# Iterated Oscillation Criteria for Delay Dynamic Equations of First Order

- M. Bohner
^{1}, - B. Karpuz
^{2}Email author and - Ö. Öcalan
^{2}

**2008**:458687

**DOI: **10.1155/2008/458687

© M. Bohner et al. 2008

**Received: **9 June 2008

**Accepted: **4 December 2008

**Published: **17 December 2008

## Abstract

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.

## 1. Introduction

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 [1], which unifies discrete and continuous analysis by a new theory called *time scale theory*.

Our results improve and extend the known results in [7, 8] to arbitrary time scales. We refer the readers to [9, 10] for some new results on the oscillation of delay dynamic equations.

*forward jump operator*and the

*graininess function*are defined by

where and . We refer the readers to [11, 12] for further results on time scale calculus.

*positively regressive*if and for all , and we write . It is well known that if , then there exists a positive function satisfying the initial value problem

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.

## 2. Main Results

We state the following lemma, which is an extension of [3, Lemma 2] and improvement of [10, Lemma 2].

Lemma 2.1.

Proof.

Letting tend to infinity, we see that (2.2) holds.

for , where .

Lemma 2.2.

where is defined in (2.3).

Proof.

which implies that , contradicting (2.11). Therefore, (2.12) holds.

Theorem 2.3.

Assume (2.1). If there exists such that (2.11) holds, then every solution of (1.9) oscillates on .

Proof.

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]).

for and , where .

Lemma 2.5.

holds, then (2.1) is true.

Proof.

In view of (2.26), taking on both sides of the above inequality, we see that (2.1) holds. Hence, the proof is done.

Theorem 2.6.

Assume that there exists such that (2.26) and (2.11) hold. Then, every solution of (1.9) is oscillatory on .

Proof.

The proof follows from Lemmas 2.1, 2.2, and 2.5.

Remark 2.7.

which indicates that (2.26) is implied by (2.1).

## 3. Particular Time Scales

Example 3.1.

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].

Example 3.2.

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].

Example 3.3.

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.

Example 3.4.

is oscillatory on . We note again that follows from (3.19).

## Authors’ Affiliations

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