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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ1_HTML.gif)
where and
, is oscillatory if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ2_HTML.gif)
holds [2, Theorem 2.3.1]. Also the corresponding result for the difference equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ3_HTML.gif)
where ,
and
, is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ4_HTML.gif)
[2, Theorem 7.5.1]. Li [3] and Shen and Tang [4, 5] improved (1.2) for (1.1) to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ7_HTML.gif)
where is defined by a similar recursion in [6], as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ12_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ13_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ15_HTML.gif)
Now, consider the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ17_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ18_HTML.gif)
Thus, for all , we can calculate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ19_HTML.gif)
and using (2.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ20_HTML.gif)
Letting tend to infinity, we see that (2.2) holds.
For the statement of our main results, we introduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ21_HTML.gif)
for , where
.
Lemma 2.2.
Let be a nonoscillatory solution of (1.9). If there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ22_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ24_HTML.gif)
for . Integrating (2.13) from
to
, where
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ25_HTML.gif)
which implies . From (2.13), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ26_HTML.gif)
and thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ27_HTML.gif)
where . Note
for
. Now define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ28_HTML.gif)
By the definition (2.17), we have for all
and all
, which yields
for all
. Then, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ29_HTML.gif)
holds for all (see also [13, Corollary 2.11]). Therefore, from (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ30_HTML.gif)
for . Integrating (2.19) from
to
, where
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ31_HTML.gif)
which implies that . Thus,
for all
, where
, and we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ32_HTML.gif)
for all . By induction, there exists
with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ33_HTML.gif)
for all . To prove now (2.12), we assume on the contrary that
. Taking
on both sides of (2.22), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ35_HTML.gif)
Now, we introduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ36_HTML.gif)
for and
, where
.
Lemma 2.5.
If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ37_HTML.gif)
holds, then (2.1) is true.
Proof.
There exists such that
(see the proof of Lemma 2.2). Then, Lemma 2.4 implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ38_HTML.gif)
which yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ41_HTML.gif)
Example 3.1.
Clearly, if and
, then (3.1) reduces to (1.6) and thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ42_HTML.gif)
by evaluating (2.10). For the general case, it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ43_HTML.gif)
for . Thus if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ45_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ46_HTML.gif)
By induction, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ47_HTML.gif)
for . Therefore, every solution of (1.3) is oscillatory on
provided that there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ49_HTML.gif)
and the exponential function takes the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ50_HTML.gif)
Therefore, one can show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ51_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ52_HTML.gif)
For the general case, for , it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ53_HTML.gif)
Therefore, if there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ54_HTML.gif)
then every solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ55_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ56_HTML.gif)
One can show that (2.10) satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ57_HTML.gif)
where (3.1) has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ58_HTML.gif)
Therefore, existence of satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ59_HTML.gif)
ensures by Theorem 2.3 that every solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F458687/MediaObjects/13662_2008_Article_1140_Equ60_HTML.gif)
is oscillatory on . We note again that
follows from (3.19).
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Bohner, M., Karpuz, B. & Öcalan, Ö. Iterated Oscillation Criteria for Delay Dynamic Equations of First Order. Adv Differ Equ 2008, 458687 (2008). https://doi.org/10.1155/2008/458687
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DOI: https://doi.org/10.1155/2008/458687