (3) does not require the leading coefficient to be always nonsingular in . So, the coefficient and the weight functions of the corresponding discrete linear Hamiltonian system do not satisfy assumption (2.1) and the positive definiteness of the weight function in [18]. Hence the Hamiltonian system considered in [18] does not include the equation we discuss in this paper.

In the present paper, we study error estimate of eigenvalues of (1)-(2) under small perturbation. By employing a variational property—the minimax theorem established in [1]—an error estimate of eigenvalues of all perturbed problems sufficiently close to problem (1)-(2) is given under certain nonsingularity conditions. The continuous dependence of eigenvalues on problems is consequently obtained from the error estimate under the nonsingularity conditions. The continuous dependence of eigenvalues on problems may not hold in general. It is under certain nonsingularity conditions that we get the related result. In addition, the minimax theorem [1, Theorem 3.5] was established in an admissible function space, which is dependent on boundary condition (2). Hence, it is difficult to apply to the case that some perturbation occurs in boundary condition (2). So we will first establish a minimax theorem in an admissible function space with a new weight function that includes the data of (1) and boundary condition (2) by [1, Theorem 3.5]. Then, employing the new minimax theorem, we study the error estimate of eigenvalues of perturbed problem. Another difficulty results from the complicated calculations since the problem is not only of higher order but also of higher dimension and needs to estimate the norms of inverses of some perturbed matrices.

The setup of this paper is as follows. In the next section, we recall some useful existing results, introduce a new suitable admissible function space, and establish a new minimax theorem in it. In Section 3, we give the main results that provide error estimates of eigenvalues of perturbed problems of (1)-(2) under certain nonsingularity conditions. Finally, We discuss two special perturbed problems in Section 4.

2. Preliminaries

In this section, we first introduce some notations and results for convenience in the following discussion, then give a suitable admissible function space, and establish a new variational property of eigenvalues for (1)-(2) in this space.

Consider the following linear space:

Obviously, . Let denote the following difference operator:

For convenience, for , we write if satisfies boundary condition (2). Denote

Lemma 1 (see, [1, Lemma 2.2]). Assume that (3) and (4) hold. Then if and only if there exists a unique vector such that
Let

In particular,

Express and in terms of : where , , , and and are matrices; and are matrices about , which are the shifts of variable of in and to the right with units, respectively. More precisely, for , ,

Obviously, is nonsingular and . In upper triangular matrix , . So if (3) holds, then and are nonsingular. By Proposition  2.1 in [1], , , , and are Hermitian matrices.

Denote where and () are matrices. Substitute (15) into (2) to get where

Next, we always assume that

Let

When (22) holds, is an -dimensional Hilbert space with the inner product by Theorem  2.3 in [1]. In this case, is the same as the admissible function space in [1].

A series of spectral results including the variational properties of eigenvalues for problem (1)-(2) have been established by Shi and Chen in [1]. We state some of these results for future use.

The following lemma is Theorem  3.1 in [1] in the special case that (22) holds.

Lemma 2. Assume that (3), (4), and (22) hold. Then problem (1)-(2) has exactly real eigenvalues (multiplicity included) arranged as

The Rayleigh quotient for the difference operator on with is defined by where and .

The following lemma is the minimax theorem—Theorem  3.5 in [1] in the special case that (22) holds.

Lemma 3. Assume that (3), (4), and (22) hold. Then, for each , , with , where denotes .

Since the perturbation of (1) and (2) affects the space , we need a new suitable admissible function space and a variational formula to apply (26).

For any , by Lemma 1 and (15) there exists a unique vector such that that is,

From above we know that , and then can be uniquely determined by . Hence, we introduce the following new admissible function space:

Since (), it follows from (3) and (22) that where

Thus, we can define an inner product on by where , . Denote its induced norm by

Obviously, is also an -dimensional Hilbert space. Note that the elements of the space are independent of (1) and boundary condition (2), which are partly put in the new weight function .

In order to establish a connection between and , we define a linear map by with determined by (28) for . Evidently, is an invertible linear map. Moreover, for any , , set and . Then, from (23), (27), and (30), we have that is, is a product-preserving map.

Next, we introduce the Rayleigh quotient corresponding to on with as follows: where and . By a direct calculation we have from (9) and (23) that where

For any , , and  , are matrices, are matrices, and

Further, it follows from (27) that where

On the basis of the above discussion, we obtain the following variational formula of eigenvalues for (1)-(2) on by Lemma 3, which plays an important role in the next section.

Theorem 4. Assume that (3), (4), and (22) hold. Then, for each , , with , where denotes .

At the end of this section, we quote two lemmas about matrices and their perturbation. For convenience, we introduce the following notation for an invertible matrix : where the norm of matrix is defined by

With the aid of [9, Corollary 7.8.2] we have immediately the following results.

Lemma 5. For any matrix , .

Lemma 6 ([17, Lemma 2.5]). Let be invertible. If a matrix satisfies then is invertible, and

3. Main Results

In this section, we discuss eigenvalues of perturbed problems sufficiently close to problem (1)-(2) and give error estimates of them.

For convenience, introduce the following notations and several constant matrices:

For any , , where denotes the minimum value of all eigenvalues of and is the same as in (30). It is evident that and .

Based on the above discussion, we know

Now, we consider the following perturbed problem of (1)-(2): where () and () are Hermitian matrices, (), and and are matrices and satisfy

In the following, we will prove that if the perturbation is sufficiently small in norm, then where has the same form of with in (18) replaced by . The matrices , , , (), () are the perturbations of the matrices , , , (), (), respectively.

Proposition 7. Let where is a nonsingular submatrix of . For any , if then(i)(52) holds, and are nonsingular, and where (ii)(53) holds, and

Proof. (i) We only prove that is invertible. The invertibility of can be similarly proved. Since we have
Thus, is invertible by Lemma 6, and
In addition, since then is nonsingular on , which, together with the invertibility of , yields that (52) holds. So (i) is proved.
(ii) Let be a submatrix of and let its position be the same as that of in . Since that is,
is invertible by Lemma 6 and, consequently, .
In addition,
Then we have
Thus,
Hence, is invertible and by Lemma 6. Further, which yields that (59) holds. The proof is complete.

Under the assumptions of Proposition 7, and are invertible. So, we can define the following inner product on corresponding to problem -: for any , , where

The corresponding induced norm is denoted by

Similarly, is also an -dimensional Hilbert space.

Under the hypotheses of Proposition 7, if further (51) holds, then, by Lemma 2, the perturbed problem - has also real eigenvalues (multiplicity included) arranged as

Notice that the multiplicity of , the th eigenvalue of -, may be different from that of the th eigenvalue of (1)-(2) in general.

Similarly, The Rayleigh quotient corresponding to the difference operator for - on with can be defined by where in which

According to the above discussion, if (51), (55), and (56) hold, then we can get the following variational formula of eigenvalues for - on in a similar way to Theorem 4: for each , , with , where denotes .

In order to obtain an error estimate of eigenvalues for the perturbed problem by applying the above variational formula of eigenvalues, we will discuss the relationship between and and then give another form of variational formula of eigenvalues for - on . Now we introduce the following linear transformation: where, for any

Obviously, is invertible, and

So, for any , we get

Thus, the variational formula of eigenvalues for - on can be restated as follows: if (51), (55), and (56) hold, then, for each , , with .

Before giving the main results, we prepare some estimates.

Lemma 8. For any , if (3) and (56) hold, then where , , , and are the same as in (54), (58), (41), and (77), respectively.

Proof. denotes the adjoint matrix of . Then, by Lemma 5, we get which yields
So,
Similarly, one gets
Hence, we have from (57) and (89) that
Similarly, we obtain
It follows from (41) and (77) that
From (91) one can get
In addition,
Therefore, we have
It is easy to get from (41) that
Inequality (86) can be obtained by a similar argument. The proof is complete.

Now, we study for any .

Proposition 9. For any , if (3) holds, then where and are defined as in (49),

Proof. For any given , it follows from (40) that where is the Euclidean norm of ; that is,
Further, from (50), (69), (85), (89), and (90) we have
This completes the proof.

Next, we study the difference between and for any .

Proposition 10. Let whereis defined as in (54). For any , if (55) and (56) hold and then where

Proof. It follows from (80) that, for any given , which, together with (33), yields that
Since it follows from Lemma 6 that
Thus,
Similarly, we have
In addition, from (59), (71), and (113) we get
Now, we are in a position to estimate . Let
Then, from (89) we obtain
With a similar argument to that for (93), we get
From (67) one has which, together with (69), implies that
Hence, it follows from (30), (71), and (121) that whereis the same as in (108). It can be easily concluded from (69) that
Therefore, from (113), (116), (122), and (123) we have
Consequently, (106) holds and the proof is complete.

The following result is about the estimate of difference betweenand.

Proposition 11. For any, in whichis defined as in (104), if (55), (56), and (105) hold, then where
and ,, andare the same as in (107), (108), and (109), respectively.

Proof. It follows from (76) and (80) that, for any , where
So we get from (40) and (129) that where
In the following we discuss , , term by term. It follows from the first relation in (132) that in which the right-hand side is a sum of three terms. Now, we calculate the first term.
From (67), (69), and (121) we have which, together with (84) and (85) in Lemma 8, implies that
In addition, from (86) and (121) we get
Hence, it follows from (134)–(137) that
Next, we study the second term in the right-hand side of (133):
Since , from (66) we have
So, which, together with (57), (89), and (91), yields that
Hence, it follows from (139)–(142) that
With a similar argument, one can obtain an estimate of the third term in the right-hand side of (133), which is the same as (143). Then, from (116), (122), (123), (133), (138), and (143), one can get
Next, we consider the second relation in (132). It is evident that
From (50) we have
It follows from the expression of that
Additionally,
Further, from (116) and (122) one has
From the third relation in (132) we get
From (57), (66), (89), and (91) we obtain
According to the expression of , we know that it has the same estimate as in (147). Thus, we have
It follows from (113) and (132) that, for any ,
Similarly, it can be concluded that
So, by the assumptions and the Hölder inequality, we have
Therefore, from (144) and (149)–(155) we obtain which, together with (49), implies that (125) holds. The proof is complete.

Now we give the main result of the present paper—an error estimate of eigenvalues of the perturbed problem -.

Theorem 12. Assume that (3), (4), (22), and (51) hold. Let where , , , , and are the same as in (104), (49), (107), and (109), respectively. For any , if (55), (56), and (105) hold, then theth eigenvalue of - and the th eigenvalue of (1)-(2) (in the increasing order as in (73) and (24), resp.) satisfy where
and , , , and are the same as in (99), (100), (126), and (127), respectively.

Proof. By Propositions 9–11, we have that, for any with ,
Since we have from (106) that which implies that ; that is, . Hence, it follows from (160) that
Therefore, for each , and for any , we get from Theorem 4 and (82) that
which, together with (83), yields that (158) holds. The proof is complete.

The following result is a direct consequence of Theorem 12.

Corollary 13. Assume that all the assumptions in Theorem 12 hold. Then each eigenvalue of problem (1)-(2) is continuously dependent on the coefficients and weight function of (1) and the coefficients of the boundary condition (2).

Remark 14. The nonsingularity assumption (22) for can be illustrated by giving examples. Since -order discrete vector boundary value problems include second-order discrete boundary value problems and the necessity of the nonsingularity assumption forhas been clarified through an example in [17]. Here we will not discuss it.

4. Two Special Cases

In this section, we consider two special perturbed problems. The error estimates will be simpler for these two special cases.

Case 1. The perturbed problem consists of (1)-; that is, only the coefficients of boundary condition (2) are perturbed, and the coefficients and weight function of (1) are invariant. Since the method of proof is similar to that of Theorem 12, only the related result is given.

Theorem 15. Assume that (3), (4), (22), and (51) hold. Let where is a nonsingular submatrix of ,
For any , if (55) holds, then theth eigenvalue of (1)- and theth eigenvalueof (1)-(2) satisfy where , , , , and are the same as in (49), (99), and (100), respectively,

Case 2. The perturbed problem consists of -(2); that is, only the coefficients and weight function of (1) are perturbed, and the coefficients of boundary condition (2) are invariant.

Since boundary condition contains the coefficients and ( and ) of equation, the coefficients are invariant in this case; then , , , , , and are invariant.

In addition, since in this case the admissible function space of perturbed problem is the same as that for the original problem, it can be directly applied instead of the space . However, since the weight function is perturbed, the inner product on for the perturbed problem changes with it. Define an inner product on for the perturbed problem by and the following induced norm

Obviously, is still an -dimensional Hilbert space with the inner product by [1, Theorem 2.3].

For convenience, we now introduce the Rayleigh quotient corresponding to the difference operator on with as follows: where is the same as in (74).

By Lemma 2, problem -(2) has also real eigenvalues (multiplicity included) arranged as

The variational property (26) of eigenvaluesfor perturbed problem -(2) on still holds, where , , , and are replaced by , , , , and , respectively.

In a similar way to the discussion in Section 3, we first discuss the relation between and and then give another form of variational formula of eigenvalues for problem -(2) on . Now we introduce the following linear transformation: for any , we have where , , , and are the same as in (31) and (71), respectively, and has the same definition as in (13) only with replaced by .

Evidently, is invertible and Hence, for any , we get

Therefore, the variational property (26) of eigenvalues for problem -(2) on still holds, where , , and are replaced by , , and , respectively.

Now, we give an error estimate of eigenvalues of the perturbed problem -(2).

Theorem 16. Assume that (3), (4), and (22) hold. Let where is the same as in (49). For any , if (56) and (105) hold, then the th eigenvalue of -(2) and the th eigenvalue of (1)-(2) (in the increasing order as in (172) and (24), resp.) satisfy where and is the same as in (109).

Proof. It follows from (25) and (171) that, for any with , where , and
It follows from (27) that that is,
So,
Hence,
Similarly,
From (69) and (88), we have
Thus, it follows from (185)–(187) that
In addition, from (174), we get which, together with (113), yields that
By the assumption , one can easily obtain
With a similar argument to that used in the proof of Proposition 11, from (174) and (184) one can get that which, together with (180) and (188)–(191), implies that
By Theorem 4, we have
This completes the proof.

Remark 17. Sincein Theorem 16 is greater than in Theorem 12, the perturbed amplitude in Theorem 16 is even bigger.

Remark 18. The error estimate of eigenvalues of the special perturbed problem -(2) can be deduced from the proof of Theorem 12. Here, we give the proof instead of using the method of the space transformation from into . The proof here is simpler and more direct.

Remark 19. The estimate obtained in Theorem 16 does not involve of (49), so we do not need to calculate the eigenvalues of matrix when Theorem 16 is applied.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the NNSF of China (Grants 11071143, 11326113, 11226160, and 11301304) and the NSF of He’nan Educational Committee (Grants 2011A110001 and 14A110001). The authors thank the referee for his valuable comments and suggestions.