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Iterated Oscillation Criteria for Delay Dynamic Equations of First Order
Advances in Difference Equations volume 2008, Article number: 458687 (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.
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.
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
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 , as
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
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
Let be a nonoscillatory solution of (1.9). If
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
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 .
Let be a nonoscillatory solution of (1.9). If there exists such that
where is defined in (2.3).
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
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.
Assume (2.1). If there exists such that (2.11) holds, then every solution of (1.9) oscillates on .
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 .
If there exists such that
holds, then (2.1) is true.
There exists such that (see the proof of Lemma 2.2). Then, Lemma 2.4 implies
In view of (2.26), taking on both sides of the above inequality, we see that (2.1) holds. Hence, the proof is done.
Assume that there exists such that (2.26) and (2.11) hold. Then, every solution of (1.9) is oscillatory on .
The proof follows from Lemmas 2.1, 2.2, and 2.5.
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
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].
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].
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
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.
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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Bohner, M., Karpuz, B. & Öcalan, Ö. Iterated Oscillation Criteria for Delay Dynamic Equations of First Order. Adv Differ Equ 2008, 458687 (2008) doi:10.1155/2008/458687
- Difference Equation
- Delay Differential Equation
- Scale Theory
- Oscillation Theory
- Delay Function