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

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.

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

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:

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 [4-9]. 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]:

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 [16-23]). 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

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.

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

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 2.1Letand.

(i) Iffis Δ-differentiable att, thenfis continuous att.

(ii) Iffandgare Δ-differentiable att, thenfgis Δ-differentiable attand

(iii) Iffandgare Δ-differentiable att, and, thenis Δ-differentiable attand

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 kth Δ-differentiable functions with rd-continuous kth derivative is denoted by .

Lemma 2.2Iff, gare rd-continuous functions on, then

(i) is rd-continuous andfhas an anti-derivative on;

(ii) for all.

(iii) (Integration by parts) .

(iv) (Hölder’s inequality [[29], Lemma 2.2(iv)]) Letwith, then

whereand.

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

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

Letand. Then

implies

We define the Wronskian by

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

Lemma 2.4Letxandybe any two solutions of (1.1). Thenis a constant in.

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

Let , , with , and let (3.1) satisfy

Then

It can be verified that the integral identity

holds for any solution of (1.1) and for any . Setting , , in (3.4) and taking its imaginary part, we obtain

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

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

By using (3.1), (3.7) can be expanded as

Moreover, setting , we have

It follows from the last relation in (3.9) that we have . By using (2.3) and (3.9), it can be verified that

It follows from the first relation in (3.9) and (3.5) that we have . Then (3.8) becomes

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

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

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) 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

(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 3.1For 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 3.2If Eq. (1.1) has two linearly independent solutions infor 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

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,

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 .

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 4.1Letandfor all. If there exists a positive Δ-differentiable functiononfor someand two positive constantsandsuch that for all,

(i) ,

(ii) ,

(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

are in . Let and be the solutions of (4.1) satisfying the following initial conditions:

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

Applying integration by parts in Lemma 2.2, by (iii) in Lemma 2.1, we get

Again applying the Hölder inequality, from condition (ii), we have

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

Theorem 4.2If

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

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)

Letbe two non-negative functions onandMbe a non-negative constant. If

then

whereis 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 4.2Letfor allandbe bounded with respect toon; that is, there exists a positive constantMsuch that

Then Eq. (1.1) is in the limit-circle case atif and only if the equation

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

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

From (4.9) and (4.12), we have

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 4.3Letfor allandbe bounded with respect toon. 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.

The authors declare that they have no competing interests.

YS supervised the study and helped the revision. CZ carried out the main results of this article and drafted the manuscript. All the authors have read and approved the final manuscript.

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).

1. Weyl, H: Über gewöhnliche Differentialglichungen mit singularitäten und die zugehörigen entwicklungen. Math. Ann.. 68, 220–269 (1910). Publisher Full Text

2. Coddington, EA, Levinson, N: Theory of Ordinary Differential Equations, McGraw-Hill, New York (1955)

3. Tichmarsh, EC: Eigenfunction Expansions, Oxford University Press, Oxford (1962)

4. Hille, E: Lectures on Ordinary Differential Equations, Addison-Wesley, London (1969)

5. Hinton, DB, Shaw, JK: On Titchmarsh-Weyl -functions for linear Hamiltonian systems. J. Differ. Equ.. 40, 316–342 (1981). Publisher Full Text

6. Hinton, DB, Shaw, JK: Hamiltonian systems of limit point or limit circle type with both endpoints singular. J. Differ. Equ.. 50, 444–464 (1983). Publisher Full Text

7. Krall, AM: theory for singular Hamiltonian systems with one singular point. SIAM J. Math. Anal.. 20, 664–700 (1989). Publisher Full Text

8. Krall, AM: A limit-point criterion for linear Hamiltonian systems. Appl. Anal.. 61, 115–119 (1996). Publisher Full Text

9. Shi, Y: On the rank of the matrix radius of the limiting set for a singular linear Hamiltonian system. Linear Algebra Appl.. 376, 109–123 (2004)

10. Atkinson, FV: Discrete and Continuous Boundary Problems, Academic Press, New York (1964)

11. Hinton, DB, Lewis, RT: Spectral analysis of second order difference equations. J. Math. Anal. Appl.. 63, 421–438 (1978). Publisher Full Text

12. Jirari, A: Second-order Sturm-Liouville difference equations and orthogonal polynomials. Mem. Am. Math. Soc.. 542, 1–136 (1995)

13. Chen, J, Shi, Y: The limit circle and limit point criteria for second-order linear difference equations. Comput. Math. Appl.. 42, 465–476 (2001). Publisher Full Text

14. Shi, Y: Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems. Linear Algebra Appl.. 416, 452–519 (2006). Publisher Full Text

15. Sun, H, Shi, Y: Limit-point and limit-circle criteria for singular second-order linear difference equations with complex coefficients. Comput. Math. Appl.. 52, 539–554 (2006). Publisher Full Text

16. Erbe, L, Hilger, S: Sturmian theory on measure chains. Differ. Equ. Dyn. Syst.. 1, 223–246 (1993)

17. Chyan, CJ, Davis, JM, Henderson, J, Yin, WKC: Eigenvalue comparisons for differential equations on a measure chain. Electron. J. Differ. Equ.. 1998, 1–7 (1998)

18. Agarwal, RP, Bohner, M, Wong, PJY: Sturm-Liouville eigenvalue problem on time scales. Appl. Math. Comput.. 99, 153–166 (1999). Publisher Full Text

19. Davidson, FA, Rynne, BP: Eigenfunction expansions in spaces for boundary value problems on time scales. J. Math. Anal. Appl.. 335, 1038–1051 (2007). Publisher Full Text

20. Guseinov, GS: Eigenfunction expansions for a Sturm-Liouville problem on time scales. Int. J. Differ. Equ. Appl.. 2, 93–104 (2007)

21. Zhang, C, Shi, Y: Eigenvalues of second-order symmetric equations on time scales with periodic and antiperiodic boundary conditions. Appl. Math. Comput.. 203, 284–296 (2008). Publisher Full Text

22. Kong, Q: Sturm-Liouville problems on time scales with separated boundary conditions. Results Math.. 52, 111–121 (2008). Publisher Full Text

23. Amster, P, De Napoli, P, Pinasco, JP: Detailed asymptotic of eigenvalues on time scales. J. Differ. Equ. Appl.. 15, 225–231 (2009). Publisher Full Text

24. Zhang, C, Zhang, LF: Classification for a class of second-order singular equations on time scales. Proc 8th International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, pp. 73–76. IEEE, Piscataway Qingdao, July 31-Aug 1, 2007. (2007)

25. Huseynov, A: Limit point and limit circle cases for dynamic equations on time scales. Hacet. J. Math. Stat.. 39, 379–392 (2010)

26. Sun, S, Bohner, M, Chen, S: Weyl-Titchmarsh theory for Hamiltonian dynamic systems. Abstr. Appl. Anal.. 2010, Article ID 514760 (2010)

27. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston (2001)

28. Lakshmikantham, V, Sivasundaram, S, Kaymakcalan, B: Dynamic Systems on Measure Chains, Kluwer Academic, Dordrecht (1996)

29. Bohner, M, Lutz, DA: Asymptotic behavior of dynamic equations on time scales. J. Differ. Equ. Appl.. 7, 21–50 (2001). Publisher Full Text

30. Hilger, S: Special functions, Laplace and Fourier transform on measure chains. Dyn. Syst. Appl.. 8(3-4), 471–488 (1999)

31. Weidmann, J: Spectral Theory of Ordinary Differential Operators, Springer, Berlin (1987)