RETRACTED
This article has been retracted as it is found to contain a substantial amount of material from the paper “Eigenvalue comparisons for boundary value problems for second order difference equations,” authored by Jun Ji and Bo Yang and it is published in “Journal of Mathematical Analysis and Applications” in 2006.

Journal of Applied Mathematics

Volume 2012 (2012), Article ID 486230, 10 pages

doi:10.1155/2012/486230

Research Article

Eigenvalue Comparisons for Second-Order Linear Equations with Boundary Value Conditions on Time Scales

Chao Zhang and Shurong Sun

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

Received 29 January 2012; Revised 22 March 2012; Accepted 22 March 2012

Academic Editor: Kai Diethelm

Copyright © 2012 Chao Zhang and Shurong Sun. 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.

Abstract.

This paper studies the eigenvalue comparisons for second-order linear equations with boundary conditions on time scales. Using results from matrix algebras, the existence and comparison results concerning eigenvalues are obtained.

1. Introduction

In this paper, we consider the eigenvalue problems for the following second-order linear equations:

with the boundary conditions

where and are parameters, and are the forward and backward jump operators, is the delta derivative, , and is a finite isolated time scale; the discrete interval is given by

We assume throughout this paper that

, , and are real-valued functions on , , on and on ;.

First we briefly recall some existing results of eigenvalues comparisons for differential and difference equations. In 1973, Travis [1] considered the eigenvalue problem for boundary value problems of higher-order differential equations. He employed the theory of -positive linear operator on a Banach space with a cone of nonnegative elements to obtain comparison results for the smallest eigenvalues. A representative set of references for these works would be Davis et al. [2], Diaz and Peterson [3], Hankerson and Henderson [4], Hankerson and Peterson [5–7], Henderson and Prasad [8], and Kaufmann [9]. However, in all the above papers, the comparison results are for the smallest eigenvalues only. The main purpose of this paper is to establish the comparison theorems for all the eigenvalues of (1.1) with (1.3) and (1.2) with (1.3).

Like the eigenvalue comparison for the boundary value problems of linear equations, this type of comparison of eigenvalues in matrix algebra is known as Weyl’s inequality [10, Corllary 6.5.]: If are Hermitian matrices, that is, , where is the conjugate transpose of and is positive semidefinite, then , where and are all eigenvalues of and . Associated with this conclusion is spectral order of operators. The spectral order has proved to be useful for solving several open problems of spectral theory and has been studied in the context of von Neumann algebras, matrix algebras, and so forth in [10–15]. Recently, Hamhalter [15] studied the spectral order in a more general setting of Jordan operator algebras, which is a generalization of the result due to Kato [13]. And as a preparatory material, he extended Olson’s characterization of the spectral order to JBW algebras [14]. Since the boundary value problems (1.1), (1.3) and (1.2), (1.3) can be rewritten into matrix equations, we employ some results from matrix algebras to establish the comparison theorems for the eigenvalues of (1.1), (1.3) and (1.2), (1.3).

This paper is organized as follows. Section 2 introduces some basic concepts and a fundamental theory about time scales, which will be used in Section 3. By some results from matrix algebras and time scales, the existence and comparison theorems of eigenvalues of boundary value problems (1.1), (1.3) and (1.2), (1.3) are obtained, which will be given in Section 3.

2. Preliminaries

In this section, some basic concepts and some fundamental results on time scales are introduced.

Let be a nonempty closed subset. Define the forward and backward jump operators by

where . We put if is unbounded above and otherwise. The graininess functions are defined by

Let be a function defined on . is said to be (delta) differentiable at provided there exists a constant such that for any , there is a neighborhood of (i.e., for some ) with

In this case, denote . If is (delta) differentiable for every , then is said to be (delta) differentiable on . If is differentiable at , then

For convenience, we introduce the following results ([16, Chapter 1], [17, Chapter 1], and [18, Lemma 1]), which are useful in this paper.

Lemma 2.1. Let and .

(i)If and are differentiable at , then is differentiable at and

(ii)If and are differentiable at , and , then is differentiable at and

3. Eigenvalue Comparisons

In the following, we will write if and are symmetric matrices and is positive semidefinite. A matrix is said to be positive if every component of the matrix is positive. We denote , and .

It follows from Lemma 2.1(ii), (2.4), and (1.4) that the boundary value problem (1.1), (1.3) can be written in the form

where

where donates  , donates, donates , donates , and donates .

And the problem (1.2), (1.3) is equivalent to the equation

where

Since the solutions of (1.1), (1.3) can be written into the form of vectors, then the nontrivial solution corresponding to is called an eigenvector.

Let be the th column of the identity matrix of order and

Define . It is easily seen that

It follows from , and (3.8) that

For any , we have

Moreover, implies . Hence, the matrix is positive definite.

Lemma 3.1. If is an eigenvalue of the boundary value problem (1.1), (1.3) and is a corresponding eigenvector, then

(i),(ii) is real and positive.

If is an eigenvalue of the boundary value problem (1.1), (1.3) and is a corresponding eigenvector, then .

Proof. (i) It follows from (H₁) and (3.2) that . Assume the contrary that , we have . Since is positive definite, then , which is a contradiction.

(ii) We can write

which implies , that is, is real. Since is positive definite and , we have .

If and , then

Hence, . This completes the proof.

Lemma 3.2. If is an eigenvalue of the boundary value problem (1.1), (1.3), then is an eigenvalue of . If is a positive eigenvalue of , then is an eigenvalue of (1.1), (1.3), respectively.

Proof. If is an eigenvalue of the boundary value problem (1.1), (1.3) and is a corresponding eigenvector, then and . Therefore,

With a similar argument, one can get that if is a positive eigenvalue of , then is an eigenvalue of (1.1), (1.3). This completes proof.

Lemma 3.3. For any , define . We have

(i);(ii) = − + .

Proof. It is easy to see that if , while if . Hence,

(i)It is seen from (3.10) and (3.15) that

(ii)It follows from (3.7) and the Sherman-Morrison updating formula [19] that

leading to

which, together with (i), further implies the result (ii). This completes the proof.

Theorem 3.4. (i) If is an eigenvalue of the boundary value problem (1.1), (1.3) and is a corresponding eigenvector, then and .

(ii) If is the smallest eigenvalue of the boundary value problem (1.1), (1.3), then there exists a positive eigenvector corresponding to .

Proof. (i) Assume the contrary that either or . By the boundary condition (1.3), we can easily deduce a contradiction .

(ii) It follows from that is the maximum eigenvalue of and the is an eigenvector corresponding to . By Lemma 3.3, we have that all the elements of are positive, then is a positive matrix. Since for all , hence, the following discussions are divided into two cases.

Case 1. If for all , then we obtain that the matrix is positive and therefore, the result follows from the Perron-Forbenius theorem [20].

Case 2. Let for some . Without loss of generality, we assume that for all and for all ; we can write as follows:

where is an matrix and is an matrix. Both and are positive matrices. is also the maximum eigenvalue of . Applying the Perron-Forbenius theorem to the positive matrix , there exists a positive vector such that . Let and . Obviously, we have

This completes the proof.

Lemma 3.5. If is an eigenvalue of the boundary value problem (1.1), (1.3), then the dimension of the null space of is 1.

Proof. Let and be any two eigenvectors of the boundary value problem (1.1), (1.3) corresponding to and define . Obviously, we have

which, together with , indicates that , that is, . Therefore, and are linearly dependent. So the dimension of the null space of is 1. This completes the proof.

Lemma 3.6. Let be the number of positive elements in the set . Then there are distinct eigenvalues of the boundary value problem (1.1), (1.3) and are the only positive eigenvalues of .

Proof. Suppose that are all eigenvalues of . Since is real and symmetric that there exists an orthogonal matrix such that

therefore, we have that

indicating that the number of positive is the same as that of positive number in which is equal to .

Suppose that for some where . Observe that in view of (3.22), which further implies that

Thus, we have two independent vectors in the null space of for , which contradicts Lemma 3.5. Thus, from Lemma 3.2, we see that gives the complete set of eigenvalues of the boundary value problem (1.1), (1.3). This completes the proof.

Theorem 3.7. Let be the number of positive elements in the set and the number of positive elements in the set . Let be the set of all eigenvalues of the boundary value problem (1.1), (1.3) and the set of all eigenvalues of the boundary value problem (1.2), (1.3). If for all , then for .

Proof. It follows from Lemma 3.6 that

are the eigenvalues of and , respectively. If for all , then , implying

By Weyl’s inequality and (3.26), we have

Finally, it is easily seen from (3.25) and (3.27) that

implying that for . This completes the proof.

Acknowledgments

Many thanks are due to Kai Diethelm (the editor) and the anonymous reviewer(s) for helpful comments and suggestions. This paper was supported by the NNSF of China (Grants nos. 11071143 and 11101241), the NNSF of Shandong Province (Grants nos. ZR2009AL003, ZR2010AL016, and ZR2011AL007), the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01), and the NSF of University of Jinan (XKY0918).

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