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Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales
Advances in Difference Equations volume 2010, Article number: 317416 (2010)
Abstract
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.
1. Introduction
In this paper we consider the following second-order symmetric equation:

with the coupled boundary conditions:

where is a time scale;
and
are real and continuous functions in
over
over
and
;
and
are the forward and backward jump operators in
,
is the delta derivative, and
;
is a constant parameter;

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.
Hence, according to [1, Theorem ], the periodic and antiperiodic boundary value problems have
real eigenvalues and they satisfy the following inequality:

where ,
denote the
th Dirichlet eigenvalues. Denote the number of point of a set
by
and introduce the following notation for
:

Furthermore, if , then

In [2], Eastham et al. considered the second-order differential equation:

with the coupled boundary condition:

where ,
,
,
, and
,
,
,
,
a.e. on
. Here
denote the set of real number, and
the space of real valued Lebesgue integrable functions on
. They obtained the following results: the coupled boundary value problem (1.7) with (1.8) has an infinite but countable number of only real eigenvalues which can be ordered to form a nondecreasing sequence:

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 [2], in their proof. Instead, we will make use of some oscillation results that are extended from the results obtained by Agarwal et al. [4] 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.
2. Preliminaries
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.
Let be a nonempty closed subset. Define the forward and backward jump operators
by

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.
Since is a nonempty bounded closed subset of
, we put
. The graininess
is 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

If for all
, then
is called an antiderivative of
on
. In this case, define the delta integral by

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
.
For convenience, we introduce the following results ([5, Chapter 1], [6, Chapter 1], and [7, Lemma ]), which are useful in the paper.
Lemma 2.1.
Let and
.
-
(i)
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)
-
(iv)
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.
Define the Wronskian by

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
].
Lemma 2.2.
For any two solutions and
of (1.1),
is a constant on
.
In [4], Agarwal et al. studied the following second-order symmetric linear equation:

with the boundary conditions:

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
(H) and
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
.
In order to study the kind of separated boundary value problem for (1.1), we now extend the above oscillation theorem to the more general equation (1.1) with

By denote the number of generalized zeros of the solution
of (1.1) with the initial conditions

in the open interval (0,1), where and
satisfy
with
and
replaced by
and
, respectively. It can be easily verified that

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.
Theorem 2.6.
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
.
Setting ,
, and
in (2.11), where
,
are elements of
in (1.2), we get the following separated boundary conditions:

The following result is a direct consequence of Theorem 2.6.
Theorem 2.7.
All the eigenvalues of (1.1) with (2.14) are simple and can be arranged as

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
Lemma 2.8.
For each ,
Proof.
The proof is similar to that of [4, Theorem ]. So the details are omitted.
Lemma 2.9.
All the eigenvalues of the coupled boundary value problem (1.1) with (1.2) are real.
Proof.
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).
Theorem 3.1.
If and
or
and
, then, for every fixed
,
, coupled boundary value problem (1.1) with (1.2) has
eigenvalues and these eigenvalues satisfy the following inequalities:

where . Furthermore, if
then

Remark 3.2.
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
Â
Before showing Theorem 3.1, we need to prove the following six propositions.
Let and
be the solutions of (1.1) satisfying the following initial conditions:

respectively. Obviously, and
are two linearly independent solutions of (1.1). By Lemma 2.2 we have

which, together with the assumption of , implies

For any fixed ,
, and all
, we define






Note that

Let

Hence, we have

and by Lemma 2.2, we get

Proposition 4.1.
For ,
is an eigenvalue of (1.1) with (1.2) if and only if

Moreover, is a multiple eigenvalue of (1.1) with (1.2) if and only if

Proof.
Since and
are linearly independent solutions of (1.1), then
is an eigenvalue of the problem (1.1) with (1.2) if and only if there exist two constants
and
, not both zero, such that
satisfies (1.2), which yields

It is evident that (4.15) has a nontrivial solution if and only if

which together with (4.3), (4.4) and implies that

It follows from the above relation and the fact that that
is an eigenvalue of (1.1) with (1.2) if and only if
satisfies

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.
Corollary 4.2.
For any ,

For and
, we consider the separated boundary problem (1.1) with (2.14). Let
, be all the eigenvalues of (1.1) with (2.14) and ordered as that in Theorem 2.7. Since
and
are all entire functions in
for each
,
is an entire functions in
. Denote

Proposition 4.3.
Assume that and
or
and
. For each
,
if
is odd, and
if
is even.
Proof.
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 and the last two relations in (4.1), we have that
has exactly
generalized zeros in
and

-
(i)
If
, then it follows from
that
(423)
By (4.3) and the first relation in (4.23) we have

By the definition of , the second relation in (4.23), and
, we get

Hence,

Noting ,
, and (4.22), we have that if
is odd, then

and if is even, then

-
(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
Next we will show In the case that
,
and
It follows from (1.1) and (2.4) that

which implies

In the other case that , then

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
By referring to and from (4.3), it follows that

Hence, noting and
, if
is odd, then

and if is even, then

This completes the proof.
Proposition 4.4.
Assume that and
or
and
. There exists a constant
such that
and
.
Proof.
Since and
are solutions of (1.1), we have

By integration, it follows from (4.1) and (4.36) that

where is used. In addition, from (4.36), we obtain

which, similarly together with (4.1) and by integration, imply that

On the other hand, it follows from Lemma 2.5 and (4.1) that for all sufficiently large ,
,
, for all
, where
and
in (2.11) are taken as
,
, and
,
, respectively, which satisfy
. So, from (4.37) and (4.39), we obtain that

and by Lemma 2.4, it implies

By Proposition 4.3, . Therefore, there exists a
such that
. This completes the proof.
Lemma 4.5.
For any one has



Proof.
Since and
are solutions of (1.1) with (4.1), then they satisfy (4.36). Differentiating (4.36) with respect to
, we have

By the variation of constants formula [5, Theorem ], we get

Further, it follows from [5, Theorem ] that

From (4.46), (4.47), and (4.11), we have

where

It follows from (4.11) and (4.13) that

where
Hence,

By (4.10) and (4.12), we have

Differentiating above relation with respect to , and with (4.12), we have

which together with (4.10) confirm (4.42).
To establish (4.43), from (4.12) and (4.10), we obtain

Thus

That is, (4.43) holds. The identity (4.44) can be verified similarly. This completes the proof.
Corollary 4.6.
If satisfies
, then
,
, and
.
Proof.
These are direct consequences of (4.43) and (4.44).
Lemma 4.7.
if and only if
for some
and
is an eigenfunction of
.
Proof.
It is directly follows from the definition of and the initial conditions (4.1).
Lemma 4.8.
Assum that or
and
is a multiple eigenvalue of (1.1) with (1.2) if and only if
.
Proof.
Assume that . By (4.15)
is a multiple eigenvalue if and only if

hence, it follows from (4.12) that

Therefore and
(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

Since and
are linearly independence solutions of (1.1), we have

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.
Lemma 4.9.
Assume or
. If
is a multiple eigenvalue of (1.1) with (1.2), then there exists
such that
.
Proof.
Assume that is a multiple eigenvalue of (1.1) with (1.2). From the proof of Lemma 4.8 we see that
. From (4.9) we have

This means that for some
This completes the proof.
Proposition 4.10.
Assume that and
or
and
.
-
(i)
Equations
and
or
hold if and only if
is a multiple eigenvalue of (1.1) with (1.2) with
or
.
-
(ii)
If
or
and
is a multiple eigenvalue of (1.1) with (1.2), then
,
.
-
(iii)
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)
and in the case of ,

Proof.
Parts (i), (ii), and (iii) follow from Lemmas 4.8 and 4.9. It follows from Propositions 4.3 and 4.4 and Corollary 4.6 that ,
with
and
when
. Hence,
with
,
. Similarly, by Proposition 4.3 and Corollary 4.6, we have

which implies

If , then all the points of
are isolated. In this case, (1.1) can be rewritten as

where

By Theorem 2.7, (1.1) with (2.14) has eigenvalues:

It follows from (4.65) that and
are two polynomials of degree
in
and
and
are two polynomials of degree
in
. Then
can be written as

where and
is a polynomial in
whose order is not larger than
. By Proposition 4.3, if
is odd, then
, and if
is even, then
. It follows that if
is odd, then
as
and if
is even, then
as
. Hence, if
is odd, then there exists a constant
such that
. Similarly, in the other case that
is even, there exists a constant
such that
, and by using Corollary 4.6, we have

Hence,

This completes the proof.
Proposition 4.11.
For any fixed ,
, each eigenvalue of (1.1) with (1.2) is simple.
Proof.
It follows from (4.46) and (4.47) that

where

Then from (4.1), , and the definition of
, we have

Thus, if , then
.
is always positive semidefinite or negative semidefinite. Consequently,
is not change sign in
. In this case,
cannot vanish unless
. Because
and
are linearly independent,
if and only if all the entries of the matrix
vanish, namely,

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.
Proposition 4.12.
If is odd,
, and
then
; if
is even,
, and
then
.
Proof.
Assume and
with
being odd. It follows from Proposition 4.1 that

As in the proof of Lemma 4.5 and by (4.11) and (4.13),

Hence,

where (4.51) is used and

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.
By Propositions 4.1–4.12 and the intermediate value theorem, the inequalities in (3.1)–(3.2) can been illustrated with the graph of (see Figures 1–3). We now give the detail proof of Theorem 3.1. By Lemma 2.9, all the eigenvalues of the coupled boundary value problem (1.1) with (1.2) are real. By Propositions 4.3–4.10,
,
for all
with
and there exists
such that
Therefore, by the continuity of
and the intermediate value theorem, (1.1) and (1.2) with
have only one eigenvalue
, (1.1) and (1.2) with
hve only one eigenvalue
, and (1.1) and (1.2) with
,
have only one eigenvalue
, and they satisfy

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).
If , then all points of
are isolated. In this case (1.1) and
can be rewritten as (4.65) and (4.68). By the same method in the proof of Proposition 4.10, that if
is even, then there exists a constant
such that
, which together with (4.62), implies that (1.1) and (1.2) with
,
,
, and
have only one eigenvalue
,
, and
, satisfying

(see Figure 2). Similarly, in the other case that is even, there exists a constant
such that
which, together with (4.62) implies that (1.1) and (1.2) with
,
,
, and
have only one eigenvalue
,
, and
, satisfying

(see Figure 3). Therefore, we get that (1.1) and (1.2) with ,
have
eigenvalues and they are real and satisfy

if is odd; and

if is even. This completes the proof.
Remark 5.1.
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
].
Example 5.2.
Consider the following three specific cases:

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.
References
Zhang C, Shi Y: Eigenvalues of second-order symmetric equations on time scales with periodic and antiperiodic boundary conditions. Applied Mathematics and Computation 2008,203(1):284-296. 10.1016/j.amc.2008.04.044
Eastham MSP, Kong Q, Wu H, Zettl A: Inequalities among eigenvalues of Sturm-Liouville problems. Journal of Inequalities and Applications 1999,3(1):25-43.
Everitt WN, Möller M, Zettl A:Discontinuous dependence of the
-th Sturm-Liouville eigenvalue. In General Inequalities, 7. Volume 123. Birkhäuser, Basel, Switzerland; 1997:145-150.
Agarwal RP, Bohner M, Wong PJY: Sturm-Liouville eigenvalue problems on time scales. Applied Mathematics and Computation 1999,99(2-3):153-166. 10.1016/S0096-3003(98)00004-6
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.
Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(1-2):3-22.
Bohner M, Lutz DA: Asymptotic behavior of dynamic equations on time scales. Journal of Difference Equations and Applications 2001,7(1):21-50. 10.1080/10236190108808261
Acknowledgments
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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Zhang, C., Sun, S. Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales. Adv Differ Equ 2010, 317416 (2010). https://doi.org/10.1155/2010/317416
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DOI: https://doi.org/10.1155/2010/317416