Oscillation Behavior of a Class of Second-Order Dynamic Equations with Damping on Time Scales

Volume 2010, Article ID 907130,15pages doi:10.1155/2010/907130

Research Article

Oscillation Behavior of a Class of Second-Order Dynamic Equations with Damping on Time Scales

Weisong Chen,

Zhenlai Han,

Shurong Sun,

and Tongxing Li

1School of Science, University of Jinan, Jinan, Shandong 250022, China

2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

3Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA

Correspondence should be addressed to Zhenlai Han,[email protected] Received 21 May 2010; Accepted 16 September 2010

Academic Editor: Guang Zhang

Copyrightq2010 Weisong Chen 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.

By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.

1. Introduction

The theory of time scales, which has recently received a lot of attraction, was introduced by Hilger in his Ph.D. Thesis in 19901in order to unify continuous and discrete analysis. The books on the subjects of time scale, that is, measure chain, by Bohner and Peterson 2,3 summarize and organize much of time scale calculus.

We are concerned with second-order nonlinear dynamic equations with damping

x^Δt_γΔ

pt x^Δt_γ

qtfx^σt 0 1.1

on a time scale T; here p and q are real-valued positive rd-continuous positive functions defined onT, andγis a quotient of odd positive integers. We assume thatfx/x^γ ≥L >0, x /0, supT∞,and definet0,∞_T: t0,∞∩T.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations 4–13. However, there are few papers dealing with the oscillation of dynamic equations with damping term14–17.

Saker 18 presented several oscillation criteria for the nonlinear second-order dynamic equation

ptx^Δt_Δ

qtfxσt 0, t∈a, b, 1.2

wherea, b∈Tanda < b.

Hassan19studied the oscillation behavior of the second-order half-linear dynamic equation

rt

x^ΔtγΔ

ptx^γt 0, 1.3

and obtained several new results.

Bohner et al.20established some oscillation criteria for the second-order nonlinear dynamic equation

x^ΔΔt qtx^Δσt pt f◦x^σ

0. 1.4

Erbe et al. 16 considered the second-order nonlinear dynamic equations with damping

rt

x^ΔtγΔ

pt

x^Δσtγ

qtfxτt 0, t∈T, 1.5 and established some sufficient conditions for oscillation of1.5.

Saker et al.17investigated the oscillation of second-order dynamic equations with damping term of the form

rtx^Δt_Δ

ptx^Δσt qtfxσt 0, t∈T, 1.6

and obtained some new oscillation criteria for1.6.

Zafer21studied the second-order nonlinear dynamic equations on time scales

y^ΔΔpty^Δqty^σ 0, t∈T, 1.7

and presented some oscillation and nonoscillation criteria. Obviously, 1.7 is the special situation of1.1.

Note that in the special case whenT R,1.1becomes the second-order nonlinear damped differential equation

xt_γ pt

xt_γ

qtfx^σt 0, t∈R, 1.8

and whenTZ,1.1becomes the second-order nonlinear damped difference equation Δ

Δxt^γ

ptΔxt^γqtfx^σt 0, t∈Z, 1.9

whereΔxt xt1−xt.

This paper is organized as follows: in Section 2, we give some preliminaries and lemmas. In Section 3, we will establish some oscillation criteria for 1.1. In Section 4, we give an example to illustrate the main results.

2. Preliminaries

It will be convenient to make the following notations:

dt:max{0, dt}, d−t:max{0,−dt}, βt:

⎧⎨

⎩

αt, 0< γ≤1, α^γt, γ >1, αt: t−t∗

t−t∗μt, Rt:ep/1−pμt, t∗.

2.1

Lemma 2.1. Assume thatxisΔ-differentiable. Then from Keller’s chain rule [2, Theorem 1.90], xt^γ_Δ

γ

1 0

hx^σt 1−hxt^γ−1x^Δtdh. 2.2

Lemma 2.2see22. Iffx −Ax^γ1/γBx,A >0, thenfxattains its maximum value at x0 γB/γ1A^γ, andfx0 γ^γ/γ1^γ1B^γ1/A^γ.

Lemma 2.3. Suppose thatxis an eventually positive solution of equation1.1, 1−ptμt>0,and

∞ t0

Δt

R^1/γt ∞. 2.3

Then there exists at_∗> t0, such that fort > t_∗,

x^Δt_γΔ

<0, x^Δt>0, x^ΔΔt<0, xt>t−t∗x^Δt, xt

x^σt > αt. 2.4

Proof. Pickt1∈t0,∞_Tsuch thatx^σt>0 ont1,∞_T. From1.1, we have x^ΔtγΔ

pt x^Δtγ

<0, t∈t1,∞_T. 2.5

So, we get

1 1−μtpt

x^Δt_γΔ

pt 1−μtpt

x^Δt_γ

<0, t∈t1,∞_T. 2.6

Therefore,

Rt

x^Δt_γΔ

<0, t∈t1,∞_T. 2.7

We claim thatx^Δt>0. If not, there existt1≥t0and a constantC <0 such that

Rt

x^Δt_γ

≤C <0, 2.8

hence

x^Δt≤ C

Rt 1/γ

. 2.9

Integrating the above inequality fromt1tot, we obtain

xt≤xt1 C^1/γ

t t1

1

R^1/γsΔs−→ −∞, t−→ ∞, 2.10 which is a contradiction. Hence,

x^Δt>0. 2.11

Obviously, by2.7and2.11, we can see that

x^Δt_γΔ

<0. 2.12

From2.11and2.12, we have

x^ΔΔt<0. 2.13

It follows from2.13that

xt> xt−xt∗ t t∗

x^ΔsΔs≥x^Δtt−t∗. 2.14

In view of2.14andx^σt xt μtx^Δt, it is easy to get that xt

x^σt > αt. 2.15

3. Main Results

In this section, we will give some new oscillation criteria for1.1.

Theorem 3.1. Assume that2.3holds. Further, suppose that 1−ptμt > 0, and there exists a positiveΔ-differentiable functionδ, such that for all sufficiently larget_∗,

lim sup

t→ ∞ t t∗

Lqsδ^σs− γ^γ

γ1_γ1B^γ1s A^γs

Δs∞, 3.1

whereAt γδ^σtβt/δt^γ1/γ,Bt δ^Δt−ptδ^σtα^γt/δt. Then every solution xof1.1oscillates ont0,∞_T.

Proof. Letxtbe a nonoscillatory solution of1.1ont0,∞_T. Without loss of generality, we assumext>0, fort≥t∗≥t0. Consider the generalized Riccati substitution

wt δt

x^Δtγ

x^γt , t≥t∗≥t0, 3.2

thenwt>0, and by the product rule and then the quotient

w^Δt δ^Δt

x^Δtγ

x^γt δ^σt

x^Δtγ

x^γt _Δ

δ^Δt

x^Δtγ

x^γt δ^σt

x^Δtγ_Δ

x^σt^γ −δ^σt

x^Δtγ

xt^γ_Δ xt^γx^σt^γ .

3.3

Using1.1and3.2, we find

w^Δt≤wtδ^Δt

δt −ptδ^σt x^Δt

xt

_γxt x^σt

_γ

−Lqtδ^σt−δ^σt

x^Δtγ

xt^γ_Δ xt^γx^σt^γ .

3.4

If 0< γ ≤1, fromLemma 2.1, we get xt^γ_Δ

≥γx^σt^γ−1x^Δt, 3.5

hence

w^Δt≤wtδ^Δt

δt −ptδ^σt x^Δt

xt

_γ xt x^σt

_γ

−Lqtδ^σt−γδ^σt

x^Δt xt

_γ1 xt x^σt.

3.6 In view ofLemma 2.3and3.2, we obtain

w^Δt≤ −Lqtδ^σt

δ^Δt

δt −ptδ^σt δtα^γt

wt− γδ^σtαt

δt^γ1/γwt^γ1/γ. 3.7 Ifγ >1, fromLemma 2.1, we get

xt^γ_Δ

≥γxt^γ−1x^Δt, 3.8

hence

w^Δt≤wtδ^Δt

δt −ptδ^σt x^Δt

xt

γxt x^σt

γ

−Lqtδ^σt−γδ^σt x^Δt

xt

_γ1xt x^σt

γ

. 3.9 In view ofLemma 2.3, we have

w^Δt≤ −Lqtδ^σt

δ^Δt

δt −ptδ^σt δtα^γt

wt− γδ^σtα^γt

δt^γ1/γwt^γ1/γ. 3.10 Therefore,

w^Δt≤ −Lqtδ^σt

δ^Δt

δt −ptδ^σt δtα^γt

wt− γδ^σtβt

δt^γ1/γwt^γ1/γ. 3.11 FromLemma 2.3, we get

w^Δt≤ −Lqtδ^σt γ^γ

γ1_γ1B^γ1t

A^γt . 3.12

Integrating the above inequality fromt∗tot, we have

t t∗

Lqsδ^σs− γ^γ

γ1_γ1B^γ1s A^γs

Δs≤wt_∗ 3.13

which leads to a contradiction to3.1. This completes the proof.

Remark 3.2. From Theorem 3.1, we can obtain different conditions for oscillation of all solutions of1.1with different choice ofδ.

Theorem 3.3. Assume that2.3holds. Further, suppose that 1−ptμt>0, and there exist positive Δ-differentiable functionsδandr, such that for all sufficiently larget∗,

lim sup

t→ ∞ t t∗

Lqsδsrs− γ^γ

γ1_γ1B^γ1s A^γs

Δs∞, 3.14

where At γrtδt/δ^σt^γ1/γ,Bt rtδ^Δt−ptδt/δ^σt r^Δt. Then every solutionxof 1.1oscillates ont0,∞_T.

Proof. Letxtbe a nonoscillatory solution of1.1ont0,∞_T. Without loss of generality, we assumext>0, fort≥t_∗≥t0. Consider the generalized Riccati substitution as in3.2. Then wt>0, and by the product rule and then the quotient

w^Δt δ^Δt

x^Δtγ

x^γt σ

δt

x^Δtγ

x^γt _Δ

δ^Δt

x^Δtγ

x^γt _σ

δt

x^Δtγ_Δ x^σt^γ −δt

x^Δtγ

xt^γ_Δ xt^γx^σt^γ ,

3.15

it follows from1.1and3.2that

w^Δt≤w^σtδ^Δt

δ^σt −ptδt

x^Δσt x^σt

_γ x^Δt x^Δσt

_γ

−Lqtδt−δt

x^Δtγ

xt^γ_Δ xt^γx^σt^γ .

3.16 If 0< γ ≤1, fromLemma 2.1, we get

xt^γ_Δ

≥γx^σt^γ−1x^Δt, 3.17 hence

w^Δt≤w^σtδ^Δt

δ^σt −ptδt

x^Δσt x^σt

_γ x^Δt x^Δσt

_γ

−Lqtδt

−γδt

x^Δσt x^σt

_γ1 x^σt

xt γ

x^Δt x^Δσt

_γ1 .

3.18

In view ofLemma 2.3, we see that

w^Δt≤ −Lqtδt

δ^Δt

δ^σt −ptδt δ^σt

w^σt− γδt

δ^σt^γ1/γw^σt^γ1/γ. 3.19

Ifγ >1, fromLemma 2.1, we get

xt^γ_Δ

≥γxt^γ−1x^Δt. 3.20

So,

w^Δt≤w^σtδ^Δt

δ^σt −ptδt

x^Δσt x^σt

γ x^Δt x^Δσt

γ

−Lqtδt

−γδt

x^Δσt x^σt

_γ1 x^σt

xt

x^Δt x^Δσt

_γ1 .

3.21

In view ofLemma 2.3, we find

w^Δt≤ −Lqtδt

δ^Δt

δ^σt −ptδt δ^σt

w^σt− γδt

δ^σt^γ1/γw^σt^γ1/γ. 3.22 Therefore,

w^Δt≤ −Lqtδt

δ^Δt

δ^σt −ptδt δ^σt

w^σt− γδt

δ^σt^γ1/γw^σt^γ1/γ. 3.23 FromLemma 2.2, we obtain

w^Δt≤ −Lqtδ^σt γ^γ

γ1_γ1B^γ1t

A^γt . 3.24

Integrating the above inequality fromt_∗tot, we get

t t∗

Lqsδsrs− γ^γ

γ1_γ1B^γ1s A^γs

Δs≤rt∗wt∗ 3.25

which leads to a contradiction to3.14. This completes the proof.

Remark 3.4. From Theorem 3.3, we can obtain different conditions for oscillation of all solutions of1.1with different choice ofδandr.

In the following, we will establish Kamenev-type oscillation criteria for1.1.

Theorem 3.5. Assume that2.3holds. Further, suppose that 1−ptμt > 0, and there exists a positiveΔ-differentiable functionδ, such that form >1 and all sufficiently larget_∗,

lim sup

t→ ∞

1 t^m

t t∗

t−s^m

Lqsδ^σs− γ^γ

γ1_γ1B^γ1s A^γs

Δs∞, 3.26

whereAt γδ^σtβt/δt^γ1/γ,Bt δ^Δt−ptδ^σtα^γt/δt. Then every solution xof1.1is oscillatory ont0,∞_T.

Proof. We may assume that1.1has a nonoscillatory solutionxtsuch thatxt>0. Define wby3.2as before, then we get3.24. From3.24, we have

Lqtδ^σt− γ^γ

γ1_γ1B^γ1t

A^γt ≤ −w^Δt. 3.27

Thus

t t∗

t−s^m

Lqsδ^σs− γ^γ

γ1_γ1B^γ1s A^γs

Δs≤ − ^t

t∗

t−s^mw^ΔsΔs. 3.28

Upon integration, we arrive at

− ^t

t∗

t−s^mw^ΔtΔs t−s^mwt|^t_t∗− ^t

t∗

t−s^m_Δs

wσtΔs. 3.29

Note thatt−s^mΔs ≤ −mt−σs^m−1 ≤ 0, t ≥ σs, m ≥ 1 see Saker11; then using 3.28, we have

t t∗

t−s^m

Lqsδ^σs− γ^γ

γ1_γ1B^γ1s A^γs

Δs≤t−t∗^mwt∗. 3.30

Therefore, 1 t^m

t t_∗

t−s^m

Lqsδ^σs− γ^γ

γ1_γ1B^γ1s A^γs

Δs≤ t−t_∗^m

t^m wt_∗. 3.31

Hence,

lim sup

t→ ∞

1 t^m

t t_∗

t−s^m

Lqsδ^σs− γ^γ

γ1_γ1B^γ1s A^γs

Δs∞, 3.32

which contradicts3.26. This completes the proof.

Theorem 3.6. Assume that2.3holds. Further, suppose that 1−ptμt > 0, and there exists a positiveΔ-differentiable functionδ, such that form >1 and all sufficiently larget_∗,

lim sup

t→ ∞

1 t^m

t t∗

t−s^m

Lqsδs− γ^γ

γ1_γ1B^γ1s A^γs

Δs∞, 3.33

whereAt γδt/δ^σt^γ1/γ,Bt δ^Δt−ptδt/δ^σt. Then every solutionxof 1.1 oscillates ont0,∞_T.

Proof. In view ofTheorem 3.3, the proof is similar to that of18, Theorem 3.2.

In the following, we will establish the Philos-type oscillation criteria for1.1.

Theorem 3.7. Assume that2.3holds. Further, suppose that 1−ptμt>0, there exists a positive Δ-differentiable functionδ,andH, h∈CrdD,R, whereD{t, s:t≥s≥t0}such that

Ht, t 0, t≥t0, Ht, s>0, t > s≥t0. 3.34

Hhas a continuous and nonpositiveΔ-partial derivativeH^Δst, swith respect to the second variable and satisfies

H^Δsσt, s Hσt, σs

δ^Δs−psδ^σsα^γs δs

−ht, s

δs Hσt, σs^γ/γ1, 3.35 and for sufficiently larget_∗,

lim sup

t→ ∞

1 Hσt, t_∗

σt t_∗

Kt, sΔs∞, 3.36

where

Kt, s LHσt, σsδ^σsqs− h−t, s^γ1 γ1_γ1

δ^σsβs_γ. 3.37

Then every solutionxof 1.1oscillates ont0,∞_T.

Proof. Letxtbe a nonoscillatory solution of1.1ont0,∞_T. Without loss of generality, we assumext>0, fort≥t∗≥t0. Definewby3.2as before, then we have3.11. From3.11, we have

Lqtδ^σt≤ −w^Δt

δ^Δt

δt −ptδ^σt δt α^γt

wt− γδ^σtβt

δt^γ1/γwt^γ1/γ. 3.38

Thus,

L

σt t∗

Hσt, σsqsδ^σsΔs≤ − ^σt

t∗

Hσt, σsw^ΔsΔs

^σt

t_∗

Hσt, σs

δ^Δs

δs −psδ^σs δsα^γs

wsΔs

− ^σt

t∗

Hσt, σsγδ^σsβs

δs^γ1/γws^γ1/γΔs.

3.39 Integrating the right side by parts, we have

− ^σt

t∗

Hσt, σsw^ΔsΔs≤Hσt, t^∗wt^{∗ σt}

t∗

H^Δsσt, swsΔs, 3.40

and then by using3.34and3.35, we arrive at

L

σt t∗

Hσt, σsqsδ^σsΔs

≤ ^σt

t∗

h−t, s

δsHσt, σs^γ/γ1ws−Hσt, σsγδ^σsβs

δs^γ1/γws^γ1/γ

Δs Hσt, t^∗wt^∗.

3.41

Define

λ γ1

γ , A^λHσt, σsγδ^σsβs

δ^λs , B^λ−1 h−t, s λ

γδ^σsβs1/λ. 3.42 By employing the inequality

λAB^λ−1−A^λ≤λ−1B^λ, λ≥1, 3.43

we obtain

h−t, s

δsHσt, σs^γ/γ1ws

−Hσt, σsγδ^σsβs

δs^γ1/γws^γ1/γ ≤ h−t, s^γ1 γ1_γ1

δ^σsβs_λ.

3.44

Therefore,

1 Hσt, t^∗

σt t∗

Kt, sΔs≤wt^∗, 3.45

which contradicts3.36. The proof is complete.

Theorem 3.8. Assume that2.3holds. Further, suppose that 1−ptμt>0, there exists a positive Δ-differentiable functionδ,andH, h ∈CrdD,R, whereD {t, s :t≥ s≥ t0}such that3.28 holds, and H has a continuous and nonpositiveΔ-partial derivativeH^Δst, swith respect to the second variable and satisfies

H^Δst, s Ht, s

δ^Δs

δs −psδ^σs δs

−ht, s

δ^σsHt, s^γ/γ1. 3.46

If for sufficiently larget∗

lim sup

t→ ∞

1 Ht, t∗

t t∗

Kt, sΔs∞, 3.47

where

Kt, s LHt, sδsqs− h₋t, s^γ1 γ1_γ1

δ^σs^γ, 3.48

then every solutionxof1.1oscillates ont0,∞_T.

Proof. In view ofTheorem 3.3, the proof is similar to16, Theorem 2.2.

Theorem 3.9. Assume that2.3holds. Further, suppose that 1−ptμt>0, and for all sufficiently larget_∗,

lim sup

t→ ∞ t−t∗^{γ ∞}

t

qsΔs > 1

L. 3.49

Then every solutionxof 1.1oscillates ont0,∞_T.

Proof. Letxtbe a nonoscillatory solution of1.1ont0,∞_T. Without loss of generality, we assumext>0, fort≥t_∗≥t0. From1.1andLemma 2.3, we get forT ≥t≥t_∗,

L

T t

qsx^γsΔs < L ^T

t

qsx^γσsΔs <

x^Δtγ

−

x^ΔTγ

<

x^Δtγ

. 3.50

LettingT → ∞, we obtain L

∞ t

qsx^γsΔs <

x^Δtγ

. 3.51

In view ofLemma 2.3, we obtain

L

∞ t

qsΔs <

x^Δt xt

γ

<

1 t−t_∗

_γ

. 3.52

Thus

lim sup

t→ ∞ t−t∗^{γ ∞}

t

qsΔs≤ 1

L, 3.53

which is a contradiction. This completes the proof.

4. Example

In this section, we will give an example to illustrate our results.

Example 4.1. Consider the second-order damped dynamic equation on time scales

x^Δt_γΔ 1

t

x^Δt_γ

tx^σt^γ0, 4.1

where

μt< t, pt 1

t, qt t, δt 1, L1, fx x^γ. 4.2

Obviously,fx/x^γ1≥L1.

It is easy to see that2.3holds. For 0< γ≤1, one has

lim sup

t→ ∞ t t^∗

s− γ^γ

γ1_γ11/sα ^γs^γ1 γα^γS_γ

Δslim sup

t→ ∞ t t^∗

s− 1

γ1_γ1 1 s^γ1α^γ2s

Δs∞, 4.3

and forγ >1,

lim sup

t→ ∞ t t^∗

s− γ^γ

γ1_γ11/sα^γs^γ1 γαs_γ

Δslim sup

t→ ∞ t t^∗

s− 1

γ1_γ1 1 s^γ1α^γs

Δs∞.

4.4 Hence, byTheorem 3.1, every solutionxof4.1is oscillatory.

Remark 4.2. It is easy to see that the results in16–21cannot be applied in4.1, and to the best of our knowledge nothing is known regarding the oscillatory behavior of1.1, so our results are new.

Acknowledgments

This research is supported by the Natural Science Foundation of China11071143, 60904024, China Postdoctoral Science Foundation funded project20080441126, 200902564, Shandong Postdoctoral funded project200802018and supported by the Natural Science Foundation of ShandongY2008A28, ZR2009AL003, also supported by University of Jinan Research Funds for DoctorsXBS0843.

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