Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 450264, 24 pages
doi:10.1155/2010/450264
Research Article
Oscillatory Behavior of Quasilinear Neutral
Delay Dynamic Equations on Time Scales
Zhenlai Han,1, 2 Shurong Sun,1, 3 Tongxing Li,1
and Chenghui Zhang2
1
School of Science, University of Jinan, Jinan, Shandong 250022, China
School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
3
Department of Mathematics and Statistics, Missouri University of Science and Technology,
Rolla, MO 65409-0020, USA
2
Correspondence should be addressed to Zhenlai Han, hanzhenlai@163.com
Received 6 December 2009; Accepted 4 March 2010
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.
By means of the averaging technique and the generalized Riccati transformation technique,
we establish some oscillation criteria for the second-order quasilinear neutral delay dynamic
equations rt|xΔ t|γ−1 xΔ tΔ q1 t|yδ1 t|α−1 yδ1 t q2 t|yδ2 t|β−1 yδ2 t 0, t ∈
t0 , ∞T , where xt yt ptyτt, and the time scale interval is t0 , ∞T : t0 , ∞ ∩ T. Our
results in this paper not only extend the results given by Agarwal et al. 2005 but also unify the
oscillation of the second-order neutral delay differential equations and the second-order 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 T 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.
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Advances in Difference Equations
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 second-order nonlinear neutral delay
dynamic equations on time scales:
rt
xt ptxt − τ
Δ γ Δ
ft, xt − δ 0
for t ∈ T,
1.1
where 0 ≤ pt < 1, γ ≥ 0 is a quotient of odd positive integer, τ, δ are positive constants,
∞
r, p ∈ Crd T, R , t0 1/rt1/γ Δt ∞, f ∈ CT × R, R such that uft, u > 0 for all nonzero
u, and there exists a nonnegative function qt defined on T satisfing |ft, u| ≥ qt|uγ |.
Agwo 35 examines the oscillation of the second-order nonlinear neutral delay
dynamic equations:
xt − rxτtΔΔ H t, xh1 t, xΔ h2 t 0
for t ∈ T.
1.2
Li et al. 36 discuss the existence of nonoscillatory solutions to the second-order
neutral delay dynamic equation of the form
ΔΔ
xt ptxτ0 t
q1 txτ1 t − q2 txτ2 t et
for t ∈ T.
1.3
Saker et al. 38, 39, Sahı́ner 40, and Wu et al. 43 consider the second-order neutral
delay and mixed-type dynamic equations on time scales:
rt
xt ptxτt
Δ γ Δ
ft, xδt 0 for t ∈ T,
1.4
∞
where 0 ≤ pt < 1, γ is a quotient of odd positive integer, r, p ∈ Crd T, R , t0 1/rt1/γ Δt ∞, τ, δ ∈ Crd T, T, τt ≤ t, limt → ∞ τt ∞, limt → ∞ δt ∞, f ∈ CT × R, R such that
uft, u > 0 for all nonzero u, and there exists a nonnegative function q defined on T satisfing
|ft, u| ≥ qt|uγ |.
Zhu and Wang 44 study existence of nonoscillatory solutions to neutral dynamic
equations on time scales:
Δ
ft, xht 0
xt ptx gt
for t ∈ T.
1.5
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3
Recently, Tripathy 42 has established some new oscillation criteria for second-order
nonlinear delay dynamic equations of the form
Δ γ Δ
qt|xt − δ|γ sgn xt − δ 0,
rt xt ptxt − τ
for t ∈ T,
1.6
where 0 ≤ pt, γ ≥ 0 is a quotient of odd positive integer, τ, δ are positive constants, r, p, q ∈
∞
Crd T, R , and t0 1/rt1/γ Δt ∞.
To the best of our knowledge, there are no results regarding the oscillation of the
solutions of the following second-order nonlinear neutral delay dynamic equations on time
scales up to now:
Δ
γ−1
α−1
rt
xΔ t
xΔ t q1 t
yδ1 t
yδ1 t
β−1
q2 t
yδ2 t
yδ2 t 0 for t ∈ t0 , ∞T ,
1.7
where xt yt ptyτt, and the time scale interval is t0 , ∞T : t0 , ∞ ∩ T.
In what follows we assume the following:
A1 α, β, and γ are positive constants with 0 < α < γ < β;
∞
A2 r, q1 , q2 ∈ Crd t0 , ∞T , R , R 0, ∞, t0 1/rt1/γ Δt ∞, r Δ t ≥ 0;
A3 p ∈ Crd t0 , ∞T , R, and −1 < p0 ≤ pt < 1, p0 constant;
A4 τ, δ1 , δ2 ∈ Crd t0 , ∞T , T, τt ≤ t, δ1 t ≤ t, δ2 t ≤ t, for t ∈ t0 , ∞T , and
limt → ∞ τt ∞, limt → ∞ δ1 t ∞, limt → ∞ δ2 t ∞.
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 second-order 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 second-order quasilinear delay differential equation and
the second-order quasilinear delay difference equation. Applications to equations to which
previously known criteria for oscillation are not applicable are given.
1
ty , ∞T ,
By a solution of 1.7, we mean a nontrivial real-valued function y ∈ Crd
1
Δ
γ−1 Δ
ty ∈ t0 , ∞T , which has the property rt|x t| x t ∈ Crd ty , ∞T and satisfying 1.7 for
t ∈ ty , ∞T . Our attention is restricted to those solutions y of 1.7 which exist on some half
line ty , ∞T with sup{|yt| : t ≥ t1 } > 0 for any t1 ∈ ty , ∞T . A solution y 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 q1 t ≡
0, 1.7 is the prototype of a wide class of nonlinear dynamic equations called Emden-Fowler
neutral delay superlinear dynamic equation:
Δ
γ−1
β−1
rt
xΔ t
xΔ t q2 t
yδ2 t
yδ2 t 0 for t ∈ t0 , ∞T ,
where xt yt ptyτt.
1.8
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Advances in Difference Equations
If q2 t ≡ 0, 1.7 is the prototype of nonlinear dynamic equations called Emden-Fowler
neutral delay sublinear dynamic equation:
Δ
α−1
rt|xΔ t|γ−1 xΔ t q1 t
yδ1 t
yδ1 t 0 for t ∈ t0 , ∞T ,
1.9
where xt yt ptyτt.
We note that if γ 1, then 1.7 becomes second-order nonlinear delay dynamic
equation on time scales:
α−1
Δ Δ
q1 t
yδ1 t
yδ1 t
rt yt ptyτt
β−1
q2 t
yδ2 t
yδ2 t 0
1.10
for t ∈ t0 , ∞T .
If pt ≡ 0, then 1.7 becomes second-order nonlinear delay dynamic equation on time
scales:
Δ
γ−1
α−1
rt
yΔ t
yΔ t q1 t
yδ1 t
yδ1 t
β−1
q2 t
yδ2 t
yδ2 t 0 for t ∈ t0 , ∞T .
1.11
If γ 1, pt ≡ 0, then 1.7 becomes second-order nonlinear delay dynamic equation
on time scales:
Δ
α−1
β−1
rtyΔ t q1 t
yδ1 t
yδ1 t q2 t
yδ2 t
yδ2 t 0
1.12
for t ∈ t0 , ∞T .
If γ 1, rt ≡ 1, pt ≡ 0, then 1.7 becomes second-order nonlinear delay dynamic
equation on time scales:
α−1
β−1
yΔΔ t q1 t
yδ1 t
yδ1 t q2 t
yδ2 t
yδ2 t 0
for t ∈ t0 , ∞T .
1.13
It is interesting to study 1.7 because the continuous version and its special cases have
several physical applications, see 1 and when t 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:
|ab| ≤
1 p 1 q
|a| |b| ,
p
q
a, b ∈ R, p > 1, q > 1,
1 1
1,
p q
1.14
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5
and the inequality
λABλ−1 − Aλ ≤ λ − 1B λ ,
1.15
λ ≥ 1,
where A and B 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 0 ≤
pt < 1 and −1 < p0 ≤ pt < 0. It will be convenient to make the following notations in
the remainder of this paper. Define
β−α β−α
,
,
θ min
β−γ γ −α
δt min{δ1 t, δ2 t},
t≥t0
γ
⎧
⎪
⎨γ 2 ,
⎪
⎩γ,
γ ≥ 1,
0 < γ < 1,
α β−γ/β−α β γ−α/β−α δt γ
q2 t 1 − pδ2 t
Q1 t θ q1 t 1 − pδ1 t
,
σt
Q2 t θ
q1 t
α β−γ/β−α q2 t
β γ−α/β−α δt γ
.
σt
2.1
We define the function space R as follows: H ∈ R provided that H is defined for t0 ≤
s ≤ σt, t, s ∈ t0 , ∞T , Ht, s ≥ 0, Hσt, t 0, H Δs σt, s −ht, sHσt, σs, and
1
t0 , ∞T , R ,
ht, s is rd-continuous function and nonnegative. For given function ρ, η ∈ Crd
we set
λt, s hσt, s −
ρΔ s
,
ρs
ρσs
Θi t, s Qi s − η s
λt, sηs,
ρs
Δ
2.2
i 1, 2.
In order to prove our main results, we will use the formula
γ Δ
xt
Δ
γx t
1
hxσt 1 − hxtγ−1 dh,
2.3
0
where xt 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 1.90.
Also, we assume the condition H − 1 < p0 ≤ pt < 0 and limt → ∞ pt p > −1, and
there exists {ck }k≥0 such that limk → ∞ ck ∞ and τck1 ck .
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Advances in Difference Equations
Lemma 2.1. Assume 0 ≤ pt < 1. If yt is an eventually positive solution of 1.7, then, there
exists a t∗ ∈ t0 , ∞T such that xt > 0, xΔ t ≥ 0 for t ∈ t∗ , ∞T . Moreover,
rt|xΔ t|γ−1 xΔ t
Δ
α
≤ −q1 t 1 − pδ1 t xδ1 t
− q2 t 1 − pδ2 t xδ2 t
β
2.4
< 0,
t ∈ t∗ , ∞T .
Lemma 2.2. Assume 0 ≤ pt < 1,
∞
β
q2 t 1 − pδ2 tδ2 t Δt ∞.
2.5
t0
If yt is an eventually positive solution of 1.7, then
xΔΔ t < 0, xt ≥ txΔ t,
xt
is strictly decreasing.
t
2.6
The proof of Lemmas 2.1 and 2.2 is similar to that of Saker et al. 39, Lemma 2.1; so it
is omitted.
Lemma 2.3. Assume that the condition H holds:
∞
q2 tδ2 tβ Δt ∞.
2.7
t0
If yt is an eventually positive solution of 1.7, then
Δ
γ−1
rt
xΔ t
xΔ t < −q1 txδ1 tα − q2 txδ2 tβ < 0,
xΔΔ t < 0, xt ≥ txΔ t,
t ∈ t∗ , ∞T ,
xt
is strictly decreasing,
t
2.8
2.9
or limt → ∞ yt 0.
Proof. Since yt is an eventually positive solution of 1.7, there exists a number t1 ∈ t0 , ∞T
such that yt > 0, yτt > 0, yδ1 t > 0 and yδ2 t > 0 for all t ∈ t1 , ∞T . In view of
1.7, we have
Δ
α
β
Δ γ−1 Δ
rt
x t
x t −q1 t yδ1 t − q2 t yδ2 t ,
t ∈ t1 , ∞T .
2.10
By −1 < p0 ≤ pt < 0, then yt > xt, we get
Δ
Δ γ−1 Δ
rt
x t
x t < −q1 txδ1 tα − q2 txδ2 tβ < 0,
t ∈ t1 , ∞T .
2.11
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Let t∗ ≥ t1 , then 2.8 holds, and rt|xΔ t|
follows that
γ−1
xΔ t is an eventually decreasing function. It
γ−1
lim rt
xΔ t
xΔ t l ≥ −∞.
2.12
t→∞
We prove that rt|xΔ t|γ−1 xΔ t is eventually positive.
Otherwise, there exists a t2 ∈ t1 , ∞T such that rt2 |xΔ t2 |γ−1 xΔ t2 c < 0, then
we have rt|xΔ t|γ−1 xΔ t ≤ rt2 |xΔ t2 |γ−1 xΔ t2 c for t ∈ t2 , ∞T , and hence xΔ t ≤
−−c1/γ 1/rt1/γ , which implies that
1/γ
xt ≤ xt2 − −c
t t2
1
rs
1/γ
Δs −→ −∞ as t −→ ∞.
2.13
Therefore, there exists d > 0, and t3 ≥ t2 such that
yt ≤ −d − ptyτt ≤ −d p0 yτt,
t ≥ t3 .
2.14
We can choose some positive integer k0 such that ck ≥ t3 , for k ≥ k0 . Thus, we obtain
yck ≤ −d p0 yτck −d p0 yck−1 ≤ −d − r0 d p02 yτck−1 −d − p0 d p02 xck−2 ≤ · · · ≤ −d − p0 d − · · · − p0k−k0 −1 d p0k−k0 yτck0 1 2.15
−d − p0 d − · · · − p0k−k0 −1 d p0k−k0 yck0 .
The above inequality implies that yck < 0 for sufficiently large k, which contradicts the fact
that yt > 0 eventually. Hence xΔ t > 0 eventually. Consequently, there are two possible
cases:
i xt > 0, eventually;
ii xt < 0, eventually.
If Case i holds, we can get
zt ≥ tzΔ t > 0,
zΔΔ t < 0,
zt
is nonincreasing.
t
2.16
Actually, by rtxΔ tγ Δ r Δ txΔ tγ rσtxΔ tγ Δ < 0, r Δ t ≥ 0, we can easily
verify that xΔ tγ Δ < 0. Using 2.3, we get
xΔ t
γ Δ
1 γ−1
σ
h xΔ 1 − hxΔ
γxΔΔ t
dh.
0
From
1
0
hxΔ σ 1 − hxΔ γ−1 dh > 0, we have that xΔΔ t is eventually negative.
2.17
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Advances in Difference Equations
Let Xt : xt − txΔ t, since X Δ t xΔ t − xΔ t σtxΔΔ t −σtxΔΔ t
is eventually positive, so Xt is eventually increasing. Therefore, Xt is either eventually
positive or eventually negative. If Xt is eventually negative, then there is a t3 ∈ t∗ , ∞T
such that Xt < 0 for t ∈ t3 , ∞T . So,
xt
t
Δ
Xt
txΔ t − xt
−
> 0,
tσt
tσt
t ∈ t3 , ∞T ,
2.18
which implies that xt/t is strictly increasing for t ∈ t3 , ∞T . Pick t4 ∈ t3 , ∞T so that
δi t ≥ δi t∗ for t ∈ t4 , ∞T , i 1, 2. Then
xδi t xδi t∗ ≥
: di > 0,
δi t
δi t∗ 2.19
so that xδi t ≥ di δi t for t ∈ t4 , ∞T .
By 2.8, we have
t γ
γ
q1 sxδ1 sα q2 sxδ2 sβ Δs,
rt xΔ t − rt4 xΔ t4 < −
2.20
t4
which implies that
γ
γ t rt4 xΔ t4 > rt xΔ t q1 sxδ1 sα q2 sxδ2 sβ Δs
≥ d1 α
t
q1 sδ1 sα Δs d2 β
t4
t
t4
2.21
q2 sδ2 sβ Δs,
t4
which contradicts 2.7. Hence, without loss of generality, there is a t∗ ≥ t0 such that Xt > 0,
that is, xt > txΔ t for t ≥ t∗ . Consequently,
xt
t
Δ
Xt
txΔ t − xt
−
< 0,
tσt
tσt
t ≥ t∗ ,
2.22
and we have that xt/t is strictly decreasing for t ≥ t∗ .
If there exists a t4 ≥ t1 such that Case ii holds, then limt → ∞ xt exists, limt → ∞ xt l ≤ 0, and we claim that limt → ∞ xt 0. Otherwise, limt → ∞ xt < 0. We can choose some
positive integer k0 such that ck ≥ t4 , for k ≥ k0 . Thus, we obtain
yck ≤ p0 yτck p0 yck−1 ≤ p02 yτck−1 p02 yck−2 ≤ · · · ≤ p0k−k0 yτck0 1 p0k−k0 yck0 ,
2.23
which implies that limk → ∞ yck 0, and limk → ∞ xck 0, which contradicts limt → ∞ xt
< 0.
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Now, we assert that yt is bounded. If it is not true, there exists {tk } with tk → ∞ as
k → ∞ such that
ytk sup ys,
lim ytk ∞.
2.24
k→∞
t0 ≤s≤tk
From τt ≤ t and
xtk ytk ptk yτtk ≥ 1 − p0 ytk ,
2.25
which implies that limk → ∞ xtk ∞, it contradicts the existence of limt → ∞ xt. Therefore,
we can assume that
lim sup yt y1 ,
lim inf yt y2 .
2.26
t→∞
t→∞
By −1 < p < 0, we get
y1 py1 ≤ 0 ≤ y2 py2 ,
2.27
thus y1 ≤ y2 , and y1 y2 . Hence, limt → ∞ yt 0. The proof is complete.
Theorem 2.4. Assume that 2.5 holds, 0 ≤ pt < 1,
⎡
1/γ ⎤
∞
β δ2 s β
1
⎦ ∞.
lim sup⎣t
q2 s 1 − pδ2 s
Δs
t→∞
rt t
s
2.28
Then 1.7 is oscillatory on t0 , ∞T .
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is
eventually positive. We shall consider only this case, since the proof when yt is eventually
negative is similar. In view of Lemmas 2.1 and 2.2, there exists a t∗ ∈ t0 , ∞T such that
xt > 0, xΔ t ≥ 0, xΔΔ t < 0, xt ≥ txΔ t, and xt/t is strictly decreasing for t ∈ t∗ , ∞T .
From 2.4 we have for T ≥ t, T, t ∈ t∗ , ∞T ,
T
β
q2 s 1 − pδ2 s xδ2 sβ Δs ≤ −
t
T
γ Δ
rs xΔ s
Δs
t
γ
γ
rt xΔ t − rT xΔ T 2.29
γ
≤ rt xΔ t ,
and hence
1
rt
∞
t
β
β
q2 s 1 − pδ2 s xδ2 s Δs
1/γ
≤ xΔ t.
2.30
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Advances in Difference Equations
So,
1/γ
β
1 ∞
β
q2 s 1 − pδ2 s xδ2 s Δs
xt ≥ tx t ≥ t
rt t
β 1/γ
β δ2 s
1 ∞
xs Δs
≥t
q2 s 1 − pδ2 s
rt t
s
Δ
≥x
β/γ
tt
1
rt
∞
t
2.31
1/γ
β δ2 s β
q2 s 1 − pδ2 s
Δs
.
s
So,
t
1
rt
∞
t
1/γ β δ2 s β
1 β/γ−1
q2 s 1 − pδ2 s
Δs
≤
.
s
xt
2.32
Now note that β/γ > 1 imply
t
1
rt
∞
t
1/γ β/γ−1
β δ2 s β
1
q2 s 1 − pδ2 s
Δs
≤
.
s
xt∗ 2.33
This contradicts 2.28. The proof is complete.
Remark 2.5. Theorem 2.4 includes results of Agarwal et al. 21, Theorem 4.4 and Han et al.
25, Theorem 3.1.
Theorem 2.6. Assume that the condition H and 2.7 hold:
⎡
⎤
∞
β 1/γ
δ
1
s
2
⎦ ∞.
q2 s
Δs
lim sup⎣t
rt t
s
t→∞
2.34
Then every solution of 1.7 either oscillates or tends to zero as t → ∞.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is
eventually positive. In view of Lemma 2.3, either limt → ∞ yt 0 or there exists a t∗ ∈ t0 , ∞T
such that xt > 0, xΔ t ≥ 0, xΔΔ t < 0, xt ≥ txΔ t, and xt/t is strictly decreasing for
t ∈ t∗ , ∞T .
Then from 2.8, we have for T ≥ t, T, t ∈ t∗ , ∞T ,
T
t
β
q2 sxδ2 s Δs ≤ −
T
t
γ Δ
rs xΔ s
Δs
γ
γ
γ
rt xΔ t − rT xΔ T ≤ rt xΔ t ,
2.35
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11
Since the rest of the proof is similar to Theorem 2.4, so we omit the detail. The proof is
complete.
1
Theorem 2.7. Assume that 2.5 holds. 0 ≤ pt < 1, let ρ, η ∈ Crd
t0 , ∞T , R ,
t
lim sup
t→∞
⎡
⎣ρσs Q1 s − η
t0
Δ
⎤
Δ γ1 ρ s rs
σs γ ⎦
Δs ∞,
−ρ η− γ
γ1 s
ρσs
γ 1
2.36
Δ
then 1.7 is oscillatory on t0 , ∞T , where ρΔ s max{ρΔ s, 0}.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is
eventually positive. In view of Lemmas 2.1 and 2.2, there exists a t∗ ∈ t0 , ∞T such that
xt > 0, xΔ t ≥ 0, xΔΔ t < 0, xt ≥ txΔ t, and xt/t is strictlydecreasing for t ∈ t∗ , ∞T .
Define the function ωt by
⎤
γ−1
rt
xΔ t
xΔ t
ηt⎦,
ωt ρt⎣
xγ t
⎡
t ∈ t∗ , ∞T .
2.37
We get
⎡
⎤
Δ γ Δ
Δ γ γ
Δ
Δ
rt
x
t
rt
x
t
x
t
ρ
t
⎢
⎥
ωΔ t ωt ρσt⎣
−
ηΔ t⎦.
ρt
xγ txσtγ
xσtγ
2.38
If γ ≥ 1, by 2.3, we get
xt
γ Δ
1
γx t hxσt 1 − hxtγ−1 dh ≥ γxtγ−1 xΔ t,
Δ
2.39
0
So, from 2.4 we have
ωΔ t ≤
α xδ1 tα
ρΔ t
ωt − ρσtq1 t 1 − pδ1 t
ρt
xσtγ
γ1
rt xΔ t
β xδ2 tβ
− ρσtq2 t 1 − pδ2 t
− γρσt
ρσtηΔ t.
xtxσtγ
xσtγ
2.40
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By Young’s inequality 1.14, we obtain that
α xδ1 tα γ − α
β xδ2 tβ
β−γ
q1 t 1 − pδ1 t
q
1
−
pδ
t
t
2
2
β−α
β−α
xσtγ
xσtγ
α xδ1 tα β−γ/β−α
β xδ2 tβ
q
≥ q1 t 1 − pδ1 t
1
−
pδ
t
t
2
2
xσtγ
xσtγ
γ−α/β−α
α β−γ/β−α β γ−α/β−α
q2 t 1 − pδ2 t
q1 t 1 − pδ1 t
×
xδ1 tα
xσtγ
β−γ/β−α xδ2 tβ
xσtγ
2.41
γ−α/β−α
α β−γ/β−α β γ−α/β−α xδt γ
q2 t 1 − pδ2 t
≥ q1 t 1 − pδ1 t
.
xσt
From xt/t being strictly decreasing, xδt/xσt ≥ δt/σt, xt/xσt ≥ t/σt, by
2.37 and 2.40, we get that
ωΔ t ≤
α β−γ/β−α β γ−α/β−α
ρΔ t
ωt − μρσt q1 t 1 − pδ1 t
q2 t 1 − pδ2 t
ρt
γ γ1/γ
1
δt γ
t
ωt
− ηt
×
− γρσt
ρσtηΔ t,
σt
ρt
rt1/γ σt
2.42
that is,
γ1/γ
γ ρΔ t
ωt
1
t
ωt − γρσt
−
ηt
.
ωΔ t ≤ −ρσt Q1 t − ηΔ t 1/γ
ρt
σt
ρt
rt
2.43
So,
ωt
Δ
Δ
Δ
− ηt
ω t ≤ −ρσt Q1 t − η t ρ tηt ρ t ρt
γ1/γ
γ ωt
1
t
− γρσt
.
ρt − ηt
1/γ
σt
rt
Δ
2.44
Using the inequality 1.15 we have
Δ γ1 2
ρ t σt γ
rt
Δ
Δ
.
ω t ≤ −ρσt Q1 t − η t ρ tηt γ
γ1 t
ρσt
γ 1
Δ
2.45
Advances in Difference Equations
13
Integrating the inequality above from t∗ to t we obtain
ωt − ωt∗ ⎛
⎞
Δ γ1 t
γ 2
ρ
s
σs
rs
⎠Δs.
≤ − ⎝ρσs Q1 s − ηΔ s − ρΔ sηs − γ
γ1 s
ρσs
t∗
γ 1
2.46
Therefore,
⎞
Δ γ1 γ 2
ρ
s
σs
rs
⎝ρσs Q1 s − ηΔ s − ρΔ sηs − ⎠Δs
γ
γ1 s
ρσs
t∗
γ 1
t
⎛
≤ ωt∗ − ωt ≤ ωt∗ ,
2.47
which contradicts 2.36.
If 0 < γ < 1, proceeding as the proof of above, we have 2.37 and 2.38. By 2.3, we
get that
γ Δ
xt
Δ
γx t
1
hxσt 1 − hxtγ−1 dh ≥ γxσtγ−1 xΔ t.
2.48
0
So, from 2.4 we have
ωΔ t ≤
α xδ1 tα
ρΔ t
ωt − ρσtq1 t 1 − pδ1 t
ρt
xσtγ
γ1
rt xΔ t
β xδ2 tβ
− ρσtq2 t 1 − pδ2 t
−
γρσt
ρσtηΔ t.
xσtxγ t
xσtγ
2.49
Since the rest of the proof is similar to that of above, so we omit the detail. The proof is
complete.
1
Theorem 2.8. Assume that the condition H and 2.7 hold, let ρ, η ∈ Crd
t0 , ∞T , R ,
t
lim sup
t→∞
⎡
⎣ρσs Q2 s − ηΔ − ρΔ η − rs
γ1
t0
γ 1
⎤
γ1 γ
ρΔ s σs ⎦
Δs ∞,
γ
s
ρσs
2.50
then every solution of 1.7 either oscillates or tends to zero as t → ∞, where ρΔ s max{ρΔ s, 0}.
14
Advances in Difference Equations
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is
eventually positive. In view of Lemma 2.3, either limt → ∞ yt 0 or there exists a t∗ ∈ t0 , ∞T
such that xt > 0, xΔ t ≥ 0, xΔΔ t < 0, xt ≥ txΔ t, and xt/t is strictly decreasing for
t ∈ t∗ , ∞T .
Since the rest of the proof is similar to Theorem 2.7, so we omit the detail. The proof is
complete.
1
Theorem 2.9. Assume that 2.5 holds. 0 ≤ pt < 1, let ρ, η ∈ Crd
t0 , ∞T , R ,
lim sup
t→∞
1
Hσt, t0 t
rs
× Hσt, σsρs Θ1 t, s − γ1
t0
γ 1
ρs
ρσs
γ σs
s
γ
|λt, s|
γ1
Δs ∞,
2.51
then 1.7 is oscillatory on t0 , ∞T .
Proof. We prove only case γ ≥ 1. The proof of case 0 < γ < 1 is similar. Proceeding as the proof
of Theorem 2.7, we have 2.37 and 2.43. Replacing t in 2.43 by s, then multiplying 2.43
by Hσt, σs, and integrating from T to t, t > T, t, T ∈ t∗ , ∞T , we have
t
Hσt, σsρσs Q1 s − ηΔ s Δs
T
≤−
t
Δ
Hσt, σsω sΔs T
−γ
t
Hσt, σsρΔ s
T
t
Hσt, σsρσs
T
1
rs1/γ
s
σs
γ ωs
Δs
ρs
ωs
− ηs
ρs
2.52
γ1/γ
Δs.
Integrating by parts and using the fact that Hσt, t 0, we get
t
Hσt, σsωΔ sΔs −Hσt, T ωT −
T
t
H Δs σt, sωsΔs.
2.53
T
So,
t
Hσt, σsρσt Q1 s − ηΔ s Δs ≤ Hσt, T ωT T
t
t
ωs
Δs
ρs
T
T
t
γ γ1/γ
1
s
ωs
− γ Hσt, σsρσs
−
ηs
Δs,
ρs
rs1/γ σs
T
−
hσt, sHσt, σsωsΔs Hσt, σsρΔ s
2.54
Advances in Difference Equations
15
that is,
t
Hσt, σsρσs Q1 s − ηΔ s Δs ≤ Hσt, T ωT T
−
t hσt, s −
T
−γ
ρΔ s
Hσt, σsωsΔs
ρs
t
Hσt, σsρσs
T
1
rs1/γ
s
σs
2.55
γ ωs
− ηs
ρs
γ1/γ
Δs.
Hence,
t
T
Hσt, σsρtΘ1 t, sΔs ≤ Hσt, T ωT t
ωs
− ηs
Δs
Hσt, σsρs|λt, s|
ρs
T
t
γ1/γ
γ ωs
1
s
− γ Hσt, σsρσs
−
ηs
Δs.
1/γ
σs
ρs
rs
T
2.56
Using the inequality 1.15 we have
1
Hσt, T t
γ 2
ρs
σs γ
rs
γ1
× Hσt, σsρs Θ1 t, s − Δs ≤ ωT .
|λt, s|
γ1 ρσs
s
T
γ 1
2.57
Set T t∗ , so,
lim sup
t→∞
t
1
Hσt, t∗ rs
× Hσt, σsρs Θ1 t, s − γ1
t∗
γ 1
This contradicts 2.51 and finishes the proof.
ρs
ρσs
γ σs
s
γ 2
|λt, s|
γ1
Δs ≤ ωt∗ .
2.58
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Advances in Difference Equations
1
t0 , ∞T , R ,
Theorem 2.10. Assume that the condition H and 2.7 hold, let ρ, η ∈ Crd
lim sup
t→∞
1
Hσt, t0 t
rs
× Hσt, σsρs Θ2 t, s − γ1
t0
γ 1
ρs
ρσs
γ σs
s
γ
|λt, s|
γ1
Δs ∞,
2.59
then every solution of 1.7 either oscillates or tends to zero as t → ∞.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is
eventually positive. In view of Lemma 2.3, either limt → ∞ yt 0 or there exists a t∗ ∈ t0 , ∞T
such that xt > 0, xΔ t ≥ 0, xΔΔ t < 0, xt ≥ txΔ t, and xt/t is strictly decreasing for
t ∈ t∗ , ∞T .
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.
1
t0 , ∞T , R ,
Theorem 2.11. Assume that 2.5 holds. 0 ≤ pt < 1, let H ∈ R, ρ, η ∈ Crd
Hσt, s
≤ ∞,
0 < inf lim inf
s≥t0 t → ∞
Hσt, t0 1
lim sup
Hσt,
t0 t→∞
t
ρs
Hσt, σsρsrs
ρσs
t0
γ σs
s
2.60
γ
|λt, s|γ1 Δs < ∞,
2.61
there exists ϕ ∈ Crd t0 , ∞T , R such that
∞
ρσs
t0
1
rs1/γ
s
σs
γ ϕs
− ηs
ρs
γ1/γ
if γ ≥ 1,
2.62
if 0 < γ < 1,
2.63
Δs ∞,
or
∞
ρσs
t0
1
rs1/γ
γ1/γ
ϕs
s
− ηs
Δs ∞,
σs ρs
Advances in Difference Equations
17
and for any T ∈ t0 , ∞T ,
lim sup
t→∞
1
Hσt, T t
rs
× Hσt, σsρs Θ1 t, s − γ1
T
γ 1
ρs
ρσs
γ σs
s
γ
|λt, s|
γ1
Δs ≥ ϕT ,
2.64
and then 1.7 is oscillatory on t0 , ∞T , where ϕt max{ϕt, 0}.
Proof. We prove only case γ ≥ 1. The proof of case 0 < γ < 1 is similar. Proceeding as the proof
of Theorem 2.9, we have 2.56 and 2.57. So we have for all t > T, t, T ∈ t∗ , ∞T ,
lim sup
t→∞
1
Hσt, T t
rs
× Hσt, σsρs Θ1 t, s − γ1
T
γ 1
ρs
ρσs
γ σs
s
γ 2
|λt, s|
γ1
Δs ≤ ωT .
2.65
By 2.64, we obtain
ϕT ≤ ωT ,
T ∈ t∗ , ∞T .
2.66
Define
1
P t Hσt, t∗ γ
Qt Hσt, t∗ t
t∗
%
% ωs
%
− ηs Δs,
Hσt, σsρs
λt, s%
ρs
t
s
Hσt, σsρσs
σs
t∗
γ
γ1/γ
ωs
− ηs
Δs.
1/γ ρs
rs
1
2.67
Then, by 2.56 and 2.64, we have that
lim infQt − P t ≤ ωt∗ t→∞
− lim sup
t→∞
1
Hσt, t∗ t
Hσt, σsρsΘ1 t, sΔs ≤ ωt∗ − ϕt∗ < ∞.
2.68
t∗
We claim that
t
ρσs
t∗
s
σs
γ
γ1/γ
ωs
−
ηs
Δs < ∞.
1/γ ρs
rs
1
2.69
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Advances in Difference Equations
Suppose, to the contrary, that
t
ρσs
t∗
s
σs
γ
γ1/γ
ωs
−
ηs
Δs ∞.
1/γ ρs
rs
1
2.70
By 2.60, there exists a positive constant k1 such that
Hσt, s
≥ k1 .
inf lim inf
s≥t0
t → ∞ Hσt, t0 2.71
Let k2 be an arbitrary positive number, then it follows from 2.70 that there exists a T1 ∈
t∗ , ∞T such that
t
ρσs
t∗
s
σs
γ
γ1/γ
ωs
k2
−
ηs
Δs ≥
,
1/γ ρs
γk1
rs
1
t ∈ T1 , ∞T .
2.72
Hence,
Qt γ
Hσt, t∗ ×
s
t
Hσt, σs
t∗
≥
τ
ρστ
στ
t∗
γ
γ1/γ Δ
ωτ
− ητ
Δτ
Δs
1/γ ρτ
rτ
1
γ
Hσt, t∗ t γ1/γ γ
s
ωτ
1
τ
Δs
×
− ητ
−H σt, s
ρστ
Δτ Δs
1/γ
στ rτ
ρτ
t∗
t∗
γ
Hσt, t∗ t γ1/γ γ
s
ωτ
1
τ
Δs
×
− ητ
−H σt, s
ρστ
Δτ Δs
1/γ ρτ
στ
rτ
T1
t∗
k2
1
≥
k1 Hσt, t∗ t T1
k2 Hσt, T1 −H Δs σt, s Δs .
k1 Hσt, t∗ 2.73
By 2.71, there exists a T2 ∈ T1 , ∞T such that Hσt, T1 /Hσt, t∗ ≥ k1 , t ∈ T2 , ∞T ,
which implies that Qt ≥ k2 . Since k2 is arbitrary, then
lim Qt ∞.
t→∞
2.74
Advances in Difference Equations
19
In view of 2.68, we consider a sequence {Tn }∞
n1 ⊂ t∗ , ∞T with limn → ∞ Tn ∞ satisfying
lim QTn − P Tn lim infQt − P t < ∞.
n→∞
2.75
t→∞
Then, there exists a constant M such that QTn − P Tn ≤ M for all sufficiently large n ∈ N.
Since 2.70 ensures that
lim QTn ∞, lim P Tn ∞
n→∞
2.76
n→∞
so,
P Tn −M
1
−1≥
>−
QTn QTn 2
or
P Tn 1
≥
QTn 2
2.77
hold for all sufficiently large n ∈ N. Therefore,
P γ1 Tn ∞.
n → ∞ Qγ Tn 2.78
lim
On the other hand, from the definition of P we can obtain, by Hölder’s inequality,
P Tn ≤
1
HσTn , t∗ T
n
×
t∗
×
T
n
t∗
s
HσTn , σsρσs
σs
γ
γ1/γ
ωs
− ηs
Δs
1/γ ρs
rs
1
γ 2
ρs
σs γ
HσTn , σsρs
rs|λTn , s|γ1 Δs
ρσs
s
γ/γ1
1/γ1
,
2.79
and, accordingly,
P γ1 Tn 1
1
≤ γ
γ
Q Tn γ HσTn , t∗ Tn
HσTn , σsρs
t∗
ρs
ρσs
γ σs
s
γ 2
rs|λTn , s|γ1 Δs.
2.80
So, because of 2.78, we get
1
1
n → ∞ γ γ HσTn , t∗ Tn
lim
t∗
HσTn , σsρs
ρs
ρσs
γ σs
s
γ 2
rs|λTn , s|γ1 Δs ∞,
2.81
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Advances in Difference Equations
which gives that
1
lim sup
t → ∞ Hσt, t∗ t
γ 2
ρs
σs γ
Hσt, σsρs
rs|λt, s|γ1 Δs ∞
ρσs
s
t∗
2.82
contradicting 2.61. Hence, 2.69 holds. In view of 2.66, from 2.69, we have
∞
ρσs
t∗
≤
1
rs1/γ
∞
ρσs
t∗
s
σs
γ 1
rs1/γ
ϕs
− ηs
ρs
s
σs
γ1/γ
Δs
2.83
γ1/γ
γ ωs
−
ηs
Δs < ∞,
ρs
which contradicts 2.62. The proof is complete.
1
Theorem 2.12. Assume that the condition H and 2.7 hold, let H ∈ R, ρ, η ∈ Crd
t0 , ∞T , R ,
there exists ϕ ∈ Crd t0 , ∞T , R such that 2.60 and 2.61, and either 2.62 or 2.63 hold, and
for any T ∈ t0 , ∞T ,
lim sup
t→∞
1
Hσt, T t
rs
× Hσt, σsρs Θ2 t, s − γ1
T
γ 1
ρs
ρσs
γ σs
s
γ
|λt, s|
γ1
Δs ≥ ϕT ,
2.84
and then every solution of 1.7 either oscillates or tends to zero as t → ∞.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is
eventually positive. In view of Lemma 2.3, either limt → ∞ yt 0 or there exists a t∗ ∈ t0 , ∞T
such that xt > 0, xΔ t ≥ 0, xΔΔ t < 0, xt ≥ txΔ t, and xt/t is strictly decreasing for
t ∈ t∗ , ∞T .
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.
1
t0 , ∞T ,
Theorem 2.13. Assume that 2.5 and 2.60 hold, 0 ≤ pt < 1, let H ∈ R, ρ, η ∈ Crd
R ,
lim inf
t→∞
1
Hσt, t0 t
Hσt, σsρsrs
t0
ρs
ρσs
γ σs
s
γ
|λt, s|γ1 Δs < ∞,
2.85
Advances in Difference Equations
21
there exists ϕ ∈ Crd t0 , ∞T , R such that 2.62 or 2.63 holds, and for any T ∈ t0 , ∞T ,
lim inf
t→∞
1
Hσt, T t
rs
× Hσt, σsρs Θ1 t, s − γ1
T
γ 1
ρs
ρσs
γ σs
s
γ
|λt, s|
γ1
Δs ≥ ϕT ,
2.86
and then 1.7 is oscillatory on t0 , ∞T .
Theorem 2.14. Assume that the condition H, 2.7, and 2.60 hold, let H ∈ R, ρ, η ∈
1
t0 , ∞T , R , there exists ϕ ∈ Crd t0 , ∞T , R such that 2.62 or 2.63 holds, and for any
Crd
T ∈ t0 , ∞T ,
lim inf
t→∞
1
Hσt, T t
rs
× Hσt, σsρs Θ2 t, s − γ1
T
γ 1
ρs
ρσs
γ σs
s
γ
|λt, s|
γ1
Δs ≥ ϕT ,
2.87
and then every solution of 1.7 either oscillates or tends to zero as t → ∞.
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 H, ρ and η.
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:
tγ |xΔ t|γ−1 xΔ t
t
δ2 t
β
Δ
σt
δ1 t
α
α−1
1 yδ1 t
yδ1 t
t
β−1
1 yδ2 t
yδ2 t 0,
t
3.1
t ∈ 1, ∞T ,
where
rt tγ ,
q1 t σt
δ1 t
α
1
,
t
q2 t t
δ2 t
β
1
,
t
xt yt p0 yτt,
0 ≤ p0 < 1, 0 < α < γ < β
3.2
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Advances in Difference Equations
with γ α β/2, τ, δ1 , δ2 ∈ Crd 1, ∞T , T, τt ≤ t, δ1 t ≤ t, δ2 t ≤ t, for t ∈ 1, ∞T , and
limt → ∞ τt ∞, limt → ∞ δ1 t ∞, limt → ∞ δ2 t ∞. Then, by Theorem 2.4, we have
∞
1
∞
α
1
rt
q1 t 1 − p0 δ1 t Δt 1
1/γ
∞
1
Δt ∞
1
1
Δt ∞,
t
α σtα
Δt ≥
1 − p0
t
∞
1 − p0
1
α t α
Δt ∞,
t
⎡
⎤
∞
β 1/γ
1/γ
∞
β 1
1
β δ2 s
⎦ lim sup
lim sup⎣t
Δs
q2 s 1 − p0
Δs
∞.
1 − p0
t→∞
rt t
s
s
t→∞
t
3.3
Then 3.1 is oscillatory on 1, ∞T .
Example 3.2. Consider the second-order delay dynamic equations on time scales:
Δ γ−1
α−1
σt α 1 γ
Δ
Δ
yδt
yδt
t x t
x t δt
t
β−1
σt β 1 yδt
yδt 0, t ∈ 1, ∞T ,
δt
t
3.4
where
rt t ,
γ
q1 t σt
δt
α
1
,
t
q2 t σt
δt
β
1
,
t
xt yt p0 yτt,
0 ≤ p0 < 1,
3.5
0<α<γ <β
with γ α β/2, τ, δ ∈ Crd 1, ∞T , T, τt ≤ t, δt ≤ t, for t ∈ 1, ∞T , and limt → ∞ τt ∞, limt → ∞ δt ∞. Then,
β−α β−α
θ min
,
β−γ γ −α
2,
α β−γ/β−α β γ−α/β−α δt γ
γ 1
Q1 t μ q1 t 1 − pδ1 t
q2 t 1 − pδ2 t
,
2 1 − p0
σt
t
3.6
let ρ 1, η 1, and by Theorem 2.7, we have
t
lim sup
t→∞
Q1 sΔs lim sup
1
Then 3.4 is oscillatory on 1, ∞.
t→∞
t
γ 1
Δs ∞.
2 1 − p0
s
1
3.7
Advances in Difference Equations
23
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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