Abstract
This paper is concerned with classification and criteria of the limit cases for singular secondorder linear equations on time scales. By the different cases of the limiting set, the equations are divided into two cases: the limitpoint and limitcircle cases just like the continuous and discrete cases. Several sufficient conditions for the limitpoint cases are established. It is shown that the limit cases are invariant under a bounded perturbation. These results unify the existing ones of secondorder singular differential and difference equations.
Keywords:
singular secondorder linear equation; time scales; limitpoint case; limitcircle case1 Introduction
In this paper, we consider classification and criteria of the limit cases for the following singular secondorder 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 Δderivative 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 wellbehaved coefficients are called regular. Otherwise, they are called singular. In 1910, Weyl [1] gave a dichotomy of the limitpoint and limitcircle cases for the following singular secondorder 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 WeylTitchmarsh theory [2,3]. Their work has been greatly developed and generalized to higherorder differential equations and Hamiltonian systems, and a classification and some criteria of limit cases were formulated [49]. Singular spectral problems of selfadjoint scalar secondorder 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 limitpoint and limitcircle cases were obtained for the following formally selfadjoint secondorder 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 WeylTitchmarsh theory of discrete linear Hamiltonian systems. Later, several sufficient conditions and sufficient and necessary conditions for the limitpoint and limitcircle cases were established for the singular secondorder 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 secondorder linear equations on time scales into two cases: limitpoint and limitcircle 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 secondorder 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 WeylTitchmarsh theory of Hamiltonian systems on time scales. It has been found that the secondorder singular differential and difference equations can be divided into limitpoint and limitcircle 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 limitpoint and limitcircle 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 limitpoint and limitcircle cases for singular secondorder linear equations on time scales is given by the geometric properties of the limiting set. Finally, several criteria of the limitpoint 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 nonempty closed set. The forward and backward jump operators are defined by
respectively, where , . A point is called rightscattered, rightdense, leftscattered, and leftdense 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 Δdifferentiable 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 Δdifferentiable for every , then f is said to be Δdifferentiable on . If f is Δdifferentiable at , then
If for all , then is called an antiderivative of f on . In this case, define the Δintegral by
For convenience, we introduce the following results ([[27], Chapter 1] and [[28], Chapter 1]), which are useful in this paper.
(i) Iffis Δdifferentiable att, thenfis continuous att.
(ii) Iffandgare Δdifferentiable att, thenfgis Δdifferentiable attand
(iii) Iffandgare Δdifferentiable att, and, thenis Δdifferentiable attand
A function f defined on is said to be rdcontinuous if it is continuous at every rightdense point in and its leftsided limit exists at every leftdense point in . The set of rdcontinuous functions is denoted by . The set of kth Δdifferentiable functions with rdcontinuous kth derivative is denoted by .
Lemma 2.2Iff, gare rdcontinuous functions on, then
(i) is rdcontinuous andfhas an antiderivative on;
(iii) (Integration by parts) .
(iv) (Hölder’s inequality [[29], Lemma 2.2(iv)]) Letwith, then
Let
A function is called regressive if
Higer [30] showed that for any given and for any given rdcontinuous and regressive g, the initial value problem
has a unique solution
Lemma 2.3 ([[27], Theorem 6.1])
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.
3 Classification
In this section, we focus on the classification of the limit cases for singular secondorder 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 halfplane 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
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 limitpoint 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 limitcircle 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 nonreal, Eq. (1.1) has at least one nontrivial solution in.
Proof In the limitcircle case, it follows from the above discussion that .
Next, we will show that in the limitpoint 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 limitcircle case. Clearly, and are linearly independent. Hence, all the solutions of Eq. (1.1) belong to for any with in the limitcircle 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 nontrivial 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 nonnegative 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 secondorder 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 limitpoint case at . If Eq. (1.1) has two linear independent solutions in for some , then Eq. (1.1) is said to be in the limitcircle case at .
4 Several criteria of the limitpoint and limitcircle cases
In this section, we establish several criteria of the limitpoint and limitcircle cases for Eq. (1.1).
We first give two criteria of the limitpoint case.
Theorem 4.1Letandfor all. If there exists a positive Δdifferentiable functiononfor someand two positive constantsandsuch that for all,
then Eq. (1.1) is in the limitpoint case at.
Proof Suppose that Eq. (1.1) is in the limitcircle 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 nontrivial solution outside of . It follows from Theorem 3.2 that Eq. (1.1) is in the limitpoint 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 limitpoint case for secondorder differential equations [[5], Chapter 9, Theorem 2.4], but also the criterion of the limitpoint case for secondorder 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 limitpoint case at.
Theorem 4.2If
then Eq. (1.1) is in the limitpoint case at.
Proof On the contrary, suppose that Eq. (1.1) is in the limitcircle 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 limitpoint case at . This completes the proof. □
Remark 4.2 Let , Theorem 4.2 is the same as that obtained by Chen and Shi for secondorder 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 nonnegative functions onandMbe a nonnegative constant. If
then
The following result shows that if Eq. (1.1) is in the limitcircle 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 limitcircle case atif and only if the equation
is in the limitcircle case at.
Proof Suppose that (4.10) is in the limitcircle case at . To show that Eq. (1.1) is in the limitcircle 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 existenceuniqueness 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 limitcircle case at .
On the other hand, using
one can easily conclude that if Eq. (1.1) is in the limitcircle case, then Eq. (4.10) is in the limitcircle 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 secondorder 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
Many thanks are due to M Victoria OteroEspinar (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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