Oscillation of certain third-order variable delay damped dynamic equations on time scales

2016-10-18LIMohan

华东师范大学学报(自然科学版) 2016年4期

LI Mo-han

(Mathematical Department，Teachers College，Eastern Liaoning University，Dandong Liaoning，118003，China)

Article ID：1000-5641(2016)04-0011-14

Oscillation of certain third-order variable delay damped dynamic equations on time scales

LI Mo-han

(Mathematical Department，Teachers College，Eastern Liaoning University，Dandong Liaoning，118003，China)

The oscillation for certain third-order nonlinear variable delay dynamic equations with damping term and nonlinear neutral term on time scales is discussed in this article. By using the generalized Riccati transformation and inequality technique，some new oscillation criteria for the equations are established. Our results extend and improve some known results in the literature. Many of the results in this paper are new for the corresponding third-order difference equations and differential equations being as special cases. Some examples are given to illustrate the importance of our results.

oscillation；delay dynamic equations；Riccati transformation；time scales；damping term

0　 Introduction

Consider third-order nonlinear variable delay damping dynamic equation

{r(t)φ([a(t)yΔ(t)]Δ)}Δ+ b(t)φ([a(t)yΔ(t)]Δ)+ P(t)F(φ(x(δ(t))))= 0，t∈T，t≥t0，(0.1)

收稿日期：2015-06
基金项目：辽宁省高等学校优秀科技人才支持计划项目(LR2013062)；国家自然科学基金(60974144)
作者简介：李默涵，男，副教授，硕士，研究方向为微分方程的理论及应用. E-mail：lmh0819@sina.com.

on time scale T，where y(t)= x(t)+ B(t)g(x(τ(t)))，φ(u)= |u|λ-1u，λ>0. Throughout this article，we assume that：

(H1)T is an arbitrary time scale with sup T = +∞，and t0∈T with t0>0，we define the time scale interval [t0，+∞)Tby [t0，+∞)T= [t0，+∞)∩T. r(t)，a(t)，B(t)，b(t)，P(t)∈Crd(T，R). g(u)，F(u)∈C(R，R)with ug(u)>0(u /= 0)and uF(u)>0(u /= 0).

(H2)τ(t)，δ(t)：T→T are delay functions with τ(t)≤t and limt→+∞τ(t)= +∞；δ(t)≤t and lim

t→+∞δ(t)= +∞.

(H3)r(t)>0 and rΔ(t)≥0；a(t)>0 and aΔ(t)≥0；0≤B(t)≤1，b(t)≥0，P(t)>0.

(H4)-b/r∈ℜ+，i.e.，-b/r：T→R is rd-continuous and such that r(t)-µ(t)b(t)>0 for all t∈[t0，+∞)T.

(H5)There exist constants 0<β≤1 and L>0，such that g(u)/u≤β(u /= 0)，F(u)/u≥L(u /= 0).

By a solution of Eq.(0.1)，we mean a nontrivial real-valued function x(t)satisfying Eq.(0.1)for t∈T. We recall that a solution x(t)of Eq.(0.1)is said to be oscillatory on t∈[t0，+∞)Tif it is neither eventually positive nor eventually negative；otherwise，the solution is said to be nonoscillatory. Eq.(0.1)is said to be oscillatory if all of its solutions are oscillatory. Our attention is restricted to those solutions x(t)of Eq.(0.1)where x(t)is not eventually identically zero.

Equations of this type arise in a number of important applications such as problems in biological population dynamics，in neural network，in quantum theory，in computer science and in control theory. Hence it is important and useful to study the oscillatory properties of solutions of Equation(0.1). Recently，there has been an increasing interest in studying the oscillatory behavior of first and second-order dynamic equations on time scales(see [1-7]). However，there are very few results regarding the oscillation of third-order equations. Among these papers dealing with the subject，we refer in particular to [8-18]，the monographs [1-2]and the references therein. Our concern is especially motivated by several recent papers such as [9-13].

Erbe et al.[9]studied third-order linear dynamic equation

xΔΔΔ(t)+ P(t)x(t)= 0，t∈T，t≥t0，(0.2)

and they established Hille and Nehari type oscillation criteria for the equation(0.2).

After that，in [10]and [11]，the authors discussed oscillatory criteria of the following equations[(xΔΔ(t))λ]Δ+ P(t)xλ(δ(t))= 0，t∈T，t≥t0，(0.3)

and

[r(t)(x(t)+ B(t)x(τ(t)))ΔΔ]Δ+ P(t)xγ(δ(t))= 0，t∈T，t≥t0，(0.4)

respectively under the condition

and obtained the result that every solution of Equations(0.3)and(0.4)oscillates or converges to zero. Where the Taylor monomials{hn(t，s)}+∞n=0are defined as follows

In [12]，the author discussed oscillatory criteria of the following equations

{r(t)[a(t)xΔ(t)]Δ}Δ+ P(t)F(x(δ(t)))= 0，t∈T，t≥t0，(0.6)

Moreover，in [13]，the author considered the oscillation for the equations {r(t)[(a(t)xΔ(t))Δ]λ}Δ+ P(t)f(x(δ(t)))= 0，t∈T，t≥t0，(0.7)

under the conditions

and obtained the result that every solution of Equation(0.7)oscillates or converges to zero.

Clearly，the results in [9-13]are inapplicable for the following functional differential equation

{t2[(tx′(t))′]3}′+ t(t - 1)2(3t2- 8t + 2)e6t-18x9(t - 2)= 0.

Therefore，the purpose of this article is to obtain new criteria for the oscillation of Equation (0.1). This topic is fairly new for dynamic equations on time scales. We should note that many of our results of this article are new for the corresponding third-order nonlinear differential and difference equations. In fact，the obtained results extend，unify and correlate many of the existing results of [8-18].

The paper is organized as follows：In Section 1，we present some lemmas which play important roles in the proofs of the main results. In Section 2，we intend to use the Riccati transformation technique to obtain some sufficient conditions which guarantee that every solution x(t)of Eq.(0.1)is either oscillatory or converges as t→+∞. In Section 3，we give some examples in order to illustrate the main results.

1　 Preliminaries

We shall employ the following lemmas.

Lemma1.1[1]Assume that x(t)is Δ-differentiable and eventually positive or eventually negative，then

Lemma1.2[3]Suppose thatrd(I，R)，where I = [t∗，+∞)，t∗>0；(2)u(t)>0，uΔ(t)>0，uΔΔ(t)≤0，t≥t∗. Then，for every k∈(0，1)，there exists a constant tk∈T，tk>t∗，such that

(1)u∈C2

Lemma1.3[1]If g∈ℜ+，i.e.，g：T→R is rd-continuous and such that 1+µ(t)g(t)>0 for all t∈[t0，+∞)T，then the initial value problem yΔ(t)=g(t)y(t)，y(t0)=y0∈R has a unique and positive solution on [t0，+∞)T，denoted by eg(t，t0). This“exponential function”satisfies the semigroup property eg(a，b)eg(b，c)= eg(a，c).

Lemma1.4[2]Assume that a and b are nonnegative real numbers，then rabr-1-ar≤(r - 1)brfor all r>1，where the equality holds if and only if a = b.

Lemma1.5[8]Assume that u(t)>0，uΔ(t)>0，uΔΔ(t)>0，uΔΔΔ(t)<0，then

Lemma1.6 Assume(H1)—(H6)hold，and let x(t)be an eventually position solution of Eq.(0.1). Then there exists t1∈[t0，+∞)T，for all t∈[t1，+∞)T，such that either

(i)y(t)>0，yΔ(t)>0，[a(t)yΔ(t)]Δ>0，{r(t)φ([a(t)yΔ(t)]Δ)}Δ<0，

(ii)y(t)>0，yΔ(t)<0，[a(t)yΔ(t)]Δ>0，{r(t)φ([a(t)yΔ(t)]Δ)}Δ<0.

Proof Since x(t)be an eventually position solution of(0.1)，then there exists t1∈[t0，+∞)Tsuch that x(t)>0，x(τ(t))>0，x(δ(t))>0 for all t∈[t1，+∞)T，thus，y(t)>0. From(0.1)we have

?r(t)φ([a(t)yΔ(t)]Δ)?Δ+ b(t)φ([a(t)yΔ(t)]Δ)≤-LP(t)(x(δ(t)))λ<0.(1.2)

Thus，by Lemma 1.3，we obtain on [t1，+∞)T，

Then there exists t3∈[t2，+∞)T，such that a(t)yΔ(t)≤a(t3)yΔ(t3)<0. Similarly，we can get

which contradicts with y(t)>0. So [a(t)yΔ(t)]Δ>0，this implies that yΔ(t)>0 or yΔ(t)<0. This completes the proof.□

Lemma1.7 Assume that(H1)-(H6)hold，and let x(t)be a solution of Eq.(0.1)which satisfies the case(ii)in Lemma 1.6，if either

holds，then lim

t→+∞x(t)= 0.

Proof Since x(t)is a solution of Eq.(0.1)which satisfies the case(ii)in Lemma 1.6，i.e.，

y(t)>0，yΔ(t)<0，[a(t)yΔ(t)]Δ>0，{r(t)φ([a(t)yΔ(t)]Δ)}Δ<0，t∈[t1，+∞)T.

Therefore，it follows that limt→+∞y(t)= c≥0. If c>0，then in view of y(t)≤x(t)+βB(t)x(τ(t))and 0≤βB(t)≤1，it is not difficult to see that there exists t2≥t1such that x(δ(t))≥c2for all t∈[t2，+∞)T. Thus，from(1.3)，we find

If(1.4)holds，then integrating(1.6)from t2to t(t∈[t2，+∞)T))，we obtain

If(1.5)holds，then integrating(1.6)from t to T(T≥t，T，t∈[t2，+∞)T)，we can get

tP(s)Δs，this implies that

In view of yΔ(t)<0，similarly，we can obtain

Integrating the above inequality from t2to t(t∈[t2，+∞)T)，we have

which contradicts with y(t)>0. So limt→+∞x(t)=0. This completes the proof.□

For the above reasons，in the following theorems，we suppose that either(1.4)or(1.5)holds.

t→+∞y(t)=0，and then，lim

2　 Main results

In this section，we establish some sufficient conditions which guarantee that every solution x(t)of(0.1)either oscillates on [t0，+∞)Tor converges as t→+∞.

Theorem2.1 Assume(H1)-(H6)hold. Furthermore，assume that one of(1.4)and(1.5) holds，if there exists a function φ(t)∈C1rd(T，(0，+∞))，such that

If case(i)in Lemma 1.6 holds，then in view of x(t)≤y(t)，we obtain

y(t)≤x(t)+βB(t)x(τ(t))≤x(t)+βB(t)y(τ(t))≤x(t)+βB(t)y(t)，

i.e.，

x(t)≥[1 -βB(t)]y(t).(2.2)

Now let

Then V(t)>0(t∈[t1，+∞)T). In view of(2.3)and(1.2)，we get for t∈[t1，+∞)T，

Next，we shall distinguish the following two cases：

(I)λ>1；(II)0<λ≤1.

Case I：λ>1. From(1.1)，we then have [(u(t))λ]Δ≥λR10[hu +(1 - h)u]λ-1uΔ(t)dh =λ(u(t))λ-1uΔ(t)，hence，

[(a(t)yΔ(t))λ]Δ≥λ(a(t)yΔ(t))λ-1(a(t)yΔ(t))Δ.

By Lemma 1.6，we have

0>{r(t)[(a(t)yΔ(t))Δ]λ}Δ= rΔ(t)[(a(t)yΔ(t))Δ]λ+ r(σ(t)){[(a(t)yΔ(t))Δ]λ}Δ，then

0>{[(a(t)yΔ(t))Δ]λ}Δ≥λ[(a(t)yΔ(t))Δ]λ-1[(a(t)yΔ(t))Δ]Δ，

hence，[(a(t)yΔ(t))Δ]Δ<0. Using the above inequality and(2.2)in(2.4)，and in view of that

a(t)yΔ(t)≤a(σ(t))yΔ(σ(t))，(a(t)yΔ(t))Δ≥(a(σ(t))yΔ(σ(t)))Δ，

we can obtain

Case II：0<λ≤1. From(1.1)，then we get

thus，

[(a(t)yΔ(t))λ]Δ≥λ[a(σ(t))yΔ(σ(t))]λ-1[a(t)yΔ(t)]Δ.

Similarly，we can obtain(2.5).

t1a(s)yΔ(s)Δs. In view of(2.7)，one can easy to see that

for all t∈[T0，+∞)T. Using the above inequality in(2.5)，we get for all t∈[T0，+∞)T，

Therefore，by(2.6)and(2.8)，∃T0∈[t0，+∞)Twith T0≥max{t2，t1/2}，such that

that is，

Now，set

Then，by Lemma 1.4，we find

Using the above inequality in(2.9)，we get

Integrating inequality(2.10)from T0to t(t∈[T0，+∞)T)，we see that

Consequently，

which contradicts(2.1).

Remark 2.1 From Theorem 2.1，we can get different conditions for oscillation of all solutions of(0.1)with different choices of φ(t). For example，φ(t)= t(k = 1/2)or φ(t)= 1(k = 1/2). Then，we have the following results respectively.

Corollary 2.2 Assume(H1)-(H6)hold. Furthermore，assume that one of(1.4)and (1.5)holds，if

Corollary 2.3 Assume(H1)-(H6)hold. Furthermore，assume that one of(1.4)and (1.5)holds，if

then every solution x(t)of Eq.(0.1)is either oscillatory or limt→+∞x(t)= 0 on [t0，+∞)T.

Next，for convenience，consider the set D ={(t，s)：t≥s≥t0，t，s∈[t0，+∞)T}. We say that a function H = H(t，s)belongs to function class Ω，denoted by H∈Ω，if H∈Crd(D，R)，which satisfies

H(t，t)= 0 for t≥t0，H(t，s)>0 for t>s≥t0，t，s∈[t0，+∞)T

and has a nonpositive continuous Δ-partial derivative HΔs(t，s)with respect to the second variable，i.e.，HΔs(t，s)∈Crdand HΔs(t，s)≤0.

Proof Suppose to the contrary that x(t)is a nonoscillatory solution of Eq.(0.1)on [t0，+∞)T. We may assume without loss of generality that x(t)>0 and x(τ(t))>0，x(δ(t))>0 for all t∈[t1，+∞)T，t1∈[t0，+∞)T. By Lemma 1.6 there are two possible cases. If case(ii)in Lemma 1.6 holds，then limt→+∞x(t)= 0. If case(i)in Lemma 1.6 holds，we proceed as in the proof of Theorem 2.1 to obtain(2.9). Then from(2.9)，we get

for all s∈[T0，+∞)T. Multiplying both sides of the above inequality by H(t，s)K(s)，and integrating with respect to s from T0to t(t∈[T0，+∞)T)，we have

Now，in Lemma 1.4，we let

From Lemma 1.4，we then get

Hence，(2.12)implies

that is，

and therefore，

Thus

contradicting(2.11). This completes the proof.□

Theorem2.5 Assume(H1)-(H6)hold. Furthermore，assume that one of(1.4)and(1.5) 0 holds，if there exist functions H∈Ω and φ(t)∈C1rd(T，(0，+∞))，such that

where function Ψ(t)is defined as in Theorem 2.1. Then every solution x(t)of Eq.(0.1)is either oscillatory or limt→+∞x(t)= 0 on [t0，+∞)T.

for all s∈[T0，+∞)T. Then it follows that

and hence，for all t≥s≥T0，

The above inequality implies that 0

φ(t)= 1，H(t，s)=(t - s)m(m≥1，t≥s≥t0，t，s∈[t0，+∞)T).

Then H(t，t)= 0 for t≥t0，and H(t，s)>0，HΔs(t，s)≤0 for t>s≥t0，t，s∈[t0，+∞)T. Then from Theorem 2.5，we can obtain the following result.

Corollary2.6 Assume(H1)-(H6)hold. Furthermore，assume that one of(1.4)and (1.5)holds，if there exists a constant m≥1，such that

Then every solution x(t)of Eq.(0.1)is either oscillatory or limt→+∞x(t)= 0 on [t0，+∞)T.

Remark 2.3 Obviously，Kamenev type oscillation criteria for second-order linear differential equation was extended to third-order nonlinear variable delay damping dynamic equations on time scales. One can easily see that the recent results cannot be applied in Eq.(0.1)，so our results are new ones.

3　 Examples

In this section，we give some examples to illustrate our main results.

Example 3.1 Consider third-order delay functional differential equation {t2[(tx′(t))′]3}′+ t(t - 1)2(3t2- 8t + 2)e6t-18x9(t - 2)= 0，t≥2.(3.1)

It is not difficult to verify that all conditions of Corollary 2.3 are satisfied. Hence，every solution of Eq.(3.1)is oscillatory or tends to zero as t→+∞. For example，it is easy to verify that x(t)= e-tis a solution of Eq.(3.1). The important point to note here is that the recent results due to [10-14]do not apply to Eq.(3.1)for the condition(0.5)or(0.8)can be a restrictive condition.

Example 3.2 Consider third-order variable delay dynamic equations with damping on time scales

+P(t)f(φ(x(δ(t))))= 0，t∈T = 2Z，t≥2.(3.2)

This is a third-order 2-difference equation，here t0=2，r(t)=t25，a(t)=t，b(t)=1t2. Now，pick λ= 37，B(t)=1-12t，τ(t)=δ(t)=t2，P(t)=1t1/7[σ(t)h2(δ(t)，t0)]3/7，g(u)= u √1+sin4u，f(u)= u[6+ln(1+u2)]，then

and so

and it is not difficult to verify thatR+∞t0P(s)Δs = +∞. Hence conditions(H1)-(H4)and(1.4)are clearly satisfied. Let m = 2，in view of β= 1，L = 6，then we have

This implies

and so conditions of Corollary 2.6 are satisfied as well. Altogether，by Corollary 2.6，we have that every solution of Eq.(3.2)is oscillatory or tends to zero as t→+∞. But the results in [8-18]are inapplicable for Eq.(3.2).

Remark 3.1 Our results in this paper not only extend and improve some known results，and show some results of [8-18]to be special examples of our results，but also unify the oscillation of third-order nonlinear delay damped differential equations and third-order nonlinear delay damped difference equations. The theorems in this paper are new even for the cases T = R and T = Z.

[References]

[1]BOHNER M，PETERSON A. Dynamic Equations on Time Scales，An Introduction with Applications[M]. Boston：Birkhauser，2001.

[2]AGARWAL R P，BOHNER M，LI W T. Nonoscillation and Oscillation：Theory for Functional Differential Equations[M]. New York：Marcel Dekker，2004.

[3]SAHINER Y. Oscillation of second order delay differential equations on time scales[J]. Nonlinear Analysis，TMA，2005，63：e1073-e1080.

[4]AGARWAL R P，BOHNER M，SAKER S H. Oscillation of second order delay dynamic equations[J]. Canadian Applied Mathematics Quarterly，2005，13(1)：1-18.

[5]ZHANG Q X，GAO L，LIU S H. Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales(II)[J]. Sci Sin Math，2011，41(10)：885-896.

[6]YANG J S. Oscillation for nth-order nonlinear neutral delay dynamic equations on time scales[J]. Mathematica Applicata，2013，26(4)：816-827.

[7]YANG J S，QIN X W，ZHANG X J. Oscillation criteria for certain second-order nonlinear neutral delay dynamic equations with damping on time scales[J]. Mathematica Applicata，2015，28(2)：439-448.

[8]ERBE L，PETERSON A，SAKER S H. Hille and Nehari type criteria for third-order dynamic equations[J]. J Math Anal Appl，2007，329(1)：112-131.

[9]ERBE L，HASSAN T S，PETERSON A. Oscillation of third order nonlinear functional dynamic equations on time scales[J]. Differ Equ Dyn Syst，2010，18：199-227.

[10]HAN Z L，LI T X，SUN S R，et al. Oscillation criteria for third order nonlinear delay dynamic equations on time scales[J]. Ann Polon Math，2010，99(2)：143-156.

[11]HAN Z L，LI T X，SUN S R，et al. Oscillation behavior of third-order neutral Emden-Fowler delay dynamic equations on time scales[J]. Adv Diff Eq，2010，2010：1-23.

[12]HAN Z L，LI T X，SUN S R，et al. Oscillation behavior of solutions of third-order nonlinear delay dynamic equations on time scales[J]. Commun Korean Math Soc，2011，26(3)：499-513.

[13]HASSAN T S. Oscillation of third order nonlinear delay dynamic equations on time scales[J]. Math Comput Model，2009，49：1573-1586.

[14]LI T X，HAN Z L，SUN Y B，et al. Asymptotic behavior of solutions for third-order half-linear delay dynamic equations on time scales[J]. J Appl Math Comput，2011，36：333-346.

[15]LI T X，HAN Z L，ZHANG C H，et al. Oscillation criteria for third-order Emden-Fowler delay dynamic equations on time scales[J]. Acta Mathematica Scientia，2012，32A(1)：222-232.

[16]YANG J S. Oscillation of third-order delay dynamic equations on time scales[J]. Chinese Quarterly Journal of Mathematics，2014，29(3)：447-456.

[17]ZHANG X J，YANG J S. Oscillation and asymptotic behaviors for third-order delay dynamic equations on time scales[J]. Journal of East China Normal University(Natural Science)，2014(3)：51-59.

[18]YANG J S. Oscillation criteria for certain third-order delay dynamic equations[J]. Advances in Difference Equations，2013，2013：178.

(责任编辑：林磊)

10.3969/j.issn.1000-5641.2016.04.002

时间尺度上一类三阶变时滞阻尼动态方程的振荡性

李默涵

(辽东学院师范学院数学系，辽宁丹东118003)

讨论了时间尺度上一类具阻尼项和非线性中立项的三阶非线性变时滞动态方程的振荡性，利用广义的Riccati变换和不等式技巧，获得了该方程的一些新的振荡准则，推广并改进了现有文献中的一些结果，本文的这些结果对于作为其特例的相应三阶差分方程和微分方程来说也是新的，最后通过例子来说明了文章中的这些结果的重要性.

振荡性；时滞动态方程；Riccati变换；时间尺度；阻尼项

O157.7 Document code：A

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