Oscillation for Second Order Nonlinear Delay Dynamic Equations on Time Scales

Advances in Difference Equations Volume 2009, Article ID 756171,13pages doi:10.1155/2009/756171

Research Article

Oscillation for Second-Order Nonlinear Delay

Dynamic Equations on Time Scales

Zhenlai Han,

_{Tongxing Li,}

_{Shurong Sun,}

and Chenghui Zhang

1_{School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China}

2_{School of science, University of Jinan, Jinan, Shandon 250022, China}

Correspondence should be addressed to Zhenlai Han,[email protected]

Received 6 December 2008; Revised 27 February 2009; Accepted 25 May 2009

Recommended by Alberto Cabada

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equationsrtxΔtγΔptfxτt 0 on a time scaleT; hereγ > 0 is a quotient of odd positive integers withr andpreal-valued positive rd-continuous functions defined onT. Our results not only extend some results established by Hassan in 2008 but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.

Copyrightq2009 Zhenlai Han 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.

1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis see Hilger1 . Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. 2 and references cited therein. A book on the subject of time scales, by Bohner and Peterson3 , summarizes and organizes much of the time scale calculus. We refer also to the last book by Bohner and Peterson4 for advances in dynamic equations on time scales. For the notation used hereinafter we refer to the next section that provides some basic facts on time scales extracted from Bohner and Peterson3 .

by continuous dynamic systems, die out, say in winter, while their eggs are incubating or dormant, and then in season again, hatching gives rise to a nonoverlapping populationsee Bohner and Peterson3 . Not only does the new theory of the so-called “dynamic equations” unify the theories of differential equations and difference equations but also it extends these classical cases to cases “in between”, for example, to the so-called q-difference equations whenT qN0 {_qt _:t ∈ N

0, q > 1}which has important applications in quantum theory

and can be applied on different types of time scales likeThN,TN2_{, andTT}

nthe space of the harmonic numbers.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to Bohner and Saker5 , Erbe6 , and Hassan7 . However, there are few results dealing with the oscillation of the solutions of delay dynamic equations on time scales8–15 .

Following this trend, in this paper, we are concerned with oscillation for the second-order nonlinear delay dynamic equations

rtxΔtγΔptfxτt 0, t∈T. 1.1

We assume thatγ >0 is a quotient of odd positive integers,r andpare positive, real-valued rd-continuous functions defined onT, τ : T → R is strictly increasing, andT :

τT ⊂ Tis a time scale,τt≤ tandτt → ∞ast → ∞,f ∈CR,Rwhich satisfies for some positive constantL,fx/xγ _{≥L,for all nonzerox.}

For oscillation of the second-order delay dynamic equations, Agarwal et al. 8

considered the second-order delay dynamic equations on time scales

xΔΔt ptxτt 0, t∈T, 1.2

and established some sufficient conditions for oscillation of1.2.

Zhang and Shanliang 15 studied the second-order nonlinear delay dynamic equations on time scales

xΔΔt ptfxt−τ 0, t∈T, 1.3

and the second-order nonlinear dynamic equations on time scales

xΔΔt ptfxσt 0, t∈T, 1.4

whereτ ∈Randt−τ ∈T,f:R → Ris continuous and nondecreasingfu> k >0, and

ufu>0 for_{u /}0, and they established the equivalence of the oscillation of1.3and1.4, from which obtained some oscillation criteria and comparison theorems for1.3. However, the results established in15 are valid only when the graininess functionμtis bounded which is a restrictive condition. Also the restrictionfu> k >0 is required.

Sahiner13 considered the second-order nonlinear delay dynamic equations on time scales

wheref:R → Ris continuous,ufu>0 for_{u /}0 and|fu| ≥L|u|,τ :T → T,τt≤tand lim_{t→ ∞}τt ∞, and he proved that if there exists aΔ-differentiable functionδtsuch that

lim sup t→ ∞

t

t0

psδσsτs σs−

σsδΔs2

4Lk2_δσ_sτs Δs∞, 1.6

then every solution of1.5oscillates. Now, we observe that the condition1.6depends on an additional constantk ∈0,1which implies that the results are not sharpsee Erbe et al.

10 .

Han et al.12 investigated the second-order Emden-Fowler delay dynamic equations on time scales

xΔΔt ptxγτt 0, t∈T, 1.7

established some sufficient conditions for oscillation of1.7, and extended the results given in8 .

Erbe et al.10 considered the general nonlinear delay dynamic equations on time scales

ptxΔtΔqtfxτt 0, t∈T, 1.8

wherep andqare positive, real-valued rd-continuous functions defined onT,τ : T → T is rd-continuous,τt≤ tandτt → ∞ast → ∞, andf ∈CR,Rsuch that satisfies for some positive constantL,|fx| ≥L|x|,xfx > 0 for all nonzerox, and they extended the generalized Riccati transformation techniques in the time scales setting to obtain some new oscillation criteria which improve the results given by Zhang and Shanliang15 and Sahiner

13 .

Clearly, 1.2,1.3, 1.5, and 1.8 are the special cases of 1.1. In this paper, we consider the second-order nonlinear delay dynamic equation on time scales1.1.

As we are interested in oscillatory behavior, we assume throughout this paper that the given time scaleTis unbounded above. We assumet0 ∈ T, and it is convenient to assume

t0>0.We define the time scale interval of the formt0,∞Tbyt0,∞T t0,∞∩T.

We shall also consider the two cases

_∞

t0

1

rt 1/γ

Δt∞, 1.9

_∞

t0

1

rt 1/γ

Δt <∞. 1.10

2. Main Results

In this section, we give some new oscillation criteria for 1.1. In order to prove our main results, we will use the formula

xtγΔγ 1

0

hxσt 1−hxt γ−1xΔtdh, 2.1

wherextis delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rulesee Bohner and Peterson3, Theorem 1.90 . Also, we need the following auxiliary result.

For convenience, we note that

dt max{0, dt}, d−t max{0,−dt}, Rt:r1/γt

t

t1

Δs r1/γ_s,

αt: Rt

Rt μt, βt:αt, 0< γ≤1; βt:α

γ_{t, γ >1.}

2.2

Lemma 2.1. Assume that1.9holds and assume further thatxtis an eventually positive solution of 1.1. Then there exists at1∈t0,∞Tsuch that

xΔt>0, xt> RtxΔt, rtxΔtγΔ<0, xt

xσ_t > αt, t∈t1,∞T. 2.3

The proof is similar to that of Hassan7, Lemma 2.1 , and so is omitted.

Lemma 2.2Chain Rule. Assume thatτ : T → Ris strictly increasing andT : τT ⊂ Tis a time scale,τσt στt. Letx : T → R. IfτΔt, and letxΔτtexist fort ∈ Tκ_{, then}

xτtΔexist, and

xτtΔxΔτtτΔt. 2.4

Proof. Let 0< ε <1 be given and defineε∗ε1|τΔt||xΔτt| −1.Note that 0< ε∗<1.

According to the assumptions, there exist neighborhoodsN1oftandN2ofτtsuch that

τσt−τs−σt−sτΔt≤ε∗|σt−s|, s∈ N1,

xστt−xr−στt−rxΔτt≤ε∗|στt−r|, r∈ N2.

PutNN1τ−1N2and lets∈ N.Thens∈ N1andτs∈ N2and

xτσt−xτs−σt−sxΔτtτΔt

xτσt−xτs−στt−τsxΔτt

στt−τs−σt−sτΔtxΔτt

xστt−xτs−στt−τsxΔτt

τσt−τs−σt−sτΔtxΔτt

≤ε∗|στt−τs|ε∗|σt−s|xΔτt

≤ε∗στt−τs−σt−sτΔt|σt−s|τΔt|σt−s|xΔτt

ε∗τσt−τs−σt−sτΔt|σt−s|τΔt|σt−s|xΔτt

≤ε∗ε∗|σt−s||σt−s|τΔt|σt−s|xΔτt

ε∗|σt−s|ε∗τΔtxΔτt

≤ε∗|σt−s|1τΔtxΔτt

ε|σt−s|.

2.6

The proof is completed.

Theorem 2.3. Assume that1.9holds andτ ∈C1

rdt0,∞T,T, τσt στt. Furthermore,

assume that there exists a positive functionδ∈C1

rdt0,∞T,Rand for all sufficiently larget1, one

has

lim sup t→ ∞

_t

t1

⎛

⎝_Lpsαγ_τsδσ_s− rτs

δΔsγ1

γ1γ1δσ_sβτsτΔ_sγ

⎞

⎠_{Δs∞. 2.7}

Then1.1is oscillatory ont0,∞T.

Proof. Suppose that 1.1 has a nonoscillatory solution xt on t0,∞T. We may assume

without loss of generality that xt > 0 and xτt > 0 for allt ∈ t1,∞_T, t1 ∈ t0,∞_T.

We shall consider only this case, since the proof whenxtis eventually negative is similar. In view ofLemma 2.1, we get2.3. Define the functionωtby

ωt δtrt

xΔtγ

From2.16, we find

ωΔt≤ −Lptδσtαγτt rτt

δΔtγ1

γ1γ1δσ_tβτtτΔ_tγ. 2.20

Integrating the inequality2.20fromt1tot, we obtain

_t

t1

⎛

⎝_Lpsαγ_τsδσ_s− rτs

δΔsγ1

γ1γ1δσ_sβτsτΔ_sγ

⎞

⎠_{Δs≤ωt1−ωt≤ωt1,}

2.21

which contradicts2.7. The proof is completed.

Remark 2.4. From Theorem 2.3, we can obtain different conditions for oscillation of all solutions of1.1with different choices ofδt.

Theorem 2.5. Assume that1.9holds andτ ∈C_rd1 t0,∞T,T, τσt στt. Furthermore, assume that there exist functionsH,h∈CrdD,R, whereD≡ {t, s:t≥s≥t0}such that

Ht, t 0, t≥t0, Ht, s>0, t > s≥t0, 2.22

andHhas a nonpositive continuousΔ-partial derivationHΔs_{t, swith respect to the second variable} and satisfies

HΔs_{σt, s Hσt, σs}δ Δ_s

δs − ht, s

δs Hσt, σs

γ/γ1_{, 2.23}

and for all sufficiently larget1,

lim sup t→ ∞

1

Hσt, t1 _σt

t1

Kt, sΔs∞, 2.24

whereδtis a positiveΔ-differentiable function and

Kt, s LHσt, σsαγτsδσsps− rτsh−t, s γ1

γ1γ1δσ_sβτsτΔ_sγ. 2.25

Then1.1is oscillatory ont0,∞T.

Proof. Suppose that 1.1 has a nonoscillatory solution xt on t0,∞_T. We may assume

Again, letλ:γ1/γ, defineA≥0 andB≥0 by

Aλ:Hσt, σsγδσsβτs τΔs rλ−1_τsδsλω

λ_s,

Bλ−1: h−t, sr

λ−1/λ_τs

λγδσ_sβτsτΔ_s1/λ,

2.29

and using the inequality

λABλ−1−Aλ≤λ−1Bλ, λ≥1, 2.30

we find

h−t, s

δs Hσt, σs

1/λ_{ωs−Hσt, σsγδ}σ_sβτs τΔs rλ−1_τsδλ_sω

λ_s

≤ rτsh−t, sγ1

γ1γ1δσ_sβτsτΔ_sγ.

2.31

Therefore, from the definition ofKt, s, we obtain

σt

t1

Kt, sΔs≤Hσt, t1ωt1, 2.32

and this implies that

1

Hσt, t1 σt

t1

Kt, sΔs≤ωt1, 2.33

which contradicts2.24. This completes the proof.

Now, we give some sufficient conditions when 1.10 holds, which guarantee that every solution of1.1oscillates or converges to zero ont0,∞T.

Theorem 2.6. Assume 1.10 holds and τ ∈ C1

rdt0,∞T,T, τσt στt. Furthermore,

assume that there exists a positive functionδ ∈ C1_rdt0,∞_T,R, for all sufficiently larget1, such

that2.7or2.22,2.23, and2.24hold. If there exists a positive functionη∈C1

rdt0,∞T,R,

ηΔt≥0,such that

_∞

t0

1

ηtrt _t

t0

ησspsΔs

1/γ

Δt∞, 2.34

Proof. We proceed as in Theorem 2.3 or Theorem 2.5, and we assume that 1.1 has a nonoscillatory solution such thatxt>0,andxτt>0, for allt∈t1,∞T.

From the proof ofLemma 2.1, we see that there exist two possible cases for the sign of xΔt. The proof whenxΔt is an eventually positive is similar to that of the proof of

Theorem 2.3orTheorem 2.5, and hence it is omitted.

Next, suppose thatxΔt<0 fort∈t1,∞T. Thenxtis decreasing and limt→ ∞xt b≥0. We assert thatb0. If not, thenxτt≥xt≥xσt≥b >0 fort∈t1,∞_T. Since

fxτt ≥ Lxτtγ, there exists a numbert2 ∈ t1,∞T such thatfxτt ≥ Lbγ for t≥t2.

Defining the functionut ηtrtxΔtγ,we obtain from1.1

uΔt ηΔtrtxΔtγησtrtxΔtγΔ

≤ −ησtptfxτt≤ −Lησtptxτtγ≤ −Lbγησtpt, t∈t2,∞_T. 2.35

Hence, fort∈t2,∞T, we have

ut≤ut2−Lbγ _t

t2

ησspsΔs≤ −Lbγ _t

t2

ησspsΔs, 2.36

because ofut2 ηt2rt2xΔt2γ <0. So, we have

xt−xt2 _t

t2

xΔsΔs≤ −L1/γb _t

t2

1

ηsrs _s

t2

ηστpτΔτ

1/γ

Δs. 2.37

By condition2.34, we getxt → −∞ast → ∞, and this is a contradiction to the fact that xt>0 fort≥t1. Thusb0 and thenxt → 0 ast → ∞. The proof is completed.

3. Application and Example

Hassan7 considered the second-order half-linear dynamic equations on time scales

rtxΔtγΔptxγt 0, t∈T, 3.1

Theorem 3.1Hassan7, Theorem 2.1 . Assume that1.9holds. Furthermore, assume that there exists a positive functionδ∈C1

rdt0,∞,Rsuch that

Theorem 3.1, and so Theorem 2.3 in this paper essentially includes results of Hassan 7, Theorem 2.1 .

Similarly,Theorem 2.5includes results of Hassan7, Theorem 2.2 , andTheorem 2.6

includes results of Hassan7, Theorem 2.4 . One can easily see that nonlinear delay dynamic equations 1.8considered by Erbe et al.10 are the special cases of1.1, and the results obtained in10 cannot be applied in1.1, and so our results are new.

Example 3.2. Consider the second-order delay dynamic equations

We conclude that3.3is oscillatory.

Acknowledgments

This research is supported by the Natural Science Foundation of China60774004, China Postdoctoral Science Foundation Funded Project 20080441126, Shandong Postdoctoral Funded Project200802018and supported by Shandong Research FundsY2008A28, and also supported by the University of Jinan Research Funds for DoctorsB0621, XBS0843.

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