Inequalities among Eigenvalues of Second-Order Difference Equations with General Coupled Boundary Conditions
© C. Zhang and S. Sun. 2009
Received: 11 February 2009
Accepted: 11 May 2009
Published: 4 June 2009
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.
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 .
Motivated by , 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 . In this paper, we will apply some results obtained by Shi and Chen  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.  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.
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 ]).
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.
Obviously, and are two linearly independent solutions of (1.1). The following lemma can be derived from [4, Proposition ].
A representation of solutions for a nonhomogeneous linear equation with initial conditions is given by the following lemma.
Lemma 2.6 (see [4, Theorem ]).
3. Main Results
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.
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.
(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
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 .
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.
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.
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
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 ].
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).
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