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Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales
Advances in Difference Equations volume 2010, Article number: 450264 (2010)
Abstract
By means of the averaging technique and the generalized Riccati transformation technique, we establish some oscillation criteria for the secondorder quasilinear neutral delay dynamic equations , , where , and the time scale interval is . Our results in this paper not only extend the results given by Agarwal et al. (2005) but also unify the oscillation of the secondorder neutral delay differential equations and the secondorder neutral delay difference equations.
1. Introduction
The theory of time scales, which has recently received 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 and references cited therein. A book on the subject of time scale, by Bohner and Peterson [2], summarizes and organizes much of the time scale calculus; we refer also the last book by Bohner and Peterson [3] for advances in dynamic equations on time scales.
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 (see Bohner and Peterson [2]).
In the last few years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales which attempts to harmonize the oscillation theory for the continuous and the discrete to include them in one comprehensive theory and to eliminate obscurity form both, for instance, the papers [4–20] and the reference cited therein.
For oscillation of delay dynamic equations on time scales, see recently papers [21–32]. However, there are very few results dealing with the oscillation of the solutions of neutral delay dynamic equations on time scales; we refer the reader to [33–44].
Agarwal et al. [33] and Saker [37] consider the secondorder nonlinear neutral delay dynamic equations on time scales:
where , is a quotient of odd positive integer, are positive constants, such that for all nonzero , and there exists a nonnegative function defined on satisfing .
Agwo [35] examines the oscillation of the secondorder nonlinear neutral delay dynamic equations:
Li et al. [36] discuss the existence of nonoscillatory solutions to the secondorder neutral delay dynamic equation of the form
Saker et al. [38, 39], Sahíner [40], and Wu et al. [43] consider the secondorder neutral delay and mixedtype dynamic equations on time scales:
where , is a quotient of odd positive integer, , such that for all nonzero , and there exists a nonnegative function defined on satisfing .
Zhu and Wang [44] study existence of nonoscillatory solutions to neutral dynamic equations on time scales:
Recently, Tripathy [42] has established some new oscillation criteria for secondorder nonlinear delay dynamic equations of the form
where , is a quotient of odd positive integer, are positive constants, and .
To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following secondorder nonlinear neutral delay dynamic equations on time scales up to now:
where , and the time scale interval is .
In what follows we assume the following:

(A1)
and are positive constants with ;
 (A2)

(A3)
and constant;

(A4)
for and .
To develop the qualitative theory of delay dynamic equations on time scales, in this paper, by using the averaging technique and the generalized Riccati transformation, we consider the secondorder nonlinear neutral delay dynamic equation on time scales (1.7) and establish several oscillation criteria. Our results in this paper not only extend the results given but also unify the oscillation of the secondorder quasilinear delay differential equation and the secondorder quasilinear delay difference equation. Applications to equations to which previously known criteria for oscillation are not applicable are given.
By a solution of (1.7), we mean a nontrivial realvalued function , , which has the property and satisfying (1.7) for . Our attention is restricted to those solutions of (1.7) which exist on some half line with for any . A solution of (1.7) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory. Equation (1.7) is called oscillatory if all solutions are oscillatory.
Equation (1.7) includes many other special important equations; for example, if , (1.7) is the prototype of a wide class of nonlinear dynamic equations called EmdenFowler neutral delay superlinear dynamic equation:
where .
If , (1.7) is the prototype of nonlinear dynamic equations called EmdenFowler neutral delay sublinear dynamic equation:
where .
We note that if , then (1.7) becomes secondorder nonlinear delay dynamic equation on time scales:
If , then (1.7) becomes secondorder nonlinear delay dynamic equation on time scales:
If , then (1.7) becomes secondorder nonlinear delay dynamic equation on time scales:
If , then (1.7) becomes secondorder nonlinear delay dynamic equation on time scales:
It is interesting to study (1.7) because the continuous version and its special cases have several physical applications, see [1] and when is a discrete variable, and include its special cases also, are important in applications.
The paper is organized as follows: In the next section we present the basic definitions and apply a simple consequence of Keller's chain rule, Young's inequality:
and the inequality
where and are nonnegative constants, devoted to the proof of the sufficient conditions for oscillation of all solutions of (1.7). In Section 3, we present some corollaries to illustrate our main results.
2. Main Results
In this section we shall give some oscillation criteria for (1.7) under the cases when and . It will be convenient to make the following notations in the remainder of this paper. Define
We define the function space as follows: provided that is defined for and is rdcontinuous function and nonnegative. For given function , we set
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 [2, Theorem ]).
Also, we assume the condition and , and there exists such that and
Lemma 2.1.
Assume . If is an eventually positive solution of (1.7), then, there exists a such that for . Moreover,
Lemma 2.2.
Assume ,
If is an eventually positive solution of (1.7), then
The proof of Lemmas 2.1 and 2.2 is similar to that of Saker et al. [39, Lemma ]; so it is omitted.
Lemma 2.3.
Assume that the condition holds:
If is an eventually positive solution of (1.7), then
or .
Proof.
Since is an eventually positive solution of (1.7), there exists a number such that , , and for all . In view of (1.7), we have
By , then , we get
Let , then (2.8) holds, and is an eventually decreasing function. It follows that
We prove that is eventually positive.
Otherwise, there exists a such that , then we have for , and hence which implies that
Therefore, there exists and such that
We can choose some positive integer such that for Thus, we obtain
The above inequality implies that for sufficiently large which contradicts the fact that eventually. Hence eventually. Consequently, there are two possible cases:

(i)
eventually;

(ii)
eventually.
If Case holds, we can get
Actually, by , we can easily verify that Using (2.3), we get
From , we have that is eventually negative.
Let since is eventually positive, so is eventually increasing. Therefore, is either eventually positive or eventually negative. If is eventually negative, then there is a such that for . So,
which implies that is strictly increasing for . Pick so that for . Then
so that for .
By (2.8), we have
which implies that
which contradicts (2.7). Hence, without loss of generality, there is a such that , that is, for . Consequently,
and we have that is strictly decreasing for .
If there exists a such that Case holds, then exists, and we claim that Otherwise, We can choose some positive integer such that for Thus, we obtain
which implies that and which contradicts
Now, we assert that is bounded. If it is not true, there exists with as such that
From and
which implies that it contradicts the existence of Therefore, we can assume that
By we get
thus and Hence, The proof is complete.
Theorem 2.4.
Assume that (2.5) holds, ,
Then (1.7) is oscillatory on .
Proof.
Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. We shall consider only this case, since the proof when is eventually negative is similar. In view of Lemmas 2.1 and 2.2, there exists a such that , and for .
From (2.4) we have for
and hence
So,
So,
Now note that imply
This contradicts (2.28). The proof is complete.
Remark 2.5.
Theorem 2.4 includes results of Agarwal et al. [21, Theorem ] and Han et al. [25, Theorem ].
Theorem 2.6.
Assume that the condition and (2.7) hold:
Then every solution of (1.7) either oscillates or tends to zero as .
Proof.
Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and is strictly decreasing for .
Then from (2.8), we have for
Since the rest of the proof is similar to Theorem 2.4, so we omit the detail. The proof is complete.
Theorem 2.7.
Assume that (2.5) holds. , let ,
then (1.7) is oscillatory on , where .
Proof.
Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemmas 2.1 and 2.2, there exists a such that , and for . Define the function by
We get
If , by (2.3), we get
So, from (2.4) we have
By Young's inequality (1.14), we obtain that
From being strictly decreasing, , by (2.37) and (2.40), we get that
that is,
So,
Using the inequality (1.15) we have
Integrating the inequality above from to we obtain
Therefore,
which contradicts (2.36).
If , proceeding as the proof of above, we have (2.37) and (2.38). By (2.3), we get that
So, from (2.4) we have
Since the rest of the proof is similar to that of above, so we omit the detail. The proof is complete.
Theorem 2.8.
Assume that the condition and (2.7) hold, let ,
then every solution of (1.7) either oscillates or tends to zero as , where .
Proof.
Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and for .
Since the rest of the proof is similar to Theorem 2.7, so we omit the detail. The proof is complete.
Theorem 2.9.
Assume that (2.5) holds. , let ,
then (1.7) is oscillatory on .
Proof.
We prove only case . The proof of case is similar. Proceeding as the proof of Theorem 2.7, we have (2.37) and (2.43). Replacing in (2.43) by , then multiplying (2.43) by , and integrating from to , , we have
Integrating by parts and using the fact that , we get
So,
that is,
Hence,
Using the inequality (1.15) we have
Set , so,
This contradicts (2.51) and finishes the proof.
Theorem 2.10.
Assume that the condition and (2.7) hold, let ,
then every solution of (1.7) either oscillates or tends to zero as .
Proof.
Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and for .
Since the rest of the proof is similar to Theorem 2.9, so we omit the detail. The proof is complete.
Following the procedure of the proof of Theorem 2.7, we can also prove the following theorem.
Theorem 2.11.
Assume that (2.5) holds. , let ,
there exists such that
or
and for any ,
and then (1.7) is oscillatory on , where .
Proof.
We prove only case . The proof of case is similar. Proceeding as the proof of Theorem 2.9, we have (2.56) and (2.57). So we have for all ,
By (2.64), we obtain
Define
Then, by (2.56) and (2.64), we have that
We claim that
Suppose, to the contrary, that
By (2.60), there exists a positive constant such that
Let be an arbitrary positive number, then it follows from (2.70) that there exists a such that
Hence,
By (2.71), there exists a such that , which implies that . Since is arbitrary, then
In view of (2.68), we consider a sequence with satisfying
Then, there exists a constant such that for all sufficiently large . Since (2.70) ensures that
so,
hold for all sufficiently large . Therefore,
On the other hand, from the definition of we can obtain, by Hölder's inequality,
and, accordingly,
So, because of (2.78), we get
which gives that
contradicting (2.61). Hence, (2.69) holds. In view of (2.66), from (2.69), we have
which contradicts (2.62). The proof is complete.
Theorem 2.12.
Assume that the condition and (2.7) hold, let , there exists such that (2.60) and (2.61), and either (2.62) or (2.63) hold, and for any ,
and then every solution of (1.7) either oscillates or tends to zero as .
Proof.
Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and for .
Since the rest of the proof is similar to Theorem 2.11, so we omit the detail. The proof is complete.
Following the procedure of the proof of Theorem 2.8, we can also prove the following theorem.
Theorem 2.13.
Assume that (2.5) and (2.60) hold, , let ,
there exists such that (2.62) or (2.63) holds, and for any ,
and then (1.7) is oscillatory on .
Theorem 2.14.
Assume that the condition , (2.7), and (2.60) hold, let , there exists such that (2.62) or (2.63) holds, and for any ,
and then every solution of (1.7) either oscillates or tends to zero as .
Remark 2.15.
As Theorem 2.7–Theorem 2.14 are rather general, it is convenient for applications to derive a number of oscillation criteria with the appropriate choice of the functions and .
3. Examples
In this section, we give some examples to illustrate our main results.
Example 3.1.
Consider the following delay dynamic equations on time scales:
where
with , for and . Then, by Theorem 2.4, we have
Then (3.1) is oscillatory on .
Example 3.2.
Consider the secondorder delay dynamic equations on time scales:
where
with , for and . Then,
let , and by Theorem 2.7, we have
Then (3.4) is oscillatory on .
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Acknowledgments
This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation funded project (20080441126, 200902564), Shandong Postdoctoral funded project (200802018), the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and also supported by University of Jinan Research Funds for Doctors (B0621, XBS0843).
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Han, Z., Sun, S., Li, T. et al. Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales. Adv Differ Equ 2010, 450264 (2010). https://doi.org/10.1155/2010/450264
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Keywords
 Nonoscillatory Solution
 Oscillation Criterion
 Delay Dynamic Equation
 Delay Difference Equation
 Riccati Transformation