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# Iterated Oscillation Criteria for Delay Dynamic Equations of First Order

*Advances in Difference Equations*
**volume 2008**, Article number: 458687 (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*.

As is well known, a first-order delay differential equation of the form

where and , is oscillatory if

holds [2, Theorem 2.3.1]. Also the corresponding result for the difference equation

where , and , is

[2, Theorem 7.5.1]. Li [3] and Shen and Tang [4, 5] improved (1.2) for (1.1) to

where

Note that (1.2) is a particular case of (1.5) with . Also a corresponding result of (1.4) for (1.3) has been given in [6, Corollary 1], which coincides in the discrete case with our main result as

where is defined by a similar recursion in [6], as

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.

Now, we consider the first-order delay dynamic equation

where , is a time scale (i.e., any nonempty closed subset of ) with , , the delay function satisfies and for all . If , then (the usual derivative), while if , then (the usual forward difference). On a time scale, the *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.

A function is called *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.

Let be a nonoscillatory solution of (1.9). If

then

where

Proof.

Since (1.9) is linear, we may assume that is an eventually positive solution. Then, is eventually nonincreasing. Let for all , where . In view of (2.1), there exists and an increasing divergent sequence such that

Now, consider the function defined by

We see that and for all . Therefore, there exists such that and for all . Clearly, is a nondecreasing divergent sequence. Then, for all , we have

and

Thus, for all , we can calculate

and using (2.3),

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

For the statement of our main results, we introduce

for , where .

Lemma 2.2.

Let be a nonoscillatory solution of (1.9). If there exists such that

then

where is defined in (2.3).

Proof.

Since (1.9) is linear, we may assume that is an eventually positive solution. Then, is eventually nonincreasing. There exists such that for all . Thus, for all . We rewrite (1.9) in the form

for . Integrating (2.13) from to , where , we get

which implies . From (2.13), we see that

and thus

where . Note for . Now define

By the definition (2.17), we have for all and all , which yields for all . Then, we see that

holds for all (see also [13, Corollary 2.11]). Therefore, from (2.13), we have

for . Integrating (2.19) from to , where , we get

which implies that . Thus, for all , where , and we see that

for all . By induction, there exists with and

for all . To prove now (2.12), we assume on the contrary that . Taking on both sides of (2.22), we get

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 nonnegative with , one has

Now, we introduce

for and , where .

Lemma 2.5.

If there exists such that

holds, then (2.1) is true.

Proof.

There exists such that (see the proof of Lemma 2.2). Then, Lemma 2.4 implies

which yields

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.

We obtain the main results of [7, 8] by letting in Theorem 2.6. In this case, we have for all . Note that (2.1) and (2.26), respectively, reduce to

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

## 3. Particular Time Scales

This section is dedicated to the calculation of on some particular time scales. For convenience, we set

Example 3.1.

Clearly, if and , then (3.1) reduces to (1.6) and thus we have

by evaluating (2.10). For the general case, it is easy to see that

for . Thus if there exists such that

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.

Let and , where . Then (3.1) reduces to (1.8). From (2.10), we have

In the second line above, the well-known inequality between the arithmetic and the geometric mean is used. In the next step, we see that

By induction, we get

for . Therefore, every solution of (1.3) is oscillatory on provided that there exists satisfying

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.

Let and , where and . This time scale is different than the well-known time scales and since for . In the present case, (3.1) reduces to

and the exponential function takes the form

Therefore, one can show

and

For the general case, for , it is easy to see that

Therefore, if there exists such that

then every solution of

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.

Let and , where is an increasing divergent sequence and . Then, the exponential function takes the form

One can show that (2.10) satisfies

where (3.1) has the form

Therefore, existence of satisfying

ensures by Theorem 2.3 that every solution of

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

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### Keywords

- Difference Equation
- Delay Differential Equation
- Scale Theory
- Oscillation Theory
- Delay Function