doi:10.1155/2011/513757
Research Article
Oscillation Criteria for
Second-Order Neutral Delay Dynamic Equations
with Mixed Nonlinearities
Ethiraju Thandapani,
1Veeraraghavan Piramanantham,
2and Sandra Pinelas
31Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India 2Department of Mathematics, Bharathidasan University, Tiruchirappalli 620 024, India
3Departamento de Matem´atica, Universidade dos Ac¸ores, 9501-801 Ponta Delgada, Azores, Portugal
Correspondence should be addressed to Sandra Pinelas,sandra.pinelas@clix.pt
Received 20 September 2010; Revised 30 November 2010; Accepted 23 January 2011 Academic Editor: Istvan Gyori
Copyrightq 2011 Ethiraju Thandapani 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.
This paper is concerned with some oscillation criteria for the second order neutral delay dynamic equations with mixed nonlinearities of the formrtutΔ qt|xτt|α−1xτt
n
i1qit|xτit|αi−1xτit 0, where t ∈ T and ut |xt ptxδtΔ|α−1xt ptxδtΔwith α1> α2> · · · > αm> α > αm1 > · · · > αn > 0. Further the results obtained here
generalize and complement to the results obtained by Han et al.2010 . Examples are provided to illustrate the results.
1. Introduction
Since the introduction of time scale calculus by Stefan Hilger in 1988, there has been great interest in studying the qualitative behavior of dynamic equations on time scales, see, for
example,1–3 and the references cited therein. In the last few years, the research activity
concerning the oscillation and nonoscillation of solutions of ordinary and neutral dynamic
equations on time scales has been received considerable attention, see, for example,4–8
and the references cited therein. Moreover the oscillatory behavior of solutions of second order differential and dynamic equations with mixed nonlinearities is discussed in 9–16 .
In 2004, Agarwal et al.5 have obtained some sufficient conditions for the oscillation
of all solutions of the second order nonlinear neutral delay dynamic equation
on time scaleT, where t ∈ T, γ is a quotient of odd positive integers such that γ ≥ 1, rt,
pt are real valued rd-continuous functions defined on T such that rt > 0, 0 ≤ pt < 1, and ft, u ≥ qt|u|γ.
In 2009, Tripathy17 has considered the nonlinear neutral dynamic equation of the
form
rtyt ptyt − τΔγΔ qtyt − δγsgn yt − δ 0, t ∈ T, 1.2
where γ > 0 is a quotient of odd positive integers, rt, qt are positive real valued
rd-continuous functions onT, pt is a nonnegative real valued rd-continuous function on T
and established sufficient conditions for the oscillation of all solutions of 1.2 using Ricatti transformation.
Saker et al.18 , S¸ah´ıner 19 , and Wu et al. 20 established various oscillation results for the second order neutral delay dynamic equations of the form
rtyt ptyτtΔγΔ ft, yδt 0, t ∈ T, 1.3
where 0 ≤ pt < 1, γ ≥ 1 is a quotient of odd positive integers, rt, pt are real valued
nonnegative rd-continuous functions onT such that rt > 0, and ft, u ≥ qt|u|γ.
In 2010, Sun et al. 21 are concerned with oscillation behavior of the second order
quasilinear neutral delay dynamic equations of the form
rtzΔtγΔ q1txατ1t q2txβτ2t 0, t ∈ T, 1.4 where zt xt ptxτ0t,γ, α, β are quotients of odd positive integers such that 0 < α <
γ < β and γ ≥ 1, rt, pt, q1t, and q2t are real valued rd-continuous functions on T.
Very recently, Han et al.22 have established some oscillation criteria for quasilinear
neutral delay dynamic equation
rtxΔtγ−1xΔ
Δ
q1tyδ1tα−1yδ1t q2tyδ2tβ−1yδ2t 0, t ∈ T, 1.5 where xt yt ptyτt, α, β, γ are quotients of odd positive integers such that 0 < α <
γ < β, rt, pt, q1t, and q2t are real valued rd-continuous functions on T.
Motivated by the above observation, in this paper we consider the following second order neutral delay dynamic equation with mixed nonlinearities of the form:
rtutΔ qt|xτt|α−1xτt n i1
qit|xτit|αi−1xτit 0, 1.6
whereT is a time scale, t ∈ T and ut |xt ptxδtΔ|α−1xt ptxδtΔ, and
By a proper solution of1.6 on t0, ∞Twe mean a function xt ∈ C1rdt0, ∞, which
has a property that rtxt ptxτtα ∈ C1
rdt0, ∞, and satisfies 1.6 on tx, ∞T.
For the existence and uniqueness of solutions of the equations of the form 1.6, refer to
the monograph 2 . As usual, we define a proper solution of 1.6 which is said to be
oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is known as nonoscillatory.
Throughout the paper, we assume the following conditions:
C1 the functions δ, τ, τi:T → T are nondecreasing right-dense continuous and satisfy
δt ≤ t, τt ≤ t, τit ≤ t with limt → ∞δt ∞, limt → ∞τt ∞, and limt → ∞τit
∞ for i 1, 2, . . . , n;
C2 pt is a nonnegative real valued rd-continuous function on T such that 0 ≤ pt < 1; C3 rt, qt and qit, i 1, 2, . . . , n are positive real valued rd-continuous functions on
T with rΔt ≥ 0;
C4 α, αi, i 1, 2, . . . , n are positive constants such that α1> α2 > · · · > αm> α > αm1>
· · · > αn > 0 n > m ≥ 1.
We consider the two possibilities
∞ t0 1 r1/αs Δs ∞, 1.7 ∞ t0 1 r1/αsΔs < ∞. 1.8
Since we are interested in the oscillatory behavior of the solutions of1.6, we may
assume that the time scaleT is not bounded above, that is, we take it as t0, ∞T {t ≥ t0: t ∈ T}.
The paper is organized as follows. InSection 2, we present some oscillation criteria
for1.6 using the averaging technique and the generalized Riccati transformation, and in
Section 3, we provide some examples to illustrate the results.
2. Oscillation Results
We use the following notations throughout this paper without further mention:
dt max{0, dt}, d−t max{0, −dt} Qt qt1− pτtα, Qit qit 1− pτit αi , i 1, 2, 3, . . . , n, κt σt t , βt τt σt, βit τit σt, zt xt ptxδt. 2.1
In this section, we obtain some oscillation criteria for1.6 using the following lemmas.
Lemma 2.1. Let αi, i 1, 2, . . . , n be positive constants satisfying
α1> α2> · · · > αm> α > αm1> · · · > αn> 0. 2.2
Then there is an n-tuple η1, η2, . . . , ηn satisfying n
i1
αiηi α 2.3
which also satisfies either
n i1 ηi< 1, 0 < ηi< 1, 2.4 or n i1 ηi 1, 0 < ηi< 1. 2.5
In the following results we use the Keller’s Chain rule1 given by
yαtΔ αyΔt 1 0 hyσt 1 − hyt α−1dh, 2.6
where y is a positive and delta differentiable function on T.
Lemma 2.2 see 23 . Let fu Bu − Auα1/α, where A > 0 and B are constants, γ is a positive
integer. Then f attains its maximum value on R at u∗ Bγ/Aγ1γ, and
max u∈Rf fu∗ γγ γ 1γ1 Bγ1 Aγ . 2.7
Lemma 2.3. Assume that 1.7 holds. If xt is an eventually positive solution of 1.6, then there
exists a T ∈ t0, ∞T such that zt > 0, zΔt > 0, and rtzΔtαΔ < 0 for t ∈ T, ∞T.
Moreover one obtains
xt ≥1− ptzt, t ≥ t1. 2.8
Since the proof ofLemma 2.3is similar to that ofLemma 2.1in6 , we omit the details.
Lemma 2.4. Assume that 1.7 and
∞
t0
hold. If xt is an eventually positive solution of 1.6, then
zΔΔt < 0, zt ≥ tzΔt, 2.10
and zt/t is strictly decreasing.
Proof. FromLemma 2.3, we havertzΔtαΔ< 0 and
rtzΔtαΔ rΔtzΔtα rσtzΔtαΔ. 2.11
Since rΔt ≥ 0, we have zΔtαΔ< 0. Now using the Keller’s Chain rule, we find that
0 <zΔtαΔ αzΔΔt
1 0
hzΔt 1 − hzΔtα−1dh 2.12
or zΔΔt < 0. Let Zt : zt − tzΔt. Clearly ZΔt −σtzΔΔt > 0. We claim that there is a t1∈ t0, ∞Tsuch that Zt > 0 on t1, ∞T. Assume the contrary, then Zt < 0 on t1, ∞T. Therefore, zt t Δ tzΔt − zt tσt − Zt tσt > 0, t ∈t1, ∞T, 2.13
which implies that zt/t is strictly increasing on t1, ∞T. Pick t2∈ t1, ∞Tso that τt ≥ τt2 and τit ≥ τit2 for t ≥ t2. Then zτt/τt ≥ zτt2/τt2 : d > 0, and zτt/τt ≥
zτt2/τt2 : di> 0, so that zτt > τt for t ≥ t2.
Using the inequality2.8 in 1.6, we have that
rtzΔtαΔ Qtzατt n i1 Qitzαiτit ≤ 0. 2.14
Now by integrating from t2to t, we have
rtzΔtα− rt2 zΔt2 α t t2 Qszατs n i1 Qiszαiτis Δs ≤ 0, 2.15
which implies that
rt2zΔt2 ≥ rtzΔt t t2 Qszατs n i1 Qiszαiτis Δs > dα t t2 QsταsΔs n i1 dαi i t t2 QisτiαisΔs 2.16
which contradicts 2.4. Hence there is a t1 ∈ t0, ∞T such that Zt > 0 on t1, ∞T. Consequently, zt t Δ tzΔt − zt tσt − Zt tσt < 0, t ∈t1, ∞T, 2.17
and we have that zt/t is strictly decreasing on t1, ∞T.
Theorem 2.5. Assume that condition 1.7 holds. Let η1, η2, . . . , ηn be n-tuple satisfying 2.3 of
Lemma 2.1. Furthermore one assumes that there exist positive delta differentiable function ρt and a
nonnegative delta differentiable function φt such that
lim sup t → ∞ t t1 ρσs ⎡ ⎣Q∗s − φΔs − ρΔs ρσs φs − κα2s α 1α1 rsρΔsα1 ρσsα1 ⎤ ⎦Δs ∞, 2.18
for all sufficiently large t1where Q∗t Qtβαt η
n i1Q ηi i tβ αiηi i t, and η n i1η −ηi i . Then
every solution of1.6 is oscillatory.
Proof. Suppose that there is a nonoscillatory solution xt of 1.6. We assume that xt is an
eventually positive for t ≥ t0since the proof for the case xt < 0 eventually is similar. From the definition of zt andLemma 2.3, there exists t1≥ t0such that, for t ≥ t1,
zt > 0, zδt > 0, zτt > 0, zτit > 0, zΔt > 0, rtzΔtαΔ≤ 0. 2.19 Define wt ρt rtzΔtα zαt φt , t ≥ t1. 2.20
Then from2.19, we have wt > 0 and
wΔt ρ Δt ρt wt ρσt rtzΔtα zαt Δ ρσtφΔt ≤ ρΔt ρt wt ρσt rtzΔtαΔ zασt − ρσtrt zΔtαzαtΔ zαtzασt ρσtφΔt. 2.21
From Keller’s chain rule, we have, fromLemma 2.1, zαtΔ≥ ⎧ ⎨ ⎩ αzα−1tzΔt, α ≥ 1, αzα−1σtzΔt, 0 < α < 1. 2.22
Using2.22 and the definition of κt in 2.21, we obtain
wΔt ≤ −ρσt zασt Qtzατt n i1 Qitzαiτit ρΔt ρt wt −αρσt r1/αt 1 καt wρtt− φtα1/α ρσtφΔt. 2.23
FromLemma 2.4, we see that zt/t is strictly decreasing on t1, ∞T, and therefore
zτit τit ≥ zσt σt 2.24 or zτit zσt ≥ τit σt, 2.25
since τit ≤ σt for all i 1, 2, . . . , n. Using 2.25 in 2.23, we have
wΔt ≤ −ρσt Qtβαt n i1 Qitβαiitzαi−ασt ρΔt ρt wt −αρσt r1/αt 1 καt wρtt− φtα1/α ρσtφΔt. 2.26
Now let uit 1/ηiQitβαiizαi−ασt, i 1, 2, . . . , n. Then 2.26 becomes
wΔt ≤ −ρσt Qtβαt n i1 ηiuit ρΔt ρt wt −αρσt r1/αt 1 καt wρtt − φtα1/α ρσtφΔt. 2.27
ByLemma 2.1and using the arithmetic-geometric inequalityni1ηiui≥ni1ηuiiin2.27, we obtain wΔt ≤ −ρσtQ∗t − φΔt ρ Δt ρt wt −αρσt r1/αt 1 καt wρtt− φtα1/α ρσtφΔt 2.28 or wΔt ≤ −ρσtQ∗t − φΔtρΔt φt ρΔt wρtt− φt −αρσt r1/αt 1 καt wρtt− φtα1/α, t ≥ t1. 2.29 Set γ α, A αρσt/r1/αt1/καt, B ρΔt , and ut
|wt/ρt − φt| and applyingLemma 2.2to2.29, we have
wΔt ≤ −ρσtQ∗t − φΔtρΔt φt 1 α 1α1 rtρΔtα1 ρσtα κ α2 t. 2.30
Now integrating2.30 from t1to t, we obtain
t t1 ρσs ⎡ ⎣Q∗s − φΔs − ρΔs ρσs φs − 1 α 1α1 rsρΔsα1 ρσsα1 κ α2 s ⎤ ⎦Δs ≤ wt1, 2.31
which leads to a contradiction to condition2.18. The proof is now complete.
By different choices of ρt and φt, we obtain some sufficient conditions for the
solutions of 1.6 to be oscillatory. For instance, ρt 1, φt 1 and ρt t, φt 1/t
inTheorem 2.5, we obtain the following corollaries:
Corollary 2.6. Assume that 1.7 holds. Furthermore assume that, for all sufficiently large T, for
T ≥ t0, lim sup t → ∞ ∞ T Q∗sΔs ∞, 2.32
Corollary 2.7. Assume that 1.7 holds. Furthermore assume that, for all sufficiently large T, for T ≥ t0, lim sup t → ∞ ∞ T σsQ∗s − rtσt α2−α tα2 Δs ∞, 2.33
where Q∗t is as inTheorem 2.5. Then every solution of 1.6 is oscillatory.
Next we establish some Philos-type oscillation criteria for1.6.
Theorem 2.8. Assume that 1.7 holds. Suppose that there exists a function H ∈ CrdD, R, where D ≡ {t, s/t, s ∈ t0, ∞Tand t > s} such that
Ht, t 0, t ≥ t0, Ht, s ≥ 0, t > s ≥ 0, 2.34
and H has a nonpositive continuous Δ-partial derivative HΔswith respect to the second variable such that HΔsσt, s Hσt, σsρ Δs ρs ht, s ρs Hσt, σs α/α1 , 2.35
and for all sufficiently large T,
lim sup t → ∞ 1 Hσt, T t T ρσsQ∗s − ht, s α1rs α 1α1 ρσsα Δs ∞, 2.36
where Q∗t is same as inTheorem 2.5. Then every solution of1.6 is oscillatory.
Proof. We proceed as in the proof ofTheorem 2.5and define wt by 2.20. Then wt > 0 and satisfies2.28 for all t ∈ t1, ∞T. Multiplying 2.28 by Hσt, σs and integrating, we obtain t t1 Hσt, σsρσsQ∗s − φΔtΔs ≤ − t t1 Hσt, σswΔsΔs t t1 Hσt, σsρ Δt ρswtΔs − t t1 Hσt, σs αρσt r1/αtρα1/αt 1 καt wρtt− φtα1/αΔs. 2.37
Using the integration by parts formula, we have t t1 Hσt, σswΔsΔs Ht, sws |tt1− t t1 HΔsσt, swsΔs −Ht, t1wt1 − t t1 HΔsσt, swsΔs. 2.38
Substituting2.38 into 2.37, we obtain
t t1 Hσt, σsρσsQ∗s − φΔtΔs ≤ Ht, t1wt1 t t1 HΔsσt, s Hσt, σsρ Δt ρs wsΔs − t t1 Hσt, σs αρσt r1/αtρα1/αt 1 καt wρtt− φtα1/αΔs. 2.39
From2.35 and 2.39, we have
t t1 Hσt, σsρσsQt, sΔs ≤ Ht, t1wt1 t t1 ht, s ρs H α/α1σt, σswsΔs − t t1 Hσt, σs αρσt r1/αtρα1/αt 1 καt wρtt− φtα1/αΔs 2.40 or t t1 Hσt, σsQt, sΔs ≤ Ht, t1wt1 t t1 ht, s ρs H α1/ασt, σsws ρs − φs Δs − t t1 Hσt, σs αρσt r1/αtρα1/αt 1 καt wρtt− φtα1/αΔs. 2.41 where Qt, s ρσsQt, s − ht, s/ρsH1/ασt, σsφs .
By setting B ht, s/ρsHα1/ασt, σs and A αρσt/r1/αtρα1/αt1/ καt inLemma 2.2, we obtain t t1 Hσt, σs ρσsQt, s − h α1t, srsκα2 t α 1α1ρασsHσt, σs Δs ≤ Ht, t1wt1, 2.42
which contradicts condition2.35. This completes the proof.
Finally in this section we establish some oscillation criteria for1.6 when the condition
1.8 holds.
Theorem 2.9. Assume that 1.8 holds and limt → ∞pt p < 1. Let η1, η2, . . . , ηn be n-tuple
satisfying2.3 ofLemma 2.1. Moreover assume that there exist positive delta differentiable functions
ρt and θt such that θΔt ≥ 0 and a nonnegative function φt with condition 2.30 for all t ≥ t1.
If ∞ t0 1 θsrs s t0 θσvQvΔv 1/α Δs ∞, 2.43
where Qt Qt ni1Qit holds, then every solution of 1.6 either oscillates or converges to
zero as t → ∞.
Proof. Assume to the contrary that there is a nonoscillatory solution xt such that xt > 0, xδt > 0, xτt > 0, and xτit > 0 for t ∈ t1, ∞Tfor some t1 ≥ t0. FromLemma 2.3we can easily see that either zΔt > 0 eventually or zΔt < 0 eventually.
If zΔt > 0 eventually, then the proof is the same as inTheorem 2.5, and therefore we consider the case zΔt < 0.
If zΔt < 0 for sufficiently large t, it follows that the limit of zt exists, say a. Clearly
a ≥ 0. We claim that a 0. Otherwise, there exists M > 0 such that zατt ≥ M and
zαiτ it ≥ M, i 1, 2, . . . , n, t ∈ t1, ∞T. From1.6 we have rtzΔtαΔ≤ −M Qt n i1 Qit −MQt. 2.44
Define the supportive function
and we have
uΔt θΔtrtzΔtα θσtrtzΔtαΔ
≤ θσtrtzΔtαΔ
−MθσtQt.
2.46
Now if we integrate the last inequality from t1to t, we obtain
ut ≤ ut1 − M t t1 θσsQsΔs 2.47 or zΔtα≤ −M 1 θtrt t t1 θσsQsΔs. 2.48
Once again integrate from t1to t to obtain
M1/α t t1 1 θsrs s t1 θσξQξΔξ 1/α Δs ≤ zt1, 2.49
which contradicts condition 2.43. Therefore limt → ∞zt 0, and there exists a positive
constant c such that zt ≤ c and xt ≤ zt ≤ c. Since xt is bounded, lim supt → ∞xt x1
and lim inft → ∞xt x2. Clearly x2≤ x1. From the definition of zt, we find that x1 px2 ≤ 0≤ x2 px1; hence x1 ≤ x2and x1 x2 0. This completes proof of the theorem.
Remark 2.10. If qit ≡ 0, i 1, 2, . . . , n, or δt t − δ, τt t − τ, and qit ≡ 0, i 1, 2, . . . , n,
thenTheorem 2.5reduces to a result obtained in20 or 24 , respecively. If pt ≡ 0, or pt ≡
0, and α 1, or pt ≡ 0, and τt τit t, i 1, 2, . . . , n, then the results established here
complement to the results of5,9,15 respectively.
3. Examples
In this section, we illustrate the obtained results with the following examples.
Example 3.1. Consider the second order delay dynamic equation
xt 1 t2xδt ΔΔ λ1 t3/2x √ tλ2 t x 5/3√tλ3 t2x 1/3√t 0, 3.1
for all t ∈ 1, ∞T. Here α 1, α1 1/3, α2 5/3, pt 1/t2, qt λ1/t3/2, q1t λ2/t, and
q2t λ3/t2. Then η1 η2 1/2. By taking ρt t, and φt 0, we obtain
lim sup t → ∞ t t1 ρσs ⎡ ⎣Q∗s − 1 α 1α1 rsρΔsα1 ρσsα1 ⎤ ⎦Δs lim sup t → ∞ t t0 λ1 s 1−1 s λ2λ3 s 1−1 s − 1 4σs Δs ≥ lim sup t → ∞ t t0 λ1 λ2λ3− 1 4 1 s− λ1 λ2λ3 s2 Δs → ∞ if λ1 λ2λ3> 1/4. 3.2
ByTheorem 2.5, all solutions of3.1 are oscillatory if λ1
λ2λ3> 1/4.
Example 3.2. Consider the second order neutral delay dynamic equation
⎛ ⎝xt 1 2xδt Δ3⎞ ⎠ Δ σ3t t4 x 3 t 2 σt t2 x 5 t 3 σt t2 x 1/3 t 3 0, 3.3
for all t ∈ 1, ∞T. Here rt 1, pt 1/2, qt σ3t/t4, τt t/2, τ
1t τ2t t/3,
α 3, α1 5, α2 1/3. FromCorollary 2.6, every solution of3.3 is oscillatory.
Acknowledgment
The authors thank the referees for their constructive suggestions and corrections which improved the content of the paper.
References
1 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨auser, Boston, Mass, USA, 2001.
2 M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkh¨auser Boston Inc., Boston, Mass, USA, 2003.
3 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
4 R. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: a survey,”
Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
5 R. P. Agarwal, D. O’Regan, and S. H. Saker, “Oscillation criteria for second-order nonlinear neutral delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 203– 217, 2004.
6 L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 505–522, 2007. 7 S. H. Saker, D. O’Regan, and R. P. Agarwal, “Oscillation theorems for second-order nonlinear neutral
8 J. Shao and F. Meng, “Oscillation theorems for second-order forced neutral nonlinear differential equations with delayed argument,” International Journal of Differential Equations, vol. 2010, Article ID
181784, 15 pages, 2010.
9 R. P. Agarwal, D. R. Anderson, and A. Zafer, “Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 977–993, 2010.
10 R. P. Agarwal and A. Zafer, “Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities,” Advances in Difference Equations, vol. 2009, Article ID 938706, 20 pages, 2009.
11 C. Li and S. Chen, “Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 504–507, 2009.
12 S. Murugadass, E. Thandapani, and S. Pinelas, “Oscillation criteria for forced second-order mixed type quasilinear delay differential equations,” Electronic Journal of Differential Equations, p. No. 73, 9, 2010.
13 Y. G. Sun and F. W. Meng, “Oscillation of second-order delay differential equations with mixed nonlinearities,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 135–139, 2009.
14 Y. G. Sun and J. S. W. Wong, “Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 549–560, 2007.
15 M. ¨Unal and A. Zafer, “Oscillation of second-order mixed-nonlinear delay dynamic equations,”
Advances in Difference Equations, vol. 2010, Article ID 389109, 21 pages, 2010.
16 Z. Zheng, X. Wang, and H. Han, “Oscillation criteria for forced second order differential equations with mixed nonlinearities,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1096–1101, 2009.
17 A. K. Tripathy, “Some oscillation results for second order nonlinear dynamic equations of neutral type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1727–e1735, 2009. 18 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.
19 Y. S¸ah´ıner, “Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales,” Advances in Difference Equations, vol. 2006, Article ID 65626, 9 pages, 2006.
20 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.
21 Y. Sun, Z. Han, T. Li, and G. Zhang, “Oscillation criteria for second-order quasilinear neutral delay dynamic equations on time scales,” Advances in Difference Equations, vol. 2010, Article ID 512437, 14 pages, 2010.
22 Z. Han, S. Sun, T. Li, and C. Zhang, “Oscillatory behavior of quasilinear neutral delay dynamic equations on time scales,” Advances in Difference Equations, vol. 2010, Article ID 450264, 24 pages, 2010.
23 G. H. Hardy, J. E. Littlewood, and G. P´olya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1952.
24 S. H. Saker, “Oscillation of second-order nonlinear neutral delay dynamic equations on time scales,”
