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Wiman’s formula for a second order dynamic equation
Advances in Difference Equations volume 2014, Article number: 61 (2014)
We derive Wiman’s asymptotic formula for the number of generalized zeros of (nontrivial) solutions of a second order dynamic equation on a time scale. The proof is based on the asymptotic representation of solutions via exponential functions on a time scale. By using the Jeffreys et al. approximation we prove Wiman’s formula for a dynamic equation on a time scale. Further we show that using the Hartman-Wintner approximation one can derive another version of Wiman’s formula. We also prove some new oscillation theorems and discuss the results by means of several examples.
Consider the equation
on , where is a time scale (a closed nonempty subset of the real numbers ℝ). Let be the number of generalized zeros of solutions of (1.1) on . The classical result of Wiman  for the continuous time scale case states that if (1.1) is oscillatory on , and is a differentiable function such that
where the symbol ∼ means that the ratio of the two quantities tends to 1 as .
In this paper under some restrictions on the graininess of the time scale and the asymptotic behavior of the coefficient , we obtain an explicit Jeffreys, Wentzel, Kramers and Brillouin (JWKB) asymptotic representation of solutions of (1.1). Using this representation we prove the analogue of Wiman’s formula for (1.1) on , which is given by
First we recall some basic definitions and notation used in time scale analysis (see [2, 3]). A time scale is an arbitrary nonempty closed subset of the real numbers. Since we are interested in the asymptotic behavior of solutions of (1.1), we will consider time scales which are unbounded above, i.e., . For we define the forward and backward jump operators by
The (forward) graininess function is defined by
For and , we define the delta derivative to be the number (provided it exists) with the property that for any , there exist a and a neighborhood of t such that
for all (see ).
A function is said to be rd-continuous provided it is continuous at right-dense points in and at each left-dense point t in the left hand limit at t exists (finite). The set of functions such that their n th delta derivative exists and is rd-continuous on is denoted by . In (1.1) we assume that and we say is a solution provided and for . We say that a complex-valued function is regressive on if for all . The set of regressive functions on will be denoted by ℛ. The set ℛ along with the addition ⊕ defined by
forms an Abelian group called the regressive group (, p.58). If the (generalized) exponential function is the unique solution of the IVP
and is given by the formula
where ln is the principal logarithmic function. The set of regressive functions on will be denoted by .
A solution of (1.1) is said to have a zero at if , and it has a node at if . A generalized zero of is defined as its zero or its node. A solution of (1.1) is said to be oscillatory if it has an infinite sequence of generalized zeros in , and nonoscillatory otherwise. Equation (1.1) is said to be oscillatory on if all of its solutions are oscillatory, and nonoscillatory otherwise.
Wiman’s result was extended by P. Hartman, Wintner , R. Potter , Z. Nehari , and H. Gingold . Many different results on the oscillation of solutions of second order differential equations appear in the book of Swanson .
After the introduction of the time scale calculus by Hilger , oscillation theorems for second order dynamic equations on a time scale have been studied by many authors (see for example [9–11] and the references therein).
2 Main results
Since the basic method of this paper uses the asymptotic representation of solutions of (1.1) we will introduce some definitions and notation that will be important in what follows.
The following lemma will be important in the sequel. We will use the following notation:
Lemma 2.1 Assume and are rd-continuous functions, on and let
for . Then , and
Proof First note that if , and
so , and hence , are well defined. Let be as in the statement of this lemma and consider
In a similar manner one can show that
Using (1.5) we get
where and are given by (2.3) and (2.4). Similarly
and consequently the formulas (2.2) hold. □
Example 2.1 Let , be as in Lemma 2.1.
If , then
If , then
Note that the function is defined from the JWKB approximation and may be chosen differently from another approximation (see Theorem 2.6).
Using the JWKB approximation method one can prove the following theorem.
Theorem 2.2 Assume that and the conditions
are satisfied for some . Then (1.1) is oscillatory if and only if
Theorem 2.3 Assume that and conditions (2.6)-(2.8) are satisfied. Then
Corollary 2.4 Assume that and conditions (2.6)-(2.8) are satisfied. Then
Example 2.2 Consider the difference equation
on the discrete time scale . We have
Conditions (2.6)-(2.8) are satisfied for sufficiently large n, and from (2.10) we get
For a continuous time scale () conditions (2.7)-(2.8) are automatically satisfied, and from Lemma 2.1, Theorem 2.2 we get the following corollary.
Corollary 2.5 Assume that is twice differentiable on and the condition
is satisfied. Then (1.3) is true, and (1.1) is oscillatory if and only if
Note that the necessary part of Corollary 2.5 is due to Leighton  under the assumption that w is a monotone function.
For monotonic functions w, Wiman’s condition (1.2) is less restrictive than (2.11) (for example , ). His proof is based on the transformation of the time variable which causes problems in the time scale setting. In this paper we give a new proof of Wiman’s formula that does not require the monotonicity of w, and it is based on an asymptotic representation of solutions of (1.1).
Example 2.3 From Corollary 2.5 it follows that the equation
is oscillatory if and only if . Indeed from it follows that both conditions (2.11) and (2.12) are satisfied. Condition (2.11) is restrictive, but without it Corollary 2.5 is not true since for the nonoscillatory equation condition (2.12) is satisfied. It is well known (Kneser ) that (2.13) is nonoscillatory if , and oscillatory if . Our condition is stronger, but it provides explicit asymptotic representations of the solutions and their derivatives as well.
Example 2.4 For the example
condition (2.11) is satisfied if , and condition (2.12) is satisfied if . For this example Leighton’s necessary criterion for oscillation  does not apply since is not monotone.
Using the Hartman-Wintner  approximation one can prove the following theorem.
Theorem 2.6 Assume that and for some the conditions
Then (1.1) is oscillatory if and only if
Theorem 2.7 Assume that and conditions (2.15)-(2.18) are satisfied. Then Nehari’s generalization of Wiman’s formula (see ) is true:
To get the asymptotic representation of solutions of the equation
we will use the following theorem.
Theorem 3.1 (, Theorem 2.4)
Let be complex-valued functions such that
Then for arbitrary constants , there exists a solution u of (3.1) that can be written in the form
where the error vector-function satisfies
where is defined as in (3.4), is the Euclidean vector (or matrix) norm: , and , are the entries of the vector .
Remark 3.1 If we seek asymptotic solutions of (3.1) in the Euler form,
then in view of , we see that the formula (3.4) becomes
where the Riccati functions , are defined by
Theorem 3.2 Assume , and condition (3.3) is satisfied with
Then for arbitrary constants , there exists a solution u of (3.1) that can be written in the form (3.6) and (3.7), with error estimate given by (3.8).
Proof In Lemma 2.1, take , where will be chosen later in this proof. Then
are regressive functions. Note that
so condition (3.2) of Theorem 3.1 is satisfied, where , .
By the quotient rule
where . Then
where we used . Using
Choosing the JWKB approximation:
Note that there is another possible choice of as a solution of the quadratic equation (the Hartman-Wintner approximation, see ). □
Proof of Theorem 2.2 Let and be defined as in the proof of Theorem 3.2. Then by Lemma 2.1 with
where and are given by (2.3) and (2.4).
Using the Euler formula we have
From Theorem 3.2, a general solution of (3.1) is of the form
Assume is a real-valued solution of (1.1) and , be arbitrary real constants. We will prove that , are also real-valued functions.
Indeed by solving the system (3.6) and (3.7) for we get
which implies , which in turn implies that , are complex conjugates of each other. Then, since , are complex conjugates, from (3.18) it follows that are real-valued functions.
Define so that
then from (3.17) we get
We extend the domain of the function
for t in the real interval , when by the formula
Note that is linear and continuous on and
In the same way one can extend the function Φ with domain to with domain the real interval . Since the extended function .
Later we will show that there exist points (which may not belong to ) such that
so the zeros of the extended solution are located at . Assuming from (2.4) we get
Further we have
and so is well defined. Indeed (3.25) is true when and when .
From (2.8) and (3.19) we see that the following limits exist:
Further, since is a continuous function on its domain, we get
and from (3.21)
Note that from the Leibniz formula (see Theorem 1.117 )
which means that is continuous and increasing (see Theorem 1.76 ), and hence it is an invertible function on , and exists for each .
To show that the solution of (3.1) has infinitely many generalized zeros on the time scale we will prove that between two zeros on ℝ of the solution there exists a generalized zero of in .
We will show that for all , for some , there exists a point between two zeros of :
We prove this by contradiction assuming that there is no such point . That is,
From (3.28) we get
Further we have the estimate
or, since ,
The last estimate contradicts (2.7) for sufficiently large n.
Further from (3.29)
which means that the point is the generalized zero. So (3.1) is oscillatory if and only if (2.9) is satisfied. □
Proof of Theorem 2.3 Assuming the number of generalized zeros of (3.1) on is given by (3.28). From (3.21) and (3.22)
Since between two zeros , there is a generalized zero , that is, , we get
when is the integral part of . □
Proof of Corollary 2.5 To deduce Corollary 2.5 from Lemma 2.1, Theorem 2.2 note that for the continuous case and conditions (2.7) and (2.8) are automatically satisfied. From (2.5) we get
so condition (2.6) simplifies to (2.11), and (2.9) becomes (2.12). □
Proofs of Theorem 2.6 and Theorem 2.7 The proofs of Theorem 2.6 and Theorem 2.7 are similar to the proofs of Theorem 2.2 and Theorem 2.3 correspondingly. The only difference is a different choice in (2.1) of the functions
where is a solution of quadratic equation . By this choice we have
and by a few different calculations, using (3.10) and (3.11), we get
For we have
In view of (2.15) in the case we have , and
so , , and hence , are well defined.
In view of the choice (3.32) we have
in view of (2.15). So is well defined. Also
The rest of the proof is similar to the earlier proof. □
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The authors would like to thank anonymous reviewers for very useful and constructive comments that helped to improve the original manuscript.
The authors declare that they have no competing interests.
Original idea to extend on a time scale Wiman’s formula by using JWKB approximation belongs to GH. The actual proofs were performed by all authors when GH spent his sabbatical at University of Nebraska-Lincoln. All authors read and approved the final manuscript.
About this article
- dynamic equation on a time scale
- oscillation theory
- number of zeros
- asymptotic representation of solutions
- Jeffreys, Wentzel, Kramers and Brillouin approximation