- Research Article
- Open access
- Published:
Oscillation Behavior of Third-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales
Advances in Difference Equations volume 2010, Article number: 586312 (2010)
Abstract
We will establish some oscillation criteria for the third-order Emden-Fowler neutral delay dynamic equations on a time scale
, where
is a quotient of odd positive integers with
,
and
real-valued positive rd-continuous functions defined on
. To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study. Some examples are considered to illustrate the main results.
1. Introduction
The study of dynamic equations on time-scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time-scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations.
Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2], Bohner and Guseinov [3], and references cited therein. A book on the subject of time-scales, by Bohner and Peterson [4], summarizes and organizes much of the time-scale calculus; see also the book by Bohner and Peterson [5] for advances in dynamic equations on time-scales.
In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of various equations on time-scales; we refer the reader to the papers [6–38]. To the best of our knowledge, it seems to have few oscillation results for the oscillation of third-order dynamic equations; see, for example, [14–16, 21, 35]. However, the paper which deals with the third-order delay dynamic equation is due to Hassan [21].
Hassan [21] considered the third-order nonlinear delay dynamic equations

where is required, and the author established some oscillation criteria for (1.1) which extended the results given in [16].
To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following third-order nonlinear neutral delay dynamic equations on time-scales up to now:

We assume that is a quotient of odd positive integers,
and
are positive real-valued rd-continuous functions defined on
such that
the delay functions
are rd-continuous functions such that
and
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
.
For the oscillation of neutral delay dynamic equations on time-scales, Mathsen et al. [26] considered the first-order neutral delay dynamic equations on time-scales

and established some new oscillation criteria of (1.3) which as a special case involve some well-known oscillation results for first-order neutral delay differential equations.
Agarwal et al. [7], ÅžahÃner [28], Saker [31], Saker et al. [33], Wu et al. [34] studied the second-order nonlinear neutral delay dynamic equations on time-scales

by means of Riccati transformation technique, the authors established some oscillation criteria of (1.4).
Saker [32] investigated the second-order neutral Emden-Fowler delay dynamic equations on time-scales

and established some new oscillation for (1.5).
Our purpose in this paper is motivated by the question posed in [26]: What can be said about higher-order neutral dynamic equations on time-scales and the various generalizations? We refer the reader to the articles [23, 24] and we will consider the particular case when the order is 3, that is, (1.2). Set By a solution of (1.2), we mean a nontrivial real-valued function
satisfying
and
and satisfying (1.2) for all
The paper is organized as follows. In Section 2, we apply a simple consequence of Keller's chain rule, devoted to the proof of the sufficient conditions which guarantee that every solution of (1.2) oscillates or converges to zero. In Section 3, some examples are considered to illustrate the main results.
2. Main Results
In this section we give some new oscillation criteria for (1.2). 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 [4, Theorem
]).
Before stating our main results, we begin with the following lemmas which are crucial in the proofs of the main results.
For the sake of convenience, we denote: for
Also, we assume that
there exists such that
and
Lemma 2.1.
Assume that holds. Further, assume that
is an eventually positive solution of (1.2). If

then there are only the following three cases for sufficiently large:
,
,
,

or
,
Proof.
Let be an eventually positive solution of (1.2). Then there exists
such that
and
for all
From (1.2) we have

Hence is strictly decreasing on
We claim that
eventually. Assume not, then there exists
such that

Then we can choose a negative and
such that

Dividing by and integrating from
to
we have

Letting then
by (2.2). Thus, there is a
such that for

Integrating the previous inequality from to
we obtain

Therefore, there exist 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 we get

It follows from this that either or
Since

which yields

If then there are two possible cases:
(1) eventually; or
(2) eventually.
If there exists a such that case (2) holds, then
exists, and
We claim that
Otherwise,
We can choose some positive integer
such that
for
Thus, we obtain

which implies that and from the definition of
we have
which contradicts
Now, we assert that
is bounded. If it is not true, there exists
with
as
such that

From

which implies that it contradicts that
Therefore, we can assume that

By we get

which implies that so
hence,
Assume that We claim that
eventually. Otherwise, we have
or
By
there exists
we can choose some positive integer
such that
for
and we obtain

which implies that and from the definition of
we have
which contradicts
or
Now, we have that
here
is finite. We assert that
is bounded. If it is not true, there exists
with
as
such that

From

which implies that it contradicts that
Therefore, we can assume that

By we get

which implies that so
hence,
This completes the proof.
In [4, Section ] the Taylor monomials
are defined recursively by

It follows from [4, Section ] that
for any time-scale, but simple formulas in general do not hold for
Lemma 2.2 (see [15, Lemma ]).
Assume that satisfies case (i) of Lemma 2.1. Then

Lemma 2.3.
Assume that is a solution of (1.2) satisfying case (i) of Lemma 2.1. If

then satisfies eventually

Proof.
Let be a solution of (1.2) such that case
of Lemma 2.1 holds for
Define

Thus

We claim that eventually. Otherwise, there exists
such that
for
Therefore,

which implies that is strictly increasing on
Pick
such that
for
Then we have

then for
By Lemma 2.2, for any
there exists
such that

Hence there exists so that

By the definition of we have that

From (1.2), we obtain

Integrating both sides of (2.35) from to
we get

which yields that

which contradicts (2.26). Hence and
is nonincreasing. The proof is complete.
Lemma 2.4.
Assume that holds and
is a solution of (1.2) which satisfies case (iii) of Lemma 2.1. If

where for
then
Proof.
Let be a solution of (1.2) such that case
of Lemma 2.1 holds for
Then
,
Next we claim that
Otherwise, there exists
such that
for all
By the definition of
we have that (2.35) holds. Integrating both sides of (2.35) from
to
we get

Integrating again from to
we have

Integrating again from to
we obtain

which contradicts (2.38), since by [23, Lemma ] and [3, Remark
], we get

Hence and completes the proof.
Theorem 2.5.
Assume that (2.2), (2.26), and (2.38) hold,
Furthermore, assume that there exists a positive function
such that for some
and for all constants

where Then every solution
of (1.2) oscillates or
Proof.
Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that
and
for all
Then by Lemma 2.1,
satisfies three cases. Assume that
satisfies case
Define the function
by

Then Using the product rule, we have

By the quotient rule, we get

By the definition of and (1.2), we obtain (2.35). From (2.35) and (2.44), we have

from (2.25) and (2.27), for any we obtain

hence by (2.48), we have

In view of from (2.1) and
of Lemma 2.1, we have

where By (2.49), we have

from we have
by
we have

so we get

by (2.44), we have

Therefore, we obtain

Integrating inequality (2.55) from to
, we obtain

which yields

for all large which contradicts (2.43). If
holds, from Lemma 2.1, then
If case
holds, by Lemma 2.4, then
The proof is complete.
Remark 2.6.
From Theorem 2.5, we can obtain different conditions for oscillation of all solutions of (1.2) with different choices of .
For example, let Now Theorem 2.5 yields the following result.
Corollary 2.7.
Assume that (2.2), (2.26), and (2.38) hold,
If

holds for some and for all constants
then every solution
of (1.2) is either oscillatory or
For example, let From Theorem 2.5, we have the following result which can be considered as the extension of the Leighton-Wintner Theorem.
Corollary 2.8.
Assume that (2.2), (2.26), and (2.38) hold, and
If

then every solution of (1.2) is either oscillatory or
In the following theorem, we present a new Kamenev-type oscillation criteria for (1.2).
Theorem 2.9.
Assume that (2.2), (2.26), and (2.38) hold,
Let
and
be as defined in Theorem 2.5. If for some
and for all constants

where and

then every solution of (1.2) oscillates or
Proof.
Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that
and
for all
Then by Lemma 2.1,
satisfies three cases. Assume that
satisfies case
We proceed as in the proof of Theorem 2.5 to get (2.54) for all
sufficiently large. Multiplying (2.54) by
and integrating from
to
we have

Integration by parts, we obtain

Next, we show that if and
then

If it is easy to see that (2.64) is an equality. If
then we get

Using the inequality

we obtain for

and from this we see that (2.64) holds. From (2.62)–(2.64), we get

Thus

which implies that

This easily leads to a contradiction of (2.60). If holds, from Lemma 2.1, then
If
holds, by Lemma 2.4, then
The proof is complete.
In the following theorem, we present a new Philos-type oscillation criteria for (1.2).
Theorem 2.10.
Assume that (2.2), (2.26), and (2.38) hold,
Let
and
be as defined in Theorem 2.5. Furthermore, assume that there exist functions
,
, where
such that

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

If for some and for all constants

where

where then every solution
of (1.2) oscillates or
Proof.
Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that
and
for all
Then by Lemma 2.1,
satisfies three cases. Assume that
satisfies case
We proceed as in the proof of Theorem 2.5 to get (2.54) for all
sufficiently large. Multiplying both sides of (2.54), with
replaced by
by
integrating with respect to
from
to
we have

Integrating by parts and using (2.71) and (2.72), we obtain

Therefore, we get

This easily leads to a contradiction of (2.73). If case holds, from Lemma 2.1, then
If case
holds, by Lemma 2.4, then
The proof is complete.
The following result can be considered as the extension of the Atkinson's theorem [39].
Theorem 2.11.
Assume that (2.2), (2.26), and (2.38) hold,
If

then every solution of (1.2) is either oscillatory or
Proof.
Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that
and
for all
Then by Lemma 2.1,
satisfies three cases. Assume that
satisfies case
Define the function

Using the product rule, (2.25) and (2.27), for any we have that

By (1.2), we have that (2.35) holds, then from (2.80), we calculate

where the last inequality is true because due to (2.1) and because

Upon integration we arrive at

from This contradicts (2.78). If case
holds, from Lemma 2.1, then
If case
holds, by Lemma 2.4, then
The proof is complete.
Theorem 2.12.
Assume that (2.2), (2.26), and (2.38) hold,
Furthermore, assume that there exists a positive function
such that for some
and for all constants

where is as defined as in Theorem 2.5. Then every solution
of (1.2) is either oscillatory or
Proof.
Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that
and
for all
Then by Lemma 2.1,
satisfies three cases. Assume
satisfies case
Define the function
as (2.44). We proceed as in the proof of Theorem 2.5 and we get (2.49). In view of
from (2.1) and
of Lemma 2.1, we have

from (2.27), there exists a constant such that
so

By (2.49), we have

Therefore, we obtain

Integrating inequality (2.88) from to
, we obtain

which yields

for all large which contradicts (2.84). If case
holds, from Lemma 2.1, then
If case
holds, by Lemma 2.4, then
The proof is complete.
Remark 2.13.
From Theorem 2.12, we can obtain different conditions for oscillation of all solutions of (1.2) with different choices of .
For example, let Now Theorem 2.12 yields the following results.
Corollary 2.14.
Assume that (2.2), (2.26), and (2.38) hold,
If

holds for some and for all constants
then every solution
of (1.2) is either oscillatory or
For example, let From Theorem 2.12, we have the following result which can be considered as the extension of the Leighton-Wintner theorem.
Corollary 2.15.
Assume that (2.2), (2.26), and (2.38) hold,
If (2.59) holds, then every solution
of (1.2) is either oscillatory or
In the following theorem, we present a new Kamenev-type oscillation criteria for (1.2).
Theorem 2.16.
Assume that (2.2), (2.26), and (2.38) hold,
Let
and
be as defined in Theorem 2.12. If for some
and for all constants

where and

then every solution of (1.2) oscillates or
The proof is similar to that of Theorem 2.9 using inequality (2.88), so we omit the details.
In the following theorem, we present a new Philos-type oscillation criteria for (1.2).
Theorem 2.17.
Assume that (2.2), (2.26), and (2.38) hold,
Let
and
be as defined in Theorem 2.12. Furthermore, assume that there exist functions
,
, where
such that (2.71) holds, and
has a nonpositive continuous
-partial derivation
with respect to the second variable and satisfies (2.72). If

holds for some and for all constants
where

where Then every solution
of (1.2) oscillates or
The proof is similar to that of the proof of Theorem 2.10 using inequality (2.88), so we omit the details.
The following result can be considered as the extension of the Belohorec's theorem [40].
Theorem 2.18.
Assume that (2.2), (2.26), and (2.38) hold
If

then every solution of (1.2) is either oscillatory or satisfies
Proof.
Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that
and
for all
Then by Lemma 2.1,
satisfies three cases. Assume that
satisfies case
From
and (2.1) we have

so

By (1.2), we have that (2.35) holds. Using (2.25) and (2.27), for any we obtain after dividing (2.35) by
for all large

So,

Upon integration we arrive at

This contradicts (2.96). If case holds, from Lemma 2.1, then
If case
holds, by Lemma 2.4, then
The proof is complete.
Remark 2.19.
One can easily see that the results obtained in [14–16, 21, 23, 24, 35] cannot be applied in (1.2), so our results are new.
3. Examples
In this section we give the following examples to illustrate our main results.
Example 3.1.
Consider the third-order neutral delay dynamic equations on time-scales

where is a quotient of odd positive integers,
Let . It is easy to see that (2.2), (2.26), and (2.38) hold. Also

Hence by Corollary 2.8, every solution of (3.1) is either oscillatory or
Example 3.2.
Consider the third-order neutral delay differential equation

Let . It is easy to see that all the conditions of Corollary 2.8 hold. Then by Corollary 2.8, every solution
of (3.3) is either oscillatory or satisfies
In fact,
is a solution of (3.3).
Example 3.3.
Consider the third-order delay dynamic equation

where is a quotient of odd positive integers.
For , we have
. Let
. It is easy to see that (2.2) and (2.38) hold, and

Hence (2.26) holds. Also

so (2.78) holds. By Theorem 2.11, every solution of (3.4) is either oscillatory or satisfies
.
Example 3.4.
Consider the third-order delay dynamic equation

where is a quotient of odd positive integers.
Let It is easy to see that (2.2), (2.26), and (2.38) hold. Also we have

Hence (2.96) holds. By Theorem 2.18, every solution of (3.7) is either oscillatory or satisfies
References
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1-2):18-56.
Agarwal RP, Bohner M, O'Regan D, Peterson A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics 2002,141(1-2):1-26. 10.1016/S0377-0427(01)00432-0
Bohner M, Guseinov GSh: Improper integrals on time scales. Dynamic Systems and Applications 2003,12(1-2):45-65.
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358.
Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
Agarwal RP, Bohner M, Saker SH: Oscillation of second order delay dynamic equations. The Canadian Applied Mathematics Quarterly 2005,13(1):1-17.
Agarwal RP, O'Regan D, Saker SH: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. Journal of Mathematical Analysis and Applications 2004,300(1):203-217. 10.1016/j.jmaa.2004.06.041
Akin-Bohner E, Hoffacker J: Oscillation properties of an Emden-Fowler type equation on discrete time scales. Journal of Difference Equations and Applications 2003,9(6):603-612. 10.1080/1023619021000053575
Akin-Bohner E, Bohner M, Saker SH: Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations. Electronic Transactions on Numerical Analysis 2007, 27: 1-12.
Bohner M, Saker SH: Oscillation of second order nonlinear dynamic equations on time scales. The Rocky Mountain Journal of Mathematics 2004,34(4):1239-1254. 10.1216/rmjm/1181069797
Bohner M: Some oscillation criteria for first order delay dynamic equations. Far East Journal of Applied Mathematics 2005,18(3):289-304.
Erbe L, Peterson A, Saker SH: Oscillation criteria for second-order nonlinear dynamic equations on time scales. Journal of the London Mathematical Society 2003,67(3):701-714. 10.1112/S0024610703004228
Erbe L, Peterson A, Saker SH: Oscillation criteria for second-order nonlinear delay dynamic equations. Journal of Mathematical Analysis and Applications 2007,333(1):505-522. 10.1016/j.jmaa.2006.10.055
Erbe L, Peterson A, Saker SH: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. Journal of Computational and Applied Mathematics 2005,181(1):92-102. 10.1016/j.cam.2004.11.021
Erbe L, Peterson A, Saker SH: Hille and Nehari type criteria for third-order dynamic equations. Journal of Mathematical Analysis and Applications 2007,329(1):112-131. 10.1016/j.jmaa.2006.06.033
Erbe L, Peterson A, Saker SH: Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation. The Canadian Applied Mathematics Quarterly 2006,14(2):129-147.
Grace SR, Agarwal RP, Bohner M, O'Regan D: Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations. Communications in Nonlinear Science and Numerical Simulation 2009,14(8):3463-3471. 10.1016/j.cnsns.2009.01.003
Han Z, Sun S, Shi B: Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2007,334(2):847-858. 10.1016/j.jmaa.2007.01.004
Han Z, Li T, Sun S, Zhang C: Oscillation for second-order nonlinear delay dynamic equations on time scales. Advances in Difference Equations 2009, 2009:-13.
Hassan TS: Oscillation criteria for half-linear dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2008,345(1):176-185. 10.1016/j.jmaa.2008.04.019
Hassan TS: Oscillation of third order nonlinear delay dynamic equations on time scales. Mathematical and Computer Modelling 2009,49(7-8):1573-1586. 10.1016/j.mcm.2008.12.011
Jia B, Erbe L, Peterson A: New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales. Computers & Mathematics with Applications 2008,56(10):2744-2756. 10.1016/j.camwa.2008.05.014
Karpuz B: Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients. Electronic Journal of Qualitative Theory of Differential Equations 2009,2009(34):1-14.
Karpuz B: Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations. Applied Mathematics and Computation 2009,215(6):2174-2183. 10.1016/j.amc.2009.08.013
Li T, Han Z, Sun S, Yang D: Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales. Advances in Difference Equations 2009, 2009:-10.
Mathsen RM, Wang Q-R, Wu H-W: Oscillation for neutral dynamic functional equations on time scales. Journal of Difference Equations and Applications 2004,10(7):651-659. 10.1080/10236190410001667968
Şahiner Y: Oscillation of second-order delay differential equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2005,63(5–7):e1073-e1080.
ÅžahÃner Y: Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales. Advances in Difference Equations 2006, 2006:-9.
Saker SH: Oscillation criteria of second-order half-linear dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005,177(2):375-387. 10.1016/j.cam.2004.09.028
Saker SH: Oscillation of nonlinear dynamic equations on time scales. Applied Mathematics and Computation 2004,148(1):81-91. 10.1016/S0096-3003(02)00829-9
Saker SH: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. Journal of Computational and Applied Mathematics 2006,187(2):123-141. 10.1016/j.cam.2005.03.039
Saker SH: Oscillation of second-order neutral delay dynamic equations of Emden-Fowler type. Dynamic Systems and Applications 2006,15(3-4):629-644.
Saker SH, Agarwal RP, O'Regan D: Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales. Applicable Analysis 2007,86(1):1-17. 10.1081/00036810601091630
Wu H-W, Zhuang R-K, Mathsen RM: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Applied Mathematics and Computation 2006,178(2):321-331. 10.1016/j.amc.2005.11.049
Yu Z-H, Wang Q-R: Asymptotic behavior of solutions of third-order nonlinear dynamic equations on time scales. Journal of Computational and Applied Mathematics 2009,225(2):531-540. 10.1016/j.cam.2008.08.017
Zhang BG, Shanliang Z: Oscillation of second-order nonlinear delay dynamic equations on time scales. Computers & Mathematics with Applications 2005,49(4):599-609. 10.1016/j.camwa.2004.04.038
Zhang BG, Deng X: Oscillation of delay differential equations on time scales. Mathematical and Computer Modelling 2002,36(11–13):1307-1318. 10.1016/S0895-7177(02)00278-9
Zhu Z-Q, Wang Q-R: Existence of nonoscillatory solutions to neutral dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2007,335(2):751-762. 10.1016/j.jmaa.2007.02.008
Atkinson FV: On second-order non-linear oscillations. Pacific Journal of Mathematics 1955, 5: 643-647.
Belohorec Å : Non-linear oscillations of a certain non-linear second-order differential equation. Matematicko-Fyzikalny ÄŒasopis: Slovenskej Akademie Vied 1962, 12: 253-262.
Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have to the present improved version of the original manuscript. 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), Shandong Research Funds (Y2008A28), and also supported by University of Jinan Research Funds for Doctors (B0621, XBS0843).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Han, Z., Li, T., Sun, S. et al. Oscillation Behavior of Third-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales. Adv Differ Equ 2010, 586312 (2010). https://doi.org/10.1155/2010/586312
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/586312