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Hindawi Publishing Corporation

Advances in Difference Equations

Volume 2010, Article ID 586312, 23 pages

doi:10.1155/2010/586312

Research Article

Oscillation Behavior of Third-Order

Neutral Emden-Fowler Delay Dynamic Equations

on Time Scales

Zhenlai Han,1,2Tongxing Li,1Shurong Sun,1,3

and Chenghui Zhang2

1School of Science, University of Jinan, Jinan, Shandong 250022, China

2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

3Department of Mathematics and Statistics, Missouri University of Science and Technology,

Rolla, MO 65409-0020, USA

Correspondence should be addressed to Shurong Sun, sshrong@163.com

Received 14 September 2009; Revised 28 November 2009; Accepted 10 December 2009

Academic Editor: Leonid Berezansky

Copyright q 2010 Zhenlai Han et al. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

We will establish some oscillation criteria for the third-order Emden-Fowler neutral delay dynamic

equations ?r?t??x?t? − a?t?x?τ?t???ΔΔ?Δ? p?t?xγ?δ?t?? ? 0 on a time scale T, where γ > 0 is a

quotient of odd positive integers with r, a, and p real-valued positive rd-continuous functions

defined on T. 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

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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

?

c?t?

?

?a?t?xΔ?t??Δ?γ?Δ

? f?t,x?τ?t??? ? 0,t ∈ T,

?1.1?

where τ?σ?t?? ? σ?τ?t?? 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:

?

r?t??x?t? − a?t?x?τ?t???ΔΔ?Δ? p?t?xγ?δ?t?? ? 0,t ∈ T.

?1.2?

We assume that γ > 0 is a quotient of odd positive integers, r, a and p are positive

real-valued rd-continuous functions defined on T such that rΔ?t? ≥ 0, 0 < a?t? ≤ a0 <

1, limt→∞a?t? ? a < 1, the delay functions τ : T → T, δ : T → T are rd-continuous

functions such that τ?t? ≤ t, δ?t? ≤ t, and limt→∞τ?t? ? limt→∞δ?t? ? ∞.

As we are interested in oscillatory behavior, we assume throughout this paper that the

given time-scale T is unbounded above. We assume t0 ∈ T and it is convenient to assume

t0> 0. We define the time-scale interval of the form ?t0,∞?Tby ?t0,∞?T? ?t0,∞? ∩ T.

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

?y?t? − r?t?y?τ?t???Δ? p?t?y?δ?t?? ? 0,t ∈ T,

?1.3?

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?, S ¸ah´ ıner ?28?, Saker ?31?, Saker et al. ?33?, Wu et al. ?34? studied the

second-order nonlinear neutral delay dynamic equations on time-scales

?

r?t???y?t? ? p?t?y?τ?t???Δ?γ?Δ? f?t,y?δ?t???? 0,t ∈ T,

?1.4?

by means of Riccati transformation technique, the authors established some oscillation

criteria of ?1.4?.

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Saker ?32? investigated the second-order neutral Emden-Fowler delay dynamic

equations on time-scales

?

a?t??y?t? ? r?t?y?τ?t???Δ?Δ? p?t?yγ?δ?t?? ? 0,

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 t−1 :? mint∈?t0,∞?T{τ?t?,δ?t?}. By a solution of

?1.2?, we mean a nontrivial real-valued function x ∈ Crd??t−1,∞?T,R? satisfying x − ax ◦ τ ∈

C2

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.

t ∈ T,

?1.5?

rd??t0,∞?T,R? and r?x−ax◦τ?ΔΔ∈ C1

rd??t0,∞?T,R?, and satisfying ?1.2? for all t ∈ ?t0,∞?T.

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

??x?t??γ?Δ? γ

?1

0

?hxσ?t? ? ?1 − h?x?t??γ−1xΔ?t?dh,

?2.1?

where x 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 1.90??.

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: z?t? ? x?t?−a?t?x?τ?t??, for t ∈ ?t0,∞?T. Also,

we assume that

?H? there exists {ck}k∈N0⊂ T such that limk→∞ck? ∞ and τ?ck?1? ? ck.

Lemma 2.1. Assume that ?H? holds. Further, assume that x is an eventually positive solution of

?1.2?. If

?∞

t0

Δt

r?t?? ∞,

?2.2?

then there are only the following three cases for t ≥ t1sufficiently large:

?i? z?t? > 0, zΔ?t? > 0, zΔΔ?t? > 0, zΔΔΔ?t? < 0,

?ii? z?t? < 0, zΔ?t? > 0, zΔΔ?t? > 0, zΔΔΔ?t? < 0, limt→∞x?t? ? 0,

or

?iii? z?t? > 0, zΔ?t? < 0, zΔΔ?t? > 0, zΔΔΔ?t? < 0, limt→∞z?t? ? l ≥ 0, limt→∞x?t? ?

l/?1 − a? ≥ 0.

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Proof. Let x be an eventually positive solution of ?1.2?. Then there exists t1 ≥ t0such that

x?t? > 0, x?τ?t?? > 0, and x?δ?t?? > 0 for all t ≥ t1. From ?1.2? we have

?

r?t?zΔΔ?t?

?Δ

? −p?t?xγ?δ?t?? < 0,t ≥ t1.

?2.3?

Hence r?t?zΔΔ?t? is strictly decreasing on ?t1,∞?T. We claim that zΔΔ?t? > 0 eventually.

Assume not, then there exists t2≥ t1such that

r?t?zΔΔ?t? < 0,t ≥ t2.

?2.4?

Then we can choose a negative c and t3≥ t2such that

r?t?zΔΔ?t? ≤ c < 0,t ≥ t3.

?2.5?

Dividing by r?t? and integrating from t3to t, we have

zΔ?t? ≤ zΔ?t3? ? c

?t

t3

Δs

r?s?.

?2.6?

Letting t → ∞, then zΔ?t? → −∞ by ?2.2?. Thus, there is a t4≥ t3such that for t ≥ t4,

zΔ?t? ≤ zΔ?t4? < 0.

?2.7?

Integrating the previous inequality from t4to t, we obtain

z?t? − z?t4? ≤ zΔ?t4??t − t4?.

?2.8?

Therefore, there exist d > 0 and t5≥ t4such that

x?t? ≤ −d ? a?t?x?τ?t?? ≤ −d ? a0x?τ?t??,t ≥ t5.

?2.9?

We can choose some positive integer k0such that ck≥ t5, for k ≥ k0. Thus, we obtain

x?ck? ≤ −d ? a0x?τ?ck?? ? −d ? a0x?ck−1? ≤ −d − a0d ? a2

? −d − a0d ? a2

? −d − a0d − ··· − ak−k0−1

0x?τ?ck−1??

d ? ak−k0

0x?ck−2? ≤ ··· ≤ −d − a0d − ··· − ak−k0−1

d ? ak−k0

0

x?ck0?.

0

0

x?τ?ck0?1??

0

?2.10?

The above inequality implies that x?ck? < 0 for sufficiently large k, which contradicts the fact

that x?t? > 0 eventually. Hence we get

zΔΔ?t? > 0.

?2.11?

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It follows from this that either zΔ?t? > 0 or zΔ?t? < 0. Since rΔ?t? ≥ 0,

?

r?t?zΔΔ?t?

?Δ? rΔ?t?zΔΔ?t? ? rσ?t?zΔΔΔ?t? < 0,

?2.12?

which yields

zΔΔΔ?t? < 0.

?2.13?

If zΔ?t? > 0, then there are two possible cases:

?1? z?t? > 0, eventually; or

?2? z?t? < 0, eventually.

If there exists a t6

limt→∞z?t? ? b ≤ 0. We claim that limt→∞z?t? ? 0. Otherwise, limt→∞z?t? ? b < 0. We

can choose some positive integer k0such that ck≥ t6, for k ≥ k0. Thus, we obtain

≥ t1 such that case ?2? holds, then limt→∞z?t? exists, and

x?ck? ≤ a0x?τ?ck?? ? a0x?ck−1? ≤ a2

? a2

0x?τ?ck−1??

x?τ?ck0?1?? ? ak−k0

0x?ck−2? ≤ ··· ≤ ak−k0

00

x?ck0?,

?2.14?

which implies that limk→∞x?ck? ? 0, and from the definition of z?t?, we have limk→∞z?ck? ?

0, which contradicts limt→∞z?t? < 0. Now, we assert that x is bounded. If it is not true, there

exists {sk}k∈N⊂ ?t6,∞?Twith sk → ∞ as k → ∞ such that

x?sk? ? sup

t0≤s≤sk

x?s?,

lim

k→∞x?sk? ? ∞.

?2.15?

From τ?t? ≤ t

z?sk? ? x?sk? − a?sk?x?τ?sk?? ≥ ?1 − a0?x?sk?,

?2.16?

which implies that limk→∞z?sk? ? ∞, it contradicts that limt→∞z?t? ? 0. Therefore, we can

assume that

limsup

t→∞

x?t? ? x1,

liminf

t→∞

x?t? ? x2.

?2.17?

By 0 ≤ a < 1, we get

x1− ax1≤ 0 ≤ x2− ax2,

?2.18?

which implies that x1≤ x2, so x1? x2, hence, limt→∞x?t? ? 0.