Wiman’s formula for a second order dynamic equation
© Erbe et al.; licensee Springer. 2014
Received: 8 November 2013
Accepted: 27 January 2014
Published: 7 February 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.
where the symbol ∼ means that the ratio of the two quantities tends to 1 as .
for all (see ).
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.
and consequently the formulas (2.2) hold. □
- (1)If , then
- (2)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.
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.
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).
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.
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.
we will use the following theorem.
Theorem 3.1 (, Theorem 2.4)
where is defined as in (3.4), is the Euclidean vector (or matrix) norm: , and , are the entries of the vector .
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).
so condition (3.2) of Theorem 3.1 is satisfied, where , .
Note that there is another possible choice of as a solution of the quadratic equation (the Hartman-Wintner approximation, see ). □
where and are given by (2.3) and (2.4).
Assume is a real-valued solution of (1.1) and , be arbitrary real constants. We will prove that , are also real-valued functions.
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.
In the same way one can extend the function Φ with domain to with domain the real interval . Since the extended function .
and so is well defined. Indeed (3.25) is true when and when .
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 .
The last estimate contradicts (2.7) for sufficiently large n.
which means that the point is the generalized zero. So (3.1) is oscillatory if and only if (2.9) is satisfied. □
when is the integral part of . □
so condition (2.6) simplifies to (2.11), and (2.9) becomes (2.12). □
so , , and hence , are well defined.
The rest of the proof is similar to the earlier proof. □
The authors would like to thank anonymous reviewers for very useful and constructive comments that helped to improve the original manuscript.
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