Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 389109, 21 pages
doi:10.1155/2010/389109
Research Article
Oscillation of Second-Order Mixed-Nonlinear
Delay Dynamic Equations
M. Ünal1 and A. Zafer2
1
2
Department of Software Engineering, Bahçeşehir University, Beşiktaş, 34538 Istanbul, Turkey
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Correspondence should be addressed to M. Ünal, munal@bahcesehir.edu.tr
Received 19 January 2010; Accepted 20 March 2010
Academic Editor: Josef Diblik
Copyright q 2010 M. Ünal and A. Zafer. 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.
New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations
on time scales by utilizing an interval averaging technique. No restriction is imposed on the
coefficient functions and the forcing term to be nonnegative.
1. Introduction
In this paper we are concerned with oscillatory behavior of the second-order nonlinear delay
dynamic equation of the form
rtxΔ t
Δ
p0 txτ0 t n
pi t|xτi t|αi −1 xτi t et,
t ≥ t0
1.1
i1
on an arbitrary time scale T, where
α1 > α2 > · · · > αm > 1 > αm1 > · · · > αn > 0,
n > m ≥ 1;
1.2
the functions r, pi , e: T → R are right-dense continuous with r > 0 nondecreasing; the delay
functions τi : T → T are nondecreasing right-dense continuous and satisfy τi t ≤ t for t ∈ T
with τi t → ∞ as t → ∞.
We assume that the time scale T is unbounded above, that is, sup T ∞ and define
the time scale interval t0 , ∞T by t0 , ∞T : t0 , ∞ ∩ T. It is also assumed that the reader is
already familiar with the time scale calculus. A comprehensive treatment of calculus on time
scales can be found in 1–3 .
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By a solution of 1.1 we mean a nontrivial real valued function x : T → R such that
1
1
T, ∞T and rxΔ ∈ Crd
T, ∞T for all T ∈ T with T ≥ t0 , and that x satisfies 1.1.
x ∈ Crd
A function x is called an oscillatory solution of 1.1 if x is neither eventually positive nor
eventually negative, otherwise it is nonoscillatory. Equation 1.1 is said to be oscillatory if
and only if every solution x of 1.1 is oscillatory.
Notice that when T R, 1.1 is reduced to the second-order nonlinear delay
differential equation
n
rtx t p0 txτ0 t pi t|xτi t|αi −1 xτi t et,
t ≥ t0
1.3
k ≥ k0 .
1.4
i1
while when T Z, it becomes a delay difference equation
ΔrkΔxk p0 kxτ0 k n
pi k|xτi k|αi −1 xτi k ek,
i1
Another useful time scale is T qN : {qm : m ∈ N and q > 1 is a real number}, which leads
to the quantum calculus. In this case, 1.1 is the q-difference equation
n
Δq rtΔq xt p0 txτ0 t pi t|xτi t|αi −1 xτi t et,
t ≥ t0 ,
1.5
i1
where Δq ft fσt − ft /μt, σt qt, and μt q − 1t.
Interval oscillation criteria are more natural in view of the Sturm comparison theory
since it is stated on an interval rather than on infinite rays and hence it is necessary to establish
more interval oscillation criteria for equations on arbitrary time scales as in T R. As far
as we know when T R, an interval oscillation criterion for forced second-order linear
differential equations was first established by El-Sayed 4 . In 2003, Sun 5 demonstrated
nicely how the interval criteria method can be applied to delay differential equations of the
form
x t pt|xτt|α−1 xτt et,
α ≥ 1,
1.6
where the potential p and the forcing term e may oscillate. Some of these interval oscillation
criteria were recently extended to second-order dynamic equations in 6–10 . Further results
on oscillatory and nonoscillatory behavior of the second order nonlinear dynamic equations
on time scales can be found in 11–23 , and the references cited therein.
Therefore, motivated by Sun and Meng’s paper 24 , using similar techniques
introduced in 17 by Kong and an arithmetic-geometric mean inequality, we give oscillation
criteria for second-order nonlinear delay dynamic equations of the form 1.1. Examples are
considered to illustrate the results.
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3
2. Main Results
We need the following lemmas in proving our results. The first two lemmas can be found in
25, Lemma 1 .
Lemma 2.1. Let {αi }, i 1, 2, . . . , n be the n-tuple satisfying α1 > α2 > · · · > αm > 1 > αm1 > · · · >
αn > 0. Then, there exists an n-tuple {η1 , η2 , . . . , ηn } satisfying
n
αi ηi 1,
n
i1
i1
ηi < 1,
0 < ηi < 1.
2.1
Lemma 2.2. Let {αi }, i 1, 2, . . . , n be the n-tuple satisfying α1 > α2 > · · · > αm > 1 > αm1 > · · · >
αn > 0. Then there exists an n-tuple {η1 , η2 , . . . , ηn } satisfying
n
αi ηi 1,
n
i1
i1
ηi 1,
0 < ηi < 1.
2.2
The next two lemmas are quite elementary via differential calculus; see 23, 25 .
Lemma 2.3. Let u, A, and B be nonnegative real numbers. Then
1/γ−1 1/γ 1−1/γ
Auγ B ≥ γ γ − 1
A B
u,
γ > 1.
2.3
Lemma 2.4. Let u, A, and B be nonnegative real numbers. Then
Cu − Duγ ≥ γ − 1 γ γ/1−γ Cγ/γ−1 D1/1−γ ,
0 < γ < 1.
2.4
The last important lemma that we need is a special case of the one given in 6 . For
completeness, we provide a proof.
Lemma 2.5. Let τ : T → T be a nondecreasing right-dense continuous function with τt ≤ t, and
1
a, b ∈ T with a < b. If x ∈ Crd
τa, b T is a positive function such that rtxΔ t is nonincreasing
on τa, b T with r > 0 nondecreasing, then
xτt τt − τa
,
≥
xσ t
σt − τa
t ∈ a, b T .
2.5
Proof. By the Mean Value Theorem 2, Theorem 1.14
xt − xτa ≥ xΔ η t − τa,
2.6
for some η ∈ τa, tT , for any t ∈ τa, b T . Since rtxΔ t is nonincreasing and rt is
nondecreasing, we have
rtxΔ t ≤ r η xΔ η ≤ rtxΔ η ,
t>η
2.7
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and so xΔ t ≤ xΔ η, t ≥ η. Now
xt − xτa ≥ xΔ tt − τa,
2.8
t ∈ τa, b T .
Define
μs : xs − s − τaxΔ s,
It follows from 2.8 that μs ≥ xτa > 0 for s ∈ τt, σt
σt
0<
τt
μs
Δs xsxσ s
σt τt
s − τa
xs
2.9
s ∈ τt, σt T , t ∈ a, bT .
Δ
Δs T
and t ∈ a, bT . Thus, we have
σt − τa τt − τa
,
−
xσ t
xτt
2.10
which completes the proof.
In what follows we say that a function Ht, s : T2 → R belongs to HT if and only if
H is right-dense continuous function on {t, s ∈ T2 : t ≥ s ≥ t0 } having continuous Δ-partial
0 for all t /
s. Note
derivatives on {t, s ∈ T2 : t > s ≥ t0 }, with Ht, t 0 for all t and Ht, s /
that in case HR , the Δ-partial derivatives become the usual partial derivatives of Ht, s. The
partial derivatives for the cases HZ and HN will be explicitly given later.
Denoting the Δ-partial derivatives H Δt t, s and H Δs t, s of Ht, s with respect to t
and s by H1 t, s and H2 t, s, respectively, the theorems below extend the results obtained
in 5 to nonlinear delay dynamic equation on arbitrary time scales and coincide with them
when H 2 t, s is replaced by Ht, s. Indeed, if we set Ht, s Ut, s, then it follows that
H1 t, s U1 t, s
Uσt, s Ut, s
H2 t, s ,
U2 t, s
Ut, σs Ut, s
.
2.11
When T R, they become
∂Ht, s ∂Ut, s/∂t
,
∂t
2 Ut, s
∂Ht, s ∂Ut, s/∂s
∂s
2 Ut, s
2.12
as in 5 . However, we prefer using H 2 t, s instead of Ut, s for simplicity.
Theorem 2.6. Suppose that for any given (arbitrarily large) T ∈ T there exist subintervals a1 , b1
and a2 , b2 T of T, ∞T , where a1 < b1 and a2 < b2 such that
pi t ≥ 0
for t ∈ a1 , b1
T
∪ a2 , b2 T , i 0, 1, 2, . . . , n,
−1l et > 0 for t ∈ al , bl T , l 1, 2,
T
2.13
where
al min τj al : j 0, 1, 2, . . . , n
2.14
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5
hold. Let {η1 , η2 , . . . ηn } be an n-tuple satisfying 2.1 of Lemma 2.1. If there exist a function H ∈ HT
and numbers cν ∈ aν , bν T such that
1
2
H cν , aν cν
aν
QtH 2 σt, aν − rtH12 t, aν Δt
1
2
H bν , cν bν
QtH bν , σt −
2
cν
rtH22 bν , t
2.15
Δt > 0
for ν 1, 2, where
Qt p0 t
n
η τi t − τi aν τ0 t − τ0 aν k0 |et|η0
pi t i
σt − τ0 aν σt − τi aν i1
n
−η
ηi i ,
k0 αi ηi
,
n
η0 1 − ηi ,
i0
2.16
i1
then 1.1 is oscillatory.
Proof. Suppose on the contrary that x is a nonoscillatory solution of 1.1. First assume that
xt and xτj t j 0, 1, 2 . . . , n are positive for all t ≥ t1 for some t1 ∈ t0 , ∞T . Choose a1
sufficiently large so that τj τj a1 ≥ t1 . Let t ∈ a1 , b1 T .
Define
wt −rt
xΔ t
,
xt
t ≥ t1 .
2.17
Using the delta quotient rule, we have
2
2
Δ
Δ
rt xΔ t
rtxΔ t
rtxΔ t xt − rt xΔ t
−
.
w t −
xtxσ t
xσ t
xtxσ t
Δ
2.18
Notice that
wt
x2 t xtxσ t xt xt μtxΔ t x2 t 1 − μt
rt − μtwt
rt
rt
2.19
which implies
rt − μtwt rt
xσ t
> 0.
xt
2.20
Hence, we obtain
Δ
rtxΔ t
w2 t
w t −
.
σ
x t
rt − μtwt
Δ
2.21
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Advances in Difference Equations
Substituting 2.21 into 1.1 yields
wΔ t n
p0 txτ0 t
xτi t
w2 t
et
− σ .
pi t|xτi t|αi −1 σ
xσ t
rt − μtwt i1
x t
x t
2.22
By assumption, we can choose a1 , b1 ≥ t1 such that pi t ≥ 0 i 1, 2, 3 . . . , n and et ≤ 0
for all t ∈ a1 , b1 T , where a1 is defined as in 2.14. Clearly, the conditions of Lemma 2.5 are
satisfied when, τ replaced with τj for each fixed j 0, 1, 2, . . . , n. Therefore, from 2.5, we
have
τj t − τj a1 x τj t
≥
,
σ
x t
σt − τj a1 t ∈ a1 , b1
2.23
T
and taking into account 2.22 yields
wΔ t ≥ p0 t
n
τ0 t − τ0 a1 w2 t
τi t − τi a1 pk t
σt − τ0 a1 rt − μtwt i1
σt − τi a1 αi
xσ tαi −1 |et|
.
xσ t
2.24
Denote
Q0∗ t
τ0 t − τ0 a1 ,
: p0 t
σt − τ0 a1 Qi∗ t
τi t − τi a1 : pi t
σt − τi a1 αi
.
2.25
From 2.24, we have
wΔ t ≥ Q0∗ t n
w2 t
|et|
Qi∗ txσ tαi −1 σ .
rt − μtwt i1
x t
2.26
Now recall the well-known arithmetic-geometric mean inequality, see 26 ,
n
n
ηi
ui ηi ≥
ui ,
i0
where η0 1 −
n
i1
2.27
i0
ηi and ηi > 0, i 1, 2, . . . , n. Setting
u0 η0 :
|et|
,
xσ t
ui ηi : Qi∗ txσ tαi −1
2.28
in 2.26 yields
wΔ t ≥ Q0∗ t n
n
w2 t
w2 t
ui ηi u0 η0 Q0∗ t ui ηi .
rt − μtwt i1
rt − μtwt i0
2.29
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From 2.29 and taking into account 2.27, we get
wΔ t ≥ Q0∗ t n
w2 t
η
ui
rt − μtwt i0 i
2.30
and hence,
wΔ t ≥ Q0∗ t η0 n
η
ηi σ
w2 t
−η |et|
−ηi ∗
αi −1 i
η0 0 σ
η
t
t
Q
x
i
rt − μtwt
x tη0 i1 i
Q0∗ t n
n
η
w2 t
−η
−η η0 0 |et|η0
ηi i Qi∗ t i xσ t−η0 j1 αj ηj −ηj rt − μtwt
i1
Q0∗ t n
η
w2 t
−η
−η η0 0 |et|η0
ηi i Qi∗ t i
rt − μtwt
i1
2.31
which yields
wΔ t ≥ Q0∗ t n
ηi τi t − τi a1 w2 t
−η
−η η0 0 |et|η0
ηi i pi t
rt − μtwt
σt − τi a1 i1
αi ηi
w2 t
,
Qt rt − μtwt
2.32
where
Qt Q0∗ t
n
ηi τi t − τi a1 −η0
−ηi η0
ηi pi t
η0 |et|
σt − τi a1 i1
αi ηi
.
2.33
Multiplying both sides of 2.32 by H 2 σt, a1 and integrating both sides of the resulting
inequality from a1 to c1 , a1 < c1 < b1 yield
c1
Δ
w tH σt, a1 Δt ≥
2
a1
c1
QtH σt, a1 Δt 2
c1
a1
a1
w2 tH 2 σt, a1 Δt.
rt − μtwt
2.34
Fix s and note that
Δt
Δt
wtH 2 t, s
H 2 σt, swΔ t H 2 t, s wt
2.35
Δ
H σt, sw t H1 t, sHσt, swt Ht, sH1 t, swt,
2
from which we obtain
Δt
H 2 σt, swΔ t wtH 2 t, s
− H1 t, sHσt, swt − Ht, sH1 t, swt. 2.36
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Therefore,
c1
Δ
w tH σt, a1 Δt 2
c1 wtH 2 t, a1 a1
Δt
Δt
a1
−
c1
H1 t, a1 Hσt, a1 wt Ht, a1 H1 t, a1 wt Δt.
a1
2.37
Notice that
c1 wtH 2 t, a1 Δt
Δt wc1 H 2 c1 , a1 − wa1 H 2 a1 , a1 wc1 H 2 c1 , a1 2.38
a1
since Ha1 , a1 0 and hence, we obtain from 2.34 that
wc1 H c1 , a1 ≥
2
c1
QtH σt, a1 Δt 2
a1
c1
a1
c1
w2 t
H 2 σt, a1 Δt
rt − μtwt
2.39
H1 t, a1 Hσt, a1 wt Ht, a1 H1 t, a1 wt Δt.
a1
On the other hand,
w2 tH 2 σt, s
wtHσt, sH1 t, s Ht, sH1 t, swt
rt − μtwt
wtHσt, s
rt − μtwt
2
rt − μtwtH1 t, s
2.40
− rt − μtwt H12 t, s − wtHσt, sH1 t, s Ht, sH1 t, swt.
Taking into account that Hσt, s Ht, s μtH1 t, s, we have
w2 tH 2 σt, a1 wtHσt, a1 H1 t, a1 Ht, a1 H1 t, a1 wt ≥ −rtH12 t, a1 .
rt − μtwt
2.41
Using this inequality in 2.39, we have
wc1 H 2 c1 , a1 ≥
c1
a1
QtH 2 σt, a1 − rtH12 t, a1 Δt.
2.42
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Similarly, by following the above calculation step by step, that is, multiplying both
sides of 2.32 this time by H 2 b1 , σs after taking into account that
Δs
H 2 t, σswΔ s wsH 2 t, s
− H2 t, sHt, σsws − Ht, sH2 t, sws, 2.43
one can easily obtain
−wc1 H 2 b1 , c1 ≥
b1
QsH 2 b1 , σs − rsH22 b1 , s Δs.
c1
2.44
Adding up 2.42 and 2.44, we obtain
0≥
1
2
H c1 , a1 c1
a1
1
2
H b1 , c1 QtH 2 σt, a1 − rtH12 t, a1 Δt
b1
c1
2.45
QtH 2 b1 , σt − rsH22 b1 , t Δt.
This contradiction completes the proof when xt is eventually positive. The proof when xt
is eventually negative is analogous by repeating the above arguments on the interval a2 , b2 T
instead of a1 , b1 T .
Corollary 2.7. Suppose that for any given (arbitrarily large) T ≥ t0 there exist subintervals a1 , b1
and a2 , b2 of T, ∞ such that
pi t ≥ 0 for t ∈ a1 , b1 ∪ a2 , b2 , i 0, 1, 2, . . . , n,
2.46
−1l et ≥ 0 for t ∈ al , bl , l 1, 2,
where al min{τj al : j 0, 1, 2, . . . , n} holds. Let {η1 , η2 , . . . , ηn } be an n-tuple satisfying 2.1
of Lemma 2.1. If there exist a function H ∈ HR and numbers cν ∈ aν , bν such that
1
2
H cν , aν cν
aν
QtH 2 t, aν − rtH12 t, aν dt
1
2
H bν , cν bν
cν
2.47
QtH 2 bν , t − rtH22 bν , t dt > 0
for ν 1, 2, where
Qt p0 t
n
η τi t − τi aν τ0 t − τ0 aν k0 |et|η0
pi t i
t − τ0 aν t − τi aν i1
n
−η
ηi i ,
k0 i0
then 1.3 is oscillatory.
n
η0 1 − ηi ,
i1
αi ηi
,
2.48
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Corollary 2.8. Suppose that for any given (arbitrarily large) T ≥ t0 there exist a1 , b1 , a2 , b2 ∈ Z with
T ≤ a1 < b1 and T ≤ a2 < b2 such that for each i 0, 1, 2, . . . , n,
pi t ≥ 0 for t ∈ {a1 , a1 1, a1 2, . . . , b1 } ∪ {a2 , a2 1, a2 2, . . . , b2 },
−1l et ≥ 0 for t ∈ {al , al 1, al 2, . . . , bl } l 1, 2,
2.49
where al min{τj al : j 0, 1, 2, . . . , n} holds. Let {η1 , η2 , . . . , ηn } be an n-tuple satisfying 2.1 of
Lemma 2.1. If there exist a function H ∈ HZ and numbers cν ∈ {aν 1, aν 2, . . . , bν − 1} such that
c
ν −1
1
H 2 c
ν , aν taν
QtH 2 t 1, aν − rtH12 t, aν 2.50
b
ν −1
1
2
QtH 2 bν , t 1 − rtH22 bν , t > 0
H bν , cν tcν
for ν 1, 2, where
H2 bν , t : Hbν , t 1 − Hbν , t,
n
ηi τi t − τi aν αi ηi
τ0 t − τ0 aν η0
Qt p0 t
k0 |et|
,
pi t
t 1 − τ0 aν t 1 − τi aν i1
H1 t, aν : Ht 1, aν − Ht, aν ,
k0 n
i0
−η
η0 1 −
ηi i ,
n
2.51
ηi ,
i1
then 1.4 is oscillatory.
Corollary 2.9. Suppose that for any given (arbitrarily large) T ≥ t0 there exist a1 , b1 , a2 , b2 ∈ N with
T ≤ a1 < b1 and T ≤ a2 < b2 such that for each i 0, 1, 2, . . . , n,
pi t ≥ 0 for t ∈ qa1 , qa1 1 , . . . , qb1 ∪ qa2 , qa2 1 , . . . , qb2 ,
l
−1 et ≥ 0
2.52
for t ∈ qal , qal 1 , . . . , qbl , l 1, 2
where qal min{τj qal : j 0, 1, 2, . . . , n} holds. Let {η1 , η2 , . . . , ηn } be an n-tuple satisfying 2.1
of Lemma 2.1. If there exist a function H ∈ Hq and numbers qcν ∈ {qaν 1 , qaν 2 , . . . , qbν −1 } such that
H
1
2
qcν , qaν
c
ν −1
maν
1
qm Q qm H 2 qm1 , qaν − r qm H12 qm , qaν
b
ν −1
qm Q q
H 2 qbν , qcν mcν
m
H 2 qbν , qm1 − r q
m
H22 qbν , qm
2.53
>0
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for ν 1, 2, where
m
H1 q , q
aν
H qm1 , qaν − H qm , qaν
,
:
q − 1 qm
Qt p0 t
bν
H2 q , q
m
H qbν , qm1 − H qbν , qm
:
,
q − 1 qm
n
τ0 t − τ0 qaν
η τi t − τi qaν pi t i
a k0 |et|η0
qt − τi qaν qt − τ0 q ν
i1
k0 n
−η
ηi i ,
η0 1 −
i0
αi ηi
,
n
ηi ,
i1
2.54
then 1.5 is oscillatory.
Notice that Theorem 2.6 does not apply if there is no forcing term, that is, et ≡ 0. In
this case we have the following theorem.
Theorem 2.10. Suppose that for any given (arbitrarily large) T ∈ T there exists a subinterval a, b
of T, ∞T , where a < b such that
pi t ≥ 0 for t ∈ a, b T ,
i 0, 1, 2, . . . , n,
T
2.55
where a min{τj a : j 0, 1, 2, . . . , n} holds. Let {η1 , η2 , . . . , ηn } be an n-tuple satisfying 2.2 in
Lemma 2.2. If there exist a function H ∈ HT and a number c ∈ a, bT such that
1
H 2 c, a
c
a
QtH 2 σt, a − rtH12 t, a Δt
1
2
H b, c
b
c
2.56
QtH 2 b, σt − rsH22 b, t2 Δt > 0,
where
Qt p0 t
n
η τi t − τi a
τ0 t − τ0 a
k0
pi t i
σt − τ0 a
σt − τi a
i1
αi ηi
,
k0 n
−η
ηi i ,
2.57
i1
then 1.1 with et ≡ 0 is oscillatory.
Proof. We will just highlight the proof since it is the same as the proof of Theorem 2.6. We
should remark here that taking et ≡ 0 and η0 0 in proof of Theorem 2.6, we arrive at
wΔ t ≥ Q0∗ t n
w2 t
ui ηi .
rt − μtwt i1
2.58
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The arithmetic-geometric mean inequality we now need is
n
n
η
ui ηi ≥
ui i ,
i1
where 1 n
i1
2.59
i1
ηi and ηi > 0, i 1, 2, . . . , n are as in Lemma 2.2.
Corollary 2.11. Suppose that for any given (arbitrarily large) T ≥ t0 there exists a subinterval a, b
of T, ∞, where T ≤ a < b with a, b ∈ R such that
pi t ≥ 0 for t ∈ a, b , i 0, 1, 2, . . . , n,
2.60
where a min{τj a : j 0, 1, 2, . . . , n} holds. Let {η1 , η2 , . . . , ηn } be an n-tuple satisfying 2.2 in
Lemma 2.2. If there exist a function H ∈ HR and a number c ∈ a, b such that
1
2
H c, a
c
a
QtH 2 t, a − rtH12 t, a dt
1
2
H b, c
b
c
2.61
QsH 2 b, t − rtH22 b, t dt > 0,
where
Qt p0 t
n
η τi t − τi a
τ0 t − τ0 a
k0
pi t i
t − τ0 a
t − τi a
i1
αi ηi
,
k0 n
−η
ηi i ,
2.62
i1
then 1.3 with et ≡ 0 is oscillatory.
Corollary 2.12. Suppose that for any given (arbitrarily large) T ≥ t0 there exists a, b ∈ Z with
T ≤ a < b such that
pi t ≥ 0
for t ∈ {a, a 1, . . . , b},
i 0, 1, 2, . . . , n,
2.63
where a min{τj a : j 0, 1, 2, . . . , n} holds. Let {η1 , η2 , . . . , ηn } be an n-tuple satisfying 2.2 in
Lemma 2.2. If there exist a function H ∈ HZ and a number c ∈ {a 1, a 2, . . . , b − 1} such that
1
c−1
H 2 c, a
ta
QtH 2 t 1, a − rtH12 t, a
b−1
1
2
QtH 2 b, t 1 − rtH22 b, t > 0,
H b, c tc
2.64
Advances in Difference Equations
13
where
H1 t, a : Ht 1, a − Ht, a,
H2 b, t : Hb, t 1 − Hb, t,
n
n
η τi t − τi a αi ηi
τ0 t − τ0 a
−η
Qt p0 t
k0
, k0 ηi i ,
pi t i
t 1 − τ0 a
t 1 − τi a
i1
i1
2.65
then 1.4 with et ≡ 0 is oscillatory.
Corollary 2.13. Suppose that for any given (arbitrarily large) T ≥ t0 there exist a, b ∈ N with
T ≤ a < b such that
pi t ≥ 0
for t ∈ qa , qa1 , . . . , qb , i 0, 1, 2, . . . , n
2.66
where qa min{τj qa : j 0, 1, 2, . . . , n} holds. Let {η1 , η2 , . . . , ηn } be an n-tuple satisfying 2.2
in Lemma 2.2. If there exist a function H ∈ HqN and a number qc ∈ {qa , qa1 , . . . , qb } such that
c−1
m 2 m1 a m 1
m
m a 2
Q
q
q
−
r
q
q
,
q
q
,
q
H
H
1
H qc , qa ma
2
2.67
b−1
2 1
2 b c qm Q qm H 2 qb , qm1 − r qm H2 qb , qm > 0,
H q , q mc
where
m
H1 q , q
a
H qm1 , qa − H qm , qa
,
:
q − 1 qm
b
H2 q , q
m
H qb , qm1 − H qb , qm
:
,
q − 1 qm
n
τ0 t − τ0 qa
η τi t − τi qa Qt p0 t
pi t i
a k0
qt − τi qa qt − τ0 q
i1
αi ηi
,
k0 n
i1
−η
ηi i ,
2.68
then 1.5 with et ≡ 0 is oscillatory.
It is obvious that Theorem 2.6 is not applicable if the functions pi t are nonpositive
for i m 1, m 2, . . . , n. In this case the theorem below is valid.
Theorem 2.14. Suppose that for any given (arbitrarily large) T ∈ T there exist subintervals a1 , b1
and a2 , b2 T of T, ∞T , where a1 < b1 and a2 < b2 such that
pi t ≥ 0 for t ∈ a1 , b1
T
∪ a2 , b2 T , i 0, 1, 2, . . . , n,
−1l et > 0 for t ∈ al , bl T , l 1, 2,
T
2.69
14
Advances in Difference Equations
where al min{τj al : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ HT , positive numbers
λi and νi satisfying
m
n
λi νi 1,
i1
2.70
im1
and numbers cν ∈ aν , bν T such that
1
2
H cν , aν cν
aν
QtH 2 σt, aν − rtH12 t, aν Δt
1
2
H bν , cν bν
cν
QtH 2 bν , σt − rtH22 bν , t Δt > 0
2.71
for ν 1, 2, where
m
τ0 t − τ0 aν τi t − τi aν 1−1/αi 1/αi
Qt p0 t
μi λi |et|
pi t
σt − τ0 aν i1
σt − τi aν n
2.72
τi t − τi aν −
,
βi νi |et|1−1/αi pi 1/αi t
σt − τi aν im1
with
μi αi αi − 11/αi −1 ,
βi αi 1 − αi 1/αi −1 ,
pi max −pi t, 0 ,
2.73
then 1.1 is oscillatory.
Proof. Suppose that 1.1 has a nonoscillatory solution. Without losss of generality, we may
assume that xt and xτi t i 0, 1, 2, . . . , n are eventually positive on a1 , b1 T when a1 is
sufficiently large. If xt is eventually negative, one may repeat the same proof step by step
on the interval a2 , b2 T .
Rewriting 1.1 for t ∈ a1 , b1 T as
m
n
Δ
rtxΔ t p0 txτ0 t pi txαi τi t λi |et| pi txαi τi t νi |et| 0
i1
im1
2.74
and applying Lemma 2.3 to each term in the first sum, we obtain
rtxΔ t
Δ
p0 txτ0 t m
μi λi |et|1−1/αi pi1/αi txτi t
i1
n
im1
pi txαi τi t νi |et| ≤ 0,
2.75
Advances in Difference Equations
15
where μi αi αi − 11/αi −1 for i 1, 2, . . . , m. Setting
wt −rt
xΔ t
xt
2.76
yields
Δ
rtxΔ t
w2 t
w t −
.
σ
x t
rt − μtwt
2.77
Δ
Substituting the above last equality into 2.75, we have
wΔ t ≥ p0 t
m
xτ0 t xτi t
μi λi |et|1−1/αi pi1/αi t σ
σ
x t
x t
i1
n
1 w2 t
.
σ
pi txαi τi t νi |et| x t im1
rt − μtwt
2.78
It follows from 2.5 that
xτ0 t τ0 t − τ0 a1 ≥
,
xσ t
σt − τ0 a1 2.79
xτi t τi t − τi a1 ≥
,
xσ t
σt − τi a1 2.80
xαi τi t
τi t − τi a1 ≥ xαi −1 τi t
.
σ
x t
σt − τi a1 2.81
Notice that the second sum in 2.78 can be written as
n
n 1 xαi τi t νi |et|
αi
p
ν
p
tx
τ
t
t
|et|
i
i
i
i
xσ t im1
xσ t
xσ t
im1
n 1
τi t − τi a1 1
νi |et|
− pi t
σt
−
τ
xτ
xτ
a
t
i 1
i
i t
im1
1−αi
2.82
,
and hence applying the Lemma 2.4 yields
n 1
τi t − τi a1 1
νi |et|
− pi t
σt
−
τ
xτ
xτ
a
t
i 1
i
i t
im1
n τi t − τi a1 βi νi |et|1−1/αi pi 1/αi t,
≥−
σt
−
τ
a
i
1
im1
1−αi
2.83
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Advances in Difference Equations
where βi αi 1 − αi 1/αi −1 and pi max{−pi t, 0} for i m 1, m 2, . . . , n. Using 2.79,
2.80, and 2.78 into 2.78, we obtain
wΔ t ≥ p0 t
m τ0 t − τ0 a1 τi t − τi a1 μi λi |et|1−1/αi pi1/αi t
σt − τ0 a1 i1 σt − τi a1 n τi t − τi a1 w2 t
.
βi νi |et|1−1/αi pi 1/αi t −
σt − τi a1 rt − μtwt
im1
2.84
Setting
m
τ0 t − τ0 a1 τi t − τi a1 1−1/αi 1/αi
Qt p0 t
μi λi |et|
pi t
σt − τ0 a1 i1
σt − τi a1 n
2.85
τi t − τi a1 −
,
βi νi |et|1−1/αi pi 1/αi t
σt − τi a1 im1
we have
wΔ t ≥ Qt w2 t
.
rt − μtwt
2.86
The rest of the proof is the same as that of Theorem 2.6 and hence it is omitted.
Corollary 2.15. Suppose that for any given (arbitrarily large) T ≥ t0 there exist subintervals a1 , b1
and a2 , b2 of T, ∞, where T ≤ a1 < b1 and T ≤ a2 < b2 such that
pi t ≥ 0 for t ∈ a1 , b1 ∪ a2 , b2 , i 0, 1, 2, . . . , n,
−1l et > 0 for t ∈ al , bl , l 1, 2,
2.87
where al min{τj al : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ HR , positive numbers
λi and νi satisfying
m
n
λi νi 1,
i1
2.88
im1
and numbers cν ∈ aν , bν such that
1
2
H cν , aν cν
aν
QtH 2 t, aν − rtH12 t, aν dt
1
H 2 bν , cν bν
cν
QtH 2 bν , t − rtH22 bν , t dt > 0
2.89
Advances in Difference Equations
17
for ν 1, 2, where
Qt p0 t
−
m
τ0 t − τ0 aν τi t − τi aν μi λi |et|1−1/αi pi1/αi t
t − τ0 aν t − τi aν i1
n
βi νi |et|
1−1/αi pi
1/αi
im1
τi t − τi aν t
t − τi aν 2.90
with
μi αi αi − 11/αi −1 ,
βi αi 1 − αi 1/αi −1 ,
pi max −pi t, 0 ,
2.91
then 1.3 is oscillatory.
Corollary 2.16. Suppose that for any given (arbitrarily large) T ≥ t0 there exist a1 , b1 , a2 , b2 ∈ Z
with T ≤ a1 < b1 and T ≤ a2 < b2 such that for each i 0, 1, 2, . . . , n,
pi t ≥ 0
for t ∈ {a1 , a1 1, . . . , b1 } ∪ {a2 , a2 1, . . . , b2 }
−1l et > 0
for t ∈ {al , al 1, . . . , bl }, l 1, 2,
2.92
where al min{τj al : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ HZ , positive numbers
λi and νi satisfying
m
n
λi νi 1,
i1
2.93
im1
and numbers cν ∈ {aν 1, aν 2, . . . , bν − 1} such that
c
ν −1
1
QtH 2 t 1, aν − rtH12 t, aν 2
H cν , aν taν
b
ν −1
2.94
1
QtH 2 bν , t 1 − rtH22 bν , t > 0
H 2 bν , cν tcν
for ν 1, 2, where
H1 t, aν : Ht 1, aν − Ht, aν ,
H2 bν , t : Hbν , t 1 − Hbν , t,
m
τ0 t − τ0 aν τi t − τi aν 1−1/αi 1/αi
μi λi |et|
Qt p0 t
pi t
t 1 − τ0 aν i1
t 1 − τi aν −
n
βi νi |et|1−1/αi pi 1/αi t
im1
τi t − τi aν t 1 − τi aν 2.95
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Advances in Difference Equations
with
μi αi αi − 11/αi −1 ,
βi αi 1 − αi 1/αi −1 ,
pi max −pi t, 0 ,
2.96
then 1.4 is oscillatory.
Corollary 2.17. Suppose that for any given (arbitrarily large) T ≥ t0 there exist a1 , b1 , a2 , b2 ∈ N
with T ≤ a1 < b1 and T ≤ a2 < b2 such that for each i 0, 1, 2, . . . , n,
pi t ≥ 0 for t ∈ qa1 , qa1 1 , . . . , qb1 ∪ qa2 , qa2 1 , . . . , qb2 ,
2.97
−1 et > 0 for t ∈ qal , qal 1 , . . . , qbl , l 1, 2,
l
where qal min{τj qal : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ Hq , positive numbers
λi and νi satisfying
m
n
λi νi 1,
i1
2.98
im1
and numbers qcν ∈ {qaν 1 , qaν 2 , . . . , qbν −1 } such that
1
2
H qcν , qaν
c
ν −1
maν
qm Q qm H 2 qm1 , qaν − rtH12 qm , qaν
b
ν −1
1
qm Q q
H 2 qbν , qcν mcν
m
H 2 qbν , qm1 − rtH22 qbν , qm
2.99
>0
for ν 1, 2, where
m
H1 q , q
aν
H qm1 , qaν − H qm , qaν
,
:
q − 1 qm
bν
H2 q , q
m
H qbν , qm1 − H qbν , qm
:
,
q − 1 qm
m
τ0 t − τ0 qaν
τi t − τi qaν
1−1/αi 1/αi
pi t
Qt p0 t
μi λi |et|
qt − τ0 qaν
qt − τi qaν
i1
−
n
βi νi |et|
1−1/αi im1
pi
1/αi
t
τi t − τi qaν
qt − τi qaν
2.100
with
μi αi αi − 11/αi −1 ,
then 1.5 is oscillatory.
βi αi 1 − αi 1/αi −1 ,
pi max −pi t, 0 ,
2.101
Advances in Difference Equations
19
3. Examples
In this section we give three examples when n 2, and α1 2, α2 1/2 in 1.1. That is, we
consider
xΔΔ t p0 txτ0 t p1 t|xτ1 t|xτ1 t p2 t|xτ1 t|−1/2 xτ2 t 0.
3.1
For simplicity we take Ht, s t − s, thus H1 t, s −H2 t, s 1. Note that η1 1/3 and
η2 2/3 by Lemma 2.2.
Example 3.1. Let A ≥ 0 and B, C > 0 be constants. Consider the differential equation
x t Axt − 1 B|xt − 2|xt − 2 C|xt − 1|−1/2 xt − 1 0.
3.2
Let a j, b j 2, and c j 1, j ∈ N.
We calculate
Qt A
t−j
t−j 1
t−j
3
1/3
2/3
√
B C 2/3 1/3
3
4
t−j 2
t−j 1
3.3
1/3
4A 9 BC2
> 27.
3.4
and see that 2.61 holds if
Since all conditions of Corollary 2.11 are satisfied, we conclude that 3.2 is oscillatory when
3.4 holds.
Example 3.2. Let A ≥ 0 and B, C > 0 be constants. Define p0 t A, p1 t B, and p2 t C
for t 10j k, k −3, −2, −1, 0, 1, 2, 3, j ≥ 1; otherwise, the functions are defined arbitrarily.
Consider the difference equation
Δ2 xt p0 txt − 1 p1 t|xt − 2|xt − 2 p2 t|xt − 1|−1/2 xt − 1 0.
3.5
Let a 10j, b 10j 3, and c 10j 1. We derive
Qt A
t − 10j
t − 10j
3 2 1/3
BC
√
1/3
3
2/3
t − 10j 2
4
t − 3j 3
t − 10j 4
3.6
and see that positivity in 2.64 satisfies if
1/3
9 BC2
48
.
A
>
√
5
435
3.7
Since all conditions of Corollary 2.12 are satisfied, we conclude that 3.5 is oscillatory if 3.7
holds.
20
Advances in Difference Equations
Example 3.3. Let A ≥ 0 and B, C > 0 be constants. Define p0 t A, p1 t B and p2 t C
for t 210jk , k −3, −2, −1, 0, 1, 2, 3, j ≥ 1; otherwise, the functions are defined arbitrarily.
Consider the q-difference equation, q 2,
Δ2q xt
t
p0 tx
2
t
p1 tx
4
x t
4
t
p2 tx
8
−1/2 t
x
8
0.
3.8
Let a 10j, b 10j 3, and c 10j 1. We have
Qt A
t − 210j
3 2 1/3
t − 210j
√
BC
1/3 .
3
10j
2/3 4t − 2
4
8t − 210j
16t − 210j
3.9
We see that 2.67 holds for all A ≥ 0 and B, C > 0. Since all conditions of Corollary 2.12 are
satisfied, we conclude that 3.8 is oscillatory if A ≥ 0 and B, C > 0 are positive.
Acknowledgments
The paper is supported in part by the Scientific and Research Council of Turkey TUBITAK
under Contract 108T688. The authors would like to thank the referees for their valuable
comments and suggestions.
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