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Volume 2011, Article ID 513757, 14 pagesdoi:10.1155/2011/513757 Research Article Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities Ethiraju

Volume 2011, Article ID 513757, 14 pages

doi:10.1155/2011/513757

Research Article

Oscillation Criteria for

Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities

Ethiraju Thandapani,1 Veeraraghavan Piramanantham,2

and Sandra Pinelas3

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

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−1 xτt 

n

i1 qi t|xτ i t| αi−1xτi t  0, where t ∈ T and ut  |xt  ptxδtΔ|α−1 xt 

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

concerning the oscillation and nonoscillation of solutions of ordinary and neutral dynamic

and the references cited therein Moreover the oscillatory behavior of solutions of second

of all solutions of the second order nonlinear neutral delay dynamic equation



r t

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

form



r t

y t  ptyt − τΔγΔ qty t − 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

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



r t

nonnegative rd-continuous functions onT such that rt > 0, and ft, u ≥ qt|u| γ

quasilinear neutral delay dynamic equations of the form



r tzΔtγΔ q1tx α τ1t  q2tx β τ2t  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.

neutral delay dynamic equation



r txΔtγ−1

xΔ

Δ

 q1ty δ1tα−1

y δ1t  q2ty δ2tβ−1

y δ2t  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−1 x τt  n

i1

q i t|xτ i t| α i−1x τ i t  0, 1.6

whereT is a time scale, t ∈ T and ut  |xt  ptxδtΔ|α−1 xt  ptxδtΔ, and

By a proper solution of1.6 on t0, ∞Twe mean a function xt ∈ C1rdt0, ∞, which

rdt0, ∞, and satisfies 1.6 on tx , ∞T

oscillatory if it is neither eventually positive nor eventually negative Otherwise it is known

as nonoscillatory

Throughout the paper, we assume the following conditions:

δt ≤ t, τt ≤ t, τ i t ≤ t with lim t → ∞ δt  ∞, lim t → ∞ τt  ∞, and lim t → ∞ τ i t 

∞ 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 q i t, i  1, 2, , n are positive real valued rd-continuous functions on

T with rΔt ≥ 0;

C4 α, α i , i  1, 2, , n are positive constants such that α1> α2 > · · · > α m > α > α m1 >

· · · > α n > 0 n > m ≥ 1.

We consider the two possibilities

t0

1

t0

1

assume that the time scaleT is not bounded above, that is, we take it as t0, ∞T {t ≥ t0: t ∈

T}

Section 3, we provide some examples to illustrate the results

2 Oscillation Results

We use the following notations throughout this paper without further mention:

Q t  qt1− pτtα

, Q i t  q i t1− pτ i tα i

, i  1, 2, 3, , n,

κ t  σ t

σ t , β i t  τ i t

2.1

Lemma 2.1 Let α i , i  1, 2, , n be positive constants satisfying

α1> α2> · · · > α m > α > α m1 > · · · > α n > 0. 2.2

Then there is an n-tuple η1, η2, , η n  satisfying

n

i1

which also satisfies either

n

i1

or

n i1



y α tΔ αyΔt

1

0

hy σ t  1 − hyt α−1

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∗  B γ /A γ1γ

, and

max

u∈R f  f u∗  γ γ

γ  1γ1 B

γ1

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 rtzΔt αΔ < 0 for t ∈ T, ∞T Moreover one obtains

Lemma 2.4 Assume that 1.7 and

t

hold If xt is an eventually positive solution of 1.6, then

and zt/t is strictly decreasing.

Proof FromLemma 2.3, we havertzΔt αΔ< 0 and



r tzΔtαΔ rΔtzΔtα  rσtzΔtαΔ. 2.11

Since rΔt ≥ 0, we have zΔt αΔ< 0 Now using the Keller’s Chain rule, we find that

0 <

zΔtαΔ αzΔΔt

1

0



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,



z t

t

Δ

 tzΔt − zt

Z t

tσ t > 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 τ i t ≥ τ i t2 for t ≥ t2 Then zτt/τt ≥ zτt2/τt2 : d > 0, and zτt/τt ≥ zτt2/τt2 : d i > 0, so that zτt > τt for t ≥ t2



r tzΔtαΔ Qtz α τt  n

i1

r tzΔtα − rt2zΔt2α

t

t2



Q sz α τs  n

i1

Q i sz α i τ i s



which implies that

r t2zΔt2 ≥ rtzΔt 

t

t2



Q sz α τs  n

i1

Q i sz α i τ i s



Δs

> d α

t

t

i1

d α i

i

t

t

Q i sτ α i

2.16

which contradicts 2.4 Hence there is a t1 ∈ t0, ∞T such that Zt > 0 on t1, ∞T Consequently,



z t

t

Δ

 tzΔt − zt

Z t

tσ t < 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, , η n  be n-tuple satisfying 2.3 of

nonnegative delta differentiable function φt such that

lim sup

t → ∞

t

t1

ρ σs

⎡

⎣Q∗s − φΔs −



ρΔs

ρ σ s φ s −

κ α2s

α  1 α1

r sρΔsα1



ρ σsα1

⎤

⎦Δs  ∞,

2.18

for all sufficiently large t1where Q∗t  Qtβ α t  ηn

i1 Q η i

i tβ α i η i

i1 η −η i

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,

z t > 0, zδt > 0, zτt > 0, zτ i t > 0, zΔt > 0, r tzΔtαΔ≤ 0.

2.19 Define

w t  ρt



r tzΔtα



wΔt  ρΔt

ρ t w t  ρσt



r tzΔtα

z α t

Δ

 ρσtφΔt

≤ ρΔt

ρ t w t  ρσt



r tzΔtαΔ

z α σt

− ρσt r t



zΔtα z α tΔ

z α tz α σt  ρσtφΔt.

2.21

From Keller’s chain rule, we have, fromLemma 2.1,

z α tΔ≥

⎧

⎨

⎩

αz α−1 tzΔt, α ≥ 1,

wΔt ≤ − ρ σt

z α σt



Q tz α τt  n

i1

Q i tz α i τ i t



ρΔt

ρ t w t

r 1/α t

1

κ α t



w ρ t t − φt

α1/α  ρσtφΔt.

2.23

z τ i t

τ i t ≥

z σt

or

z τ i t

z σt ≥

τ i t

since τ i t ≤ σt for all i  1, 2, , n Using 2.25 in 2.23, we have



Q tβ α t  n

i1

Q i tβ α i

i tz α i −α σt



ρΔt

ρ t w t

r 1/α t

1

κ α t



w ρ t t − φt

α1/α  ρσtφΔt.

2.26

Now let u i t  1/η i Q i tβ α i

i z α i −α σt, i  1, 2, , n Then 2.26 becomes

wΔt ≤ −ρσt



Q tβ α t  n

i1

η i u i t



ρΔt

ρ t w t

r 1/α t

1

κ α t



w ρ t t − φt

α1/α  ρσtφΔt.

2.27

ByLemma 2.1and using the arithmetic-geometric inequalityn

i1 η i u i≥n

i1 η u i

i in2.27, we obtain

wΔt ≤ −ρσtQ∗t − φΔt ρΔt

ρ t w t

r 1/α t

1

κ α t



w ρ t t − φt

α1/α  ρσtφΔt

2.28

or

wΔt ≤ −ρσtQ∗t − φΔtρΔt

φ t

ρΔt





w ρ t t − φt

 −αρ r 1/α σt t

1

κ α t



w ρ t t − φt

α1/α , t ≥ t1. 2.29

wΔt ≤ −ρσtQ∗t − φΔtρΔt

α  1 α1

r tρΔtα1





ρ σtα κ α2

t

t1

ρ σ s

⎡

⎣Q∗s − φΔs −



ρΔs

ρ σ s φ s −

1

α  1 α1

r sρΔsα1



ρ σsα1 κ α2s

⎤

⎦Δs ≤ wt1,

2.31

By different choices of ρt and φt, we obtain some sufficient conditions for the

Corollary 2.6 Assume that 1.7 holds Furthermore assume that, for all sufficiently large T, for

T ≥ t0,

lim sup

t → ∞

T

where Q∗t is as in Theorem 2.5 Then every solution of 1.6 is oscillatory

Corollary 2.7 Assume that 1.7 holds Furthermore assume that, for all sufficiently large T, for

T ≥ t0,

lim sup

t → ∞

T



σ sQ∗s − r tσt α

2−α

where Q∗t is as in Theorem 2.5 Then every solution of 1.6 is oscillatory

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

and H has a nonpositive continuous Δ-partial derivative HΔs with respect to the second variable such that

HΔs σt, s  Hσt, σs ρΔs

ρ s 

h t, s

and for all sufficiently large T,

lim sup

t → ∞

1

t

T



ρ σ sQ∗s − ht, s α1 r s

α  1 α1

ρ σ sα



where Q∗t is same as in Theorem 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

H σt, σsρ σ sQ∗s − φΔtΔs

≤ −

t

t1

t

t1

H σt, σs ρΔt

ρ s w tΔs

−

t

t

r 1/α tρ α1/α t

1

κ α t



w ρ t t − φt

α1/α Δs.

2.37

Using the integration by parts formula, we have

t

t1

H σt, σswΔsΔs  Ht, sws | t

t1−

t

t1

HΔs σt, swsΔs

 −Ht, t1wt1 −

t

t1

HΔs σt, swsΔs.

2.38

t

t1

H σt, σsρ σ sQ∗s − φΔtΔs

≤ Ht, t1wt1

 t

t1



HΔs σt, s  Hσt, σs ρΔt

ρ s



−

t

t1

r 1/α tρ α1/α t

1

κ α t



w ρ t t − φt

α1/α Δs.

2.39

t

t1

H σt, σsρ σ sQt, sΔs

≤ Ht, t1wt1



t

t1

h t, s

ρ s H α/α1 σt, σswsΔs

− t

t1

r 1/α tρ α1/α t

1

κ α t



w ρ t t − φt

α1/α Δs

2.40

or

t

t1

≤ Ht, t1wt1



t

t1

h t, s

ρ s H α1/α σt, σs



w ρ s s − φs

Δs

− t

t1

r 1/α tρ α1/α t

1

κ α t



w ρ t t − φt

α1/α Δs.

2.41

where Qt, s  ρ σ sQt, s − ht, s/ρsH 1/α σt, σsφs

By setting B  ht, s/ρsH α1/α σt, σs and A  αρσt/r 1/α tρ α1/α t1/

t

t1

H σt, σs



2

t

α  1 α1 ρ α σsHσt, σs



Δs

≤ Ht, t1wt1,

2.42

1.8 holds

Theorem 2.9 Assume that 1.8 holds and limt → ∞ pt  p < 1 Let η1, η2, , η n  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

θ srs

s

t0

1/α

where Qt  Qt n

i1 Q i t 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 xτ i t > 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 z α τt ≥ M and

z α i τ i t ≥ M, i  1, 2, , n, t ∈ t1, ∞T From1.6 we have



r tzΔtαΔ≤ −M



i1

Q i t



Define the supportive function

and we have

uΔt  θΔtrtzΔtα  θσtr tzΔtαΔ

≤ θσtr tzΔtαΔ

 −MθσtQt.

2.46

u t ≤ ut1 − M

t

t1

or



zΔtα ≤ −M 1

θ trt

t

t1

M 1/α

t

t1

 1

θ srs

s

t1

1/α

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 q i t ≡ 0, i  1, 2, , n, or δt  t − δ, τt  t − τ, and q i t ≡ 0, i  1, 2, , n,

0, and α  1, or pt ≡ 0, and τt  τ i t  t, i  1, 2, , n, then the results established here

3 Examples

In this section, we illustrate the obtained results with the following examples

Example 3.1 Consider the second order delay dynamic equation



x t  1

t2x δt

ΔΔ

t 3/2 x√

t

λ2

t x

5/3√

t

λ3

t2x 1/3√

t

for all t ∈ 1, ∞T Here α  1, α1 1/3, α2 5/3, pt  1/t2, qt  λ1/t 3/2 , 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

ρ σ s

⎡

α  1 α1

r sρΔsα1



ρ σ sα1

⎤

⎦Δs

 lim sup

t → ∞

t

t0



λ1

s



s



λ2λ3

s



s



4σs



Δs

≥ lim sup

t → ∞

t

t0



λ1λ2λ3−1

4

 1

s− λ1λ2λ3

s2



Δs

→ ∞ if λ1λ2λ3> 1/4.

3.2

Example 3.2 Consider the second order neutral delay dynamic equation

⎛

x t  1

2x δt

Δ3⎞

⎠

Δ

σ3t

t4 x3



t

2

σ t

t2 x5



t

3

σ t

t2 x 1/3



t

3

for all t ∈ 1, ∞T Here rt  1, pt  1/2, qt  σ3t/t4, τt  t/2, τ1t  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

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Corollary 2.7 Assume that 1.7 holds Furthermore assume that, for all sufficiently large T, for< /p>

T... 2002.

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delay dynamic equations, ” Journal of Mathematical Analysis and Applications,... 1/3√

t

for all t ∈ 1, ∞T Here α  1, α1