Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 952932, 16 pages
doi:10.1155/2012/952932
Research Article
Interval Oscillation Criteria of Second-Order
Nonlinear Dynamic Equations on Time Scales
Yang-Cong Qiu1, 2 and Qi-Ru Wang2
1
2
Department of Humanities and Education, Shunde Polytechnic, Foshan, Guangdong 528333, China
School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou,
Guangdong 510275, China
Correspondence should be addressed to Qi-Ru Wang, mcswqr@mail.sysu.edu.cn
Received 7 May 2012; Revised 22 July 2012; Accepted 7 August 2012
Academic Editor: Yuriy Rogovchenko
Copyright q 2012 Y.-C. Qiu 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.
Using functions in some function classes and a generalized Riccati technique, we establish interval
oscillation criteria for second-order nonlinear dynamic equations on time scales of the form
Δ
ptψxtxΔ t ft, xσt 0. The obtained interval oscillation criteria can be applied
to equations with a forcing term. An example is included to show the significance of the results.
1. Introduction
In this paper, we study the second-order nonlinear dynamic equation
ptψxtxΔ t
Δ
ft, xσt 0,
1.1
on a time scale T.
Throughout this paper we will assume that
C1 p ∈ Crd T, 0, ∞;
C2 ψ ∈ CR, 0, η, where η is an arbitrary positive constant;
C3 f ∈ CT × R, R.
Preliminaries about time scale calculus can be found in 1–3 and hence we omit them
here. Without loss of generality, we assume throughout that sup T ∞.
Definition 1.1. A solution xt of 1.1 is said to have a generalized zero at t∗ ∈ T if
xt∗ xσt∗ ≤ 0, and it is said to be nonoscillatory on T if there exists t0 ∈ T such that
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Discrete Dynamics in Nature and Society
xtxσt > 0 for all t > t0 . Otherwise, it is oscillatory. Equation 1.1 is said to be oscillatory
if all solutions of 1.1 are oscillatory. It is well-known that either all solutions of 1.1 are
oscillatory or none are, so 1.1 may be classified as oscillatory or nonoscillatory.
The theory of time scales, which has recently received a lot of attention, was introduced
by Hilger in his Ph.D. thesis 4 in 1988 in order to unify continuous and discrete analysis, see
also 5. In recent years, there has been much research activity concerning the oscillation and
nonoscillation of solutions of dynamic equations on time scales, for example, see 1–27 and
the references therein. In Došlý and Hilger’s study 10, the authors considered the secondorder dynamic equation
Δ
ptxΔ t qtxσt 0,
1.2
and gave necessary and sufficient conditions for the oscillation of all solutions on unbounded
time scales. In Del Medico and Kong’s study 8, 9, the authors employed the following
Riccati transformation:
ut ptxΔ t
,
xt
1.3
and gave sufficient conditions for Kamenev-type oscillation criteria of 1.2 on a measure
chain. And in Yang’s study 27, the author considered the interval oscillation criteria of
solutions of the differential equation
ptx t qtfxt gt.
1.4
In Wang’s study 24, the author considered second-order nonlinear differential equation
atψxtk x t ptk x t qtfxt 0,
t ≥ t0 ,
1.5
used the following generalized Riccati transformations:
ψxtkx t
Rt ,
fxt
ψxtkx t
Rt ,
vt φtat
xt
vt φtat
t ≥ t0 ,
1.6
t ≥ t0 ,
where φ ∈ C1 t0 , ∞, R , R ∈ Ct0 , ∞, R, and gave new oscillation criteria of 1.5.
In Huang and Wang’s study 16, the authors considered second-order nonlinear
dynamic equation on time scales
Δ
ptxΔ t ft, xσt 0.
1.7
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3
By using a similar generalized Riccati transformation which is more general than 1.3
ut AtptxΔ t
Bt,
xt
1.8
1
1
where A ∈ Crd
T, R \ {0}, B ∈ Crd
T, R, the authors extended the results in Del Medico and
Kong 8, 9 and Yang 27, and established some new Kamenev-type oscillation criteria and
interval oscillation criteria for equations with a forcing term.
In this paper, we will use functions in some function classes and a similar generalized
Riccati transformation as 1.8 and was used in 24, 25 for nonlinear differential equations,
and establish interval oscillation criteria for 1.1 in Section 2. Finally in Section 3, an example
is included to show the significance of the results.
For simplicity, throughout this paper, we denote a, b T a, b, where a, b ∈ R,
and a, b, a, b, a, b are denoted similarly.
2. Main Results
In this section, we establish interval criteria for oscillation of 1.1. Our approach to oscillation
problems of 1.1 is based largely on the application of the Riccati transformation.
Let D0 {s ∈ T : s ≥ 0} and D {t, s ∈ T2 : t ≥ s ≥ 0}. For any function ft, s:
2
T → R, denote by f1Δ and f2Δ the partial derivatives of f with respect to t and s, respectively.
For E ⊂ R, denote by Lloc E the space of functions which are integrable on any compact
subset of E. Define
1
1
D0 , R \ {0}, Bs ∈ Crd
D0 , R ,
A, B A, B : As ∈ Crd
ηAsps ± μsBs > 0, s ∈ D0 ;
H∗ Ht, s ∈ C1 D, R : Ht, t 0, Ht, s > 0, H2Δ t, s ≤ 0, t > s ≥ 0 ;
2.1
H∗ Ht, s ∈ C1 D, R : Ht, t 0, Ht, s > 0, H1Δ t, s ≥ 0, t > s ≥ 0 ;
H H∗
H∗ .
These function classes will be used throughout this paper. Now, we are in a position to give
our first lemma.
Lemma 2.1. Assume that (C1)–(C3) hold and that there exist c1 < b1 < c2 < b2 , α ≥ 1, functions
≡ 0 for t ∈ c1 , b1 c2 , b2 ,
q, g ∈ Crd T, R such that qt ≥ 0 /
gt
≤ 0,
t ∈ c1 , b1 ,
≥ 0,
t ∈ c2 , b2 ,
α−1 gt
f t, y
≥ qty −
,
y
y
2.2
2.3
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Discrete Dynamics in Nature and Society
0. If xt is a solution of 1.1 such that xt > 0 on c1 , σb1 for all t ∈ c1 , b1 c2 , b2 and y /
(or xt < 0 on c2 , σb2 ), for any A, B ∈ A, B one defines
ut At
ptψxtxΔ t
Bt,
xt
2.4
on ci , bi , i 1, 2, and Φ1 t Aσ tqt − Bt/AtΔ , Aσ t Aσt. Then for any
A, B ∈ A, B, H ∈ H∗ , and M1 t, · ∈ L0, ρt, one has
Ψ1 ci , bi ≤ Hbi , ci uci ,
i 1, 2,
2.5
where Φ2 s Aσ sαα−11−α/α qs1/α |gs|1−1/α −Bs/AsΔ for α > 1, Φ2 s Φ1 s
for α 1, and
Ψ1 ci , bi bi
Hbi , σsΦ2 sΔs −
ρbi M1 bi , sΔs
ci
ci
H2Δ bi , ρbi ηA ρbi p ρbi − μ ρbi B ρbi ,
i 1, 2,
M1 t, s
2
Ht, sAsBsHt, σsAσ sBsηAspsHt, sAsΔs
.
4Ht, σsAs min As ηAsps−μsBs , Aσ s ηAspsμsBs
2.6
Proof. Suppose that xt is a solution of 1.1 such that xt > 0 on c1 , σb1 . First,
μu − μB Apψx μ
ApψxxΔ
xσ
Apψx Apψx
> 0.
x
x
2.7
Hence, we always have
μu − μB ηAp ≥ μu − μB Apψx > 0,
2.8
Apψx
Apψx
x
≥
.
xσ μu − μB Apψx μu − μB ηAp
2.9
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5
Then differentiating 2.4 and using 1.1, it follows that
Δ
u A
Δ
pψxxΔ
x
A
σ
pψxxΔ
x
Δ
BΔ
2
Δ
pψxxΔ x − pψx xΔ
AΔ
σ
BΔ
u − B A
A
xxσ
Δ 2
σ
x
x
AΔ
AΔ
Δ
σ ft, x σ
uB −
B−A
− A pψx
.
A
A
xσ
x2 xσ
2.10
i α > 1. Noting that gt ≤ 0 on c1 , b1 , from 2.10, we have
Δ 2
Δ
g x
x
AΔ
σ B
σ
σ α−1
σ
− A pψx
uA
u ≤
−A
qx σ
2
A
A
x
x xσ
Δ
Δ 2
Δ
x
x
AΔ
1−α/α σ 1/α 1−1/α
σ B
σ
g
uA
≤
− αα − 1
A q
− A pψx
A
A
x2 xσ
≤
2.11
Aσ
AΔ
u − B2
u−
− Φ2 .
A
A μu − μB ηAp
That is, for α > 1,
Atu2 t − Aσ t AtBt ηAΔ tAtpt ut Aσ tB2 t
u t Φ2 t ≤ 0.
At μtut − μtBt ηAtpt
2.12
Δ
ii For α 1, from 2.10, we have
Δ 2
Δ
g x
x
AΔ
σ B
σ
σ
uA
u ≤
−A
q − A pψx
σ
2
A
A
xσ
x
x
Δ
≤
Δ
Δ 2
x
A
u − Aσ pψx
A
x2
x
Aσ
xσ
B
A
Δ
2.13
−q .
Then 2.12 also holds.
From i and ii above, we see that 2.12 holds for α ≥ 1. For simplicity in the
following, we let Hσ Hb1 , σs, H Hb1 , s, H2Δ H2Δ b1 , s, and omit the arguments in
the integrals. For s ∈ T,
Hσ − H H2Δ μ.
2.14
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Discrete Dynamics in Nature and Society
Since H2Δ ≤ 0 on D, we see that Hσ ≤ H. Multiplying 2.12, where t is replaced by s, by Hσ ,
and integrating it with respect to s from c1 to b1 , we obtain
b1
Hσ Φ2 Δs ≤ −
Au2 − Aσ AB ηAΔ Ap u Aσ B2
Δs.
Hσ u Hσ
A μu − μB ηAp
b1 c1
c1
Δ
2.15
Noting that Ht, t 0, by the integration by parts formula, we have
b1
Hσ Φ2 Δs ≤ Hb1 , c1 uc1 c1
b1 c1
≤ Hb1 , c1 uc1 ρb1 c1
H2Δ u
b1
ρb1 H2Δ u
Au2 − Aσ AB ηAΔ Ap u Aσ B2
Δs
− Hσ
A μu − μB ηAp
H2Δ uΔs
Au2 − Aσ AB ηAΔ Ap u
Δs.
− Hσ
A μu − μB ηAp
2.16
Since H2Δ ≤ 0 on D, from 2.8, we see that
b1
ρb1 H2Δ uΔs H2Δ b1 , ρb1 u ρb1 μ ρb1 ≤
−H2Δ
b1 , ρb1 ηA ρb1 p ρb1 − μ ρb1 B ρb1 .
For s ∈ c1 , ρb1 , and us ≤ 0, we have
H2Δ u
Au2 − Aσ AB ηAΔ Ap u
− Hσ
A μu − μB ηAp
−
−
HAB Hσ Aσ B ηApHAΔ
H
u2 u
μu − μB ηAp
A ηAp − μB
μu2
HAB Hσ Aσ B ηApHAΔ
μu − μB ηAp
A ηAp − μB
2.17
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Hσ Aσ ηAp μB 2 HAB Hσ Aσ B ηApHAΔ
u
≤−
2 u A ηAp − μB
A ηAp − μB
⎡
⎤2
ηAp − μB HAB Hσ Aσ B ηApHAΔ
Hσ Aσ ηAp μB ⎢
⎥
−
⎦
2 ⎣u −
σ ηAp μB
2H
A
σ
A ηAp − μB
2
HAB Hσ Aσ B ηApHAΔ
4Hσ Aσ A ηAp μB
2
HAB Hσ Aσ B ηApHAΔ
≤
M1 .
4Hσ A min A ηAp − μB , Aσ ηAp μB
2.18
For s ∈ c1 , ρb1 , and us > 0, we have
H2Δ u
Au2 − Aσ AB ηAΔ Ap u
− Hσ
A μu − μB ηAp
2
HAB Hσ Aσ B ηApHAΔ
H
u−
−
μu − μB ηAp
2HA
2
HAB Hσ Aσ B ηApHAΔ
4HA2 μu − μB ηAp
2
HAB Hσ Aσ B ηApHAΔ
≤
M1 .
4Hσ A min A ηAp − μB , Aσ ηAp μB
2.19
Therefore, for s ∈ c1 , ρb1 , we have
H2Δ u
Au2 − Aσ AB ηAΔ Ap u
≤ M1 .
− Hσ
A μu − μB ηAp
2.20
Then from 2.16, 2.17, and 2.20, we obtain that 2.5 holds for i 1.
If xt < 0 on c2 , σb2 , then we see that gt ≥ 0 on c2 , b2 and
Δ 2
Δ
x
g
x
AΔ
σ B
σ
σ α−1
σ
uA
u ≤
−A
.
− A pψx
q|x |
σ
A
A
|x |
x2 xσ
Δ
Following the steps above, we have that 2.5 holds for i 2. The proof is complete.
Next, we have the second lemma.
2.21
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Lemma 2.2. Assume that (C1)–(C3) hold, and that there exist a1 < c1 < a2 < c2 , α ≥ 1, functions
≡ 0 for t ∈ a1 , c1 a2 , c2 and
q, g ∈ Crd T, R such that qt ≥ 0 /
gt
≤ 0,
t ∈ a1 , c1 ,
≥ 0,
t ∈ a2 , c2 ,
2.22
0. If xt is a solution of 1.1 such that xt > 0
and 2.3 holds for all t ∈ a1 , c1 a2 , c2 and y /
on a1 , σc1 or xt < 0 on a2 , σc2 ), define ut as in 2.4 on ai , ci , i 1, 2. Then for any
A, B ∈ A, B, H ∈ H∗ , M2 ·, t ∈ Lloc σt, ∞, one has
Ψ2 ai , ci ≤ −Hci , ai uci ,
i 1, 2,
2.23
where Φ2 is defined as before, and
Ψ2 ai , ci ci
Hσs, ai Φ2 sΔs −
ci
σai ai
M2 s, ai Δs
Hσai , ai Aσ ai Bai − ηpai H1Δ ai , ai Aσ ai ,
Aai i 1, 2,
M2 s, t
2
Hs, tAsBs Hσs, tAσ sBs ηAspsHs, tAsΔs
4Hs, tAs min As ηAsps − μsBs , Aσ s ηAsps μsBs
2.24
Proof. Suppose that xt is a solution of 1.1 such that xt > 0 on a1 , σc1 . For simplicity
in the following, we let Hσ Hσs, a1 , H Hs, a1 , H1Δ H1Δ s, a1 , and omit
the arguments in the integrals. Multiplying 2.12, where t is replaced by s, by Hσ , and
integrating it with respect to s from a1 to c1 and then using the integration by parts formula
we have that
c1
a1
Hσ Φ2 Δs
≤ −
c1 Hσ uΔ
a1
Hσ
Au2 − Aσ AB ηAΔ Ap u Aσ B2
Δs
A μu − μB ηAp
≤ − Hc1 , a1 uc1 σa c 1
a1
1
σa1 H1Δ u
−
Hσ
Au2 − Aσ AB ηAΔ Ap u
Δs.
A μu − μB ηAp
2.25
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For s ∈ a1 , c1 ,
Hσ − H1Δ μ H .
2.26
Hence,
Au2 − Aσ AB ηAΔ Ap u
Δs
−
A μu − μB ηAp
Hσ Aσ B ηApH AΔ uμ A μu − μB ηAp
σa1 a1
H1Δ u
Hσ
2.27
sa1
≤ ηpa1 H1Δ a1 , a1 Aσ a1 Hσa1 , a1 Aσ a1 Ba1 .
Aa1 Furthermore, for s ∈ σa1 , c1 , and us ≤ 0,
H1Δ u
−
Hσ
−
−
Au2 − Aσ AB ηAΔ Ap u
A μu − μB ηAp
H AB Hσ Aσ B ηApH AΔ
H
u2 u
μu − μB ηAp
A ηAp − μB
H AB Hσ Aσ B ηApH AΔ
μu2
μu − μB ηAp
A ηAp − μB
Hσ Aσ ηAp μB 2 H AB Hσ Aσ B ηApH AΔ
≤−
u
2 u A ηAp − μB
A ηAp − μB
⎡
ηAp − μB
Hσ Aσ ηAp μB ⎢
−
2 ⎣u −
A ηAp − μB
H AB Hσ Aσ B ηApH AΔ
4Hσ Aσ A ηAp μB
⎤2
H AB Hσ Aσ B ηApH AΔ
⎥
⎦
2Hσ Aσ ηAp μB
2
2
H AB Hσ Aσ B ηApH AΔ
≤
M2 .
4H A min A ηAp − μB , Aσ ηAp μB
2.28
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For s ∈ σa1 , c1 , and us > 0,
H1Δ u
−
Hσ
Au2 − Aσ AB ηAΔ Ap u
A μu − μB ηAp
2
H AB Hσ Aσ B ηApH AΔ
H
−
u−
μu − μB ηAp
2H A
2
H AB Hσ Aσ B ηApH AΔ
4H A2 μu − μB ηAp
2
H AB Hσ Aσ B ηApH AΔ
≤
M2 .
4H A min A ηAp − μB , Aσ ηAp μB
2.29
Hence, for s ∈ σa1 , c1 , we have
H1Δ u
−
Hσ
Au2 − Aσ AB ηAΔ Ap u
≤ M2 .
A μu − μB ηAp
2.30
From 2.25, 2.27, and 2.30, we have that 2.23 holds for i 1.
If xt < 0 on a2 , σc2 , then we see that gt ≥ 0 on a2 , c2 . Following the steps
above, we have that 2.23 holds for i 2. The proof is complete.
Theorem 2.3. Assume that (C1)–(C3) and the following two conditions hold:
C4 For any T ≥ t0 , there exist T ≤ a1 < b1 ≤ a2 < b2 , α ≥ 1, functions q, g ∈ Crd T, R such
that qt ≥ 0 /
≡ 0 for t ∈ a1 , b1 a2 , b2 ,
gt
≤ 0,
≥ 0,
and 2.3 holds for all t ∈ a1 , b1 t ∈ a1 , b1 ,
t ∈ a2 , b2 ,
2.31
a2 , b2 and y /
0.
C5 There exist ci ∈ ai , bi , i 1, 2, A, B ∈ A, B, H ∈ H, M1 t, · ∈ L0, ρt,
M2 ·, t ∈ Lloc σt, ∞ such that for i 1, 2,
1
1
Ψ1 ci , bi Ψ2 ai , ci > 0,
Hbi , ci Hci , ai 2.32
where M1 , M2 , Ψ1 ci , bi and Ψ2 ai , ci are defined as before.
Then 1.1 is oscillatory.
Proof. Suppose that xt is a nonoscillatory solution of 1.1 which is eventually positive, say
xt > 0 when t ≥ T ≥ t0 for some T depending on the solution xt. From the assumption
C4, we can choose a1 , b1 ≥ T so that gt ≤ 0 on the interval I a1 , b1 with a1 < b1 .
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From Lemmas 2.1 and 2.2, we see that 2.5 and 2.23 hold for i 1. By dividing 2.5
and 2.23 by Hb1 , c1 and Hc1 , a1 , respectively, and then adding them, we obtain a
contradiction to assumption 2.32 with i 1.
When xt is eventually negative, we choose a2 , b2 ≥ T so that gt ≥ 0 on a2 , b2 to
reach a similar contradiction. Hence, every solution of 1.1 has at least one generalized zero
in a1 , b1 or a2 , b2 .
Pick a sequence {Tj } ⊂ T such that Tj ≥ T and Tj → ∞ as j → ∞. By assumption,
for each j ∈ N there exists aj , bj , cj ∈ R such that Tj ≤ aj < cj < bj and 2.32 holds, where
a, b, and c are replaced by aj , bj , and cj , respectively. Hence, every solution xt has at least
one generalized zero tj ∈ aj , bj . Noting that tj > aj ≥ Tj , j ∈ N, we see that every solution
has arbitrarily large generalized zeros. Thus, 1.1 is oscillatory. The proof is complete.
Corollary 2.4. Assume that (C1)–(C4) hold and that
(C6) there exist ci ∈ ai , bi , i 1, 2, A, B ∈ A, B, H ∈ H, M1 t, · ∈ L0, ρt,
M2 ·, t ∈ Lloc σt, ∞ such that for i 1, 2,
Ψ1 ci , bi > 0,
2.33
Ψ2 ai , ci > 0,
2.34
where M1 , M2 , Ψ1 ci , bi and Ψ2 ai , ci are defined as before. Then 1.1 is oscillatory.
Proof. By 2.33 and 2.34, we get 2.32. Therefore, 1.1 is oscillatory by Theorem 2.3. The
proof is complete.
When q ∈ Crd T, R , gt ≡ 0, α 1, we have the following corollary.
Corollary 2.5. Assume that (C1)–(C3) hold and that there exists a function q ∈ Crd T, R such
that uft, u ≥ qtu2 . Also, suppose that there exist A, B ∈ A, B, H ∈ H, M1 t, · ∈
L0, ρt, M2 ·, t ∈ Lloc σt, ∞ such that for any l ∈ T
t
lim sup
t→∞
Hσs, lΦ1 sΔs −
l
t
M2 s, lΔs
σl
Hσl, lAσ lBl
− ηplH1Δ l, lAσ l Al
t
ρt
lim sup
Ht, σsΦ1 sΔs −
M1 t, sΔs
t→∞
l
> 0,
l
> 0.
t, ρt ηA ρt p ρt − μ ρt B ρt
Δ
H2
Then 1.1 is oscillatory.
2.35
2.36
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Proof. When C3 holds and there exists a function q ∈ Crd T, R such that uft, u ≥ qtu2 ,
it follows that C4 holds for gt ≡ 0 and α 1. Now Φ1 s Φ2 s. For any T ≥ t0 , let a1 T .
In 2.35, we choose l a1 . Then there exists c1 > a1 such that
Ψ2 a1 , c1 > 0.
2.37
In 2.36, we choose l c1 . Then there exists b1 > c1 such that
Ψ1 c1 , b1 > 0.
2.38
Combining 2.37 and 2.38 we obtain 2.32 with i 1.
Next, in 2.35 we choose l a2 b1 . Then there exists c2 > a2 such that
Ψ2 a2 , c2 > 0.
2.39
In 2.36, we choose l c2 . Then there exists b2 > c2 such that
Ψ1 c2 , b2 > 0.
2.40
Combining 2.39 and 2.40 we obtain 2.32 with i 2. The conclusion thus follows from
Theorem 2.3. The proof is complete.
3. Example
In this section, we will show the application of our oscillation criteria in an example. The
example is to demonstrate Theorem 2.3.
Example 3.1. Consider the equation
Δ
2 x2 σt
sin xt
π
Δ
3
pt 2 cos 2xt x t
cos t 0, 3.1
qtx σt
16
1 x2 t
1 x2 σt
where p ∈ Crd T, 0, η0 , t ∈ T, ψxt 2 cos 2xt sin xt/1 x2 t,
⎧
π
⎪
cos t,
⎪
⎪
⎪
16
⎪
⎪
√
⎪
⎪
⎪
2 2
⎪
⎪
⎪
⎨ 8 t − 32n − 12,
qt π
⎪
⎪
⎪
− cos t,
⎪
⎪
16
⎪
⎪
⎪
√
⎪
⎪
⎪2 2
⎪
⎩
t − 32n − 28,
8
and gt − cosπ/16t. So we have η 4.
t ∈ 32n, 32n 12,
t ∈ 32n 12, 32n 16,
t ∈ 32n 16, 32n 28,
t ∈ 32n 28, 32n 32, n ∈ N0 ,
3.2
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For any T > 0, there exists n ∈ N0 such that 32n > T . Let α 3, a1 32n, b1 32n8, c1 32n4, a2 32n16, b2 32n24, c2 32n20, A, B 1, 0, Ht, s t−s2 ,
we have
gt
≤ 0,
≥ 0,
t ∈ 32n, 32n 8,
3.3
t ∈ 32n 16, 32n 24.
i Consider T R ,
3
Ψ2 a1 , c1 ≥ √
3
4
32n4
32n
s − 32n2 cos
π
s ds −
16
32n4
32n
4η0 s − 32n2
s − 32n2
ds
√
192 6 32 2
π
8π
−
32
− 16η0 ,
π3
π
3 32n20
Ψ2 a2 , c2 ≥ − √
s − 32n − 162 cos s ds
3
16
4 32n16
32n20
4η0 s − 32n − 162
−
ds
s − 32n − 162
32n16
√
192 6 32 2
π
8π
−
32
− 16η0 ,
π3
32n8
4η0 32n 8 − s2
π
3 32n8
2
s
ds
−
Ψ1 c1 , b1 ≥ √
cos
ds
8
−
s
32n
3
16
4 32n4
32n 8 − s2
32n4
√
√
192 6 32 2
−π
8π
−
32
2
−
1
− 16η0 ,
π3
π
3 32n24
Ψ1 c2 , b2 ≥ − √
32n 24 − s2 cos s ds
3
16
4 32n20
32n24
4η0 32n 24 − s2
−
ds
32n 24 − s2
32n20
√
√
192 6 32 2
−π
2
−
1
− 16η0 .
8π
−
32
π3
3.4
So for i 1, 2, we have
√
√ 1
1
192 6 32 π − 2 2 − 2η0 .
Ψ1 ci , bi Ψ2 ai , ci ≥
3
Hbi , ci Hci , ai π
3.5
√
√
√
√
When 0 < η0 < 96 6 32/π 3 π −2 2 ≈ 1.728, we have 192 6 32/π 3 π −2 2−2η0 > 0,
so 2.32 holds, which means
By Theorem 2.3, we have that 3.1 is oscillatory.
√
√ that C5 holds.
However, when η0 ≥ 96 6 32/π 3 π − 2 2, we do not know whether 3.1 is oscillatory.
14
Discrete Dynamics in Nature and Society
2 Consider T N0 ,
32n3
%
% 2k − 64n 12
π
3 32n3
2
k
−
η
Ψ2 a1 , c1 ≥ √
cos
− 4η0
1
−
32n
k
0
3
16
4 k32n
k − 32n2
k32n1
π
3π
889
3
π
9
cos
16
cos
−
η0 ,
√
1
4
cos
3
16
8
16
36
4
32n19
%
% 2k − 64n − 32 12
π
3 32n19
2
k
−
η
cos
− 4η0
Ψ2 a2 , c2 ≥ − √
1
−
32n
−
16
k
0
3
16
4 k32n16
k − 32n − 162
k32n17
π
3π
889
3
π
9
cos
16
cos
−
η0 ,
√
1
4
cos
3
16
8
16
36
4
32n6
%
% 64n 16 − 2k − 12
π
3 32n7
2
Ψ1 c1 , b1 ≥ √
k
−
η
cos
− 4η0
8
−
k
−
1
32n
0
3
2
16
4 k32n4
k32n4 32n 8 − k − 1
5π
3π
889
3
π
4
cos
cos
−
η0 ,
√
9
cos
3
4
16
8
36
4
%
π
3 32n23
Ψ1 c2 , b2 ≥ − √
32n 24 − k − 12 cos k
3
16
4 k32n20
− η0
32n22
%
64n 48 − 2k − 12
k32n20
32n 24 − k − 12
− 4η0
5π
3π
3
π
889
4
cos
cos
η0 .
√
9
cos
−
3
4
16
8
36
4
3.6
So we have
1
1
Ψ1 ci , bi Ψ2 ai , ci Hbi , ci Hci , ai 3
π
3π
π
≥ √
9
cos
16
cos
1
4
cos
3
16
8
16
16 4
5π
3π
889
π
cos
η0 ,
−
9 cos 4 cos
4
16
8
288
3.7
i 1, 2.
√
3
When 0 < η0 < 54/889 41 4 cosπ/16 9 cosπ/8
16 cos3π/16 9 cosπ/4 √
3
4 cos5π/16 cos3π/8 ≈ 1.359, we have 3/16 41 4 cosπ/16 9 cosπ/8 16 cos3π/16 9 cosπ/4 4 cos5π/16 cos3π/8 − 889/288η0 > 0, so 2.32
holds, which means that C5
holds. By Theorem 2.3, we have that 3.1 is oscillatory.
√
3
However, when η0 ≥ 54/889 41 4 cosπ/16 9 cosπ/8 16 cos3π/16 9 cosπ/4 4 cos5π/16 cos3π/8, we do not know whether 3.1 is oscillatory.
Discrete Dynamics in Nature and Society
15
Acknowledgments
The authors sincerely thank the referees for their valuable comments and useful suggestions
that have led to the present improved version of the original paper. This project was
supported by the NNSF of China nos. 10971231, 11071238, 11271379 and the NSF of
Guangdong Province of China no. S2012010010552.
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