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Inequalities among Eigenvalues of Second-Order Difference Equations with General Coupled Boundary Conditions
Advances in Difference Equations volume 2009, Article number: 347291 (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.
1. Introduction
Consider the second-order difference equation

with the general coupled boundary condition

where is an integer,
is the forward difference operator:
,
is the backward difference operator:
, and
and
are real numbers with
for
,
for
, and
is the spectral parameter; the interval
is the integral set
;
,
is a constant parameter;
,

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
.
We first briefly recall some relative existing results of eigenvalue problems for difference equations. Atkinson [1, Chapter , Section
] discussed the boundary conditions

when he investigated the recurrence formula

where ,
,
,
and
are real numbers, subject to
and

He remarked that all the eigenvalues of the boundary value problem (1.4) and (1.5) are real, and they may not be all distinct. If and
, he viewed the boundary conditions (1.4) as the periodic boundary conditions for (1.5). Shi and Chen [2] investigated the more general boundary value problem


where ,
, and
are
Hermitian matrices;
and
are nonsingular;
for
;
and
are
matrices. Moreover,
and
satisfy rank
and the self-adjoint condition
[2, Lemma
]. A series of spectral results was obtained. We will remark that the boundary condition (1.8) includes the coupled boundary condition (1.2) when
, and the boundary conditions (1.4) when (1.6) holds. Agarwal and Wong studied existence of minimal and maximal quasisolutions of a second-order nonlinear periodic boundary value problem [3, Section
]. In 2005, Wang and Shi [4] considered (1.1) with the periodic and antiperiodic boundary conditions. They found out the following results (see [4, Theorems
and
]): the periodic and antiperiodic boundary value problems have exactly
real eigenvalues
and
, respectively, which satisfy

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
Equation (1.1) can be rewritten as the recurrence formula

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 ]).
Let and
be any solutions of (1.1). Then the Wronskian

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 setting ,
,
,

shifting the whole interval left by one unit, and using
, (1.1) and (1.2) are written as (1.7) and (1.8), respectively. It is evident that
and
. Hence, the boundary condition (1.2) is self-adjoint by [2, Lemma
]. In addition, it follows from (2.3) and
that

By noting that , we get
. Therefore, by [2, Theorem
], the problem (1.1) and (1.2) has exactly
real eigenvalues. This completes the proof.
Let be the solution of (1.1) with the initial conditions

Consider the sequence

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.
Now we consider (1.1) with the following separated boundary conditions:

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.
In [9], Agarwal et al. studied the following boundary value problem on time scales:

with the boundary conditions

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.
The eigenvalues of the boundary value problem are

with

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.
The boundary value problem (1.1) and (2.7) has   real and simple eigenvalues as
and
real and simple eigenvalues as
, which can be arranged in the increasing order

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
.
Let and
be the solutions of (1.1) satisfying the following initial conditions:

respectively. By Lemma 2.1 and using , we have

Obviously, and
are two linearly independent solutions of (1.1). The following lemma can be derived from [4, Proposition
].
Lemma 2.5.
Let be the eigenvalues of (1.1) and (2.7) with
and be arranged as (2.12). Then,
is an eigenfunction of the problem (1.1) and (2.7) with respect to
, that is, for
,
is a nontrivial solution of (1.1) satisfying

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 ]).
For any and for any
, the initial value problem

has a unique solution , which can be expressed as

where .
3. Main Results
Let and
be defined in Section 2, let
be the eigenvalues of the separated boundary value problem (1.1) with (2.7), and let
be the eigenvalues of the coupled boundary value problem (1.1) and (1.2) and arranged in the nondecreasing order

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.
Assume that or
. Then, for every fixed
,
, one has the following inequalities:

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

where

Moreover, is a multiple eigenvalue of (1.1) and (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) and (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 (3.6) has a nontrivial solution if and only if

which, together with (2.14) and , implies that

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.
For any ,

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.
(i) If is an eigenfunction of the problem (1.1) and (2.7) respect to
then
. By Lemma 2.3 and the initial conditions (2.13), we have that if
then the sequence
,
exhibits
changes of sign and

Case 1.
If then it follows from
that

By (2.14) and the first relation in (3.11), for each ,
, we have

By the definition of , (3.11), and

Hence,

Noting ,
, and (3.10), we have that if
is odd then

and if is even then

Case 2.
If then it follows from (2.7) and (2.14) that for each
,
,

From (2.15) and by the definition of , we get

Hence, noting ,
, and by Lemma 2.5, we have that if
is odd, then

and if is even, then

(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

where and
is a certain constant for
. Then

where is a polynomial in
whose degree is not larger than
. Clearly, as
,
since
. By the first part of this proposition,
. So there exists a constant
such that
.
-
(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.
Since and
are both polynomials in
, so is
. Denote

Proposition 3.6.
Assume that or
. Equations
and
or
hold if and only if
is a multiple eigenvalue of (1.1) and (1.2) with
or
. If
or
for some
, then
is a simple eigenvalue of (1.1) and (1.2) with
or
and for every
, with
one has:

Proof.
Since and
are solutions of (1.1), we have


Differentiating (3.25) and (3.26) with respect to , respectively, yields that

It follows from (2.13) that

Thus, by Lemma 2.6 and from (3.27)–(3.28), we have

It follows from (3.29) that

Hence, not indicating explicitly, we get

where

which is symmetric for any . Then, we have

Hence, if or
, we get from (3.33) that
. Then, for any fixed
with
or
, the matrix
is positive semidefinite or negative semidefinite. Therefore, for such a
,
cannot vanish unless
for all
. Because
and
are linearly independent,
is identically zero if and only if all the entries of the matrix
vanish, namely,

which, together with and
, implies

Then by Proposition 3.3, is a multiple eigenvalue of (1.1) and (1.2) with
. In addition, (3.34), together with
and
, implies

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

Therefore, from (3.37) and by the definition of , we have

and consequently, not indicating explicitly, we have

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.
We first prove the first result. Suppose that is odd,
, and
. Then
is a multiple eigenvalue of (1.1) and (1.2) with
by Proposition 3.6. Then by Proposition 3.3, (3.5) holds for
and
, that is,

Differentiating with respect to
two times, we get

Differentiating (2.14) with respect to two times and from (3.40), we get

which, together with (3.41), implies that

On the other hand, it follows from (3.29) and (2.14) that, not indicating explicitly,

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

(see Figure 2). Similarly, in the other case that is even, 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

(see Figure 3). Therefore, we get that (1.1) and (1.2) with ,
has
eigenvalues and it is real and satisfies

This completes the proof.
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
].
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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).
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Zhang, C., Sun, S. Inequalities among Eigenvalues of Second-Order Difference Equations with General Coupled Boundary Conditions. Adv Differ Equ 2009, 347291 (2009). https://doi.org/10.1155/2009/347291
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DOI: https://doi.org/10.1155/2009/347291
Keywords
- Difference Operator
- Nontrivial Solution
- Positive Semidefinite
- Recurrence Formula
- Hermitian Matrice