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,[email protected] Received 20 September 2010; Revised 30 November 2010; Accepted 23 January 2011 Academic Editor: Istvan Gyori
Copyrightq2011 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 formrtutΔqt|xτt|α−1xτt n
i1qit|xτit|αi−1xτit 0, where t ∈ T and ut |xt ptxδtΔ|α−1xt ptxδtΔwithα1> 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 in9–16 .
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
rtyt ptyt−τΔγΔ
f
t, yt−δ
0 1.1
on time scaleT, wheret ∈ T,γ is a quotient of odd positive integers such thatγ ≥ 1,rt, ptare real valued rd-continuous functions defined onTsuch thatrt>0, 0≤pt<1, and ft, u≥qt|u|γ.
In 2009, Tripathy17 has considered the nonlinear neutral dynamic equation of the form
rt
yt ptyt−τΔγΔ
qtyt−δγsgnyt−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,ptis a nonnegative real valued rd-continuous function on T and established sufficient conditions for the oscillation of all solutions of1.2using Ricatti transformation.
Saker et al.18 , S¸ah´ıner19 , and Wu et al.20 established various oscillation results for the second order neutral delay dynamic equations of the form
rt
yt ptyτtΔγΔ f
t, 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 onTsuch thatrt>0, andft, 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
rt
zΔtγΔ
q1txατ1t q2txβτ2t 0, t∈T, 1.4
wherezt xt ptxτ0t,γ,α,βare quotients of odd positive integers such that 0< α <
γ < βandγ ≥1,rt,pt,q1t, andq2tare real valued rd-continuous functions onT.
Very recently, Han et al.22 have established some oscillation criteria for quasilinear neutral delay dynamic equation
rtxΔtγ−1xΔ
Δq1tyδ1tα−1yδ1t q2tyδ2tβ−1yδ2t 0, t∈T, 1.5 wherext yt ptyτt,α, β, γare quotients of odd positive integers such that 0< α <
γ < β,rt,pt,q1t, andq2tare real valued rd-continuous functions onT.
Motivated by the above observation, in this paper we consider the following second order neutral delay dynamic equation with mixed nonlinearities of the form:
rtutΔqt|xτt|α−1xτt n
i1
qit|xτit|αi−1xτit 0, 1.6
whereTis a time scale,t ∈Tandut |xt ptxδtΔ|α−1xt ptxδtΔ, and this includes all the equations1.1–1.5as special cases.
By a proper solution of1.6ont0,∞Twe mean a functionxt∈C1rdt0,∞,which has a property that rtxt ptxτtα ∈ C1rdt0,∞, 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:
C1the functionsδ, τ, τi:T → Tare nondecreasing right-dense continuous and satisfy δt≤t,τt≤t,τit≤twith limt→ ∞δt ∞, limt→ ∞τt ∞, and limt→ ∞τit
∞fori1,2, . . . , n;
C2ptis a nonnegative real valued rd-continuous function onTsuch that 0≤pt<1;
C3rt, qtandqit,i1,2, . . . , nare positive real valued rd-continuous functions on TwithrΔt≥0;
C4α,αi,i1,2, . . . , nare positive constants such thatα1> α2 >· · ·> αm> α > αm1 >
· · ·> αn >0n > 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 scaleTis not bounded above, that is, we take it ast0,∞T{t≥t0:t∈ T}.
The paper is organized as follows. InSection 2, we present some oscillation criteria for1.6using 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 qt
1−pτtα
, Qit qit
1−pτitαi
, i1,2,3, . . . , n,
κt σt
t , βt τt
σt, βit τit
σt, zt xt ptxδt.
2.1
In this section, we obtain some oscillation criteria for1.6using the following lemmas.
Lemma 2.1is an extension of Lemma 1 of13 .
Lemma 2.1. Letαi,i1,2, . . . , nbe positive constants satisfying
α1> α2>· · ·> αm> α > αm1>· · ·> αn>0. 2.2 Then there is ann-tupleη1, η2, . . . , ηnsatisfying
n i1
αiηiα 2.3
which also satisfies either
n i1
ηi<1, 0< ηi<1, 2.4
or
n i1
ηi1, 0< ηi<1. 2.5
In the following results we use the Keller’s Chain rule1 given by yαtΔ
αyΔt 1
0
hyσt 1−hytα−1
dh, 2.6
whereyis a positive and delta differentiable function onT.
Lemma 2.2see23 . Letfu Bu−Auα1/α, whereA >0 andBare constants,γis a positive integer. Thenfattains its maximum value onRatu∗ Bγ/Aγ1γ, and
maxu∈Rffu∗ γγ
γ1γ1Bγ1
Aγ . 2.7
Lemma 2.3. Assume that1.7holds. Ifxtis an eventually positive solution of 1.6, then there exists a T ∈ t0,∞T such that zt > 0, zΔt > 0, andrtzΔtαΔ < 0 fort ∈ T,∞T. Moreover one obtains
xt≥
1−pt
zt, t≥t1. 2.8
Since the proof ofLemma 2.3is similar to that ofLemma 2.1in6 , we omit the details.
Lemma 2.4. Assume that1.7and ∞
t0
ταsQsΔs∞ 2.9
hold. Ifxtis an eventually positive solution of1.6, then
zΔΔt<0, zt≥tzΔt, 2.10
andzt/tis strictly decreasing.
Proof. FromLemma 2.3, we havertzΔtαΔ<0 and rt
zΔtαΔ
rΔt zΔtα
rσt
zΔtαΔ
. 2.11
SincerΔt≥0, we havezΔtαΔ<0. Now using the Keller’s Chain rule, we find that
0<
zΔtαΔ
αzΔΔt 1
0
hzΔt 1−hzΔtα−1
dh 2.12
orzΔΔt<0. LetZt:zt−tzΔt. ClearlyZΔt −σtzΔΔt>0. We claim that there is at1∈t0,∞Tsuch thatZt>0 ont1,∞T. Assume the contrary, thenZt<0 ont1,∞T. Therefore,
zt t
Δ tzΔt−zt
tσt −Zt
tσt >0, t∈t1,∞T, 2.13 which implies thatzt/tis strictly increasing ont1,∞T. Pickt2∈t1,∞Tso thatτt≥τt2 andτit≥ τit2fort ≥ t2. Thenzτt/τt ≥ zτt2/τt2 : d >0, andzτt/τt ≥ zτt2/τt2 :di>0, so thatzτt> τtfort≥t2.
Using the inequality2.8in1.6, we have that
rt
zΔtαΔ
Qtzατt n
i1
Qitzαiτit≤0. 2.14
Now by integrating fromt2tot, we have
rt
zΔtα
−rt2
zΔt2α
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
QsταsΔsn
i1
diαi 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 thatzt/tis strictly decreasing ont1,∞T.
Theorem 2.5. Assume that condition1.7holds. Letη1, η2, . . . , ηnben-tuple satisfying2.3of Lemma 2.1. Furthermore one assumes that there exist positive delta differentiable functionρtand a nonnegative delta differentiable functionφtsuch that
lim sup
t→ ∞
t
t1
ρσs
⎡
⎣Q∗s−φΔs−
ρΔs
ρσs φs− κα2s α1α1
rs
ρΔsα1
ρσsα1
⎤
⎦Δs∞, 2.18 for all sufficiently larget1whereQ∗t Qtβαt ηn
i1Qηiitβiαiηit, andηn
i1η−ηi i. Then every solution of1.6is oscillatory.
Proof. Suppose that there is a nonoscillatory solutionxtof1.6. We assume thatxtis an eventually positive fort≥t0since the proof for the casext<0 eventually is similar. From the definition ofztandLemma 2.3, there existst1≥t0such that, fort≥t1,
zt>0, zδt>0, zτt>0, zτit>0, zΔt>0,
rt
zΔtαΔ
≤0.
2.19
Define
wt ρt
rt zΔtα
zαt φt
, t≥t1. 2.20
Then from2.19, we havewt>0 and
wΔt ρΔt
ρt wt ρσt
rt zΔtα
zαt Δ
ρσtφΔt
≤ ρΔt
ρt wt ρσt
rt
zΔtαΔ 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
Using2.22and the definition ofκtin2.21, we obtain
wΔt≤ −ρσt zασt
Qtzατt n
i1
Qitzαiτit
ρΔt
ρt wt
−αρσt r1/αt
1 καt
wt ρt −φt
α1/αρσtφΔt.
2.23
FromLemma 2.4, we see thatzt/tis strictly decreasing ont1,∞T, and therefore zτit
τit ≥ zσt
σt 2.24
or
zτit zσt ≥ τit
σt, 2.25
sinceτit≤σtfor alli1,2, . . . , n. Using2.25in2.23, we have
wΔt≤ −ρσt
Qtβαt n
i1
Qitβαiitzαi−ασt
ρΔt
ρt wt
−αρσt r1/αt
1 καt
wt ρt −φt
α1/αρσtφΔt.
2.26
Now letuit 1/ηiQitβαiizαi−ασt, i1,2, . . . , n. Then2.26becomes
wΔt≤ −ρσt
Qtβαt n
i1
ηiuit
ρΔt
ρt wt
−αρσt r1/αt
1 καt
wt ρt −φt
α1/αρσtφΔt.
2.27
ByLemma 2.1and using the arithmetic-geometric inequalityn
i1ηiui≥n
i1ηuiiin2.27, we obtain
wΔt≤ −ρσt
Q∗t−φΔt
ρΔt
ρt wt
−αρσt r1/αt
1 καt
wt ρt −φt
α1/αρσtφΔt
2.28
or
wΔt≤ −ρσt
Q∗t−φΔt
ρΔt
φt
ρΔt
wt ρt −φt
−αρσt r1/αt
1 καt
wt ρt −φt
α1/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≤ −ρσt
Q∗t−φΔt
ρΔt
φt 1 α1α1
rt
ρΔtα1
ρσtα κα2t. 2.30
Now integrating2.30fromt1tot, we obtain t
t1
ρσs
⎡
⎣Q∗s−φΔs−
ρΔs
ρσs φs− 1
α1α1
rs
ρΔsα1
ρσsα1 κα2s
⎤
⎦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 1.6to be oscillatory. For instance,ρt 1,φt 1 andρt 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 largeT, for T ≥t0,
lim sup
t→ ∞
∞
T
Q∗sΔs∞, 2.32
whereQ∗tis as inTheorem 2.5. Then every solution of 1.6is oscillatory.
Corollary 2.7. Assume that 1.7 holds. Furthermore assume that, for all sufficiently largeT, for T ≥t0,
lim sup
t→ ∞
∞
T
σsQ∗s−rtσtα2−α
tα2 Δs∞, 2.33
whereQ∗tis as inTheorem 2.5. Then every solution of 1.6is oscillatory.
Next we establish some Philos-type oscillation criteria for1.6.
Theorem 2.8. Assume that1.7holds. Suppose that there exists a functionH ∈CrdD,R, where D≡ {t, s/t,s∈t0,∞Tandt > s}such that
Ht, t 0, t≥t0, Ht, s≥0, t > s≥0, 2.34
andHhas a nonpositive continuousΔ-partial derivativeHΔswith respect to the second variable such that
HΔsσt, s Hσt, σsρΔs
ρs ht, s
ρs Hσt, σsα/α1, 2.35
and for all sufficiently largeT,
lim sup
t→ ∞
1
Hσt, T
t
T
ρσsQ∗s− ht, sα1rs α1α1
ρσsα
Δs∞, 2.36
whereQ∗tis same as inTheorem 2.5. Then every solution of1.6is oscillatory.
Proof. We proceed as in the proof ofTheorem 2.5and definewtby2.20. Then wt > 0 and satisfies2.28for allt ∈ t1,∞T. Multiplying 2.28byHσt, σsand integrating, we obtain
t
t1
Hσt, σsρσs
Q∗s−φΔt Δs
≤ − t
t1
Hσt, σswΔsΔs
t
t1
Hσt, σsρΔt
ρswtΔs
− t
t1
Hσt, σs αρσt
r1/αtρα1/αt 1 καt
wt ρt −φt
α1/αΔs.
2.37
Using the integration by parts formula, we have t
t1
Hσt, σswΔsΔsHt, sws|tt1− t
t1
HΔsσt, swsΔs
−Ht, t1wt1− t
t1
HΔsσt, swsΔs.
2.38
Substituting2.38into2.37, we obtain t
t1
Hσt, σsρσs
Q∗s−φΔt Δs
≤Ht, t1wt1
t
t1
HΔsσt, s Hσt, σsρΔt ρs
wsΔs
− t
t1
Hσt, σs αρσt
r1/αtρα1/αt 1 καt
wt ρt −φt
α1/αΔs.
2.39
From2.35and2.39, we have t
t1
Hσt, σsρσsQt, sΔs
≤Ht, t1wt1
t
t1
ht, s
ρs Hα/α1σt, σswsΔs
− t
t1
Hσt, σs αρσt
r1/αtρα1/αt 1 καt
wt ρt −φt
α1/αΔs
2.40
or
t
t1
Hσt, σsQt, sΔs
≤Ht, t1wt1
t
t1
ht, s
ρs Hα1/ασt, σs ws
ρs −φs
Δs
− t
t1
Hσt, σs αρσt
r1/αtρα1/αt 1 καt
wt ρt −φt
α1/αΔs.
2.41
whereQt, s ρσsQt, s−ht, s/ρsH1/ασt, σsφs .
By settingB ht, s/ρsHα1/ασt, σsandA αρσt/r1/αtρα1/αt1/
καtinLemma 2.2, we obtain t
t1
Hσt, σs
ρσsQt, s− hα1t, srsκα2t α1α1ρασsHσt, σs
Δs
≤Ht, t1wt1,
2.42
which contradicts condition2.35. This completes the proof.
Finally in this section we establish some oscillation criteria for1.6when the condition 1.8holds.
Theorem 2.9. Assume that1.8holds and limt→ ∞pt p < 1. Letη1, η2, . . . , ηnben-tuple satisfying2.3ofLemma 2.1. Moreover assume that there exist positive delta differentiable functions ρtandθtsuch thatθΔt≥0 and a nonnegative functionφtwith condition2.30for allt≥t1. If
∞
t0
1
θsrs
s
t0
θσvQvΔv 1/α
Δs∞, 2.43
whereQt Qt n
i1Qitholds, then every solution of 1.6either oscillates or converges to zero ast → ∞.
Proof. Assume to the contrary that there is a nonoscillatory solutionxtsuch thatxt >0, xδt>0,xτt>0, andxτit>0 fort∈t1,∞Tfor somet1 ≥t0. FromLemma 2.3we can easily see that eitherzΔt>0 eventually orzΔt<0 eventually.
IfzΔt>0 eventually, then the proof is the same as inTheorem 2.5, and therefore we consider the casezΔt<0.
IfzΔt<0 for sufficiently large t, it follows that the limit ofztexists, saya. Clearly a ≥ 0. We claim that a 0. Otherwise, there exists M > 0 such thatzατt ≥ Mand zαiτit≥M, i1,2, . . . , n, t∈t1,∞T. From1.6we have
rt
zΔtαΔ
≤ −M
Qt n
i1
Qit
−MQt. 2.44
Define the supportive function
ut θtrt
zΔtα
, t∈t1,∞T, 2.45
and we have
uΔt θΔtrt zΔtα
θσt
rt
zΔtαΔ
≤θσt
rt
zΔtαΔ
−MθσtQt.
2.46
Now if we integrate the last inequality fromt1tot, we obtain
ut≤ut1−M t
t1
θσsQsΔs 2.47
or
zΔtα
≤ −M 1 θtrt
t
t1
θσsQsΔs. 2.48
Once again integrate fromt1totto 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 constantcsuch thatzt≤candxt≤zt≤c. Sincextis bounded, lim supt→ ∞xt x1
and lim inft→ ∞ xt x2. Clearlyx2≤x1. From the definition ofzt, we find thatx1px2 ≤ 0≤x2px1; hencex1 ≤x2andx1x20. This completes proof of the theorem.
Remark 2.10. Ifqit≡0,i1,2, . . . , n, orδt t−δ,τt t−τ, andqit≡0,i1,2, . . . , n, thenTheorem 2.5reduces to a result obtained in20 or24 , respecively. Ifpt≡0, orpt≡ 0, andα1, orpt≡0, andτt 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√ t
λ2
t x5/3√ t
λ3
t2x1/3√ t
0, 3.1
for allt∈1,∞T. Hereα1,α11/3,α25/3,pt 1/t2,qt λ1/t3/2,q1t λ2/t, and q2t λ3/t2. Thenη1η21/2. By takingρt t, andφt 0, we obtain
lim sup
t→ ∞
t
t1
ρσs
⎡
⎣Q∗s− 1 α1α1
rs
ρΔsα1 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
→ ∞ifλ1
λ2λ3>1/4.
3.2
ByTheorem 2.5, all solutions of3.1are oscillatory ifλ1
λ2λ3>1/4.
Example 3.2. Consider the second order neutral delay dynamic equation
⎛
⎝
xt 1
2xδt Δ
3⎞
⎠
Δ
σ3t t4 x3
t
2 σt
t2 x5 t
3 σt
t2 x1/3 t
3 0, 3.3
for allt∈1,∞T. Herert 1,pt 1/2,qt σ3t/t4,τt t/2,τ1t τ2t t/3, α3,α15,α21/3. FromCorollary 2.6, every solution of3.3is oscillatory.
Acknowledgment
The authors thank the referees for their constructive suggestions and corrections which improved the content of the paper.
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