## Boundary Value Problems

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# Classification and criteria of the limit cases for singular second-order linear equations on time scales

Boundary Value Problems20122012:103

DOI: 10.1186/1687-2770-2012-103

Accepted: 4 September 2012

Published: 19 September 2012

## Abstract

This paper is concerned with classification and criteria of the limit cases for singular second-order linear equations on time scales. By the different cases of the limiting set, the equations are divided into two cases: the limit-point and limit-circle cases just like the continuous and discrete cases. Several sufficient conditions for the limit-point cases are established. It is shown that the limit cases are invariant under a bounded perturbation. These results unify the existing ones of second-order singular differential and difference equations.

### Keywords

singular second-order linear equation time scales limit-point case limit-circle case

## 1 Introduction

In this paper, we consider classification and criteria of the limit cases for the following singular second-order linear equation:
(1.1)

where , q, and w are real and piecewise continuous functions on , and for all ; is the spectral parameter; is a time scale with and ; and are the forward and backward jump operators in ; is the Δ-derivative of y; and .

The spectral problems of symmetric linear differential operators and difference operators can both be divided into two cases. Those defined over finite closed intervals with well-behaved coefficients are called regular. Otherwise, they are called singular. In 1910, Weyl [1] gave a dichotomy of the limit-point and limit-circle cases for the following singular second-order linear differential equation:
(1.2)
where q is a real and continuous function on , is the spectral parameter. Later, Titchmarsh, Coddington, Levinson etc. developed his results and established the Weyl-Titchmarsh theory [2, 3]. Their work has been greatly developed and generalized to higher-order differential equations and Hamiltonian systems, and a classification and some criteria of limit cases were formulated [49]. Singular spectral problems of self-adjoint scalar second-order difference equations over infinite intervals were firstly studied by Atkinson [10]. His work was followed by Hinton, Jirari etc. [11, 12]. In 2001, some sufficient and necessary conditions and several criteria of the limit-point and limit-circle cases were obtained for the following formally self-adjoint second-order linear difference equations with real coefficients [13]:
(1.3)

where and Δ are the backward and forward difference operators respectively, namely and ; , , and are real numbers with for and  for ; λ is a complex spectral parameter. In 2006, Shi [14] established the Weyl-Titchmarsh theory of discrete linear Hamiltonian systems. Later, several sufficient conditions and sufficient and necessary conditions for the limit-point and limit-circle cases were established for the singular second-order linear difference equation with complex coefficients (see [15]).

In the past twenty years, a lot of effort has been put into the study of regular spectral problems on time scales (see [1623]). But singular spectral problems have started to be considered only quite recently. In 2007, we employed Weyl’s method to divide the following singular second-order linear equations on time scales into two cases: limit-point and limit-circle cases [24]:
where q is real and continuous on , is the spectral parameter. By using the similar method, Huseynov [25] studied the classification of limit cases for the following singular second-order linear equations on time scales:
as well as of the form
(1.4)

where (or ) and q are real and piecewise continuous functions in (or ), for all t, and is the spectral parameter. Obviously, let and , then (1.1) is the same as (1.4). In 2010, by using the properties of the Weyl matrix disks, Sun [26] established the Weyl-Titchmarsh theory of Hamiltonian systems on time scales. It has been found that the second-order singular differential and difference equations can be divided into limit-point and limit-circle cases. We wonder whether the classification of the limit cases holds on time scales. In the present paper, we extend these results obtained in [24] to Eq. (1.1) and establish several sufficient conditions and sufficient and necessary conditions for the limit-point and limit-circle cases for Eq. (1.1).

This paper is organized as follows. In Section 2, some basic concepts and a fundamental theory about time scales are introduced. In Section 3, a family of nested circles which converge to a limiting set is constructed. The dichotomy of the limit-point and limit-circle cases for singular second-order linear equations on time scales is given by the geometric properties of the limiting set. Finally, several criteria of the limit-point case are established, and the invariance of the limit cases is shown under a bounded perturbation for the potential function q in Section 4.

## 2 Preliminaries

In this section, some basic concepts and fundamental results on time scales are introduced.

Let be a non-empty closed set. The forward and backward jump operators are defined by
respectively, where , . A point is called right-scattered, right-dense, left-scattered, and left-dense if , and separately. Denote if is unbounded above and otherwise. The graininess is defined by
Let f be a function defined on . f is said to be Δ-differentiable at provided there exists a constant a such that, for any , there is a neighborhood U of t (i.e., for some ) with
In this case, denote . If f is Δ-differentiable for every , then f is said to be Δ-differentiable on . If f is Δ-differentiable at , then
(2.1)
If for all , then is called an anti-derivative of f on . In this case, define the Δ-integral by

For convenience, we introduce the following results ([[27], Chapter 1] and [[28], Chapter 1]), which are useful in this paper.

Lemma 2.1 Let and .
1. (i)

If f is Δ-differentiable at t, then f is continuous at t.

2. (ii)
If f and g are Δ-differentiable at t, then fg is Δ-differentiable at t and

3. (iii)
If f and g are Δ-differentiable at t, and , then is Δ-differentiable at t and

A function f 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 . The set of rd-continuous functions is denoted by . The set of k th Δ-differentiable functions with rd-continuous k th derivative is denoted by .

Lemma 2.2 If f, g are rd-continuous functions on , then
1. (i)

is rd-continuous and f has an anti-derivative on ;

2. (ii)

for all .

3. (iii)

(Integration by parts) .

4. (iv)
(Hölder’s inequality [[29], Lemma 2.2(iv)]) Let with , then

where and .

Let
A function is called regressive if
Higer [30] showed that for any given and for any given rd-continuous and regressive g, the initial value problem
has a unique solution
(2.2)

Lemma 2.3 ([[27], Theorem 6.1])

Let and . Then
implies
We define the Wronskian by
(2.3)

The following result is a direct consequence of the Lagrange identity [[27], Theorem 4.30].

Lemma 2.4 Let x and y be any two solutions of (1.1). Then is a constant in .

## 3 Classification

In this section, we focus on the classification of the limit cases for singular second-order linear equations on time scales.

Let and be the two solutions of (1.1) satisfying the following initial conditions:
respectively. Since their Wronskian is identically equal to 1, these two solutions form a fundamental solution system of (1.1). We form a linear combination of and
(3.1)
Let , , with , and let (3.1) satisfy
(3.2)
Then
(3.3)
It can be verified that the integral identity
(3.4)
holds for any solution of (1.1) and for any . Setting , , in (3.4) and taking its imaginary part, we obtain
(3.5)
So

It follows from (3.2) and that . Hence, the denominator in (3.3) is not equal to zero, and consequently, m is well defined.

Next, we will show that (3.3) describes a circle for any fixed b. It follows from (3.1) and (3.2) that
and
By (3.4) and the above two relations, we have
(3.6)
which implies that m lies in the upper half-plane if . It follows from (3.2) that
which, together with , yields that
It is equivalent to
(3.7)
By using (3.1), (3.7) can be expanded as
(3.8)
Moreover, setting , we have
(3.9)
It follows from the last relation in (3.9) that we have . By using (2.3) and (3.9), it can be verified that
(3.10)
It follows from the first relation in (3.9) and (3.5) that we have . Then (3.8) becomes
(3.11)
which implies that (3.3) forms a circle as k varies. It is evident that the center of is
It follows from Lemma 2.4 and (3.10) that
From (3.11), (3.9), (2.3), and (3.5) we have that the radius of is
(3.12)

Let denote the closed disk bounded by . We are going to show that the circle sequence is nested.

Set
From the first relation in (3.9), we have
Similarly,
So, it follows from (3.6) that
(3.13)
In the case of , the point is interior to the circle if . This shows that if and only if
Let with and consider the corresponding disks and . For any , we have
Hence, . This yields that . Therefore, is nested. Consequently, there are the following two alternatives:
1. (1)
as . In this case there is one point which is common to all the disks , . This is called the limit-point case. It follows from (3.12) that this case occurs if and only if
(3.14)

2. (2)

as . In this case there is a disk contained in all the disks , . This is called the limit-circle case. It follows from (3.12) that this case occurs if and only if the integral in (3.14) is convergent, i.e., .

Theorem 3.1 For every non-real , Eq. (1.1) has at least one non-trivial solution in .

Proof In the limit-circle case, it follows from the above discussion that .

Next, we will show that in the limit-point case. Let with and choose any . Then as and uniformly converges to on any finite interval , . Since the sequence is bounded from above and its upper bound is denoted by , then for ,
Hence, by the uniform convergence of , we have

for all ω. Therefore, . This completes the proof. □

Remark 3.1 Similar to the proof of Theorem 3.1, it can be easily verified that for any with in the limit-circle case. Clearly, and are linearly independent. Hence, all the solutions of Eq. (1.1) belong to for any with in the limit-circle case.

Remark 3.2 It follows from (3.14) and Theorem 3.1 that Eq. (1.1) has exactly one linearly independent solution in in the limit point case for any with .

Theorem 3.2 If Eq. (1.1) has two linearly independent solutions in for some , then this property holds for all .

Proof Suppose that Eq. (1.1) has two linearly independent solutions in for . Then and are in . For briefness, denote
For any , let be an arbitrary non-trivial solution of (1.1), and let be the solution of (1.1) with and with the initial values
From the variation of constants [[27], Theorem 3.73], we have
(3.15)
Replacing t with in (3.15) and using (ii) of Lemma 2.2, we obtain
which implies by the Hölder inequality in Lemma 2.2 that
It follows from the inequality
where A, B, C are non-negative numbers, that
Integrating the two sides of the above inequality with respect to t from a to , we get
which yields that
Hence,
(3.16)
The constant a can be chosen in advance so large that

It follows from (3.16) that and hence . Therefore, all the solutions of Eq. (1.1) are in . The proof is complete. □

At the end of this section, from the above discussions we present the classification of the limit cases for singular second-order linear equations over the infinite interval on time scales.

Definition 3.1 If Eq. (1.1) has only one linear independent solution in for some , then Eq. (1.1) is said to be in the limit-point case at . If Eq. (1.1) has two linear independent solutions in for some , then Eq. (1.1) is said to be in the limit-circle case at .

## 4 Several criteria of the limit-point and limit-circle cases

In this section, we establish several criteria of the limit-point and limit-circle cases for Eq. (1.1).

We first give two criteria of the limit-point case.

Theorem 4.1 Let and for all . If there exists a positive Δ-differentiable function on for some and two positive constants and such that for all ,
1. (i)

,

2. (ii)

,

3. (iii)

,

then Eq. (1.1) is in the limit-point case at .

Proof Suppose that Eq. (1.1) is in the limit-circle case at . By Theorem 3.2, all the solutions of
(4.1)
are in . Let and be the solutions of (4.1) satisfying the following initial conditions:
(4.2)
It is evident that and are two linearly independent solutions of (4.1) in . By Lemma 2.4, for all . Hence, we have
It follows from the Hölder inequality and assumption (iii) that
are divergent. Suppose
From (4.1) and assumption (i), we have
(4.3)
Applying integration by parts in Lemma 2.2, by (iii) in Lemma 2.1, we get
(4.4)
Again applying the Hölder inequality, from condition (ii), we have
(4.5)
where
Since
it follows from (4.3)-(4.5) that

It follows from the assumption that as . From the above relation and for all , we have that is ultimately positive. Therefore, as ; and consequently, does not belong to . This contradicts the assumption that all the solutions of (4.1) are in . Then Eq. (4.1) has at least one non-trivial solution outside of . It follows from Theorem 3.2 that Eq. (1.1) is in the limit-point case at . This completes the proof. □

Remark 4.1 Since and are two special time scales, Theorem 4.1 not only contains the criterion of the limit-point case for second-order differential equations [[5], Chapter 9, Theorem 2.4], but also the criterion of the limit-point case for second-order difference equation (1.3) [[15], Theorem 3.3].

The following corollary is a direct consequence of Theorem 4.1 by setting for .

Corollary 4.1 If , , is bounded below in , and , then Eq. (1.1) is in the limit-point case at .

Theorem 4.2 If
(4.6)

then Eq. (1.1) is in the limit-point case at .

Proof On the contrary, suppose that Eq. (1.1) is in the limit-circle case at . Let and be two linearly independent solutions of (1.1) in satisfying the initial conditions (4.2). By Lemma 2.4, we have
which, together with (2.1), implies that
So, we get
which implies
(4.7)
where . By the Hölder inequality and the assumption that , one has
Hence, it follows from (4.7) that

which is a contradiction to the assumption (4.6). Therefore, Eq. (1.1) is in the limit-point case at . This completes the proof. □

Remark 4.2 Let , Theorem 4.2 is the same as that obtained by Chen and Shi for second-order difference equations [[13], Corollary 3.1].

Next, we study the invariance of the limit cases under a bounded perturbation for the potential function q. Let and in [[27], Theorem 2.4(i)]. It follows from [[27], Theorem 2.36(i)], [[27], Theorem 2.39(i)], and [[27], Theorem 2.4(i)] that we have the following lemma, which is useful in the subsequent discussion.

Lemma 4.1 (Gronwall’s inequality)

Let be two non-negative functions on and M be a non-negative constant. If
(4.8)
then

where is defined as in (2.2).

The following result shows that if Eq. (1.1) is in the limit-circle case, so is it under a bounded perturbation for the potential function q.

Lemma 4.2 Let for all and be bounded with respect to on ; that is, there exists a positive constant M such that
(4.9)
Then Eq. (1.1) is in the limit-circle case at if and only if the equation
(4.10)

is in the limit-circle case at .

Proof Suppose that (4.10) is in the limit-circle case at . To show that Eq. (1.1) is in the limit-circle case, it suffices to show that each solution (4.1) is in by Theorem 3.2.

Let and be two solutions of the equation
(4.11)

satisfying the initial conditions (4.2). Then , are two linearly independent solutions in by Theorem 3.2.

Let be any solution of (4.1). Then
where . By the variation of constants [[27], Theorem 3.73] there exist two constants α and β such that
Hence, replacing t by and by (ii) in Lemma 2.2, we get
(4.12)
From (4.9) and (4.12), we have
(4.13)
Since , are solutions of Eq. (4.11), which satisfy the initial conditions (4.2), it follows from the existence-uniqueness theorem that for all . Let
From (4.13), we have
It follows from (i) of Lemma 2.2 that . By Lemma 4.1, we have

which implies that . Hence, ; and consequently, Eq. (1.1) is in the limit-circle case at .

On the other hand, using

one can easily conclude that if Eq. (1.1) is in the limit-circle case, then Eq. (4.10) is in the limit-circle case. This completes the proof. □

Theorem 4.3 Let for all and be bounded with respect to on . Then the limit cases for Eq. (1.1) are invariant.

Remark 4.3 Lemma 4.2 extends the related result [[13], Lemma 2.4] for the singular second-order difference equation to the time scales. In addition, let in Lemma 4.2, then we can directly prove [[31], Theorem 6.1] with the similar method.

## Declarations

### Acknowledgements

Many thanks are due to M Victoria Otero-Espinar (the editor) and the anonymous reviewers for helpful comments and suggestions. This research was supported by the NNSF of China (Grant 11071143 and 11101241), the NNSF of Shandong Province (Grant ZR2009AL003, ZR2010AL016, and ZR2011AL007), the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01), and the NSF of University of Jinan (XKY0918).

## Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Jinan
(2)
Department of Mathematics, Shandong University

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