- Research Article
- Open Access
Iterated Oscillation Criteria for Delay Dynamic Equations of First Order
© M. Bohner et al. 2008
- Received: 9 June 2008
- Accepted: 4 December 2008
- Published: 17 December 2008
We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations. Examples, some of which coincide with well-known results on particular time scales, are provided to illustrate the applicability of our results.
- Difference Equation
- Delay Differential Equation
- Scale Theory
- Oscillation Theory
- Delay Function
Oscillation theory on and has drawn extensive attention in recent years. Most of the results on have corresponding results on and vice versa because there is a very close relation between and . This relation has been revealed by Hilger in , which unifies discrete and continuous analysis by a new theory called time scale theory.
where and , and it is called the exponential function and denoted by . Some useful properties of the exponential function can be found in [11, Theorem 2.36].
The setup of this paper is as follows: while we state and prove our main result in Section 2, we consider special cases of particular time scales in Section 3.
The proof is an immediate consequence of Lemmas 2.1 and 2.2.
We need the following lemmas in the sequel.
Lemma 2.4 (see [7, Lemma 2]).
holds, then (2.1) is true.
The proof follows from Lemmas 2.1, 2.2, and 2.5.
which indicates that (2.26) is implied by (2.1).
then every solution of (1.1) is oscillatory on . Note that (3.4) implies . Otherwise, we have for . This result for the differential equation (1.1) is a special case of Theorem 2.3 given in Section 2, and it is presented in [3, Theorem 1], [4, Corollary 1], and [5, Corollary 1].
Note that (3.8) implies that . Otherwise, we would have for . This result for the difference equation (1.3) is a special case of Theorem 2.3 given in Section 2, and a similar result has been presented in [6, Corollary 1].
is oscillatory on . Clearly, (3.14) ensures . This result for the -difference equation (3.15) is a special case of Theorem 2.3 given in Section 2, and it has not been presented in the literature thus far.
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