# Oscillation Criteria for Second-Order Nonlinear Dynamic Equations on Time Scales

Authors: Shao-Yan Zhang; Qi-Ru Wang
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

This paper is concerned with oscillation of second-order nonlinear dynamic equations of the form(r(t)((y(t)+p(t)y(τ(t)))Δ)γ)Δ+f1(t,y(δ1(t)))+f2(t,y(δ2(t)))=0 on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria.


---

## Body

## 1. Introduction

The theory of time scales was introduced by Stefan Hilger 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 theories of differential equations and difference equations but also extend these classical cases to cases “in between", for example, to the so-calledq-difference equations. A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ with the topology and ordering inherited form ℝ, and the cases when this time scale is equal to ℝ or to the integers ℤ represent the classical theories of differential and difference equations. Of course many other interesting time scales exist, and they give rise to plenty of applications. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales, we refer the reader to [1–14].In 2006, Wu et al. [1] considered the second-order nonlinear neutral dynamic equation with variable delays
(1.1)(r(t)((y(t)+p(t)y(τ(t)))Δ)γ)Δ+f(t,y(δ(t)))=0,t∈𝕋,
where γ≥1 is a quotient of odd positive integers. In 2007, Saker et al. [2] also discussed (1.1) for an odd positive integer γ≥1. In 2010, Zhang and Wang [3] extended and complemented some results in [1, 2] for γ≥1 and gave some new results for 0<γ<1. In 2011, Saker [4] considered (1.1) in different conditions. In 2010, Sun et al. [5] considered the second-order quasiliner neutral delay dynamic equation
(1.2)(r(t)((y(t)+p(t)y(τ(t)))Δ)γ)Δ+q1xα(τ1(t))+q2xβ(τ2(t))=0,t∈𝕋,
where γ, α, and β are quotients of odd positive integers with 0<α<γ<β.In this paper, we study the second-order nonlinear dynamic equation(1.3)(r(t)((y(t)+p(t)y(τ(t)))Δ)γ)Δ+f1(t,y(δ1(t)))+f2(t,y(δ2(t)))=0,
on a time scale 𝕋, where p∈Crd(𝕋,[0,1)), fi∈C(𝕋×ℝ,ℝ),i=1,2,γ>0 is a quotient of odd positive integers.The paper is organized as follows. In the next section, we give some preliminaries and lemmas. In Section3, we will use the Riccati transformation technique to prove our main results. In Section 4, we present two examples to illustrate our results.


## 2. Preliminaries and Lemmas

For convenience, we recall some concepts related to time scales. More details can be found in [6].Definition 2.1.
Let𝕋 be a time scale, for t∈𝕋 the forward jump operator is defined by σ(t):=inf{s∈𝕋:s>t}, the backward jump operator by ρ(t):=sup{s∈𝕋:s<t}, and the graininess function by μ(t):=σ(t)-t, where inf∅:=sup𝕋 and sup∅:=inf𝕋. If σ(t)>t, t is said to be right-scattered, otherwise, it is right-dense. If ρ(t)<t, t is said to be left-scattered, otherwise, it is left-dense. The set 𝕋κ is defined as follows. If 𝕋 has a left-scattered maximum m, then 𝕋κ=𝕋-{m}, otherwise, 𝕋κ=𝕋.Definition 2.2.
For a functionf:𝕋→ℝ and t∈𝕋κ, one defines the delta-derivative fΔ(t) of f(t) to be the number (provided it exists) with the property that given any ε>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩𝕋 for some δ) such that
(2.1)|[f(σ(t))-f(s)]-fΔ(t)[σ(t)-s]|≤ε|σ(t)-s|,∀s∈U.
We say that f is delta-differentiable (or in short, differentiable) on 𝕋κ provided fΔ(t) exists, for all t∈𝕋κ.It is easily seen that iff is continuous at t∈𝕋 and t is right-scattered, then f is differentiable at t with
(2.2)fΔ(t)=f(σ(t))-f(t)μ(t).
Moreover, if t is right-dense then f is differential at t if the limit
(2.3)lims→tf(t)-f(s)t-s
exists as a finite number. In this case
(2.4)fΔ(t)=lims→tf(t)-f(s)t-s.
In addition, if fΔ≥0, then f is nondecreasing. A useful formula is
(2.5)fσ(t)=f(t)+μ(t)fΔ(t),wherefσ(t):=f(σ(t)).
We will make use of the following product and quotient rules for the derivative of the product fg and the quotient f/g (where ggσ≠0) of two differentiable functions f and g:
(2.6)(fg)Δ=fΔg+fσgΔ=fgΔ+fΔgσ,(fg)Δ=fΔg-fgΔggσ.Definition 2.3.
Letf:𝕋→ℝ be a function, f is called right-dense continuous (rd-continuous) if it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋. A function F:𝕋→ℝ is called an antiderivative of f provided FΔ(t)=f(t) holds for all t∈𝕋k. By the antiderivative, the Cauchy integral of f is defined as ∫ab‍f(s)Δs=F(b)-F(a), and ∫a∞‍f(s)Δs=limt→∞∫at‍f(s)Δs.
LetCrd(𝕋,ℝ) denote the set of all rd-continuous functions mapping 𝕋 to ℝ. It is shown in [6] that every rd-continuous function has an antiderivative. An integration by parts formula is
(2.7)∫ab‍f(t)gΔ(t)Δt=[f(t)g(t)]|ab-∫ab‍fΔ(t)gσ(t)Δt.
In (1.3), we assume that 𝕋 is a time scale and(h1)
τ(t),δi(t)∈Crd(𝕋,𝕋), limt→∞τ(t)=∞, τ(t)≤t, limt→∞δi(t)=∞, and t≤δi(t), i=1,2,(h2)
r(t)∈Crd(𝕋,ℝ+), ∫∞(1/r(t))1/γΔt=∞,  p(t)∈Crd(𝕋,[0,1)), where ℝ+=(0,∞),(h3)
fi(t,u):𝕋×ℝ→ℝ is continuous function such that ufi(t,u)>0 for all u≠0, there exist qi(t)∈Crd(𝕋,ℝ+) (i=1,2), quotients of odd positive integers α and β such that |uf1(t,u)|≥q1(t)|u|α+1, |uf2(t,u)|≥q2(t)|u|β+1, and 0<α<γ<β.Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume throughout that the time scale𝕋 under consideration satisfies inf𝕋=t0 and sup𝕋=∞. For T∈𝕋, let [T,∞)𝕋:={t∈𝕋:t≥T}. Throughout this paper, these assumptions will be supposed to hold. Let τ*(t)=min{τ(t),δ1(t),δ2(t)}, T0=min{τ*(t):t≥t0} and τ-1*(t)=sup{s≥t0:τ*(s)≤t}. Clearly τ-1*(t)≥t for t≥T0, τ-1*(t) is nondecreasing and coincides with the inverse of τ*(t) when the latter exists.By a solution of (1.3), we mean a nontrivial real-valued function y(t) which has the properties [y(t)+p(t)y(τ(t))]∈Crd1[τ-1*(t0),∞) and r(t)([y(t)+p(t)y(τ(t))])Δ)γ∈Crd1[τ-1*(t0),∞). Our attention is restricted to those solutions of (1.3) that exist on some half line [ty,∞) and satisfy sup{|y(t)|:t≥t1}>0 for any t1≥ty. A solution y(t) of (1.3) 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.For convenience, we use the notationx(σ(t))=xσ(t), x(δi(t))=xδi(t)(i=1,2) and xΔ(σ(t))=(xΔ(t))σ, and set
(2.8)x(t):=y(t)+p(t)y(τ(t)).
Then (1.3) becomes
(2.9)(r(t)(xΔ(t))γ)Δ+f1(t,y(δ1(t)))+f2(t,y(δ2(t)))=0.
Now, we give the first lemma. Set
(2.10)RT(t)=∫Tt‍Δs(r(s))1/γ.Lemma 2.4.
Let conditions(h1)–(h3) hold. If y(t) is an eventually positive solution of (1.3), then there exists T∈𝕋 sufficiently large such that x(t)>0,xΔ(t)>0, (r(t)(xΔ(t))γ)Δ<0, x(t)>RT(t)r1/γ(t)xΔ(t), and (xδi(t)/xσ(t))>(RT(t)r1/γ(t))/(RT(t)r1/γ(t)+μ(t)) (i=1,2) for t∈[T,∞)𝕋.Proof.
Ify(t) is an eventually positive solution of (1.3), then by (h1) there exists a T∈[t0,∞)𝕋 such that
(2.11)y(t)>0,y(τ(t))>0,y(δi(t))>0,fort≥T,i=1,2.
From (2.8), (1.3), and (h2), we see that x(t)≥y(t). Also by (1.3) and (h3), we have
(2.12)(r(t)(xΔ(t))γ)Δ≤-q1(t)yα(δ1(t))-q2(t)yβ(δ2(t))<0,fort≥T,
which implies that r(t)(xΔ(t))γ is decreasing on [T,∞)𝕋.
We claim thatr(t)(xΔ(t))γ>0 on [T,∞)𝕋. Assume not, there is a t1∈[T,∞)𝕋 such that r(t1)(xΔ(t1))γ<0. Since r(t)(xΔ(t))γ≤r(t1)(xΔ(t1))γ for t≥t1, we have
(2.13)xΔ(t)≤(r(t1))1/γxΔ(t1)(1r(t))1/γ.
Integrating the inequality above form t1 to t(≥t1), by (h2) we get
(2.14)x(t)≤x(t1)+(r(t1))1/γxΔ(t1)∫t1t‍(1r(s))1/γΔs⟶-∞(t⟶∞),
and this contradicts the fact that x(t)>0, for all t≥T. Thus we have r(t)(xΔ(t))γ>0 on [T,∞)𝕋 and so xΔ(t)>0 on [T,∞)𝕋.
Note that(2.15)x(t)>x(t)-x(T)=∫Tt‍xΔ(s)=∫Tt‍(r(s)(xΔ(s))γ)1/γr1/γ(s)Δs>(r(t)(xΔ(t))γ)1/γ∫Tt‍1r1/γ(s)Δs=RT(t)r1/γ(t)xΔ(t),
we have
(2.16)xσ(t)x(t)=x(t)+μ(t)xΔ(t)x(t)=1+μ(t)xΔ(t)x(t)<1+μ(t)1RT(t)r1/γ(t)=RT(t)r1/γ(t)+μ(t)RT(t)r1/γ(t).
Since δi(t)≥t and xΔ(t)>0, we get
(2.17)xδi(t)xσ(t)=xδi(t)x(t)·x(t)xσ(t)≥x(t)xσ(t)>RT(t)r1/γ(t)RT(t)r1/γ(t)+μ(t),i=1,2.
The proof is complete.Remark 2.5.
Byx(t)≥y(t) on [T,∞)𝕋, xΔ>0, (1.3), (2.8) and (h1)–(h3), we get
(2.18)0≥(r(t)(xΔ(t))γ)Δ+q1(t)[y(δ1(t))]α+q2(t)[y(δ2(t))]β=(r(t)(xΔ(t))γ)Δ+q1(t)[x(δ1(t))-p(δ1(t))y(τ(δ1(t)))]α+q2(t)[x(δ2(t))-p(δ2(t))y(τ(δ2(t)))]β≥(r(t)(xΔ(t))γ)Δ+q1(t)[x(δ1(t))-p(δ1(t))x(τ(δ1(t)))]α+q2(t)[x(δ2(t))-p(δ2(t))x(τ(δ2(t)))]β≥(r(t)(xΔ(t))γ)Δ+q1(t)[1-p(δ1(t))]α(x(δ1(t)))α+q2(t)[1-p(δ2(t))]β(x(δ2(t)))β.Lemma 2.6 (see [3]).
Letg(u)=Bu-Au(γ+1)/γ, where A>0 and B are constants, γ is a quotient of odd positive integers. Then g attains its maximum value on ℝ at u*=(Bγ/(A(γ+1)))γ, and
(2.19)maxu∈ℝg=g(u*)=γγ(γ+1)γ+1Bγ+1Aγ.Lemma 2.7 (see [11]).
x and z are delta-differentiable on 𝕋. For x≠0 and any t∈𝕋, one has
(2.20)xΔ(t)(z2(t)x(t))Δ=(zΔ(t))2-x(t)xσ(t)[(z(t)x(t))Δ]2.


## 3. Main Results

In this section, by employing the Riccati transformation technique we will establish oscillation criteria for (1.3) in two cases: γ≥1 and 0<γ<1. Set
(3.1)Q(s)=(q1(s)(1-p(δ1(s)))α)(β-γ)/(β-α)·(q2(s)(1-p(δ2(s)))β)(γ-α)/(β-α),Q1(s)=(RT(s)r1/γ(s)RT(s)r1/γ(s)+μ(s))γz(s)Q(s),Q2(s)=(RT(s)r1/γ(s)RT(s)r1/γ(s)+μ(s))γ(z2(s))σQ(s),C(t,s)=HsΔ(t,s)+H(t,s)zΔ(s)z(σ(s)),A(s)=zΔ(s)z(σ(s))+HsΔ(t,s)H(t,s),(I)γ≥1.Theorem 3.1.
Assume that(h1)–(h3) hold and γ≥1. Furthermore, assume that there exists a positive rd-continuous Δ-differentiable function z(t) such that for all sufficiently large T∈𝕋,
(3.2)limsupt→∞∫Tt‍[Q1(s)-1(γ+1)γ+1r(s)(zΔ(s))γ+1zγ(s)]Δs=∞,
then (1.3) is oscillatory.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive (note that in the case when y(t) is eventually negative, the proof is similar, since the substitution Y(t)=-y(t) transforms (1.3) into the same form). Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0, and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8). Define the function w(t) by the Riccati substitution
(3.3)w(t):=z(t)r(t)(xΔ(t))γxγ(t),fort≥T,
then w(t)>0 and
(3.4)wΔ(t)=(r(t)(xΔ(t))γ)Δ(z(t)xγ(t))+(r(t)(xΔ(t))γ)σ(z(t)xγ(t))Δ=(r(t)(xΔ(t))γ)Δ(z(t)xγ(t))+(r(t)(xΔ(t))γ)σ[zΔ(t)xγ(t)-z(t)(xγ(t))Δxγ(t)(xσ(t))γ].
By (1.3), xΔ(t)>0, and (2.18), we obtain
(3.5)(r(t)(xΔ(t))γ)Δxγ(t)≤-q1(t)[1-p(δ1(t))]α(xδ1(t))α(xσ(t))γ-q2(t)[1-p(δ2(t))]β(xδ2(t))β(xσ(t))γ.
Noting that 0<α<γ<β, we have
(3.6)β-γβ-α<1,γ-αβ-α<1.
By Young’s inequality
(3.7)aχb1-χ≤χa+(1-χ)b,0<χ<1,
with
(3.8)χ=β-γβ-α,a=q1(t)[1-p(δ1(t))]α(xδ1(t))α(xσ(t))γ,b=q2(t)[1-p(δ2(t))]β(xδ2(t))β(xσ(t))γ,
we have
(3.9)q1(t)[1-p(δ1(t))]α(xδ1(t))α(xσ(t))γ+q2(t)[1-p(δ2(t))]β(xδ2(t))β(xσ(t))γ≥β-γβ-αq1(t)[1-p(δ1(t))]α(xδ1(t))α(xσ(t))γ+γ-αβ-αq2(t)[1-p(δ2(t))]β(xδ2(t))β(xσ(t))γ≥(q1(t)[1-p(δ1(t))]α(xδ1(t))α(xσ(t))γ)(β-γ)/(β-α)·(q2(t)[1-p(δ2(t))]β(xδ2(t))β(xσ(t))γ)(γ-α)/(β-α)=(q1(t)[1-p(δ1(t))]α)(β-γ)/(β-α)·(q2(t)[1-p(δ2(t))]β)(γ-α)/(β-α)·(xδ1(t))(αβ-αγ)/(β-α)·(xδ2(t))(βγ-βα)/(β-α)(xσ(t))γ.
By γ=((αβ-αγ)/(β-α))+((βγ-βα)/(β-α)) and Lemma 2.4, we get
(3.10)q1(t)[1-p(δ1(t))]α(xδ1(t))α(xσ(t))γ+q2(t)[1-p(δ2(t))]β(xδ2(t))β(xσ(t))γ≥(q1(t)[1-p(δ1(t))]α)(β-γ)/(β-α)·(q2(t)[1-p(δ2(t))]β)(γ-α)/(β-α)·(xδ1(t)xσ(t))(αβ-αγ)/(β-α)·(xδ2(t)xσ(t))(βγ-βα)/(β-α)>(RT(t)r1/γ(t)RT(t)r1/γ(t)+μ(t))γQ(t).
In view of xΔ(t)>0 and (3.5)–(3.10), for all t≥T, we obtain
(3.11)wΔ(t)<-z(t)(RT(t)r1/γ(t)RT(t)r1/γ(t)+μ(t))γQ(t)+wσ(t)zΔ(t)zσ(t)-(r(t)(xΔ(t))γ)σz(t)(xγ(t))Δxγ(t)(xσ(t))γ=-Q1(t)+wσ(t)zΔ(t)zσ(t)-(r(t)(xΔ(t))γ)σz(t)(xγ(t))Δxγ(t)(xσ(t))γ.
Using γ≥1, Lemma 2.4 and the Keller’s chain rule, we get
(3.12)(xγ(t))Δ=γ[∫01‍(x(t)+hμ(t)xΔ(t))γ-1dh]xΔ(t)=γxΔ(t)∫01‍((1-h)x(t)+hxσ(t))γ-1dh≥γxΔ(t)∫01‍((1-h)x(t)+hx(t))γ-1dh=γxγ-1(t)xΔ(t).
Also from Lemma 2.4 and σ(t)≥t, we have
(3.13)r(t)(xΔ(t))γ≥r(σ(t))(xΔ(σ(t)))γ.
By (3.11)–(3.13), we get
(3.14)wΔ(t)<-Q1(t)+wσ(t)zΔ(t)zσ(t)-(r(t)(xΔ(t))γ)σz(t)γxγ-1(t)xΔ(t)xγ(t)(xσ(t))γ≤-Q1(t)+wσ(t)zΔ(t)zσ(t)-z(t)γr1/γ(t)(rσ(t))(γ+1)/γ(xΔ(σ(t)))γ+1(xσ(t))γ+1=-Q1(t)+wσ(t)zΔ(t)zσ(t)-z(t)γr1/γ(t)(zσ(t))(γ+1)/γ(wσ(t))(γ+1)/γ.
Setting
(3.15)B=zΔ(t)zσ(t),A=z(t)γr1/γ(t)(zσ(t))(γ+1)/γ,u=wσ(t),
then by Lemma 2.6, from (3.14) we obtain that for all t≥T,
(3.16)wΔ(t)<-Q1(t)+1(γ+1)γ+1r(t)(zΔ(t))γ+1zγ(t).
Integrating the above inequality from T to t(≥T), we get
(3.17)∫Tt‍[Q1(s)-1(γ+1)γ+1r(s)(zΔ(s))γ+1zγ(s)]Δs<w(T)-w(t)<w(T).
Taking lim sup on both sides of the above inequality as t→∞, we obtain a contradiction to condition (3.2). The proof is complete.The following theorem gives new oscillation criteria for (1.3) which can be considered as the extension of Philos-type oscillation criterion. Define D={(t,s)∈𝕋2:t≥s≥0} and
(3.18)ℋ*={H(t,s)∈C1(D,ℝ+):H(t,t)=0,H(t,s)>0,HsΔ(t,s)≥0,fort>s≥0}.Theorem 3.2.
Assume that(h1)–(h3) hold and γ≥1. Furthermore, assume that there exist a positive rd-continuous Δ-differentiable function z(t) and a function H∈ℋ* such that for all sufficiently large T∈𝕋,
(3.19)limsupt→∞1H(t,T)∫Tt‍[H(t,s)Q1(s)-Cγ+1(t,s)r(s)(z(σ(s)))γ+1Hγ(t,s)(γ+1)γ+1zγ(s)]Δs=∞,
then (1.3) is oscillatory.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive. Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0, and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8). Define w(t) as in (3.3). Proceeding as in the proof of Theorem 3.1, we can get (3.14). From (3.14), for function H∈ℋ* and all t≥T we have
(3.20)∫Tt‍H(t,s)Q1(s)Δs<-∫Tt‍H(t,s)wΔ(s)Δs+∫Tt‍H(t,s)wσ(s)zΔ(s)zσ(s)Δs-∫Tt‍H(t,s)z(s)γr1/γ(s)(zσ(s))(γ+1)/γ(wσ(s))(γ+1)/γΔs.
Using the integration by parts formula (2.7), we obtain
(3.21)-∫Tt‍H(t,s)wΔ(s)Δs=-H(t,s)w(s)|Tt+∫Tt‍HsΔ(t,s)wσ(s)Δs=H(t,T)w(T)+∫Tt‍HsΔ(t,s)wσ(s)Δs.
It follows that
(3.22)∫Tt‍H(t,s)Q1(s)Δs<H(t,T)w(T)+∫Tt‍[HsΔ(t,s)+H(t,s)zΔ(s)zσ(s)]wσ(s)Δs-∫Tt‍H(t,s)z(s)γr1/γ(s)(zσ(s))(γ+1)/γ(wσ(s))(γ+1)/γΔs=H(t,T)w(T)+∫Tt‍C(t,s)wσ(s)Δs-∫Tt‍H(t,s)z(s)γr1/γ(s)(zσ(s))(γ+1)/γ(wσ(s))(γ+1)/γΔs.
Setting
(3.23)B=C(t,s),A=H(t,s)z(s)γr1/γ(s)(zσ(s))(γ+1)/γ,u=wσ(s),
by Lemma 2.6 we obtain that for all t≥T,
(3.24)∫Tt‍H(t,s)Q1(s)Δs<H(t,T)w(T)+∫Tt‍[C(t,s)]γ+1(zσ(s))γ+1r(s)Hγ(t,s)(γ+1)γ+1zγ(s)Δs.
That is,
(3.25)1H(t,T)∫Tt‍[H(t,s)Q1(s)-[C(t,s)]γ+1r(s)(zσ(s))γ+1Hγ(t,s)(γ+1)γ+1zγ(s)]Δs<w(T).
Taking lim sup on both sides of the above inequality as t→∞, we obtain a contradiction to condition (3.19). The proof is complete.Theorem 3.3.
Assume that(h1)–(h3) hold and γ≥1. Then (1.3) is oscillatory if for all sufficiently large T∈𝕋,
(3.26)limsupt→∞∫Tt‍Q(s)RTγ(s)Δs=∞.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive. Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0, and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8). Set ϕ(t)=r1/γ(t)xΔ(t). By Lemma 2.4, we get ϕ>0,(ϕγ)Δ<0. Using γ≥1 and the keller’s chain rule, we get
(3.27)(ϕγ(t))Δ=γ[∫01‍(ϕ(t)+hμ(t)ϕΔ(t))γ-1dh]ϕΔ(t)=γ[∫01‍((1-h)ϕ(t)+hϕσ(t))γ-1dh]ϕΔ(t)<0.
So we have ϕΔ(t)<0 and there is a constant L>0 such that ϕ(t)≤L for t≥T. Then (1.3) becomes (ϕγ)Δ(t)+f1(t,y(δ1(t)))+f2(t,y(δ2(t)))=0. By (2.18), we have
(3.28)0≥(ϕγ(t))Δ(ϕσ(t))γ+q1(t)(1-p(δ1(t)))α(xδ1(t))α(ϕσ(t))γ+q2(t)(1-p(δ2(t)))β(xδ2(t))β(ϕσ(t))γ.
Similar to the proof of (3.10), we get
(3.29)0≥(ϕγ(t))Δ(ϕσ(t))γ+Q(t)(xδ1(t)ϕσ(t))(αβ-αγ)/(β-α)·(xδ2(t)ϕσ(t))(βγ-βα)/(β-α).
Using the Keller’s chain rule and ϕΔ(t)<0, we get ϕσ≤ϕ and
(3.30)(ϕγ(t))Δ≥γ(ϕσ(t))γ-1ϕΔ(t).
From δ1,2(t)≥t and xΔ(t)>0, it follows that
(3.31)0≥γϕΔ(t)ϕσ(t)+Q(t)(xδ1(t)ϕσ(t))(αβ-αγ)/(β-α)·(xδ2(t)ϕσ(t))(βγ-βα)/(β-α)≥γϕΔ(t)L+Q(t)(x(t)ϕσ(t))γ≥γϕΔ(t)L+Q(t)(x(t)ϕ(t))γ.
By Lemma 2.4, we get
(3.32)x(t)ϕ(t)=x(t)r1/γ(t)xΔ(t)>RT(t).
It follows that
(3.33)0>γϕΔ(t)L+Q(t)RTγ(t).
Integrating the above inequality from T to t(≥T), we obtain
(3.34)∫Tt‍Q(s)RTγ(s)Δs<-γL∫Tt‍ϕΔ(s)Δs=γL(ϕ(T)-ϕ(t))<γLϕ(T).
Taking lim sup on both sides of the above inequality as t→∞, we obtain a contradiction to condition (3.26). The proof is complete.Theorem 3.4.
Assume that(h1)–(h3) hold and γ≥1. Furthermore, assume that there exists a positive rd-continuous Δ-differentiable function z(t) such that for all sufficiently large T∈𝕋,
(3.35)limsupt→∞∫Tt‍[Q1(s)-r1/γ(s)(zΔ(s))24γ(RT(s))γ-1z(s)]Δs=∞,
then (1.3) is oscillatory.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive. Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0, and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8). Define w(t) as in (3.3). By (2.6), we obtain
(3.36)wΔ(t)=zΔ(t)[r(t)(xΔ(t))γxγ(t)]σ+z(t)[r(t)(xΔ(t))γxγ(t)]Δ=zΔ(t)zσ(t)wσ(t)+z(t)[(r(t)(xΔ(t))γ)Δxγ(t)-r(t)(xΔ(t))γ(xγ(t))Δxγ(t)(xσ(t))γ]=zΔ(t)zσ(t)wσ(t)+z(t)[(r(t)(xΔ(t))γ)Δ(xσ(t))γ-rσ(t)((xΔ(t))σ)γ(xγ(t))Δxγ(t)(xσ(t))γ].
By Lemma 2.4, σ(t)≥t, and (3.5)–(3.13), for all t≥T we obtain
(3.37)wΔ(t)<-Q1(t)+wσ(t)zΔ(t)zσ(t)-z(t)rσ(t)(xΔ(σ(t)))γγxγ-1(t)xΔ(t)xγ(t)(xσ(t))γ=-Q1(t)+wσ(t)zΔ(t)zσ(t)-γz(t)rσ(t)[(xΔ(σ(t)))γ(xσ(t))γ]2(xσ(t))γxΔ(t)x(t)(xΔ(σ(t)))γ<-Q1(t)+wσ(t)zΔ(t)zσ(t)-γz(t)(wσ(t)zσ(t))2xγ-1(t)(xΔ(t))γ-1r(t)<-Q1(t)+wσ(t)zΔ(t)zσ(t)-γz(t)1r(t)(wσ(t)zσ(t))2(RT(t)r1/γ(t))γ-1.
It follows that
(3.38)wΔ(t)<-Q1(t)+zΔ(t)zσ(t)wσ(t)-γz(t)(RT(t))γ-1r1/γ(t)(zσ(t))2(wσ(t))2.
By completing the square, we have
(3.39)wΔ(t)<-Q1(t)+r1/γ(t)(zΔ(t))24γ(RT(t))γ-1z(t).
Integrating the above inequality from T to t(≥T), we get
(3.40)∫Tt‍[Q1(s)-r1/γ(s)(zΔ(s))24γ(RT(s))γ-1z(s)]Δs<w(T)-w(t)<w(T).
Taking lim sup on both sides of the above inequality as t→∞, we obtain a contradiction to condition (3.35). The proof is complete.Theorem 3.5.
Assume that(h1)–(h3) and γ≥1 hold. Furthermore, assume that there exist a positive rd-continuous Δ-differentiable function z(t) and a function H∈ℋ* such that for all sufficiently large T∈𝕋,
(3.41)limsupt→∞1H(t,T)∫Tt‍[H(t,s)Q1(s)-H(t,s)A2(s)r1/γ(s)(zσ)2(s)4γ(RT(s))γ-1z(s)]Δs=∞,
then (1.3) is oscillatory.Proof.
By (3.39), the proof is similar to Theorems 3.2 and 3.4, so we omit it.Theorem 3.6.
Assume that(h1)–(h3) hold and γ≥1. Furthermore, assume that there exists a rd-continuous Δ-differentiable function z(t) such that for all sufficiently large T∈𝕋,
(3.42)limsupt→∞∫Tt‍[Q2(s)-1γ(zΔ(s))2r1/γ(s)(RT(s))1-γ]Δs=∞,
then (1.3) is oscillatory.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive. Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0, and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8). Define the function v(t) by the Riccati substitution
(3.43)v(t):=z2(t)r(t)(xΔ(t))γxγ(t),fort≥T,
then v(t)>0. From (3.5) and (3.10), it follows that for all t≥T(3.44)vΔ(t)=[r(t)(xΔ(t))γ]Δ(z2(t)xγ(t))σ+r(t)(xΔ(t))γ(z2(t)xγ(t))Δ≤-Q2(t)+r(t)(xΔ(t))γ(xγ(t))Δ(xγ(t))Δ(z2(t)xγ(t))Δ.
By (3.12) and Lemma 2.7, we obtain
(3.45)vΔ(t)≤-Q2(t)+r(t)(xΔ(t))γ(xγ(t))Δ[(zΔ(t))2-xγ(t)(xσ(t))γ((z(t)xγ(t))Δ)2]<-Q2(t)+r(t)(xΔ(t))γ(zΔ(t))2(xγ(t))Δ<-Q2(t)+r(t)(xΔ(t))γ(zΔ(t))2γxγ-1(t)xΔ(t).
It follows from Lemma 2.4 that
(3.46)vΔ(t)<-Q2(t)+1γ(zΔ(t))2r1/γ(t)(RT(t))1-γ.
Integrating the above inequality from T to t(≥T), we get
(3.47)∫Tt‍[Q2(s)-1γ(zΔ(s))2r1/γ(s)(RT(s))1-γ]Δs<v(T)-v(t)<v(T).
Taking lim sup on both sides of the above inequality as t→∞, we obtain a contradiction to condition (3.42). The proof is complete.From Theorem3.6, we can establish different sufficient conditions for the oscillation of (1.3) by using different choices of z(t). For instance, if z(t)=1 or z(t)=t, we have the following results.Corollary 3.7.
Assume that(h1)–(h3) hold and γ≥1. Then (1.3) is oscillatory if for all sufficiently large T∈𝕋,
(3.48)limsupt→∞∫Tt‍(RT(s)r1/γ(s)RT(s)r1/γ(s)+μ(s))γQ(s)Δs=∞.Corollary 3.8.
Assume that(h1)–(h3) hold and γ≥1. Then (1.3) is oscillatory if for all sufficiently large T∈𝕋,
(3.49)limsupt→∞∫Tt‍[Q3(s)-(1s+σ(s))2r1/γ(s)1γ(RT(s))1-γ]Δs=∞,
where Q3(s)=(RT(s)r1/γ(s)/(RT(s)r1/γ(s)+μ(s)))γσ(s)Q(s).
(
II
)
0
<
γ
<
1.
Theorem3.1′. Assume that  (h1)–(h3) hold and  0<γ<1. Furthermore, assume that there exists a positive  rd-continuous  Δ-differentiable function  z(t) such that for all sufficiently large T∈𝕋,
(3.50)limsupt→∞∫Tt‍[Q1(s)-1(γ+1)γ+1r(s)(zΔ(s))γ+1zγ(s)]Δs=∞,
then (1.3) is oscillatory.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive. Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0 and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8). Define w(t) as in (3.3). Proceeding as in the proof of Theorem 3.1, we get
(3.51)wΔ(t)=(r(t)(xΔ(t))γ)Δ(z(t)xγ(t))+(r(t)(xΔ(t))γ)σ[zΔ(t)xγ(t)-z(t)(xγ(t))Δxγ(t)(xσ(t))γ].
Using 0<γ<1, Lemma 2.4 and the Keller’s chain rule, we get
(3.52)(xγ(t))Δ=γ[∫01‍(x(t)+hμ(t)xΔ(t))γ-1dh]xΔ(t)≥γxΔ(t)∫01‍((1-h)xσ(t)+hxσ(t))γ-1dh=γ(xσ(t))γ-1xΔ(t).
By (3.10), (3.51), and (3.52), we get
(3.53)wΔ(t)<-Q1(t)+wσ(t)zΔ(t)zσ(t)-(r(t)(xΔ(t))γ)σz(t)γ(xσ(t))γ-1xΔ(t)xγ(t)(xσ(t))γ.
Since
(3.54)-(r(t)(xΔ(t))γ)σz(t)γ(xσ(t))γ-1xΔ(t)xγ(t)(xσ(t))γ=-(rσ(t))(γ+1)/γ((xΔ(t))σ)γ+1z(t)γxΔ(t)xγ(t)xσ(t)(rσ(t))1/γ(xΔ(t))σ(For(3.13))≤-(rσ(t))(γ+1)/γ((xΔ(t))σ)γ+1z(t)γxΔ(t)xγ(t)xσ(t)r1/γ(t)xΔ(t)<-z(t)γr1/γ(t)(zσ(t))(γ+1)/γ(wσ(t))(γ+1)/γ,
it follows that
(3.55)wΔ(t)<-Q1(t)+wσ(t)zΔ(t)zσ(t)-z(t)γr(1/γ)(t)(zσ(t))(γ+1)/γ(wσ(t))(γ+1)/γ.
It is easy to see (3.55) is of the same form as (3.14). The following is similar to the proof of Theorem 3.1 and hence omitted.Forγ∈(0,1), Theorem 3.2 also holds. Its proof is similar to those of Theorems  3.1′ and 3.2.Theorem3.2′. Assume that(h1)–(h3)hold and0<γ<1. Furthermore, assume that there exist a positiverd-continuousΔ-differentiable functionz(t)and a functionH∈ℋ*such that for all sufficiently largeT∈𝕋,
(3.56)limsupt→∞1H(t,T)∫Tt‍[H(t,s)Q1(s)-Cγ+1(t,s)r(s)(z(σ(s)))γ+1Hγ(t,s)(γ+1)γ+1zγ(s)]Δs=∞,then (1.3) is oscillatory.Theorem3.3′. Assume that(h1)–(h3)hold and0<γ<1. Then (1.3) is oscillatory if for all sufficiently largeT∈𝕋,(3.57)limsupt→∞∫Tt‍Q(s)RTγ(s)Δs=∞.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive. Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0, and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8), ϕ is defined as in Theorem 3.3. Similar to the proof of Theorem 3.3, we get
(3.58)0≥(ϕγ(t))Δϕγ(t)+Q(t)(xδ1(t)ϕ(t))(αβ-αγ)/(β-α)·(xδ2(t)ϕ(t))(βγ-βα)/(β-α).
Using the Keller’s chain rule, 0<γ<1, and ϕΔ(t)<0, we get
(3.59)(ϕγ(t))Δ=γ[∫01‍(ϕ(t)+hμ(t)ϕΔ(t))γ-1dh]ϕΔ(t)≥γ[∫01‍((1-h)ϕ(t)+hϕ(t))γ-1dh]ϕΔ(t)=γϕγ-1(t)ϕΔ(t).
From δ1,2(t)≥t and xΔ(t)>0, it follows that
(3.60)0≥γϕΔ(t)L+Q(t)(x(t)ϕ(t))γ.
It is easy to see (3.31) is of the same form as (3.60). The following is similar to the proof of Theorem 3.3 and hence omitted.Last, we give a theorem which holds for allγ>0, a quotient of add positive integers.Theorem 3.9.
Assume that(h1)–(h3) hold. Furthermore, assume that there exists a positive rd-continuous Δ-differentiable function z(t) such that for all sufficiently large T∈𝕋,
(3.61)limsupt→∞∫Tt‍[Q1(s)-zΔRTγ(s)]Δs=∞,
then (1.3) is oscillatory.Proof.
Suppose to the contrary thaty(t) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that y(t) is eventually positive. Then, by (h1)–(h3) there exists T≥t0 sufficiently large such that y(t)>0, y(τ(t))>0, y(δ1,2(t))>0, and Lemma 2.4 holds for t≥T, where x(t) is defined by (2.8). Define w(t) as in (3.3). Then, w(t)>0 and because (1/xγ)Δ=-((xγ)Δ/xγ(xγ)σ)<0 we get
(3.62)wΔ(t)=[z(t)·r(t)(xΔ(t))γ]Δ1xγ(t)+[z(t)·r(t)(xΔ(t))γ]σ(1xγ(t))Δ<[zΔ(t)·(r(t)(xΔ(t))γ)σ+z(t)(r(t)(xΔ(t))γ)Δ]1xγ(t).
From (3.5), (3.10), and (3.13), we obtain
(3.63)wΔ(t)<[zΔ(t)·r(t)(xΔ(t))γ+z(t)(r(t)(xΔ(t))γ)Δ]1xγ(t)<zΔ(t)r(t)(xΔ(t)x(t))γ-Q1(t)<zΔ(t)RTγ(t)-Q1(t).
Integrating the above inequality from T to t(≥T), we get
(3.64)∫Tt‍[Q1(s)-zΔ(s)RTγ(s)]Δs<w(T)-w(t)<w(T).
Taking lim sup on both sides of the above inequality as t→∞, we obtain a contradiction to condition (3.61). The proof is complete.


## 4. Examples

In this section, we give two examples to illustrate our main results. To obtain the conditions for oscillation we will use the following facts:(4.1)∫1∞‍Δssγ=∞if0≤γ≤1,∫1∞‍Δssγ<∞,ifγ>1.
We first give an example to show Theorems 3.1 and  3.1′.Example 4.1.
Consider the equation(4.2)(1(t+σ(t))γ((y(t)+p(t)y(τ(t)))Δ)γ)Δ+(σ(t))2γ(1-p(δ1(t)))αt2γ+1yα(δ1(t))+(σ(t))2γ(1-p(δ2(t)))βt2γ+1yβ(δ2(t))=0,t∈𝕋,
where 𝕋=[1,∞) is a time scale, p(t) satisfies (h2), τ(t) and δ1,2(t) satisfy (h1), r(t)=1/(t+σ(t))γ, and γ is a quotient of odd positive integers.
We chooseq1(t)=(σ(t))2γ/((1-p(δ1(t)))αt2γ+1), q2(t)=(σ(t))2γ/((1-p(δ2(t)))βt2γ+1), and z=1, then zΔ=0, ∫‍(s+σ(s))Δs=t2+c, and ∫1∞(Δt/r1/γ(t))‍=∞. For any sufficiently large T∈𝕋 and s>T, there exists a constant k>0 sufficiently large such that
(4.3)(RT(s)r1/γ(s)RT(s)r1/γ(s)+μ(s))γ=((s2-T2)(1/(s+σ(s)))(s2-T2)(1/(s+σ(s)))+σ(s)-s)γ>(s2kσ2(s))γ,limsupt→∞∫Tt‍[Q1(s)-1(γ+1)γ+1r(s)(zΔ(s))γ+1zγ(s)]Δs≥limsupt→∞k-γ∫Tt‍1sΔs=∞.
Hence, by Theorems 3.1 and  3.1′, (4.2) is oscillatory.The second example illustrates Corollary3.7.Example 4.2.
Consider the equation(4.4)(y(t)+y(τ(t))δ-1(t))ΔΔ+σ(t)t2y1/3(δ(t))+σ(t)t2y5/3(δ(t))=0,t∈𝕋,
where 𝕋=[1,∞) is a time scale, γ=1, α=1/3, β=5/3, δ(t), and τ(t) satisfy (h1), δ(t) has an inverse function δ-1(t), and p(t)=1/δ-1(t) satisfies (h2).
We chooseq1(t)=q2(t)=σ(t)/t2 and z=1. For any sufficiently large T∈𝕋 and s>T, there exists a constant k>0 sufficiently large such that
(4.5)RT(s)r1/γ(s)RT(s)r1/γ(s)+μ(s)=s-Ts-T+σ(s)-s>skσ(s),limsupt→∞∫Tt‍(RT(s)r1/γ(s)RT(s)r1/γ(s)+μ(s))γQ(s)Δs≥limsupt→∞∫Tt‍1ks(1-1s)2Δs=limsupt→∞1k∫Tt‍[1s-2s2+1s3]Δs=∞.
Hence, by Corollary 3.7, (4.4) is oscillatory.



---

*Source: 743469-2012-11-26.xml*