Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales
© C. Zhang and S. Sun. 2010
Received: 28 January 2010
Accepted: 5 May 2010
Published: 1 June 2010
We consider coupled boundary value problems for second-order symmetric equations on time scales. Existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only unify the existing ones of coupled boundary value problems for second-order symmetric differential equations but also contain more complicated time scales.
The boundary condition (1.2) contains the two special cases: the periodic and antiperiodic 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 Equation (1.1) with (1.2) is called a coupled boundary value problem.
In the present paper, we try to extend these results on time scales. We shall remark that Eastham et al. employed continuous eigenvalue branch which studied in , in their proof. Instead, we will make use of some oscillation results that are extended from the results obtained by Agarwal et al.  to prove the existence of eigenvalues of (1.1) with (1.2) and compare the eigenvalues as varies.
This paper is organized as follows. Section 2 introduces some basic concepts and fundamental theory about time scales and gives some properties of eigenvalues of a kind of separated boundary value problem for (1.1) which will be used in Section 4. Our main result has been introduced in Section 3. Section 4 pays attention to prove some propositions, by which one can easily obtain the existence and the comparison result of eigenvalues of the coupled boundary value problems (1.1) with (1.2). By using these propositions, we give the proof of our main result in Section 5.
In this section, some basic concepts and some fundamental results on time scales are introduced. Next, the eigenvalues of the kind of separated boundary value problem for (1.1) and the oscillation of their eigenfunction are studied. Finally, the reality of the eigenvalues of the coupled boundary value problems for (1.1) is shown.
where , . A point is called right-scattered, right-dense, left-scattered, and left-dense if , and respectively.
We assume throughout the paper that if 0 is right-scattered, then it is also left-scattered, and if 1 is left-scattered, then it is also right-scattered.
Moreover, a function defined on is said to be rd-continuous if it is continuous at every right-dense point in and its left-sided limit exists at every left-dense point in .
If is differentiable at , then is continuous at .
- (ii)If and are differentiable at , then is differentiable at and(26)
- (iii)If and are differentiable at , and , then is differentiable at and(27)
If is rd-continuous on , then it has an antiderivative on .
Now, we turn to discuss some properties of solutions of (1.1) and eigenvalues of its boundary value problems.
where is the set of twice differentiable functions with rd-continuous second derivative. The following result can be derived from the Lagrange Identity [5, Theorem ].
For any two solutions and of (1.1), is a constant on .
where is continuous; ; satisfy that if is right-scattered, then it is also left-scattered; and if is left-scattered, then it is also right-scattered. A solution of (2.9) is said to have a node at if . A generalized zero of is defined as its zero or its node. Without loss of generality, they assumed that and in (2.10) satisfy
and obtained the following oscillation result.
Lemma 2.3 (see [4, Theorem ]).
The eigenvalues of (2.9) with (2.10) may be arranged as and an eigenfunction corresponding to has exactly -generalized zeros in the open interval .
which is independent of .
Lemma 2.4 (see [1, Lemma ]).
Let be the solution of (1.1) with (2.12). Then is strictly decreasing in for each whenever
Lemma 2.5 (see [1, Lemma ]).
If there exists such that , then for all
With a similar argument to that used in the proof of [4, Theorem ], one can show the following result.
All the eigenvalues of (1.1) with (2.11) are simple and can be arranged as and an eigenfunction corresponding to has exactly -generalized zeros in the open interval , where satisfy that if is right-scattered, then it is also left-scattered; if is left-scattered, then it is also right-scattered. Furthermore, the number of its eigenvalues is equal to .
The following result is a direct consequence of Theorem 2.6.
and an eigenfunction corresponding to has exactly -generalized zeros in Furthermore, the number of its eigenvalues is equal to .
For convenience, we shall write if
For each ,
The proof is similar to that of [4, Theorem ]. So the details are omitted.
All the eigenvalues of the coupled boundary value problem (1.1) with (1.2) are real.
The proof is similar to that of [1, Lemma ]. So the details are omitted.
3. Main Result
In this section we state our main results: general inequalities among eigenvalues of coupled boundary value problem of (1.1) with (1.2).
If and or and , 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 .
4. The Characteristic Function
Before showing Theorem 3.1, we need to prove the following six propositions.
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 (4.16) are zero. Hence, is a multiple eigenvalue of (1.1) with (1.2) if and only if (4.15) holds. This completes the proof.
The following result is a direct consequence of the first result of Proposition 4.1.
Assume that and or and . For each , if is odd, and if is even.
- (i)If , then it follows from that(423)
- (ii)If , then it is noted that is eigenvalue of (1.1) with (2.14) if and only if Hence, is an eigenfunction with respect to By Theorem 2.7, has exactly generalized zeros in and(429)
Hence for all
Further, by the existence and uniqueness theorem of solutions of initial value problems for (1.1) [5, Theorem ], we obtain that
With a similar argument from above, we get sgn
This completes the proof.
Assume that and or and . There exists a constant such that and .
By Proposition 4.3, . Therefore, there exists a such that . This completes the proof.
which together with (4.10) confirm (4.42).
That is, (4.43) holds. The identity (4.44) can be verified similarly. This completes the proof.
If satisfies , then , , and .
These are direct consequences of (4.43) and (4.44).
if and only if for some and is an eigenfunction of .
It is directly follows from the definition of and the initial conditions (4.1).
Assum that or and is a multiple eigenvalue of (1.1) with (1.2) if and only if .
(i)Suppose that is a multiple eigenvalue of (1.1) with (1.2). Then and By (4.42), .
(ii)Suppose that is an eigenvalue of (1.1) with (1.2) and . Then by (4.14), . From (4.43) and (4.44) we get
It follows from and that . Thus, is a multiple eigenvalue of (1.1) with (1.2).
The case can be established by replacing by in the above argument. This completes the proof.
Assume or . If is a multiple eigenvalue of (1.1) with (1.2), then there exists such that .
This means that for some This completes the proof.
Equations and or hold if and only if is a multiple eigenvalue of (1.1) with (1.2) with or .
If or and is a multiple eigenvalue of (1.1) with (1.2), then , .
If or for some , , then is a simple eigenvalue of (1.1) with (1.2) with or .
- (iv)Moreover, for every , , with one has(461)
This completes the proof.
For any fixed , , each eigenvalue of (1.1) with (1.2) is simple.
Suppose that is an eigenvalue of the problem (1.1) with (1.2) and fix with . By Proposition 4.1, we have , then , and the matrix is positive definite or negative definite. Hence, or for , since and are linearly independent.
If is a multiple eigenvalue of problem (1.1) with (1.2), then (4.15) holds by Proposition 4.1. By using (4.15), it can be easily verified that (4.74) holds; that is, all the entries of the matrix are zeros. Then , which is contrary to . Hence, is a simple eigenvalue of (1.1) and (1.2). This completes the proof.
If is odd, , and then ; if is even, , and then .
by the Holder inequality [8, Lemma (iv)]. Therefore . Since and are linearly independent, which proves the first conclusion, the second conclusion can be shown similarly. This completes the proof.
5. Proofs of the Main Results
Proof of Theorem 3.1.
Similarly, by Propositions 4.1, 4.3, and 4.10, the continuity of and the intermediate value theorem, reaches , ( , ), and exactly one time between any two consecutive eigenvalues of the separated boundary value problem (1.1) with (2.14). Hence, (1.1) and (1.2) with , , , and have only one eigenvalue between any two consecutive eigenvalues of (1.1) with (2.14), respectively. In addition, by Propositions 4.10 and 4.12, if or and , then is not only an eigenvalue of (1.1) with (2.14) but also a multiple eigenvalue of (1.1) and (1.2) with and .
If , then it follows from the above discussion that (1.1) and (1.2) with , have infinitely many eigenvalues, and they are real and satisfy (3.1).
In the continuous case: , , by Theorem 3.1, the coupled boundary value problems (1.1) and (1.2) have infinitely many eigenvalues: for , ; for ; for , and they satisfy inequality (3.1). This result is the same as that obtained by Eastham et al. for second-order differential equations [2, Theorem ].
It is evident that and then in these three cases. By Theorem 3.1, the coupled boundary value problems (1.1) and (1.2) have infinitely many real eigenvalues and they satisfy the inequality (3.1). Obviously, the above three cases are not continuous and not discrete. So the existing results are not available now.
By Remark 5.1 and Example 5.2, our result in Theorem 3.1 not only extends the results in the discrete cases but also contains more complicated time scales.
Many thanks to A. Pankov (the Editor) and the anonymous reviewer(s) for helpful comments and suggestions. This research was supported by the Natural Science Foundation of Shandong Province (Grant Y2008A28) (Grant ZR2009AL003) and the Natural Science Fund Project of the University of Jinan (Grant XKY0918).
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