Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 743469, 20 pages
doi:10.1155/2012/743469
Research Article
Oscillation Criteria for Second-Order Nonlinear
Dynamic Equations on Time Scales
Shao-Yan Zhang1 and Qi-Ru Wang2
1
Department of Mathematics, Guangdong University of Finance, 527 Yingfu Lu,
Guangdong, Guangzhou 510520, China
2
School of Mathematics & Computational Science, Sun Yat-Sen University,
135 Xinguang Xi Lu, Guangdong, Guangzhou 510275, China
Correspondence should be addressed to Qi-Ru Wang, mcswqr@mail.sysu.edu.cn
Received 31 August 2012; Accepted 15 October 2012
Academic Editor: Zhanbing Bai
Copyright q 2012 S.-Y. Zhang and Q.-R. Wang. 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.
This paper is concerned with oscillation of second-order nonlinear dynamic equations of the
γ Δ
form rtyt ptyτtΔ f1 t, yδ1 t f2 t, yδ2 t 0 on time scales. By using
a generalized Riccati technique and integral averaging techniques, we establish new oscillation
criteria which handle some cases not covered by known criteria.
1. Introduction
The theory of time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order
to unify continuous and discrete analysis. Not only can this theory of the so-called “dynamic
equations” unify theories of differential equations and difference equations but also extend
these classical cases to cases “in between”, for example, to the so-called q-difference
equations. A time scale T is an arbitrary nonempty closed subset of the real numbers R with
the topology and ordering inherited form R, and the cases when this time scale is equal to
R or to the integers Z represent the classical theories of differential and difference equations.
Of course many other interesting time scales exist, and they give rise to plenty of applications.
In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales, we refer the reader to
1–14.
In 2006, Wu et al. 1 considered the second-order nonlinear neutral dynamic equation
with variable delays
rt
yt ptyτt
Δ γ Δ
f t, yδt 0,
t ∈ T,
1.1
2
Abstract and Applied Analysis
where γ ≥ 1 is a quotient of odd positive integers. In 2007, Saker et al. 2 also discussed 1.1
for an odd positive integer γ ≥ 1. In 2010, Zhang and Wang 3 extended and complemented
some results in 1, 2 for γ ≥ 1 and gave some new results for 0 < γ < 1. In 2011, Saker 4
considered 1.1 in different conditions. In 2010, Sun et al. 5 considered the second-order
quasiliner neutral delay dynamic equation
rt
yt ptyτt
Δ γ Δ
q1 xα τ1 t q2 xβ τ2 t 0,
t ∈ T,
1.2
where γ, α, and β are quotients of odd positive integers with 0 < α < γ < β.
In this paper, we study the second-order nonlinear dynamic equation
rt
yt ptyτt
Δ γ Δ
f1 t, yδ1 t f2 t, yδ2 t 0,
1.3
on a time scale T, where p ∈ Crd T, 0, 1, fi ∈ CT × R, R, i 1, 2, γ > 0 is a quotient of odd
positive integers.
The paper is organized as follows. In the next section, we give some preliminaries and
lemmas. In Section 3, we will use the Riccati transformation technique to prove our main
results. In Section 4, we present two examples to illustrate our results.
2. Preliminaries and Lemmas
For convenience, we recall some concepts related to time scales. More details can be found in
6.
Definition 2.1. Let T be a time scale, for t ∈ T the forward jump operator is defined by σt :
inf{s ∈ T : s > t}, the backward jump operator by ρt : sup{s ∈ T : s < t}, and the
graininess function by μt : σt − t, where inf ∅ : sup T and sup ∅ : inf T. If σt > t, t is
said to be right-scattered, otherwise, it is right-dense. If ρt < t, t is said to be left-scattered,
otherwise, it is left-dense. The set Tκ is defined as follows. If T has a left-scattered maximum
m, then Tκ T − {m}, otherwise, Tκ T.
Definition 2.2. For a function f : T → R and t ∈ Tκ , one defines the delta-derivative f Δ t of
ft to be the number provided it exists with the property that given any ε > 0, there is a
neighborhood U of t i.e., U t − δ, t δ ∩ T for some δ such that
fσt − fs − f Δ tσt − s ≤ ε|σt − s|,
∀s ∈ U.
2.1
We say that f is delta-differentiable or in short, differentiable on Tκ provided f Δ t exists,
for all t ∈ Tκ .
It is easily seen that if f is continuous at t ∈ T and t is right-scattered, then f is
differentiable at t with
f Δ t fσt − ft
.
μt
2.2
Abstract and Applied Analysis
3
Moreover, if t is right-dense then f is differential at t if the limit
ft − fs
s→t
t−s
2.3
lim
exists as a finite number. In this case
ft − fs
.
s→t
t−s
f Δ t lim
2.4
In addition, if f Δ ≥ 0, then f is nondecreasing. A useful formula is
f σ t ft μtf Δ t,
where f σ t : fσt.
2.5
We will make use of the following product and quotient rules for the derivative of the product
0 of two differentiable functions f and g:
fg and the quotient f/g where gg σ /
fg
Δ
f Δ g f σ g Δ fg Δ f Δ g σ ,
Δ
f
f Δ g − fg Δ
.
g
gg σ
2.6
Definition 2.3. Let f : T → R be a function, f is called right-dense continuous rd-continuous
if it is continuous at right-dense points in T and its left-sided limits exist finite at left-dense
points in T. A function F : T → R is called an antiderivative of f provided F Δ t ft
b
holds for all t ∈ Tk . By the antiderivative, the Cauchy integral of f is defined as a fsΔs ∞
t
Fb − Fa, and a fsΔs limt → ∞ a fsΔs.
Let Crd T, R denote the set of all rd-continuous functions mapping T to R. It is
shown in 6 that every rd-continuous function has an antiderivative. An integration by parts
formula is
b
a
b
ftg Δ tΔt ftgt a −
b
f Δ tg σ tΔt.
2.7
a
In 1.3, we assume that T is a time scale and
h1 τt, δi t ∈ Crd T, T, limt → ∞ τt ∞, τt ≤ t, limt → ∞ δi t ∞, and t ≤ δi t,
i 1, 2,
∞
h2 rt ∈ Crd T, R , 1/rt1/γ Δt ∞, pt ∈ Crd T, 0, 1, where R 0, ∞,
0, there
h3 fi t, u : T × R → R is continuous function such that ufi t, u > 0 for all u /
exist qi t ∈ Crd T, R i 1, 2, quotients of odd positive integers α and β such
that |uf1 t, u| ≥ q1 t|u|α1 , |uf2 t, u| ≥ q2 t|u|β1 , and 0 < α < γ < β.
Since we are interested in the oscillatory and asymptotic behavior of solutions near
infinity, we assume throughout that the time scale T under consideration satisfies inf T t0
and sup T ∞. For T ∈ T, let T, ∞T : {t ∈ T : t ≥ T }. Throughout this paper, these
4
Abstract and Applied Analysis
assumptions will be supposed to hold. Let τ ∗ t min{τt, δ1 t, δ2 t}, T0 min{τ ∗ t : t ≥
∗
∗
∗
t sup{s ≥ t0 : τ ∗ s ≤ t}. Clearly τ−1
t ≥ t for t ≥ T0 , τ−1
t is nondecreasing
t0 } and τ−1
∗
and coincides with the inverse of τ t when the latter exists.
By a solution of 1.3, we mean a nontrivial real-valued function yt which has
1
∗
τ−1
t0 , ∞ and rtyt ptyτtΔ γ ∈
the properties yt ptyτt ∈ Crd
1
∗
τ−1
t0 , ∞. Our attention is restricted to those solutions of 1.3 that exist on some half
Crd
line ty , ∞ and satisfy sup{|yt| : t ≥ t1 } > 0 for any t1 ≥ ty . A solution yt of 1.3 is said to
be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called
nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.
For convenience, we use the notation xσt xσ t, xδi t xδi t i 1, 2 and
Δ
x σt xΔ tσ , and set
xt : yt ptyτt.
2.8
γ Δ
rt xΔ t
f1 t, yδ1 t f2 t, yδ2 t 0.
2.9
Then 1.3 becomes
Now, we give the first lemma. Set
t
RT t T
Δs
rs1/γ
2.10
.
Lemma 2.4. Let conditions h1 –h3 hold. If yt is an eventually positive solution of 1.3, then
Δ
there exists T ∈ T sufficiently large such that xt > 0, xΔ t > 0, rtxΔ tγ < 0, xt >
1/γ
Δ
δi
σ
1/γ
1/γ
RT tr tx t, and x t/x t > RT tr t/RT tr t μt (i 1, 2) for t ∈
T, ∞T .
Proof. If yt is an eventually positive solution of 1.3, then by h1 there exists a T ∈ t0 , ∞T
such that
yt > 0,
yτt > 0,
yδi t > 0,
for t ≥ T, i 1, 2.
2.11
From 2.8, 1.3, and h2 , we see that xt ≥ yt. Also by 1.3 and h3 , we have
γ Δ
rt xΔ t
≤ −q1 tyα δ1 t − q2 tyβ δ2 t < 0,
for t ≥ T,
2.12
which implies that rtxΔ tγ is decreasing on T, ∞T .
We claim that rtxΔ tγ > 0 on T, ∞T . Assume not, there is a t1 ∈ T, ∞T such that
rt1 xΔ t1 γ < 0. Since rtxΔ tγ ≤ rt1 xΔ t1 γ for t ≥ t1 , we have
Δ
1/γ
x t ≤ rt1 1
x t1 rt
Δ
1/γ
.
2.13
Abstract and Applied Analysis
5
Integrating the inequality above form t1 to t ≥ t1 , by h2 we get
1/γ
xt ≤ xt1 rt1 Δ
x t1 t t1
1
rs
1/γ
Δs −→ −∞ t −→ ∞,
2.14
and this contradicts the fact that xt > 0, for all t ≥ T . Thus we have rtxΔ tγ > 0 on
T, ∞T and so xΔ t > 0 on T, ∞T .
Note that
xt > xt − xT t
xΔ s T
t
T
γ 1/γ t
> rt xΔ t
T
γ 1/γ
rs xΔ s
r 1/γ s
Δs
2.15
1
Δs RT tr 1/γ txΔ t,
r 1/γ s
we have
xΔ t
xσ t xt μtxΔ t
1 μt
xt
xt
xt
RT tr 1/γ t μt
1
< 1 μt
.
RT tr 1/γ t
RT tr 1/γ t
2.16
Since δi t ≥ t and xΔ t > 0, we get
xt
RT tr 1/γ t
xδi t xδi t xt
·
≥
>
,
xσ t
xt xσ t xσ t RT tr 1/γ t μt
i 1, 2.
2.17
The proof is complete.
Remark 2.5. By xt ≥ yt on T, ∞T , xΔ > 0, 1.3, 2.8 and h1 –h3 , we get
γ Δ
α
β
q1 t yδ1 t q2 t yδ2 t
0 ≥ rt xΔ t
γ Δ
α
rt xΔ t
q1 t xδ1 t − pδ1 tyτδ1 t
β
q2 t xδ2 t − pδ2 tyτδ2 t
γ Δ
α
≥ rt xΔ t
q1 t xδ1 t − pδ1 txτδ1 t
β
q2 t xδ2 t − pδ2 txτδ2 t
γ Δ
α
≥ rt xΔ t
q1 t 1 − pδ1 t xδ1 tα
β
q2 t 1 − pδ2 t xδ2 tβ .
2.18
6
Abstract and Applied Analysis
Lemma 2.6 see 3. Let gu Bu − Auγ1/γ , where A > 0 and B are constants, γ is a quotient of
odd positive integers. Then g attains its maximum value on R at u∗ Bγ/Aγ 1γ , and
γγ
Bγ1
γ1 Aγ .
γ 1
max g gu∗ u∈R
2.19
Lemma 2.7 see 11. x and z are delta-differentiable on T. For x /
0 and any t ∈ T, one has
Δ
x t
z2 t
xt
Δ
2
2
zt Δ
σ
z t − xtx t
.
xt
Δ
2.20
3. Main Results
In this section, by employing the Riccati transformation technique we will establish oscillation criteria for 1.3 in two cases: γ ≥ 1 and 0 < γ < 1. Set
α β−γ/β−α β γ−α/β−α
· q2 s 1 − pδ2 s
,
Qs q1 s 1 − pδ1 s
γ
RT sr 1/γ s
Q1 s zsQs,
RT sr 1/γ s μs
γ
σ
RT sr 1/γ s
2
Q2 s z
Qs,
s
RT sr 1/γ s μs
Ct, s HsΔ t, s Ht, s
zΔ s
,
zσs
As 3.1
H Δ t, s
zΔ s
s
,
zσs
Ht, s
I γ ≥ 1.
Theorem 3.1. Assume that h1 –h3 hold and γ ≥ 1. Furthermore, assume that there exists a
positive rd-continuous Δ-differentiable function zt such that for all sufficiently large T ∈ T,
⎡
γ1 ⎤
rs zΔ s
⎦Δs ∞,
lim sup ⎣Q1 s − γ1
zγ s
t→∞
T
γ 1
t
1
3.2
then 1.3 is oscillatory.
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive note that in the case when yt is
eventually negative, the proof is similar, since the substitution Y t −yt transforms 1.3
into the same form. Then, by h1 –h3 there exists T ≥ t0 sufficiently large such that yt > 0,
Abstract and Applied Analysis
7
yτt > 0, yδ1,2 t > 0, and Lemma 2.4 holds for t ≥ T , where xt is defined by 2.8.
Define the function wt by the Riccati substitution
wt :
γ
ztrt xΔ t
,
xγ t
3.3
for t ≥ T,
then wt > 0 and
γ Δ zt γ σ zt Δ
Δ
rt x t
w t rt x t
xγ t
xγ t
γ Δ zt γ σ zΔ txγ t − ztxγ tΔ
Δ
Δ
.
rt x t
rt x t
xγ t
xγ txσ tγ
Δ
Δ
3.4
By 1.3, xΔ t > 0, and 2.18, we obtain
γ Δ
rt xΔ t
xγ t
≤ −
α α
q1 t 1 − pδ1 t xδ1 t
xσ tγ
β β
q2 t 1 − pδ2 t xδ2 t
−
.
xσ tγ
3.5
Noting that 0 < α < γ < β, we have
β−γ
< 1,
β−α
γ −α
< 1.
β−α
3.6
By Young’s inequality
aχ b1−χ ≤ χa 1 − χ b,
3.7
0 < χ < 1,
with
β−γ
χ
,
β−α
a
α α
q1 t 1 − pδ1 t xδ1 t
xσ tγ
β β
q2 t 1 − pδ2 t xδ2 t
,
b
xσ tγ
,
3.8
8
Abstract and Applied Analysis
we have
α α
q1 t 1 − pδ1 t xδ1 t
β β
q2 t 1 − pδ2 t xδ2 t
xσ tγ
xσ tγ
α α
β β
β − γ q1 t 1 − pδ1 t xδ1 t
γ − α q2 t 1 − pδ2 t xδ2 t
≥
β−α
β−α
xσ tγ
xσ tγ
α α β−γ/β−α ⎛
β β ⎞γ−α/β−α
q1 t 1 − pδ1 t xδ1 t
q2 t 1 − pδ2 t xδ2 t
⎠
≥
·⎝
xσ tγ
xσ tγ
α β−γ/β−α β γ−α/β−α
q1 t 1 − pδ1 t
· q2 t 1 − pδ2 t
·
xδ1 t
αβ−αγ/β−α δ βγ−βα/β−α
· x 2 t
xσ tγ
.
3.9
By γ αβ − αγ/β − α βγ − βα/β − α and Lemma 2.4, we get
α α
q1 t 1 − pδ1 t xδ1 t
xσ tγ
β β
q2 t 1 − pδ2 t xδ2 t
xσ tγ
α β−γ/β−α β γ−α/β−α
≥ q1 t 1 − pδ1 t
· q2 t 1 − pδ2 t
·
>
xδ1 t
xσ t
αβ−αγ/β−α ·
RT tr 1/γ t
RT tr 1/γ t μt
xδ2 t
xσ t
βγ−βα/β−α
3.10
γ
Qt.
In view of xΔ t > 0 and 3.5–3.10, for all t ≥ T , we obtain
Δ
w t < − zt
RT tr 1/γ t
RT tr 1/γ t μt
γ
Qt wσ t
zΔ t
zσ t
γ σ ztxγ tΔ
− rt xΔ t
xγ txσ tγ
− Q1 t wσ t
γ σ ztxγ tΔ
zΔ t Δ
x
.
−
rt
t
zσ t
xγ txσ tγ
3.11
Abstract and Applied Analysis
9
Using γ ≥ 1, Lemma 2.4 and the Keller’s chain rule, we get
Δ
x t γ
γ
1
Δ
xt hμtx t
γ−1
dh xΔ t
0
γxΔ t
1
3.12
1 − hxt hxσ tγ−1 dh
0
Δ
≥ γx t
1
1 − hxt hxtγ−1 dh γxγ−1 txΔ t.
0
Also from Lemma 2.4 and σt ≥ t, we have
γ
γ
rt xΔ t ≥ rσt xΔ σt .
3.13
By 3.11–3.13, we get
γ σ ztγxγ−1 txΔ t
zΔ t Δ
x
−
rt
t
zσ t
xγ txσ tγ
γ1
ztγ r σ tγ1/γ xΔ σt
zΔ t
σ
−
≤ −Q1 t w t σ
z t r 1/γ t
xσ tγ1
wΔ t < −Q1 t wσ t
−Q1 t wσ t
3.14
ztγ
zΔ t
−
wσ tγ1/γ .
zσ t r 1/γ tzσ tγ1/γ
Setting
B
zΔ t
,
zσ t
A
ztγ
r 1/γ tzσ tγ1/γ
,
u wσ t,
3.15
then by Lemma 2.6, from 3.14 we obtain that for all t ≥ T ,
γ1
rt zΔ t
.
w t < −Q1 t γ1
zγ t
γ 1
1
Δ
3.16
Integrating the above inequality from T to t≥ T , we get
⎡
γ1 ⎤
rs zΔ s
⎣Q1 s − ⎦Δs < wT − wt < wT .
γ1
zγ s
T
γ 1
t
1
3.17
Taking lim sup on both sides of the above inequality as t → ∞, we obtain a contradiction to
condition 3.2. The proof is complete.
10
Abstract and Applied Analysis
The following theorem gives new oscillation criteria for 1.3 which can be considered
as the extension of Philos-type oscillation criterion. Define D {t, s ∈ T2 : t ≥ s ≥ 0} and
H∗ Ht, s ∈ C1 D, R : Ht, t 0, Ht, s > 0, HsΔ t, s ≥ 0, for t > s ≥ 0 .
3.18
Theorem 3.2. Assume that h1 –h3 hold and γ ≥ 1. Furthermore, assume that there exist a
positive rd-continuous Δ-differentiable function zt and a function H ∈ H∗ such that for
all sufficiently large T ∈ T,
1
lim sup
Ht,
T
t→∞
t T
Cγ1 t, srszσsγ1
Ht, sQ1 s −
Δs ∞,
γ1 γ
H γ t, s γ 1
z s
3.19
then 1.3 is oscillatory.
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive. Then, by h1 –h3 there exists
T ≥ t0 sufficiently large such that yt > 0, yτt > 0, yδ1,2 t > 0, and Lemma 2.4 holds for
t ≥ T , where xt is defined by 2.8. Define wt as in 3.3. Proceeding as in the proof of
Theorem 3.1, we can get 3.14. From 3.14, for function H ∈ H∗ and all t ≥ T we have
t
Ht, sQ1 sΔs < −
T
t
Ht, swΔ sΔs t
T
−
Ht, swσ s
T
t
Ht, s
T
zsγ
zΔ s
Δs
zσ s
γ1/γ
r 1/γ szσ sγ1/γ
wσ s
3.20
Δs.
Using the integration by parts formula 2.7, we obtain
−
t
Δ
Ht, sw sΔs T
−Ht, sws|tT
Ht, T wT t
T
t
T
HsΔ t, swσ sΔs
3.21
HsΔ t, swσ sΔs.
It follows that
t
Ht, sQ1 sΔs < Ht, T wT T
t T
−
t
Ht, s
T
t
T
zΔ s
wσ sΔs
Ht, s σ
z s
zsγ
r 1/γ szσ sγ1/γ
Ht, T wT −
HsΔ t, s
t
wσ sγ1/γ Δs
Ct, swσ sΔs
T
Ht, szsγ
r 1/γ szσ sγ1/γ
wσ sγ1/γ Δs.
3.22
Abstract and Applied Analysis
11
Setting
B Ct, s,
A
Ht, szsγ
r 1/γ szσ sγ1/γ
u wσ s,
,
3.23
by Lemma 2.6 we obtain that for all t ≥ T ,
t
Ht, sQ1 sΔs < Ht, T wT t
T
T
Ct, sγ1 zσ sγ1 rs
Δs.
γ1 γ
H γ t, s γ 1
z s
3.24
That is,
1
Ht, T t Ct, sγ1 rszσ sγ1
Δs < wT .
Ht, sQ1 s −
γ1 γ
T
H γ t, s γ 1
z s
3.25
Taking lim sup on both sides of the above inequality as t → ∞, we obtain a contradiction to
condition 3.19. The proof is complete.
Theorem 3.3. Assume that h1 –h3 hold and γ ≥ 1. Then 1.3 is oscillatory if for all sufficiently
large T ∈ T,
t
lim sup
t→∞
T
γ
3.26
QsRT sΔs ∞.
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive. Then, by h1 –h3 there exists
T ≥ t0 sufficiently large such that yt > 0, yτt > 0, yδ1,2 t > 0, and Lemma 2.4 holds for
t ≥ T , where xt is defined by 2.8. Set φt r 1/γ txΔ t. By Lemma 2.4, we get φ > 0,
φγ Δ < 0. Using γ ≥ 1 and the keller’s chain rule, we get
φ t
γ
Δ
γ
1
Δ
φt hμtφ t
γ−1
dh φΔ t
0
γ
1
1 − hφt hφ t
σ
γ−1
3.27
Δ
dh φ t < 0.
0
So we have φΔ t < 0 and there is a constant L > 0 such that φt ≤ L for t ≥ T . Then 1.3
becomes φγ Δ t f1 t, yδ1 t f2 t, yδ2 t 0. By 2.18, we have
Δ
α
β
φγ t
α xδ1 t
β xδ2 t
0≥ γ q1 t 1 − pδ1 t σ γ q2 t 1 − pδ2 t σ γ .
φσ t
φ t
φ t
3.28
12
Abstract and Applied Analysis
Similar to the proof of 3.10, we get
αβ−αγ/β−α βγ−βα/β−α
Δ
φγ t
xδ1 t
xδ2 t
·
.
0≥ γ Qt
φσ t
φσ t
φσ t
3.29
Using the Keller’s chain rule and φΔ t < 0, we get φσ ≤ φ and
γ−1 Δ
φγ tΔ ≥ γ φσ t
φ t.
3.30
From δ1,2 t ≥ t and xΔ t > 0, it follows that
γφΔ t
Qt
0≥ σ
φ t
xδ1 t
φσ t
αβ−αγ/β−α ·
xδ2 t
φσ t
βγ−βα/β−α
γφΔ t
xt γ
Qt
≥
L
φσ t
γφΔ t
xt γ
Qt
≥
.
L
φt
3.31
By Lemma 2.4, we get
xt
xt
> RT t.
φt r 1/γ txΔ t
3.32
It follows that
0>
γφΔ t
γ
QtRT t.
L
3.33
Integrating the above inequality from T to t≥ T , we obtain
t
T
γ
QsRT sΔs < −
γ
L
t
T
φΔ sΔs γ
γ
φT − φt < φT .
L
L
3.34
Taking lim sup on both sides of the above inequality as t → ∞, we obtain a contradiction to
condition 3.26. The proof is complete.
Theorem 3.4. Assume that h1 –h3 hold and γ ≥ 1. Furthermore, assume that there exists a
positive rd-continuous Δ-differentiable function zt such that for all sufficiently large T ∈ T,
2 t r 1/γ s zΔ s
Δs ∞,
Q1 s −
lim sup
4γRT sγ−1 zs
t→∞
T
then 1.3 is oscillatory.
3.35
Abstract and Applied Analysis
13
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive. Then, by h1 –h3 there exists
T ≥ t0 sufficiently large such that yt > 0, yτt > 0, yδ1,2 t > 0, and Lemma 2.4 holds
for t ≥ T , where xt is defined by 2.8. Define wt as in 3.3. By 2.6, we obtain
γ σ
γ Δ
rt xΔ t
rt xΔ t
w t z t
zt
xγ t
xγ t
Δ
Δ
⎤
Δ γ γ
Δ γ Δ γ
Δ
x
−
rt
x
rt
x
t
t
t
x
t
z t
⎥
⎢
σ wσ t zt⎣
⎦
z t
xγ txσ tγ
⎡
Δ
3.36
⎤
σ γ γ
Δ γ Δ σ
γ
Δ
σ
Δ
−
r
x
rt
x
t
x
t
t
t
x
t
z t
⎥
⎢
σ wσ t zt⎣
⎦.
z t
xγ txσ tγ
⎡
Δ
By Lemma 2.4, σt ≥ t, and 3.5–3.13, for all t ≥ T we obtain
γ
r σ t xΔ σt γxγ−1 txΔ t
zΔ t
−
zt
zσ t
xγ txσ tγ
Δ
γ 2
x σt
zΔ t
xσ tγ xΔ t
σ
σ
− γztr t
− Q1 t w t σ
γ
γ
z t
xσ t
xt xΔ σt
wΔ t < − Q1 t wσ t
σ 2
zΔ t
xγ−1 t
w t
− γzt
< − Q1 t wσ t σ
γ−1
z t
zσ t
rt
xΔ t
< − Q1 t wσ t
3.37
σ 2 γ−1
1
zΔ t
w t
RT tr 1/γ t
−
γzt
.
σ
σ
z t
rt z t
It follows that
wΔ t < −Q1 t γztRT tγ−1 σ
zΔ t σ
w t −
w t2 .
σ
2
1/γ
σ
z t
r tz t
3.38
By completing the square, we have
Δ
w t < −Q1 t 2
r 1/γ t zΔ t
4γRT tγ−1 zt
.
3.39
Integrating the above inequality from T to t≥ T , we get
t T
2 r 1/γ s zΔ s
Δs < wT − wt < wT .
Q1 s −
4γRT sγ−1 zs
3.40
14
Abstract and Applied Analysis
Taking lim sup on both sides of the above inequality as t → ∞, we obtain a contradiction to
condition 3.35. The proof is complete.
Theorem 3.5. Assume that h1 –h3 and γ ≥ 1 hold. Furthermore, assume that there exist a
positive rd-continuous Δ-differentiable function zt and a function H ∈ H∗ such that for
all sufficiently large T ∈ T,
1
lim sup
Ht,
T
t→∞
t Ht, sA2 sr 1/γ szσ 2 s
Δs ∞,
Ht, sQ1 s −
4γRT sγ−1 zs
T
3.41
then 1.3 is oscillatory.
Proof. By 3.39, the proof is similar to Theorems 3.2 and 3.4, so we omit it.
Theorem 3.6. Assume that h1 –h3 hold and γ ≥ 1. Furthermore, assume that there exists a rdcontinuous Δ-differentiable function zt such that for all sufficiently large T ∈ T,
t lim sup
t→∞
T
1 Δ 2 1/γ
1−γ
z s r sRT s
Q2 s −
Δs ∞,
γ
3.42
then 1.3 is oscillatory.
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive. Then, by h1 –h3 there exists
T ≥ t0 sufficiently large such that yt > 0, yτt > 0, yδ1,2 t > 0, and Lemma 2.4 holds
for t ≥ T , where xt is defined by 2.8. Define the function vt by the Riccati substitution
γ
z2 trt xΔ t
,
vt :
xγ t
3.43
for t ≥ T,
then vt > 0. From 3.5 and 3.10, it follows that for all t ≥ T
γ
v t rt xΔ t
Δ
Δ
≤ −Q2 t z2 t
xγ t
γ
rt xΔ t
xγ tΔ
σ
γ
rt xΔ t
xγ tΔ
z2 t
xγ t
z2 t
xγ t
Δ
3.44
Δ
.
By 3.12 and Lemma 2.7, we obtain
⎤
Δ γ ⎡
Δ 2
2
rt
x
t
zt
⎣ zΔ t − xγ txσ tγ
⎦
vΔ t ≤ −Q2 t xγ t
xγ tΔ
< −Q2 t Δ
rt x t
γ 2
z t
xγ tΔ
Δ
< −Q2 t γ 2
rt x t zΔ t
Δ
γxγ−1 txΔ t
.
3.45
Abstract and Applied Analysis
15
It follows from Lemma 2.4 that
vΔ t < −Q2 t 1 Δ 2 1/γ
z t r tRT t1−γ .
γ
3.46
Integrating the above inequality from T to t ≥ T , we get
t 1 Δ 2 1/γ
z s r sRT s1−γ Δs < vT − vt < vT .
Q2 s −
γ
T
3.47
Taking lim sup on both sides of the above inequality as t → ∞, we obtain a contradiction to
condition 3.42. The proof is complete.
From Theorem 3.6, we can establish different sufficient conditions for
√ the oscillation
of 1.3 by using different choices of zt. For instance, if zt 1 or zt t, we have the
following results.
Corollary 3.7. Assume that h1 –h3 hold and γ ≥ 1. Then 1.3 is oscillatory if for all sufficiently
large T ∈ T,
t lim sup
t→∞
T
RT sr 1/γ s
RT sr 1/γ s μs
γ
QsΔs ∞.
3.48
Corollary 3.8. Assume that h1 –h3 hold and γ ≥ 1. Then 1.3 is oscillatory if for all sufficiently
large T ∈ T,
t
lim sup
t→∞
⎡
⎣Q3 s −
T
1
!
√
s σs
2
⎤
1
r 1/γ s RT s1−γ ⎦Δs ∞,
γ
3.49
where Q3 s RT sr 1/γ s/RT sr 1/γ s μsγ σsQs.
II 0 < γ < 1.
Theorem 3.1 . Assume that h1 –h3 hold and 0 < γ < 1. Furthermore, assume that there exists a
positive rd-continuous Δ-differentiable function zt such that for all sufficiently large T ∈ T,
⎡
γ1 ⎤
rs zΔ s
⎦Δs ∞,
lim sup ⎣Q1 s − γ1
zγ s
t→∞
T
γ 1
t
1
3.50
then 1.3 is oscillatory.
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive. Then, by h1 –h3 there exists
T ≥ t0 sufficiently large such that yt > 0, yτt > 0, yδ1,2 t > 0 and Lemma 2.4 holds
16
Abstract and Applied Analysis
for t ≥ T , where xt is defined by 2.8. Define wt as in 3.3. Proceeding as in the proof of
Theorem 3.1, we get
γ Δ zt γ σ zΔ txγ t − ztxγ tΔ
Δ
.
rt x t
w t rt x t
xγ t
xγ txσ tγ
Δ
Δ
3.51
Using 0 < γ < 1, Lemma 2.4 and the Keller’s chain rule, we get
Δ
x t γ
γ
1
Δ
xt hμtx t
γ−1
dh xΔ t
0
≥ γxΔ t
1
1 − hxσ t hxσ tγ−1 dh
3.52
0
γxσ tγ−1 xΔ t.
By 3.10, 3.51, and 3.52, we get
wΔ t < −Q1 t wσ t
γ σ ztγxσ tγ−1 xΔ t
zΔ t Δ
x
−
rt
.
t
zσ t
xγ txσ tγ
3.53
Since
γ σ
− rt xΔ t
ztγxσ tγ−1 xΔ t
xγ txσ tγ
r σ tγ1/γ
σ γ1
ztγxΔ t
−
σ
xγ txσ tr σ t1/γ xΔ t
σ γ1
ztγxΔ t
r σ tγ1/γ xΔ t
For 3.13 ≤ −
xγ txσ tr 1/γ txΔ t
<−
xΔ t
ztγ
r 1/γ tzσ tγ1/γ
3.54
wσ tγ1/γ ,
it follows that
wΔ t < −Q1 t wσ t
ztγ
zΔ t
−
wσ tγ1/γ .
zσ t r 1/γt zσ tγ1/γ
3.55
It is easy to see 3.55 is of the same form as 3.14. The following is similar to the proof of
Theorem 3.1 and hence omitted.
For γ ∈ 0, 1, Theorem 3.2 also holds. Its proof is similar to those of Theorems 3.1 and
3.2.
Abstract and Applied Analysis
17
Theorem 3.2 . Assume that h1 –h3 hold and 0 < γ < 1. Furthermore, assume that there exist
a positive rd-continuous Δ-differentiable function zt and a function H ∈ H∗ such that for all
sufficiently large T ∈ T,
1
lim sup
Ht,
T
t→∞
t Cγ1 t, srszσsγ1
Ht, sQ1 s −
Δs ∞,
γ1 γ
T
H γ t, s γ 1
z s
3.56
then 1.3 is oscillatory.
Theorem 3.3 . Assume that h1 –h3 hold and 0 < γ < 1. Then 1.3 is oscillatory if for all sufficiently large T ∈ T,
t
lim sup
t→∞
T
γ
3.57
QsRT sΔs ∞.
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive. Then, by h1 –h3 there exists
T ≥ t0 sufficiently large such that yt > 0, yτt > 0, yδ1,2 t > 0, and Lemma 2.4 holds
for t ≥ T , where xt is defined by 2.8, φ is defined as in Theorem 3.3. Similar to the proof of
Theorem 3.3, we get
αβ−αγ/β−α βγ−βα/β−α
Δ
φγ t
xδ1 t
xδ2 t
Qt
·
.
0≥
φγ t
φt
φt
3.58
Using the Keller’s chain rule, 0 < γ < 1, and φΔ t < 0, we get
Δ
φ t γ
γ
1
Δ
φt hμtφ t
γ−1
dh φΔ t
0
≥γ
1
1 − hφt hφt
γ−1
dh φΔ t
3.59
0
γφγ−1 tφΔ t.
From δ1,2 t ≥ t and xΔ t > 0, it follows that
0≥
γφΔ t
xt γ
Qt
.
L
φt
3.60
It is easy to see 3.31 is of the same form as 3.60. The following is similar to the proof of
Theorem 3.3 and hence omitted.
Last, we give a theorem which holds for all γ > 0, a quotient of add positive integers.
18
Abstract and Applied Analysis
Theorem 3.9. Assume that h1 –h3 hold. Furthermore, assume that there exists a positive rdcontinuous Δ-differentiable function zt such that for all sufficiently large T ∈ T,
t lim sup
t→∞
Q1 s −
T
zΔ
γ
RT s
Δs ∞,
3.61
then 1.3 is oscillatory.
Proof. Suppose to the contrary that yt is a nonoscillatory solution of 1.3. Without loss of
generality, we may assume that yt is eventually positive. Then, by h1 –h3 there exists
T ≥ t0 sufficiently large such that yt > 0, yτt > 0, yδ1,2 t > 0, and Lemma 2.4 holds
for t ≥ T , where xt is defined by 2.8. Define wt as in 3.3. Then, wt > 0 and because
1/xγ Δ −xγ Δ /xγ xγ σ < 0 we get
γ σ 1 Δ
1
Δ
x
zt
·
rt
t
xγ t
xγ t
γ σ
γ Δ 1
.
< zΔ t · rt xΔ t
zt rt xΔ t
xγ t
γ
wΔ t zt · rt xΔ t
Δ
3.62
From 3.5, 3.10, and 3.13, we obtain
Δ
Δ
Δ
w t < z t · rt x t
Δ
< z trt
xΔ t
xt
γ
γ
γ Δ 1
zt rt x t
xγ t
− Q1 t <
Δ
zΔ t
γ
RT t
3.63
− Q1 t.
Integrating the above inequality from T to t≥ T , we get
t zΔ s
Q1 s − γ
Δs < wT − wt < wT .
RT s
T
3.64
Taking lim sup on both sides of the above inequality as t → ∞, we obtain a contradiction to
condition 3.61. The proof is complete.
4. Examples
In this section, we give two examples to illustrate our main results. To obtain the conditions
for oscillation we will use the following facts:
∞
1
Δs
∞ if 0 ≤ γ ≤ 1,
sγ
∞
1
Δs
< ∞,
sγ
We first give an example to show Theorems 3.1 and 3.1 .
if γ > 1.
4.1
Abstract and Applied Analysis
19
Example 4.1. Consider the equation
Δ γ
1
yt
ptyτt
t σtγ
Δ
σt2γ
yα δ1 t
α
1 − pδ1 t t2γ1
σt2γ
β
β 2γ1 y δ2 t 0,
1 − pδ2 t t
4.2
t ∈ T,
where T 1, ∞ is a time scale, pt satisfies h2 , τt and δ1,2 t satisfy h1 , rt 1/t σtγ , and γ is a quotient of odd positive integers.
We choose q1 t σt2γ /1 − pδ1 tα t2γ1
σt2γ /1 − pδ2 tβ t2γ1 ,
∞, q2 t 1/γ
Δ
2
and z 1, then z 0, s σsΔs t c, and 1 Δt/r t ∞. For any sufficiently
large T ∈ T and s > T , there exists a constant k > 0 sufficiently large such that
γ γ
γ 2
s − T 2 1/s σs
RT sr 1/γ s
s2
>
,
kσ 2 s
s2 − T 2 1/s σs σs − s
RT sr 1/γ s μs
⎡
γ1 ⎤
t
t
rs zΔ s
1
1
−γ
⎣
⎦
Q1 s − lim sup
Δs ∞.
Δs ≥ lim sup k
γ1
γ s
z
t→∞
t→∞
T
T s
γ 1
4.3
Hence, by Theorems 3.1 and 3.1 , 4.2 is oscillatory.
The second example illustrates Corollary 3.7.
Example 4.2. Consider the equation
yτt ΔΔ σt 1/3
σt
yt −1
2 y δt 2 y5/3 δt 0,
t
t
δ t
t ∈ T,
4.4
where T 1, ∞ is a time scale, γ 1, α 1/3, β 5/3, δt, and τt satisfy h1 , δt has an
inverse function δ−1 t, and pt 1/δ−1 t satisfies h2 .
We choose q1 t q2 t σt/t2 and z 1. For any sufficiently large T ∈ T and s > T ,
there exists a constant k > 0 sufficiently large such that
s
s−T
RT sr 1/γ s
>
,
RT sr 1/γ s μs s − T σs − s kσs
γ
t t
RT sr 1/γ s
1
1 2
1
−
lim sup
QsΔs
≥
lim
sup
Δs
s
RT sr 1/γ s μs
t→∞
t→∞
T
T ks
1 t 1 2
1
lim sup
− 2 3 Δs ∞.
s
s
t→∞ k T s
Hence, by Corollary 3.7, 4.4 is oscillatory.
4.5
20
Abstract and Applied Analysis
Acknowledgment
This project was supported by the NNSF of China nos. 10971231, 11071238, and 11271379.
References
1 H.-W. Wu, R.-K. Zhuang, and R. M. Mathsen, “Oscillation criteria for second-order nonlinear neutral
variable delay dynamic equations,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 321–331,
2006.
2 S. H. Saker, R. P. Agarwal, and D. O’Regan, “Oscillation results for second-order nonlinear neutral
delay dynamic equations on time scales,” Applicable Analysis, vol. 86, no. 1, pp. 1–17, 2007.
3 S.-Y. Zhang and Q.-R. Wang, “Oscillation of second-order nonlinear neutral dynamic equations on
time scales,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2837–2848, 2010.
4 S. H. Saker, “Oscillation criteria for a second-order quasilinear neutral functional dynamic equation
on time scales,” Nonlinear Oscillations, vol. 13, no. 3, pp. 407–428, 2011.
5 Y. B. Sun, Z. L. Han, T. X. Li, and G. R. Zhang, “Oscillation criteria for secondorder quasilinear neutral
delay dynamic equations on time scales,” Advances in Difference Equations, Article ID 512437, 14 pages,
2010.
6 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkhäuser, Boston, Mass, USA, 2001.
7 A. Del Medico and Q. Kong, “Kamenev-type and interval oscillation criteria for second-order linear
differential equations on a measure chain,” Journal of Mathematical Analysis and Applications, vol. 294,
no. 2, pp. 621–643, 2004.
8 R. M. Mathsen, Q.-R. Wang, and H.-W. Wu, “Oscillation for neutral dynamic functional equations on
time scales,” Journal of Difference Equations and Applications, vol. 10, no. 7, pp. 651–659, 2004.
9 Y. Şahiner, “Oscillation of second-order delay differential equations on time scales,” Nonlinear
Analysis, Theory, Methods and Applications, vol. 63, no. 5–7, pp. e1073–e1080, 2005.
10 Z.-Q. Zhu and Q.-R. Wang, “Frequency measures on time scales with applications,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 398–409, 2006.
11 E. Akin-Bohner, M. Bohner, and S. H. Saker, “Oscillation criteria for a certain class of second order
Emden-Fowler dynamic equations,” Electronic Transactions on Numerical Analysis, vol. 27, pp. 1–12,
2007.
12 Z.-Q. Zhu and Q.-R. Wang, “Existence of nonoscillatory solutions to neutral dynamic equations on
time scales,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 751–762, 2007.
13 H. Huang and Q.-R. Wang, “Oscillation of second-order nonlinear dynamic equations on time scales,”
Dynamic Systems and Applications, vol. 17, no. 3-4, pp. 551–570, 2008.
14 Z.-H. Yu and Q.-R. Wang, “Asymptotic behavior of solutions of third-order nonlinear dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 531–540,
2009.
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