Luhong Ye

(School of Math.and Computational Sciences,Anqing Teachers College, Anqing 246011,Anhui,E-mail:leaf07@163.com)

OSCILLATION CRITERIA FOR SECOND-ORDER NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES??

Luhong Ye

(School of Math.and Computational Sciences,Anqing Teachers College, Anqing 246011,Anhui,E-mail:leaf07@163.com)

In this paper,we consider a class of second-order neutral delay dynamic equations on a time scale T.By means of Riccati transformation technique,we establish some new oscillation criteria in two different conditions.The obtained results enrich the well-known oscillation results for some dynamic equations.

oscillation;second-order neutral delay dynamic equations;time scales

2000 Mathematics Subject Classification 34K40;34K11;34C10

Ann.of Appl.Math.

31:2(2015),236-245

1　Introduction

The theory of time scales,which has recently received a lot of attentions,was introduced by Stefan Hilgan in his Ph.D.Thesis in 1988 in order to unify continuous and discrete analysis.Not only can this theory of the so-called dynamic equations unify the theories of differential equations and difference equations but also it is able to extend these classical cases to cases in between,e.g.,to the so-called q-difference equations.A time scale T is an arbitrary closed subset of the reals,and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist,and they give rise to many applications.

In recent years there has been much research activity concerning the oscillation and nonoscillation of solutions to different types of dynamic equations on time scales.We refers the readers to the papers[1-5]and the references cited therein.

In this paper,we are concerned with oscillation of solutions to the second-order neutral delay dynamic equation

on a time scale T,where Z(t)=x(t)+p(t)x(δ(t)).

To study asymptotic behavior of solutions,we suppose that the time scale T under consideration is not bounded above,that is,it is a time scale interval of the form[t0,∞) with t0∈T.

Throughout this paper,the following conditions are assumed to hold:

(H1)γ≥1 is an odd positive integer,r(t)＞0,q(t)≥0 and 0≤p(t)≤1 are real-valued rd-continuous positive functions defined on T;

(H2)δ:T→T is an rd-continuous function such that δ(t)≤t,δ(t)→∞as t→∞;

(H3)τ:T→T is an rd-continuous and increasing function such that τ(t)≤t,τ(t)→∞as t→∞;

(H4)f,ψ∈C(R,R),xf(x)＞0 and ψ(x)＞0 for x≠0,and there exist positive constants k and L such that f(x)/(|x|γ-1x)≥k and ψ(x)≤L-1for x≠0.

We consider the two following cases

By a solution to(1.1),we mean a nontrivial real-valued function x∈[tx,∞),tx≥t0, which has the properties x(t)+p(t)x(δ(t))∈[tx,∞)and r(t)ψ(x(t))|ZΔ(t)|γ-1ZΔ(t)∈[tx,∞),tx≥t0,and satisfies(1.1)for all t≥tx.Our attention is restricted to those solutions to(1.1)which exist on some half line[tx,∞)and satisfy sup{|x(t)|:t≥t1}＞0 for any t1≥tx.A solution x(t)to(1.1)is said to be oscillatory if it is neither eventually positive nor eventually negative.Otherwise it is called nonoscillatory.The equation itself is called oscillatory if all its solutions are oscillatory.

In this paper we use the Riccati transformation technique to obtain several oscillation criteria for(1.1)when(1.2)or(1.3)holds on time scales.We apply a simple consequence of Keller’s chain rule,and the inequality

where A and B are nonnegative constants,to derive sufficient conditions for oscillation of all solutions to(1.1).

2　Some Preliminaries on Time Scales

A time scale T is an arbitrary nonempty closed subset of the real numbers R.On any time scale T,we define the forward and backward jump operators by

A point t∈T,t＞inf T,is said to be left-dense if ρ(t)=t,right-dense if t＜sup T and σ(t)=t,left-scattered if ρ(t)＜t and right-scattered if σ(t)＞t.The graininess functionμ for a time scale T is defined byμ(t):=σ(t)-t.

A function f:T→R is called an rd-continuous function provided that it is continuous at right-dense points in T and its left-sided limits exist(finite)at left-dense points in T.The set of rd-continuous functions f:T→R is denoted by Crd=Crd(T)=Crd(T,R).

The set of functions f:T→ R that are differentiable and whose derivatives are rdcontinuous function is denoted byThe function H(t,s)is an rd-continuous function if H is an rd-continuous function in t and s.

A function p:T→ R is called positively regressive(we write p∈R+)if it is an rdcontinuous function and satisfies 1+μ(t)p(t)＞0 for all t∈T.

We say that f is increasing,decreasing,nondecreasing,and nonincreasing on[a,b]if t1,t2∈[a,b]and t2＞t1imply f(t2)＞f(t1),f(t2)＜f(t1),f(t2)≥f(t1),and f(t2)≤f(t1), respectively.

Let f be a differentiable function on[a,b].Then f is increasing,decreasing,nondecreasing,and nonincreasing on[a,b]if fΔ(t)＞0,fΔ(t)＜0,fΔ(t)≥0,and fΔ(t)≤0 for all t∈[a,b),respectively.

For a function f:T→R(the range R of f may be actually replaced by any Banach space)the(delta)derivative is defined by

if f is continuous at t and t is right-scattered.If t is not right-scattered then the derivative is defined by

provided that this limit exists.

A function f:[a,b]→R is said to be differentiable if its derivative exists,and a useful formula is

We make use of the following product and quotient rules for the derivative of the product f g and the quotient f/g(where ggσ≠0)of two differentiable functions f and g

For t0,b∈T,and a differentiable function f,the Cauchy integral of fΔis defined by

An integration by parts formula reads

and infinite integral is defined as

3　Main Results

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

which is a simple consequence of Keller’s chain rule.

For simplicity,we introduce the following notations:

Now we state and prove the main results.

Theorem 3.1Let(1.2)hold.Assume that there exists a positive rd-continuous Δ-differentiable function α(t)such that

where(αΔ(s))+=max{0,αΔ(s)}.Then every solution to(1.1)is oscillatory on[t0,∞).

Proof Suppose that(1.1)has a nonoscillatory solution x(t).Without loss of generality, we may assume that x(t)＞0,x(τ(t))＞0 and x(δ(t))＞0 for all t≥t1≥t0,then Z(t)＞0, in view of(1.1)we have

and this implies that r(t)ψ(x(t))|ZΔ(t)|γ-1ZΔ(t)is an eventually decreasing function.We first show that ZΔ(t)＞0.Otherwise,suppose that there exists an integer t2≥t1such that r(t1)ψ(x(t1))(ZΔ(t1))γ=-c＜0,then we have

for t≥t2≥t1,hence

In view of(1.2),we have

This contradicts the fact that Z(t)＞0 for all t≥t1.So ZΔ(t)＞0 holds.

Since x(t)≤Z(t),we see that

Define a function w(t)by

then w(t)＞0.Using(2.5)and(2.6),we have

In view of(3.3),(3.6)and(3.8),we have

Using ZΔ(t)＞0 and the Keller’s chain rule,we obtain

Then for t≥t2sufficiently large,we have

It follows from(3.9),(3.11)and(3.12)that

where(αΔ(t))+=max{0,αΔ(t)}.

Since ZΔ(t)＞0,we have Z(τ(σ(t)))≥Z(τ(t)),which implies that

where λ=(γ+1)/γ.

Set

Using inequality(1.4)we have

Thus,from(3.13)and(3.15)we obtain

Integrating(3.16)from t2to t we obtain

which yields

for all large t,which contradicts(3.2).The proof is completed.

Remark 3.1From Theorem 3.1,we can establish different sufficient conditions for oscillation of(1.1)by different choices of α(t).For instance,if α(t)=t and α(t)=1 for t≥t1,we have the following results respectively.

Corollary 3.1 Assume that(1.2)holds.If

then every solution to(1.1)is oscillatory on[t0,∞).

Corollary 3.2 Assume that(1.2)holds.If

then every solution to(1.1)is oscillatory on[t0,∞).

Example 3.1 Consider the following equation

where T is a time scale and γ＞1,τ and δ are nonnegative constants.

Here r(t)=tγ-1,p(t)=1/(t+τ),τ(t)=t-τ,δ(t)=t-δ,q(t)=t-2(1-1/t)-γand f(x)=xγ.Then

so(1.2)holds.

By choosing α(s)=s and L=k=1,we have

then every solution to the equation is oscillatory by Theorem 3.1.

Let D0≡{(t,s)∈T2:t＞s≥t0}and D≡{(t,s)∈T2:t≥s≥t0}.The function H∈Crd(D,R)is said to belong to the class R if

(i)H(t,t)=0,t≥t0,H(t,s)＞0 on D0;

(ii)H has a continuous Δ-partial derivative HΔs(t,s)on D0with respect to the second variable.(H is an rd-continuous function if H is an rd-continuous function in t and s.)

Theorem 3.2Let(1.2)hold.Assume that α(t)is as defined in Theorem 3.1 and H:D→R are rd-continuous functions such that H belongs to the class R and

where C(t,s)=HΔs(t,s)+H(t,s)(αΔ(s))+/ασ.Then every solution to(1.1)is oscillatory on[t0,∞).

Proof Suppose that x(t)is a nonoscillatory solution to(1.1).Without loss of generality, we may assume x(t)＞0 for all t≥t1≥t0,then x(τ(t)),x(δ(t))and Z(t)are positive.We proceed as in the proof of Theorem 3.1 to prove that there exists a t2≥t1such that(3.13) holds for t≥t2.From(3.13),we have

Using integration by parts formula(2.7),we have

where H(t,t)=0.Substituting(3.20)into(3.19),we get

Therefore,by(1.4)we have

where C(t,s)=HΔs(t,s)+H(t,s)(αΔ(s))+/ασ.

Then for all t≥t2,we have

That is,

which implies that

for all large t,which contradicts(3.18).Then every solution to(1.1)is oscillatory.The proof is completed.

As an immediate consequence of Theorem 3.2,we have the following corollary.

Corollary 3.3Assume that(1.2)holds and α(t)is as defined in Theorem 3.1.Let assumption(3.18)in Theorem 3.2 be replaced by

where C(t,s)is defined as in Theorem 3.2.Then every solution to(1.1)is oscillatory on [t0,∞).

By choosing suitable functions H and h we can establish some oscillation criteria for (1.1)on different types of time scales.If H(t,s)=(t-s)mwith m＞1,H belongs to the class R.Now,we claim that

then we can obtain the following Kamenev-type oscillation criteria for(1.1).

Corollary 3.4 Let(1.2)hold.Assume that α(t)is as defined in Theorem 3.1.If for m＞1

where K(t,s)=(t-s)m(αΔ(s))+/ασ-m(t-σ(s))m-1.Then every solution to(1.1)is oscillatory on[t0,∞).

Theorem 3.3 Let(1.3)hold.If

Then(1.1)is oscillatory.

Proof Suppose that(1.1)has a nonoscillatory solution such that x(t),x(τ(t))and x(σ(t))are all positive for all t≥t1≥t0,then Z(t)＞0.From the proof of Theorem 3.1, we see that ZΔ(t)＞0 or ZΔ(t)＜0.

Case 1 ZΔ(t)＞0 for t≥t1,then it is the case of Theorem 3.1 by choosing α(t)= πγ(τ(t)).Thus the proof of Theorem 3.1 goes through,and we get a contradiction by(3.28). Hence,ZΔ(t)＞0 for t≥t1can’t occur.

Case 2 ZΔ(t)＜0 for t≥t1.Let

Obviously,u(t)＜0 on[t1,∞).Note that r(t)ψ(x(t))|ZΔ(t)|γ-1ZΔ(t)is nonincreasing,then we have

Dividing the above by(r(s)ψ(x(s)))1/γand integrating it over[t,μ]gives that

Since ZΔ(t)＜0 and(H4),we can obtain

Lettingμ→∞in the above inequality,we have

So we get

Since ZΔ(t)＜0 and Z(τ(t))≥Z(t),the above inequality follows that

Therefore,

Thus,differentiating(3.30)and using(3.4)and(3.6),we have

Multiplying(3.31)by πγ(t)and integrating it over[t1,t]gives

for t≥t1.Note that the absolute value of the integrand of the first integral in the above inequality can be estimated by Young’s inequality as follows:

Thus,(3.32)follows from the above inequality that

By(3.29),πγ(t)u(t)→-∞ as t→∞,which contradicts the fact that πγ(t)u(t)≥-L-1. This completes the proof.

Remark 3.2 Theorem 3.3 gives an oscillatory criterion for equation(1.1).But Y.Sahiner[4]and S.H.Saker[1]didn’t consider the oscillation under the condition R∞t0r-1/γ(s)Δs＜∞,and L.Erbe et al.[3]gave no oscillatory criteria under this condition.So our results complement their results.

References

[1]S.H.Saker,Oscillation of second-order nonlinear neutral delay dynamic equations on time scales,Journal of Computational and Applied Mathematics,187(2006),123-141.

[2]S.H.Saker,Oscillation of nonlinear dynamic equations on time scales,Appl.Math.Comput., 148(2004),81-91.

[3]L.Erbe,A.Peterson,S.H.Saker,Oscillation criteria for second-order nonlinear delay dynamic equations,J.Math.Anal.Appl.,330(2007),1317-1337.

[4]Y.Sahiner,Oscillation of second-order delay differential equations on time scales,Nonlinear Analysis,63(2005),1073-1080.

[5]B.G.Zhang,Z.S.Lang,Oscillation of second-order nonlinear delay dynamic equations on time scales,Computers and Mathematics with Applications,49(2005),599-609.

[6]H.J.Li,Oscillation criteria for second order linear differential equations,J.Math.Anal.Appl., 194(1995),312-321.

[7]R.P.Agarwal,M.Bohner,S.H.Saker,Oscillation of second order delay dynamic equations, Can.Appl.Math.Q,13(2005),1-18.

(edited by Liangwei Huang)

?This project was supported by the Youth Foundation of Anqing Teachers College(KJ201107), the General Foundation of the Education Department of Anhui Province(AQKJ2014B010).

?Manuscript August 22,2014;Revised March 31,2015