doi:10.1155/2011/513757

*Research Article*

**Oscillation Criteria for**

**Second-Order Neutral Delay Dynamic Equations**

**with Mixed Nonlinearities**

**Ethiraju Thandapani,**

**1**

_{Veeraraghavan Piramanantham,}

_{Veeraraghavan Piramanantham,}

**2**

**and Sandra Pinelas**

**3**

*1 _{Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India}*

*2*

_{Department of Mathematics, Bharathidasan University, Tiruchirappalli 620 024, India}*3 _{Departamento 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

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 form*rtut*Δ*qt*|*xτt*|*α*−1_{x}_{}_{τ}_{}_{t}

*n*

*i*1*qit*|*xτit*|*αi−*1*xτit* 0, where *t* ∈ T and *ut* |*xt* *ptxδt*Δ|*α*−1*xt*

*ptxδt*Δwith*α*1*> α*2*>*· · ·*> αm> α > αm*1*>*· · ·*> 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

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 onTsuch 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*−*τ*Δ*γ*Δ*qtyt*−*δγ*sgn*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 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

*rtyt* *ptyτt*Δ*γ*Δ*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 onTsuch 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

*rtz*Δ*tγ*Δ*q*1*txατ*1*t* *q*2*txβτ*2*t* 0*,* *t*∈T*,* 1.4

where*zt* *xt* *ptxτ*0*t*,*γ*,*α*,*β*are quotients of odd positive integers such that 0*< α <*

*γ < β*and*γ* ≥1,*rt*,*pt*,*q*1*t*, and*q*2*t*are real valued rd-continuous functions onT.
Very recently, Han et al.22 have established some oscillation criteria for quasilinear
neutral delay dynamic equation

*rtx*Δ*tγ*−1*x*Δ

Δ

*q*1*tyδ*1*tα*−1*yδ*1*t* *q*2*tyδ*2*tβ*−1*yδ*2*t* 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*,*q*1*t*, and*q*2*t*are 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:

*rtut*Δ*qt*|*xτt*|*α*−1*xτt*

*n*

*i*1

*qit*|*xτit*|*αi−1xτit* 0*,* 1.6

By a proper solution of1.6on*t*0*,*∞_{T}we mean a function*xt*∈*C*1_{rd}*t*0*,*∞*,*which
has a property that *rtxt* *ptxτtα* ∈ *C*1

rd*t*0*,*∞*,* 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:

*C*1the functions*δ, τ, τi*:T → Tare nondecreasing right-dense continuous and satisfy

*δt*≤*t*,*τt*≤*t*,*τit*≤*t*with lim*t*→ ∞*δt* ∞, lim*t*→ ∞*τt* ∞, and lim*t*→ ∞*τit*

∞for*i*1*,*2*, . . . , n*;

*C*2*pt*is a nonnegative real valued rd-continuous function onTsuch that 0≤*pt<*1;

*C*3*rt, qt*and*qit*,*i*1*,*2*, . . . , n*are positive real valued rd-continuous functions on

Twith*r*Δ*t*≥0;

*C*4*α*,*αi*,*i*1*,*2*, . . . , n*are positive constants such that*α*1*> α*2 *>*· · ·*> αm> α > αm*1 *>*
· · ·*> αn* *>*0*n > m*≥1.

We consider the two possibilities

_{∞}

*t*0
1

*r*1*/α*_{}_{s}_{} Δ*s*∞*,* 1.7

_{∞}

*t*0
1

*r*1*/α*_{}_{s}_{}Δ*s <*∞*.* 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 as*t*0*,*∞T{*t*≥*t*0:*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−*pτt*α*,* *Qit* *qit*

1−*pτit*
α*i*

*,* *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 for1.6using the following lemmas.

**Lemma 2.1.** *Letαi,i*1*,*2*, . . . , nbe positive constants satisfying*

*α*1*> α*2*>*· · ·*> αm> α > αm*1*>*· · ·*> αn>*0*.* 2.2

*Then there is ann-tupleη*1*, η*2*, . . . , ηnsatisfying*

*n*

*i*1

*αiηiα* 2.3

*which also satisfies either*

*n*

*i*1

*ηi<*1*,* 0*< ηi<*1*,* 2.4

*or*

*n*

*i*1

*ηi*1*,* 0*< ηi<*1*.* 2.5

In the following results we use the Keller’s Chain rule1 given by

*yαt*Δ*αy*Δ*t*

_{1}

0

*hyσt* 1−*hyt*α−1*dh,* 2.6

where*y*is a positive and delta diﬀerentiable function onT.

**Lemma 2.2**see23 . *Letfu* *Bu*−*Auα*1*/α _{, where}_{A >}*

_{0}

_{and}_{B}_{are constants,}_{γ}_{is a positive}*integer. Thenfattains its maximum value on*R

*atu*∗

*Bγ*1

_{/A}γ_{}

*γ*

_{, and}max

*u*∈R*ffu*∗

*γγ*

*γ*1*γ*1

*Bγ*1

*Aγ* *.* 2.7

**Lemma 2.3.** *Assume that*1.7*holds. Ifxtis an eventually positive solution of* 1.6*, then there*
*exists a* *T* ∈ *t*0*,*∞T *such that* *zt* *>* 0*,* *z*Δ*t* *>* 0*, andrtz*Δ*tα*Δ *<* 0 *fort* ∈ *T,*∞T*.*
*Moreover one obtains*

*xt*≥1−*ptzt,* *t*≥*t*1*.* 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*

_{∞}

*t*0

which contradicts 2.4. Hence there is a *t*1 ∈ *t*0*,*∞_{T} such that *Zt* *>* 0 on *t*1*,*∞_{T}.
*Lemma 2.1. Furthermore one assumes that there exist positive delta diﬀerentiable functionρtand a*
*nonnegative delta diﬀerentiable functionφtsuch that*

lim sup
*every solution of*1.6*is oscillatory.*

ByLemma 2.1and using the arithmetic-geometric inequality*n _{i}*

_{}

_{1}

*ηiui*≥

*ni*1

*ηuii*in2.27, we

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 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 largeT, for*

**Corollary 2.7.** *Assume that* 1.7 *holds. Furthermore assume that, for all suﬃciently largeT, for*

*T* ≥*t*0*,*

lim sup

*t*→ ∞

_{∞}

*T*

*σsQ*∗*s*−*rtσt*

*α*2_{−}_{α}

*tα*2 Δ*s*∞*,* 2.33

*whereQ*∗*tis 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 functionH* ∈*C*rdD*,*R*, where*
D≡ {*t, s/t,s*∈*t*0*,*∞T*andt > s*}*such that*

*Ht, t* 0*,* *t*≥*t*0*,* *Ht, s*≥0*,* *t > s*≥0*,* 2.34

*andHhas a nonpositive continuous*Δ*-partial derivativeH*Δ*s _{with respect to the second variable such}*

*that*

*H*Δ*s _{σ}_{t}_{, s}*

_{H}_{σ}_{t}_{, σ}_{s}ρΔ_{}_{s}_{}

*ρs*
*ht, s*

*ρs* *Hσt, σs*

*α/α*1_{,}_{}_{2.35}_{}

*and for all suﬃciently largeT,*

lim sup

*t*→ ∞

1

*Hσt, T*

_{t}

*T*

*ρσsQ*∗*s*− *ht, s*

*α*1_{r}_{}_{s}_{}

*α*1*α*1*ρσ*_{}_{s}_{}*α*

Δ*s*∞*,* 2.36

*whereQ*∗*tis same as inTheorem 2.5. Then every solution of*1.6*is oscillatory.*

*Proof.* We proceed as in the proof ofTheorem 2.5and define*wt*by2.20. Then *wt* *>* 0
and satisfies2.28for all*t* ∈ *t*1*,*∞T. Multiplying 2.28by*Hσt, σs*and integrating,
we obtain

_{t}

*t*1

*Hσt, σsρσsQ*∗*s*−*φ*Δ*t*Δ*s*

≤ −

*t*

*t*1

*Hσt, σsw*Δ*s*Δ*s*

*t*

*t*1

*Hσt, σsρ*

Δ_{}_{t}_{}

*ρswt*Δ*s*

−

_{t}

*t*1

*Hσt, σs* *αρσt*
*r*1*/α*_{}_{t}_{}* _{ρ}α*1

*/α*

_{}

_{t}_{}

1

*κα*_{}_{t}_{}

*w _{ρ}*

_{}

_{t}t_{}−

*φt*

*α*1*/α*

Δ*s.*

By setting*B* *ht, s/ρsHα*1*/α*_{}_{σ}_{}_{t}_{}_{, σ}_{}_{s}_{}_{and}_{A}_{αρ}_{}_{σ}_{}_{t}_{}_{/}_{}* _{r}*1

*/α*

_{}

_{t}_{}

*1*

_{ρ}α*/α*

_{}

_{t}_{}

_{1}

_{/}*κα*_{}_{t}_{}_{in}_{Lemma 2.2}_{, we obtain}

_{t}

*t*1

*Hσt, σs*

*ρσsQt, s*− *h*

*α*1_{}_{t, s}_{}_{r}_{}_{s}_{}* _{κ}α*2

*t*

*α*1*α*1*ρα*_{}_{σ}_{}_{s}_{}_{H}_{}_{σ}_{}_{t}_{}_{, σ}_{}_{s}_{}

Δ*s*

≤*Ht, t*1*wt*1*,*

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 that*1.8*holds and limt*→ ∞*pt* *p <* 1*. Let* *η*1*, η*2*, . . . , ηnben-tuple*
*satisfying*2.3*ofLemma 2.1. Moreover assume that there exist positive delta diﬀerentiable functions*

*ρtandθtsuch thatθ*Δ*t*≥0*and a nonnegative functionφtwith condition*2.30*for allt*≥*t*1*.*
*If*

_{∞}

*t*0

1

*θsrs*

_{s}

*t*0

*θσvQv*Δ*v*

1*/α*

Δ*s*∞*,* 2.43

*whereQt* *Qt* *ni*1*Qitholds, then every solution of* 1.6*either oscillates or converges to*
*zero ast* → ∞*.*

*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*∈*t*1*,*∞Tfor some*t*1 ≥*t*0. FromLemma 2.3we
can easily see that either*z*Δ*t>*0 eventually or*z*Δ*t<*0 eventually.

If*z*Δ*t>*0 eventually, then the proof is the same as inTheorem 2.5, and therefore we
consider the case*z*Δ*t<*0.

If*z*Δ*t<*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*zα*_{}_{τ}_{}_{t}_{} _{≥} _{M}_{and}

*zαi _{τ}*

*it*≥*M, i*1*,*2*, . . . , n, t*∈*t*1*,*∞T. From1.6we have

*rtz*Δ*tα*Δ≤ −*M*

*Qt*

*n*

*i*1

*Qit*

−*MQt.* 2.44

Define the supportive function

and we have

*u*Δ*t* *θ*Δ*trtz*Δ*t*α*θσtrtz*Δ*t*αΔ

≤*θσtrtz*Δ*t*αΔ

−*MθσtQt.*

2.46

Now if we integrate the last inequality from*t*1to*t*, we obtain

*ut*≤*ut*1−*M*

_{t}

*t*1

*θσsQs*Δ*s* 2.47

or

*z*Δ*tα*≤ −*M* 1
*θtrt*

_{t}

*t*1

*θσsQs*Δ*s.* 2.48

Once again integrate from*t*1to*t*to obtain

*M*1*/α*

_{t}

*t*1

1

*θsrs*

_{s}

*t*1

*θσξQξ*Δ*ξ*

1*/α*

Δ*s*≤*zt*1*,* 2.49

which contradicts condition 2.43. Therefore lim*t*→ ∞*zt* 0, and there exists a positive
constant*c*such that*zt*≤*c*and*xt*≤*zt*≤*c*. Since*xt*is bounded, lim sup_{t}_{→ ∞}*xt* *x*1
and lim inf*t*→ ∞ *xt* *x*2. Clearly*x*2≤*x*1. From the definition of*zt*, we find that*x*1*px*2 ≤
0≤*x*2*px*1; hence*x*1 ≤*x*2and*x*1*x*20. 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 or24 , 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

*t*2*xδt*

ΔΔ

*λ*1

*t*3*/*2*x*

_{√}

*tλ*2
*t* *x*

5*/*3√_{t}_{}*λ*3

*t*2*x*

for all*t*∈1*,*∞_{T}. Here*α*1,*α*11*/*3,*α*25*/*3,*pt* 1*/t*2,*qt* *λ*1*/t*3*/*2,*q*1*t* *λ*2*/t*, and

*Example 3.2.* Consider the second order neutral delay dynamic equation

⎛

The authors thank the referees for their constructive suggestions and corrections which improved the content of the paper.

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