Classification and criteria of the limit cases for singular second-order linear equations on time scales
© Zhang and Shi; licensee Springer 2012
Received: 10 April 2012
Accepted: 4 September 2012
Published: 19 September 2012
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.
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 .
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 ).
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.
If f is Δ-differentiable at t, then f is continuous at t.
- (ii)If f and g are Δ-differentiable at t, then fg is Δ-differentiable at t and
- (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 .
is rd-continuous and f has an anti-derivative on ;
for all .
(Integration by parts) .
- (iv)(Hölder’s inequality [, Lemma 2.2(iv)]) Let with , then
where and .
Lemma 2.3 ([, Theorem 6.1])
The following result is a direct consequence of the Lagrange identity [, Theorem 4.30].
Lemma 2.4 Let x and y be any two solutions of (1.1). Then is a constant in .
In this section, we focus on the classification of the limit cases for singular second-order linear equations on time scales.
It follows from (3.2) and that . Hence, the denominator in (3.3) is not equal to zero, and consequently, m is well defined.
Let denote the closed disk bounded by . We are going to show that the circle sequence is nested.
- (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)
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 .
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 .
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.
then Eq. (1.1) is in the limit-point case at .
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].
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 .
then Eq. (1.1) is in the limit-point case at .
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 [, 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)
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.
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.
satisfying the initial conditions (4.2). Then , are two linearly independent solutions in by Theorem 3.2.
which implies that . Hence, ; and consequently, Eq. (1.1) is in the limit-circle case at .
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 [, 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.
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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