Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales

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:

(11)

with the coupled boundary conditions:

(12)

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;

(13)

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:

(14)

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

(15)

Furthermore, if , then

(16)

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

(17)

with the coupled boundary condition:

(18)

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:

(19)

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

(21)

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

(22)

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

(23)

In this case, denote . If is (delta) differentiable for every , then is said to be (delta) differentiable on . If is differentiable at , then

(24)

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

(25)

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

(28)

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:

(29)

with the boundary conditions:

(210)

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

(211)

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

(212)

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

(213)

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:

(214)

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

(215)

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:

(31)

where . Furthermore, if then

(32)

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  

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:

(41)

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

(42)

which, together with the assumption of , implies

(43)

For any fixed , , and all , we define

(44)

(45)

(46)

(47)

(48)

(49)

Note that

(410)

Let

(411)

Hence, we have

(412)

and by Lemma 2.2, we get

(413)

Proposition 4.1.

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

(414)

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

(415)

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

(416)

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

(417)

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

(418)

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

(419)

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 ,

(420)

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

(421)

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

(422)

• (i) If , then it follows from that

  

  (423)

By (4.3) and the first relation in (4.23) we have

(424)

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

(425)

Hence,

(426)

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

(427)

and if is even, then

(428)

• (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

(430)

which implies

(431)

In the other case that , then

(432)

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

(433)

Hence, noting and , if is odd, then

(434)

and if is even, then

(435)

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

(436)

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

(437)

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

(438)

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

(439)

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

(440)

and by Lemma 2.4, it implies

(441)

By Proposition 4.3, . Therefore, there exists a such that . This completes the proof.

Lemma 4.5.

For any one has

(442)

(443)

(444)

Proof.

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

(445)

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

(446)

Further, it follows from [5, Theorem ] that

(447)

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

(448)

where

(449)

It follows from (4.11) and (4.13) that

(450)

where

Hence,

(451)

By (4.10) and (4.12), we have

(452)

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

(453)

which together with (4.10) confirm (4.42).

To establish (4.43), from (4.12) and (4.10), we obtain

(454)

Thus

(455)

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

(456)

hence, it follows from (4.12) that

(457)

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

(458)

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

(459)

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

(460)

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 ,

(462)

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

(463)

which implies

(464)

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

(465)

where

(466)

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

(467)

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

(468)

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

(469)

Hence,

(470)

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

(471)

where

(472)

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

(473)

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,

(474)

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

(475)

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

(476)

Hence,

(477)

where (4.51) is used and

(478)

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

(51)

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

(52)

(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

(53)

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

(54)

if is odd; and

(55)

if is even. 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 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:

(56)

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.