- Research Article
- Open Access

# Inequalities among Eigenvalues of Second-Order Difference Equations with General Coupled Boundary Conditions

- Chao Zhang
^{1}Email author and - Shurong Sun
^{1}

**2009**:347291

https://doi.org/10.1155/2009/347291

© C. Zhang and S. Sun. 2009

**Received:**11 February 2009**Accepted:**11 May 2009**Published:**4 June 2009

## Abstract

This paper studies general coupled boundary value problems for second-order difference equations. Existence of eigenvalues is proved, numbers of their eigenvalues are calculated, and their relationships between the eigenvalues of second-order difference equation with three different coupled boundary conditions are established.

## Keywords

- Difference Operator
- Nontrivial Solution
- Positive Semidefinite
- Recurrence Formula
- Hermitian Matrice

## 1. Introduction

The boundary condition (1.2) contains the periodic and antiperiodic boundary conditions. In fact, (1.2) is the periodic boundary condition in the case where and , the identity matrix, and (1.2) is the antiperiodic condition in the case where and .

These results are similar to those about eigenvalues of periodic and antiperiodic boundary value problems for second-order ordinary differential equations (cf. [5–8]).

Motivated by [4], we compare the eigenvalues of the eigenvalue problem (1.1) with the coupled boundary condition (1.2) as varies and obtain relationships between the eigenvalues in the present paper. These results extend the above results obtained in [4]. In this paper, we will apply some results obtained by Shi and Chen [2] to prove the existence of eigenvalues of (1.1) and (1.2) to calculate the number of these eigenvalues, and to apply some oscillation results obtained by Agarwal et al. [9] to compare the eigenvalues as varies.

This paper is organized as follows. Section 2 gives some preliminaries including existence and numbers of eigenvalues of the coupled boundary value problems, and some properties of eigenvalues of a kind of separated boundary value problem, which will be used in the next section. Section 3 pays attention to comparison between the eigenvalues of problem (1.1) and (1.2) as varies.

## 2. Preliminaries

Clearly, is a polynomial in with real coefficients since and are all real. Hence, all the solutions of (1.1) are entire functions of . Especially, if , is a polynomial of degree in for . However, if and , is a polynomial of degree in for .

We now prepare some results that are useful in the next section. The following lemma is mentioned in [4, Theorem ].

Lemma 2.1 ([4, Theorem ]).

is a constant on .

Theorem 2.2.

If then the coupled boundary value problem (1.1) and (1.2) has exactly real eigenvalues.

Proof.

By noting that , we get . Therefore, by [2, Theorem ], the problem (1.1) and (1.2) has exactly real eigenvalues. This completes the proof.

If for some , then, we get from (2.1) that and have opposite signs. Hence, we say that sequence (2.6) exhibits a change of sign if for some , or for some . A general zero of the sequence (2.6) is defined as its zero or a change of sign.

where are entries of . It follows from (2.1) that the separated boundary value problem (1.1) with (2.7) has a unique solution, and the separated boundary value problem will be used to compare the eigenvalues of (1.1) and (1.2) as varies in the next section.

where is a time scale, and are the forward and backward jump operators in , is the delta derivative, and ; is continuous; ; with . They obtained some useful oscillation results. With a similar argument to that used in the proof of [9, Theorem ], one can show the following result.

Lemma 2.3.

where and are real and continuous functions in over over are arranged as and an eigenfunction corresponding to has exactly generalized zeros in the open interval .

By setting , , the above boundary value problem can be written as (1.1) with (2.7), then we have the following result.

Lemma 2.4.

Let be the solution of (1.1) with the separated boundary conditions (2.7). Then sequence (2.6) exhibits no changes of sign for , exactly changes of sign for , and changes of sign for .

Obviously, and are two linearly independent solutions of (1.1). The following lemma can be derived from [4, Proposition ].

Lemma 2.5.

Moreover, if is odd, and if is even, for .

A representation of solutions for a nonhomogeneous linear equation with initial conditions is given by the following lemma.

Lemma 2.6 (see [4, Theorem ]).

where .

## 3. Main Results

Clearly, denotes the eigenvalue of the problem (1.1) and (1.2) with , and denotes the eigenvalue of the problem (1.1) and (1.2) with . We now present the main results of this paper.

Theorem 3.1.

Remark 3.2.

If or , a similar result can be obtained by applying Theorem 3.1 to . In fact, for and for . Hence, the boundary condition (1.2) in the cases of or and , can be written as condition (1.2), where is replaced by for and for , and is replaced by .

Before proving Theorem 3.1, we prove the following five propositions.

Proposition 3.3.

Proof.

Then (3.3) follows from the above relation and the fact that . On the other hand, (1.1) has two linearly independent solutions satisfying (1.2) if and only if all the entries of the coefficient matrix of (3.6) are zero. Hence, is a multiple eigenvalue of (1.1) and (1.2) if and only if (3.5) holds. This completes the proof.

The following result is a direct consequence of the first result of Proposition 3.3.

Corollary 3.4.

Proposition 3.5.

Assume that or . Then one has the following results.

- (i)
For each , , if is odd, and if is even.

- (ii)
There exists a constant such that .

- (iii)
If the boundary value problem (1.1) and (2.7) has exactly eigenvalues then there exists a constant such that and , where is odd, and there exists a constant such that and , where is even.

Proof.

Case 1.

Case 2.

(ii) By the discussions in the first paragraph of Section 2, is a polynomial of degree in , is a polynomial of degree in , is a polynomial of degree in , and is a polynomial of degree in . Further, can be written as

- (iii)
It follows from the first part of this proposition that if is odd, and if is even, . By (3.22), if is odd, as ; if is even, as . Hence, if is odd, there exists a constant such that ; if is even, there exists a constant such that . This completes the proof.

Proposition 3.6.

Proof.

Then by Proposition 3.3, is a multiple eigenvalue of (1.1) and (1.2) with . Conversely, from (3.35) or (3.36), it can be easily verified that (3.34) holds, then . It follows again from (3.35) or (3.36) that or . Thus and or if and only if is a multiple eigenvalue of (1.1) and (1.2) with or .

Further, for every fixed with or , not indicating explicitly, (3.33) implies that

for every fixed with or .

Suppose that or for some ( ), we have From the above discussions again, is a simple eigenvalue of (1.1) and (1.2) with or , and is not identically zero for .

For this ( ), (3.39) implies that , and from Proposition 3.5 (i), (ii) that , . Hence, , where . It follows from Proposition 3.5 (i) that and , where . By Proposition 3.5 (i), (iii), and there exists such that if is odd, and and there exists such that if is even. Hence, where . This completes the proof.

Proposition 3.7.

For any fixed , , each eigenvalue of (1.1) and (1.2) is simple.

Proof.

Fix , with . Suppose that is an eigenvalue of the problem (1.1) and (1.2). By Proposition 3.3, we have . It follows from (3.33) that and the matrix is positive definite or negative definite. Hence, for or for since and are linearly independent.

If is a multiple eigenvalue of problem (1.1) and (1.2), then (3.5) holds by Proposition 3.3. By using (3.5), it can be easily verified that (3.34) holds, that is, all the entries of the matrix are zero. Then for , which is contrary to for . Hence, is a simple eigenvalue of (1.1) and (1.2). This completes the proof.

Proposition 3.8.

Assume that or . If is odd, , and then ; if is even, , and then for

Proof.

Since and are linearly independent on , the above relation implies that by Hölder's inequality, which proves the first conclusion.

The second conclusion can be shown similarly. Hence, the proof is complete.

Finally, we turn to the proof of Theorem 3.1.

Proof of Theorem 3.1.

By Propositions 3.3–3.8, and the intermediate value theorem, one can obtain the graph of (see Figure 1), which implies the results of Theorem 3.1. We now give its detailed proof.

By Propositions 3.3–3.6, , for all with and there exists such that Therefore, by the continuity of and the intermediate value theorem, (1.1) and (1.2) with has only one eigenvalue , (1.1) and (1.2) with has only one eigenvalue , and (1.1) and (1.2) with , has only one eigenvalue , and they satisfy

Similarly, by Propositions 3.3–3.6, the continuity of and the intermediate value theorem, reaches , ( , ), and exactly one time, respectively, between any two consecutive eigenvalues of the separated boundary value problem (1.1) with (2.7). Hence, (1.1) and (1.2) with ; , ; has only one eigenvalue between any two consecutive eigenvalues of (1.1) with (2.7), respectively. In addition, by Proposition 3.6, if or and , then is not only an eigenvalue of (1.1) with (2.7) but also a multiple eigenvalue of (1.1) and (1.2) with and .

By Proposition 3.5 (i), if is odd, and if is even, . It follows (3.22) that if is odd, then as and if is even, then as . Hence, if is odd, then there exists a constant such that , which, together with Proposition 3.6, implies that (1.1) and (1.2) with ; , ; has only one eigenvalue , , and , satisfying

Remark 3.9.

Let , that is, , . Then . In this case, Propositions 3.5 and 3.8 are the same as those mentioned in [4, Propositions , 3.3–3.5], respectively, and most of the results of Proposition 3.6 are the same as the results of [4, Proposition ].

## Declarations

### Acknowledgments

Many thanks to Johnny Henderson (the editor) and the anonymous reviewers for helpful comments and suggestions. This research was supported by the Natural Scientific Foundation of Shandong Province (Grant Y2007A27), (Grant Y2008A28), and the Fund of Doctoral Program Research of University of Jinan (B0621).

## Authors’ Affiliations

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