Oscillation Criteria for Second-order Neutral Dynamic Equations with Mixed Nonlinearities on Time Scales?

1 Introduction

In this paper,we are concerned with oscillation of the second order neutral dynamic equations with mixed nonlinearities

on an above-unbounded time scalesT,wherez(t)=x(t)+p(t)x(τ(t)).Throughout the paper,we shall impose the following conditions:

(i)r∈Crd(T,R+),whereR+=(0,∞),andp∈Crd(T,[0,1)),ψ(x)∈Crd(T,R+)forx?=0,and there exists a positive constantLsuch thatψ(x)≤L?1forx?=0;

(ii)fi(t,y):T×R→Ris continuous function such thatyfi(t,y)>0 for ally?=0,and there existsqi∈Crd(T,R+)(i=0,1,···,n),quotients of positive integersβisuch that

and

(iii)δi∈Crd(T,T),δi(t)≤t,i=0,1,···,n,τ∈Crd(T,R+),τ(t)≤t,=∞.

We will restrict our attention only to the solutions of(1)which exist on some[T0,∞)Tand satisfy sup{|x(t)|:t∈[T0,∞)T}>0 for anyT0∈[t0,∞)T.A solution of(1),means a nontrivial real-valued functionxwhich has the propertyand satisfies(1)on the intervalt∈[t0,∞)T.About the existence and uniqueness of solutions to dynamic equations,we refer the reader to[1].As usual,a solution of(1)is called nonoscillatory if it is either eventually positive or eventually negative,and otherwise it is called oscillatory.Equation(1)is said to be oscillatory if all its solutions are oscillatory.The reader can refer to[2]for necessary time scale definitions and notation used throughout this paper.

Recently,there has been an increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of various second order neutral dynamic equations.For example,([3–12])and the references cited therein.Wuet al[13]in 2006 studied the second order nonlinear neutral dynamic equation of the form

on time scalesT,whereα≥1 is a quotient of odd positive integers,andr,q∈Crd(T,R+).

Their results were further extended by Sunet al[14]in 2010 to the second order quasiliner neutral dynamic equation

whereγ,α,βare quotients of odd positive integers with 0<α<γ<β.In 2011,Yang and Xu[15]concerned with the oscillation of the second order quasilinear neutral delay denamic equation

on an arbitrary time scaleT,wherez(t)=x(t)+p(t)x(τ(t)),α,β>0 are constants.Their results were further extended by Zhang and Wang[16]in 2012 to the equation of the form

where

Most of papers in the references cited therein have a commonly used assumption that is ∫

although several authors studied the oscillation of(1)and its particular cases where

In particular,whenT=R,Ye and Xu[17]obtained several oscillation theorems for the particular cases of(1)under the assumptions that

However,as pointed out in[7],under the case

Lemma 2.3 to Lemma 2.5 in[17]have some mistakes.Therefore,the main results of Xu[17]are not completed.

On the basis of the ideas exploited by[7],we derive some new oscillation results for(1)under both the cases

Our results improve and extend the oscillation criteria established in[3–16]and complement and correct some known results in[17].Two examples are provided to illustrate the relevance of new theorems.

2 Some lemmas

For convenience,we use the notations

We begin with the following lemmas.

Lemmas 2.1[18]IfXandYare nonnegative,then forλ>1,

where the equality holds if and only ifX=Y.

Lemmas 2.2[4]For any givenn-tuple{β1,β2,···,βn}satisfying

there is ann-tuple{η1,η2,···,ηn}such that

Ifn=2 andm=1,it turns out that

The proof is similar to that of[15].

Lemmas 2.3Assume that equation(1)has a positive solutionxon[t0,∞)T,and then for sufficiently larget,

3 The case π(t0)= ∞

First we begin with the oscillatory property of equation(1)under the condition

Theorem 3.1Let(6)holds.If there exists a functionandn-tuple{η1,η2,···,ηn}satisfying Lemma 2.2 such that for allT1,T2∈[t0,∞)TwithT2>T1,

wherethen equation(1)is oscillatory.

ProofLetxbe a nonoscillatory solution of(1).Without loss of generality,we assume that there exists at1∈[t0,∞)Tsuch that fort∈[t1,∞)T,

and then

In view of(1),(8)and(i)–(iii),we can conclude that fort∈[t1,∞)T,

and this implies thatr(t)ψ(x(t))|z?(t)|α?1z?(t)is a decreasing function fort≥t1.Hence,we can have either

Now,we claim that(11)is true.If not,then

Noting thatr(t)ψ(x(t))|z?(t)|α?1z?(t)is decreasing,we have

so that

Integrating the above equation fromt2tot,we then obtain

which impliesz(t)→?∞,and this is a contradiction toz(t)>0.Thus(11)holds.Then there exists at3≥t1such thatz?(t)>0.This implies thatz(t)is strictly increasing on[t3,∞)T.Base on this result and(iii),we conclude fort∈[t3,∞)T,

and then fori=0,1,···,n,t∈[t3,∞)T,we have

Noting that(11)andz?(t)>0 hold,we have fort∈[t3,∞)T,

so that fort∈[t3,∞)T,

Consider the Riccati substitution

and then we haveW(t)>0 fort∈[t3,∞)T.By product and quotient rules(see[2]),in view of(1),(i)–(iii),(15),(17)and Lemma 2.3,we can conclude that fort∈[t3,∞)T,

whereσis the forward jump operator on time scaleT.

In view of the arithmetic-geometric mean inequality,see[18],

whereη1,η2,···,ηnare chosen to satisfy Lemma 2.2.Now returning to(18)and substituting

into(18),we obtain

Employing the P¨atzsche chain rule[2]and using the fact thatz(t)is strictly increasing ont∈[t3,∞)T,we have fort∈[t3,∞)T,

Noting thatz(t)is increasing on[t3,∞)T,we getz(t)≤zσ(t)fort∈[t3,∞)T,and then

Sinceσ(t)≥tonT,fromz?(t)>0 and noting thatr(t)ψ(x(t))(z?(t))αis decreasing on[t3,∞)T,one concludes that fort∈[t3,∞)T,

From(19),(21)and(22),it follows that

In view of(16),we get

Integrating both sides of(23)fromT>t3tot≥Tleads to

Lettingt→∞in the above inequality,the result obtained contradicts(7).This completes the proof.

Choosingφ(t)=1 in Theorem 3.1,we can obtain the following convenient oscillation criterion.

Corollary 3.1Let(6)hold.If there exists ann-tuple{η1,η2,···,ηn}satisfying Lemma 2.2 such that

then equation(1)is oscillatory.

According to Theorem 3.1,by further applying Young’s inequality we have the following theorem.

Theorem 3.2Let(6)hold.If there exists a functionandn-tuple{η1,η2,···,ηn}satisfying Lemma 2.2 such that for allT1,T2∈[t0,∞)TwithT>t0,

then equation(1)is oscillatory.

ProofLetxbe a nonoscillatory solution of(1).Without loss of generality,we assume thatxis eventually positive.Then there is at3≥t0such that(8)–(11)and(15)hold.Define the functionWas in(17),proceeding as in the proof of Theorem 3.1,we see that(18)–(22)hold.From(17)and(19),one concludes that fort∈[t3,∞)T,

In view of(21)and(22),we have fort∈[t3,∞)T,

Substituting(27)into(26),we get fort∈[t3,∞)T,

Taking

andλ=then by Lemma 2.1 and(28)we obtain fort∈[t3,∞)T,

Integrating both sides of the above inequality fromT>t3tot≥Tleads to

Lettingt→∞in the last inequality,the result so obtained contradicts(25).This completes the proof.

Similarly,using the equalitywe have the following theorem.

Theorem 3.3Let(6)hold.If there exists a functionandn-tuple{η1,η2,···,ηn}satisfying Lemma 2.2 such that for allT1,T2∈[t0,∞)TwithT2>T1,

for someα≥1,then equation(1)is oscillatory.

Remark 3.1Letn=1,ψ(x(t))=1,Theorem 3.2 and Theorem 3.3 become respectively[16].

4 The case π(t0)< ∞

Next,we will consider the oscillatory problem of equation(1)under the condition

Theorem 4.1Let(30)hold.Assume that

If there exists ann-tuple(η1,η2,···,ηn)satisfying Lemma 2.2 such that

hold,and then every solutionxof equation(1)oscillates or

ProofSuppose to the contrary thatxis an eventually positive solution of(1).In view of(1),we know thatr(t)ψ(x(t))|z?(t)|α?1z?(t)is a decreasing function fort≥T.Consequently,it is easy to conclude that there exist two possible cases of the sign ofz?(t),namely

Case 1z?(t)>0 fort≥T,and then we are back to the case of Theorem 3.2 by choosingφ(t)=πα(t).Thus the proof of Theorem 3.2 goes through,and we may get a contradiction by(32).Hencez?(t)>0 fort≥T,do not hold.

Case 2z?(t)<0 fort≥T1.Define

Noting

thenr(t)ψ(x(t))|z?(t)|α?1z?(t)is decreasing,so we get

Dividing both sides of(35)byand integrating fromttol,we then have

Sincez?(t)<0 and(ii),we have

Lettingl→∞in the above inequality,we obtain

that is

Hence by(34),we get

Next in view of(34),we obtain

In view of(20)and note thatz?(t)<0,fort≥T1,we have

and

In view of(38)andW(σ(t))≤W(t)<0,we have

Noting thatp?(t)≥0 andτ?(t)≥0,we see thatx?(t)≤0,t≥T1,so by

we get

and obtain the condition(41)which plays a key role in the subsequence.

Recall the arithmetic-geometric mean inequality[18]

whereη1,η2,···,ηnare chosen to satisfy Lemma 2.2.Now returning to(40)and substituting

into(40)and noting that(41),we have

and we have

multiplying(43)byπα(t)and integrating it fromT1totimplies

According to the Young inequality

we have

which implies that

Therefore,it follows from(44)that

By(33),we obtainπα(t)W(t)→?∞,t→∞,which contradicts the fact thatThis completes the proof.

Remark 4.1WhenT=R,Theorem 4.1 complement and correct the result in[17].

Next,we will give the following oscillation criteria which is dif f erent from Theorem 4.1.

Theorem 4.2Let(30)and(31)hold.Assume that there exists ann-tuple(η1,η2,···,ηn)satisfying Lemma 2.2 such that(32)for allT1≥t0.If

then every solutionxof equation(1)oscillates or

ProofSuppose to the contrary thatxis an eventually positive solution of(1).In view of(1),we know thatr(t)ψ(x(t))|z?(t)|α?1z?(t)is a decreasing function fort≥T.Consequently,it is easy to conclude that there exist two possible cases of the sign ofz?(t),namely

Case 1z?(t)>0 fort≥T.Then we are back to the case of Theorem 3.2,and we can get a contradiction by(32).Hence,z?(t)>0,fort≥T,do not hold.

Case 2z?(t)<0 fort≥T1.By

we get(35),and integrating(35)fromttoland lettingl→∞,we then have

Hence it holds that

where

From(1),we obtain

On the other hand,similar to the proof of Theorem 4.1,noting that(41),(46)and(47),we have

Integrating the above inequality fromT1tot,we have

Dividing both sides of(48)byand integrating fromT1tot,noting thatψ(x(t))≤L?1,we obtain

which contradicts(45).This completes the proof.

5 Applications

In this section,we provide two examples to illustrate our main results.

Example 5.1Consider the equation

where

and takingm=1,we find that(i)–(iii)are satisfied.By Lemma 2.2,we chooseAccording to the direct computation,we have

Hence,(6)and(25)are satisfied.By Corollary 3.1,equation(49)is oscillatory.

Example 5.2Consider the following equation

where

Noting

and takingm=2,we find that(i)–(iii)are satisfied.By Lemma 2.2,we chooseandL=1.According to the direct computation,we have

and

Hence,(32)and(33)are satisfied.By Theorem 4.1,then every solutionxof equation(50)oscillates or

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