Inequalities among Eigenvalues of Second-Order Difference Equations with General Coupled Boundary Conditions

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

(1.1)

with the general coupled boundary condition

(1.2)

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; ,

(1.3)

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

(1.4)

when he investigated the recurrence formula

(1.5)

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

(1.6)

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

(1.7)

(1.8)

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

(1.9)

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

(2.1)

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

(2.2)

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 , , ,

(2.3)

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

(2.4)

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

(2.5)

Consider the sequence

(2.6)

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:

(2.7)

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:

(2.8)

with the boundary conditions

(2.9)

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

(2.10)

with

(2.11)

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

(2.12)

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:

(2.13)

respectively. By Lemma 2.1 and using , we have

(2.14)

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

(2.15)

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

(2.16)

has a unique solution , which can be expressed as

(2.17)

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

(3.1)

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:

(3.2)

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

(3.3)

where

(3.4)

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

(3.5)

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

(3.6)

It is evident that (3.6) has a nontrivial solution if and only if

(3.7)

which, together with (2.14) and , implies that

(3.8)

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 ,

(3.9)

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

(3.10)

Case 1.

If then it follows from that

(3.11)

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

(3.12)

By the definition of , (3.11), and

(3.13)

Hence,

(3.14)

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

(3.15)

and if is even then

(3.16)

Case 2.

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

(3.17)

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

(3.18)

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

(3.19)

and if is even, then

(3.20)

(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

(3.21)

where and is a certain constant for . Then

(3.22)

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

(3.23)

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:

(3.24)

Proof.

Since and are solutions of (1.1), we have

(3.25)

(3.26)

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

(3.27)

It follows from (2.13) that

(3.28)

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

(3.29)

It follows from (3.29) that

(3.30)

Hence, not indicating explicitly, we get

(3.31)

where

(3.32)

which is symmetric for any . Then, we have

(3.33)

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,

(3.34)

which, together with and , implies

(3.35)

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

(3.36)

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

(3.37)

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

(3.38)

and consequently, not indicating explicitly, we have

(3.39)

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,

(3.40)

Differentiating with respect to two times, we get

(3.41)

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

(3.42)

which, together with (3.41), implies that

(3.43)

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

(3.44)

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

(3.45)

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

(3.46)

(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

(3.47)

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

(3.48)

This completes the proof.

Figure 1

The graph of .

Figure 2

The graph of in the case that is odd.

Figure 3

The graph of in the case that is even.

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 ].