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
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  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 . 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 :
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  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 ).
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 :
where q is real and continuous on , is the spectral parameter. By using the similar method, Huseynov  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  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  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.
(i) Iffis Δ-differentiable att, thenfis continuous att.
(ii) Iffandgare Δ-differentiable att, thenfgis 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 .
(iv) (Hölder’s inequality [, Lemma 2.2(iv)]) Letwith, then
Higer  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 ([, Theorem 6.1])
We define the Wronskian by
The following result is a direct consequence of the Lagrange identity [, Theorem 4.30].
In this section, we focus on the classification of the limit cases for singular second-order linear equations on time scales.
It can be verified that the integral identity
Next, we will show that (3.3) describes a circle for any fixed b. It follows from (3.1) and (3.2) that
By (3.4) and the above two relations, we have
It is equivalent to
By using (3.1), (3.7) can be expanded as
It follows from Lemma 2.4 and (3.10) that
From the first relation in (3.9), we have
So, it follows from (3.6) that
(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., .
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 ,
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.
From the variation of constants [, Theorem 3.73], we have
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
which yields that
The constant a can be chosen in advance so large that
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.
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
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 [, Chapter 9, Theorem 2.4], but also the criterion of the limit-point case for second-order difference equation (1.3) [, Theorem 3.3].
which, together with (2.1), implies that
So, we get
Hence, it follows from (4.7) that
Remark 4.2 Let , Theorem 4.2 is the same as that obtained by Chen and Shi for second-order difference equations [, Corollary 3.1].
Next, we study the invariance of the limit cases under a bounded perturbation for the potential function q. Let and in [, Theorem 2.4(i)]. It follows from [, Theorem 2.36(i)], [, Theorem 2.39(i)], and [, Theorem 2.4(i)] that we have the following lemma, which is useful in the subsequent discussion.
Lemma 4.1 (Gronwall’s inequality)
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.
where . By the variation of constants [, Theorem 3.73] there exist two constants α and β such that
From (4.9) and (4.12), we have
From (4.13), we have
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. □
Remark 4.3 Lemma 4.2 extends the related result [, 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 [, 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).
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