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Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales
Advances in Difference Equations volume 2009, Article number: 756171 (2009)
Abstract
By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations on a time scale
; here
is a quotient of odd positive integers with
and
real-valued positive rd-continuous functions defined on
. Our results not only extend some results established by Hassan in 2008 but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.
1. Introduction
The theory of time scales, which has recentlyreceived a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis (see Hilger [1]). Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [3], summarizes and organizes much of the time scale calculus. We refer also to the last book by Bohner and Peterson [4] for advances in dynamic equations on time scales. For the notation used hereinafter we refer to the next section that provides some basic facts on time scales extracted from Bohner and Peterson [3].
A time scale is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to plenty of applications, among them the study of population dynamic models which are discrete in season (and may follow a difference scheme with variable step-size or often modeled by continuous dynamic systems), die out, say in winter, while their eggs are incubating or dormant, and then in season again, hatching gives rise to a nonoverlapping population (see Bohner and Peterson [3]). Not only does the new theory of the so-called "dynamic equations" unify the theories of differential equations and difference equations but also it extends these classical cases to cases "in between", for example, to the so-called
-difference equations when
(which has important applications in quantum theory) and can be applied on different types of time scales like
, and
the space of the harmonic numbers.
In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to Bohner and Saker [5], Erbe [6], and Hassan [7]. However, there are few results dealing with the oscillation of the solutions of delay dynamic equations on time scales [8–15].
Following this trend, in this paper, we are concerned with oscillation for the second-order nonlinear delay dynamic equations

We assume that is a quotient of odd positive integers,
and
are positive, real-valued rd-continuous functions defined on
,
is strictly increasing, and
is a time scale,
and
as
,
which satisfies for some positive constant
,
for all nonzero
.
For oscillation of the second-order delay dynamic equations, Agarwal et al. [8] considered the second-order delay dynamic equations on time scales

and established some sufficient conditions for oscillation of (1.2).
Zhang and Shanliang [15] studied the second-order nonlinear delay dynamic equations on time scales

and the second-order nonlinear dynamic equations on time scales

where and
,
is continuous and nondecreasing
, and
for
, and they established the equivalence of the oscillation of (1.3) and (1.4), from which obtained some oscillation criteria and comparison theorems for (1.3). However, the results established in [15] are valid only when the graininess function
is bounded which is a restrictive condition. Also the restriction
is required.
Sahiner [13] considered the second-order nonlinear delay dynamic equations on time scales

where is continuous,
for
and
,
,
and
, and he proved that if there exists a
-differentiable function
such that

then every solution of (1.5) oscillates. Now, we observe that the condition (1.6) depends on an additional constant which implies that the results are not sharp (see Erbe et al. [10]).
Han et al. [12] investigated the second-order Emden-Fowler delay dynamic equations on time scales

established some sufficient conditions for oscillation of (1.7), and extended the results given in [8].
Erbe et al. [10] considered the general nonlinear delay dynamic equations on time scales

where and
are positive, real-valued rd-continuous functions defined on
,
is rd-continuous,
and
as
, and
such that satisfies for some positive constant
,
,
for all nonzero
, and they extended the generalized Riccati transformation techniques in the time scales setting to obtain some new oscillation criteria which improve the results given by Zhang and Shanliang [15] and Sahiner [13].
Clearly, (1.2), (1.3), (1.5), and (1.8) are the special cases of (1.1). In this paper, we consider the second-order nonlinear delay dynamic equation on time scales (1.1).
As we are interested in oscillatory behavior, we assume throughout this paper that the given time scale is unbounded above. We assume
, and it is convenient to assume
We define the time scale interval of the form
by
.
We shall also consider the two cases


The paper is organized as follows. In Section 2, we intend to use the Riccati transformation technique, a simple consequence of Keller's chain rule, and an inequality to obtain some sufficient conditions for oscillation of all solutions of (1.1). In Section 3, we give an example in order to illustrate the main results.
2. Main Results
In this section, we give some new oscillation criteria for (1.1). In order to prove our main results, we will use the formula

where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller's chain rule (see Bohner and Peterson [3, Theorem  1.90]). Also, we need the following auxiliary result.
For convenience, we note that

Lemma 2.1.
Assume that (1.9) holds and assume further that is an eventually positive solution of (1.1). Then there exists a
such that

The proof is similar to that of Hassan [7, Lemma  2.1], and so is omitted.
Lemma 2.2 (Chain Rule).
Assume that is strictly increasing and
is a time scale,
. Let
. If  
, and let
exist for
, then
exist, and

Proof.
Let be given and define
Note that
According to the assumptions, there exist neighborhoods
of
and
of
such that

Put and let
Then
and
and

The proof is completed.
Theorem 2.3.
Assume that (1.9) holds and . Furthermore, assume that there exists a positive function
and for all sufficiently large
, one has

Then (1.1) is oscillatory on .
Proof.
Suppose that (1.1) has a nonoscillatory solution on
. We may assume without loss of generality that
and
for all
. We shall consider only this case, since the proof when
is eventually negative is similar. In view of Lemma 2.1, we get (2.3). Define the function
by

Then . In view of (1.1) and (2.8) we get

When , using (2.1) and (2.4), we have

So, by (2.9)

hence, we get

and using Lemma 2.1, by we find
, by
we get
and so we obtain

When , using (2.1) and (2.4), we have

So, we get

So by the definition of , we obtain, for
,

where . Define
and
by

Then, using the inequality

yields

From (2.16), we find

Integrating the inequality (2.20) from to
, we obtain

which contradicts (2.7). The proof is completed.
Remark 2.4.
From Theorem 2.3, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of .
Theorem 2.5.
Assume that (1.9) holds and . Furthermore, assume that there exist functions
,
, where
such that

and has a nonpositive continuous
-partial derivation
with respect to the second variable and satisfies

and for all sufficiently large ,

where is a positive
-differentiable function and

Then (1.1) is oscillatory on .
Proof.
Suppose that (1.1) has a nonoscillatory solution on
. We may assume without loss of generality that
and
for all
. We proceed as in the proof of Theorem 2.3, and we get (2.16). Then from (2.16) with
replaced by
, we have

Multiplying both sides of (2.26), with replaced by
, by
, integrating with respect to
from
to
, we get

Integrating by parts and using (2.22) and (2.23), we obtain

Again, let , define
and
by

and using the inequality

we find

Therefore, from the definition of , we obtain

and this implies that

which contradicts (2.24). This completes the proof.
Now, we give some sufficient conditions when (1.10) holds, which guarantee that every solution of (1.1) oscillates or converges to zero on .
Theorem 2.6.
Assume (1.10) holds and . Furthermore, assume that there exists a positive function
, for all sufficiently large
, such that (2.7) or (2.22), (2.23), and (2.24) hold. If there exists a positive function
,
such that

then every solution of (1.1) is either oscillatory or converges to zero on .
Proof.
We proceed as in Theorem 2.3 or Theorem 2.5, and we assume that (1.1) has a nonoscillatory solution such that , for all
.
From the proof of Lemma 2.1, we see that there exist two possible cases for the sign of . The proof when
is an eventually positive is similar to that of the proof of Theorem 2.3 or Theorem 2.5, and hence it is omitted.
Next, suppose that for
. Then
is decreasing and
. We assert that
. If not, then
for
. Since
, there exists a number
such that
for
.
Defining the function we obtain from (1.1)

Hence, for , we have

because of . So, we have

By condition (2.34), we get as
, and this is a contradiction to the fact that
for
. Thus
and then
as
. The proof is completed.
3. Application and Example
Hassan [7] considered the second-order half-linear dynamic equations on time scales

where is a quotient of odd positive integers, and
and
are positive, real-valued rd-continuous functions defined on
, and he established some new oscillation criteria of (3.1). For example
Theorem 3.1 (Hassan [7, Theorem 2.1]).
Assume that (1.9) holds. Furthermore, assume that there exists a positive function such that

Then (3.1) is oscillatory on .
We note that (1.1) becomes (3.1) when , and Theorem 2.3 becomes Theorem 3.1, and so Theorem 2.3 in this paper essentially includes results of Hassan [7, Theorem  2.1].
Similarly, Theorem 2.5 includes results of Hassan [7, Theorem  2.2], and Theorem 2.6 includes results of Hassan [7, Theorem  2.4]. One can easily see that nonlinear delay dynamic equations (1.8) considered by Erbe et al. [10] are the special cases of (1.1), and the results obtained in [10] cannot be applied in (1.1), and so our results are new.
Example 3.2.
Consider the second-order delay dynamic equations

where is a quotient of odd positive integers,
is a time scale,
Let Then condition (1.9) holds,
, and we can find
such that
for
Take
By Theorem 2.3, we obtain

We conclude that (3.3) is oscillatory.
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Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project (20080441126), Shandong Postdoctoral Funded Project (200802018) and supported by Shandong Research Funds (Y2008A28), and also supported by the University of Jinan Research Funds for Doctors (B0621, XBS0843).
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Han, Z., Li, T., Sun, S. et al. Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales. Adv Differ Equ 2009, 756171 (2009). https://doi.org/10.1155/2009/756171
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DOI: https://doi.org/10.1155/2009/756171
Keywords
- Dynamic Equation
- Positive Function
- Nonoscillatory Solution
- Oscillation Criterion
- Partial Derivation