# Oscillation Theorems for Second-Order Half-Linear Advanced Dynamic Equations on Time Scales

Authors: Shuhong Tang; Tongxing Li; Ethiraju Thandapani
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation(r(t)(xΔ(t))γ)Δ+p(t)xγ(g(t))=0 on an arbitrary time scale 𝕋 with sup 𝕋=∞, where g(t)≥t and ∫to∞(Δs/(1r/γ(s)))<∞. Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify the oscillation of the second-order half-linear advanced differential equation and the second-order half-linear advanced difference equation. Three examples are included to illustrate the main results.


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## Body

## 1. Introduction

The study of dynamic equations on time scales, which has recently received a lot of attention, was introduced by Hilger [1] in order to unify continuous and discrete analysis. Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and the references cited therein. For an excellent introduction to the calculus on time scales; see Bohner and Peterson [3]. Further information on working with dynamic equations on time scales can be found in [4].The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus, that is, when𝕋=ℝ; 𝕋=ℕ and 𝕋=qℕ0={qt:t∈ℕ0}, where q>1. Many other interesting time scales exist, and they give rise to many applications; see [3]. Dynamic equations on a time scale have an enormous potential for applications such as in population dynamics. For example, it can model insect populations that are continuous while in season, die out in, say, winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population; see [3].The theory of oscillations is an important branch of the applied theory of dynamic equations related to the study of oscillatory phenomena in technology and natural and social sciences. In recent years, there has been much research activity concerning the oscillation of solutions of various dynamic equations on time scales, we refer the reader to the papers [5–18] and the references therein.We are concerned with the oscillation of the second-order half-linear advanced dynamic equation(1.1)(r(t)(xΔ(t))γ)Δ+p(t)xγ(g(t))=0,
on a time scale 𝕋 unbounded above, where γ>0 is the quotient of odd positive integers, r and p are real-valued rd-continuous positive functions defined on 𝕋, g∈Crd(𝕋,𝕋), g(t)≥t.Since we are interested in oscillatory behavior, we assume throughout this paper that the given time scale𝕋 is unbounded above. We define the time scale interval of the form [t0,∞)𝕋 by [t0,∞)𝕋∶=[t0,∞)∩𝕋.By a solution of (1.1), we mean a nontrivial real-valued function x which has the properties x(t)∈Crd1([t0,∞)𝕋,ℝ),r(t)(xΔ(t))γ∈Crd1([t0,∞)𝕋,ℝ) and satisfying (1.1) for all t∈[t0,∞)𝕋. We consider only those solutions x of (1.1) which satisfy sup{|x(t)|:t∈[T,∞)𝕋}>0 for all T∈[t0,∞)𝕋. We assume that (1.1) possesses such a solution. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [t0,∞)𝕋; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.We note that if𝕋=ℝ, (1.1) becomes the second-order advanced differential equation (1.2)(r(t)(x′(t))γ)′+p(t)xγ(g(t))=0.
A special case of (1.2) is (1.3)(r(t)x′(t))′+p(t)x(t)=0.
For the oscillation of (1.3); see [19–21]. Willett [19] gave a new version of Leighton’s criterion and obtained the following oscillation criteria: if (1.4)∫t0∞1r(t)dt<∞,∫t0∞p(t)(∫t∞dsr(s))2dt=∞,
then every solution of (1.3) is oscillatory. Later, Li [20] obtained that (1.5)(t2x′(t))′+λx(t)=0
is oscillatory when λ>1/4.The discrete analog of (1.3) is the second-order advanced difference equation (1.6)Δ(r(n)Δx(n))+q(n)x(n)=0.
Budinčević [22] proved that if (1.7)∑n=k0∞1r(n)<∞,∑n=k0∞1r(n)∑k=k0nq(k)=∞,
then every solution x of (1.6) is either oscillatory or else x(n)→0 as n→∞.Regarding the oscillation of (1.1), Agarwal et al. [5], Grace et al. [7], Saker [8], and Hassan [9] studied (1.1) when τ(t)=t, that is, (1.8)(r(t)(xΔ(t))γ)Δ+p(t)xγ(t)=0,
and established some oscillation criteria for the case when (1.9)∫t0∞Δtr1/γ(t)=∞.
Furthermore, the authors obtained some sufficient conditions which guarantee that every solution x of (1.8) oscillates or limt→∞x(t)=0 under the case when (1.10)∫t0∞Δtr1/γ(t)<∞.
Saker [14] studied the oscillation of the dynamic equation (1.11)(r(t)xΔ(t))Δ+p(t)x(σ(t))=0
for the cases when (1.9) and (1.10) hold with γ=1. Very recently, Hassan [15] has investigated the oscillation of (1.1) under the conditions (1.12)∫t0∞Δtr1/γ(t)=∞or∫t0∞[1r(t)∫t0tp(s)(∫g(s)∞Δur1/γ(u))γΔs]1/γΔt=∞.
Now a problem is how to determine the oscillatory behavior of (1.1) when (1.13)∫t0∞[1r(t)∫t0∞p(s)(∫g(s)∞Δur1/γ(u))γΔs]1/γΔt<∞.In this paper, we will establish some new oscillation criteria for (1.1) under the case when (1.10) holds. Our results can be applied when (1.13) holds. The paper is organized as follows. In Section 2, we present some basic definitions and useful results from the theory of calculus on time scales. In Section 3, we shall establish several new oscillation criteria for (1.1).Remark 1.1.
All functional inequalities considered in this note are assumed to hold eventually, that is, they are satisfied for allt large enough.


## 2. Preliminary Results

A time scale𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above, that is, it is a time scale interval of the form [t0,∞)𝕋. On any time scale, we define the forward and backward jump operators by (2.1)σ(t)∶=inf⁡{s∈T∣s>t},ρ(t)∶=sup⁡{s∈T∣s<t}.A pointt∈𝕋 is said to be left dense if ρ(t)=t, right dense if σ(t)=t, left scattered if ρ(t)<t, and right scattered if σ(t)>t. The graininess μ of the time scale is defined by μ(t)∶=σ(t)-t.For a functionf:𝕋→ℝ (the range ℝ of f may actually be replaced with any Banach space), the (delta) derivative is defined by (2.2)fΔ(t)=f(σ(t))-f(t)σ(t)-t
if f is continuous at t and t is right scattered. If t is not right scattered, then the derivative is defined by (2.3)fΔ(t)=lim⁡s→t+f(σ(t))-f(s)t-s=lim⁡s→t+f(t)-f(s)t-s,
provided that this limit exists.A functionf:𝕋→ℝ is said to be rd continuous if it is continuous at each right-dense point and if there exists a finite left limit in all left-dense points. The set of rd-continuous functions f:𝕋→ℝ is denoted by Crd(𝕋,ℝ). f is said to be differentiable if its derivative exists. The set of functions f:𝕋→ℝ that are differentiable and whose derivative is rd-continuous function is denoted by Crd1(𝕋,ℝ).The derivative and the shift operatorσ are related by the formula (2.4)fσ(t)=f(σ(t))=f(t)+μ(t)fΔ(t).Letf be a real-valued function defined on an interval [a,b]. We say that f is increasing, decreasing, nondecreasing, and nonincreasing on [a,b] if t1,t2∈[a,b] and t2>t1 imply 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.We will make use of the following product and quotient rules for the derivative of the productfg and the quotient f/g (where g(t)g(σ(t))≠0) of two differentiable functions f and g(2.5)(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)),(fg)Δ(t)=fΔ(t)g(t)-f(t)gΔ(t)g(t)g(σ(t)).Fora,b∈𝕋, and a differentiable function f, the Cauchy integral of fΔ is defined by (2.6)∫abfΔ(t)Δt=f(b)-f(a).
The integration by parts formula reads (2.7)∫abfΔ(t)g(t)Δt=f(b)g(b)-f(a)g(a)-∫abfσ(t)gΔ(t)Δt,
and infinite integrals are defined as (2.8)∫a∞f(s)Δs=lim⁡t→∞∫atf(s)Δs.


## 3. Main Results

In this section, by employing the Riccati transformation technique, we establish several oscillation criteria for (1.1). To prove the main theorems, we will use the following formula: (3.1)(xγ(t))Δ=γ∫01[hxσ(t)+(1-h)x(t)]γ-1dhxΔ(t),
which is a simple consequence of Keller’s chain rule [3, Theorem 1.90].Set(3.2)d+(t)∶=max⁡{0,d(t)},R(t)∶=∫t∞Δsr1/γ(s),
and we assume that there exists a positive real-valued Δ-differentiable function m such that (3.3)m(t)r1/γ(t)R(t)+mΔ(t)≤0.In order to prove the main results conveniently, we give the following known result.Theorem A (see [12, Theorem 2.1]).
Assume that (1.9) holds. Further, assume that there exists a positive real-valued Δ-differentiable function δ such that
(3.4)limsup⁡t→∞∫Tt[δ(s)p(s)-r(s)((δΔ(s))+)γ+1(γ+1)γ+1δγ(s)]Δs=∞
holds for all sufficiently large T. Then every solution of (1.1) is oscillatory.Remark 3.1.
From the proof of [12, Theorem 2.1], if we let x be an eventually positive solution of (1.1), then xΔ(t)>0 due to condition (1.9). Hence, we can get a contradiction to (3.4) when xΔ(t)>0 occurs.Theorem 3.2.
Assume that (1.10) holds, g(t)≥σ(t) and γ≤1. Furthermore, assume that there exists a positive real-valued Δ-differentiable function δ such that (3.4) holds for all sufficiently large T. If
(3.5)limsup⁡t→∞∫Tt[p(s)(m(g(s))m(σ(s)))γRγσ(s)-(γγ+1)γ+11Rσ(s)r1/γ(s)]Δs=∞,
then (1.1) is oscillatory.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we assume that x(t)>0 and x(g(t))>0 for t∈[t0,∞)𝕋. In view of (1.1), we obtain
(3.6)(r(t)(xΔ(t))γ)Δ=-p(t)xγ(g(t))<0,t∈[t0,∞)T.
Hence, r(t)(xΔ(t))γ is an eventually strictly decreasing function, and there exists a t1∈[t0,∞)𝕋 such that xΔ(t)>0,t∈[t1,∞)𝕋 or xΔ(t)<0,t∈[t1,∞)𝕋.
Case 1.
Assume thatxΔ(t)>0,t∈[t1,∞)𝕋. From Theorem A, we can obtain a contradiction to (3.4).
Case 2.
Assume thatxΔ(t)<0,t∈[t1,∞)𝕋. Define the function ω by
(3.7)ω(t)=r(t)(xΔ(t))γxγ(t).
Then, ω(t)<0 for t∈[t1,∞)𝕋. By (3.6), we get
(3.8)xΔ(s)≤r1/γ(t)r1/γ(s)xΔ(t),s∈[t,∞)T.
Integrating it from t to l, we have
(3.9)x(l)≤x(t)+r1/γ(t)xΔ(t)∫tlΔsr1/γ(s),l∈[t,∞)T.
Taking l→∞ in the last inequality, we get
(3.10)x(t)+r1/γ(t)xΔ(t)R(t)≥0,t∈[t1,∞)T.
Thus, we obtain
(3.11)r1/γ(t)R(t)xΔ(t)x(t)≥-1.
By (3.7) and (3.11), we have (3.12)-1≤Rγ(t)ω(t)≤0.
On the other hand, it follows from (3.11) that
(3.13)xΔ(t)x(t)≥-1r1/γ(t)R(t).
Then, we have
(3.14)(x(t)m(t))Δ=xΔ(t)m(t)-x(t)mΔ(t)m(t)mσ(t)≥-x(t)m(t)mσ(t)[m(t)r1/γ(t)R(t)+mΔ(t)]≥0.
Thus, x(t)/m(t) is nondecreasing. Hence we obtain
(3.15)x(g(t))x(σ(t))≥m(g(t))m(σ(t)),sinceg(t)≥σ(t).Δ-differentiating (3.7) and using (3.6), we obtain
(3.16)ωΔ(t)≤-p(t)(m(g(t))m(σ(t)))γ-r(t)(xΔ(t))γ(xγ(t))Δxγ(t)xγ(σ(t)).
In view of Keller’s chain rule [3, Theorem 1.90], we see that (3.17)(xγ(t))Δ≤γxγ-1(t)xΔ(t),sinceγ≤1.
Thus, (3.16) yields
(3.18)ωΔ(t)≤-p(t)(m(g(t))m(σ(t)))γ-γr(t)(xΔ(t))γ+1x(t)xγ(σ(t)).
On the other hand, from xΔ(t)<0, we have x(t)≥xσ(t) and (3.19)-γr(t)(xΔ(t))γ+1x(t)xγ(σ(t))≤-γ(1r(t))1/γω(γ+1)/γ(t).
Hence by (3.18), we have
(3.20)ωΔ(t)+p(t)(m(g(t))m(σ(t)))γ+γr-1/γ(t)ω(γ+1)/γ(t)≤0,t∈[t1,∞)T.
Multiplying (3.20) by Rγσ(t), we obtain(3.21)Rγσ(t)ωΔ(t)+p(t)(m(g(t))m(σ(t)))γRγσ(t)+γRγσ(t)r-1/γ(t)ω(γ+1)/γ(t)≤0,t∈[t1,∞)T.
Integrating it from t1 to t, we get
(3.22)∫t1tRγσ(s)ωΔ(s)Δs+∫t1tp(s)(m(g(s))m(σ(s)))γRγσ(s)Δs+γ∫t1tRγσ(s)r-1/γ(s)ω(γ+1)/γ(s)Δs≤0.
Integrating by parts, we have (3.23)∫t1tRγσ(s)ωΔ(s)Δs=Rγ(t)ω(t)-Rγ(t1)ω(t1)-∫t1t(Rγ(s))Δω(s)Δs.
From Keller’s chain rule [3, Theorem 1.90], we obtain (3.24)(Rγ(t))Δ=γ∫01[hRσ(t)+(1-h)R(t)]γ-1dhRΔ(t).
Note that RΔ(t)=-(1/r(t))1/γ<0, we get (3.25)-∫t1t(Rγ(s))Δω(s)Δs≥γ∫t1t(1r(s))1/γ(Rσ(s))γ-1ω(s)Δs
due to (3.24) and γ≤1. By (3.22), (3.23), and (3.25), we see that
(3.26)Rγ(t)ω(t)-Rγ(t1)ω(t1)+∫t1tp(s)(m(g(s))m(σ(s)))γRγσ(s)Δs+γ∫t1t(1r(s))1/γ(Rσ(s))γ-1ω(s)Δs+γ∫t1tRγσ(s)r-1/γ(s)ω(γ+1)/γ(s)Δs≤0.
Set p∶=(γ+1)/γ,q∶=γ+1, (3.27)A∶=-(γ+1)γ/(γ+1)(Rγσ(t)r1/γ(t))γ/(γ+1)ω(t),B∶=γγ+1(γ+1)1/(γ+1)(1r1/γ(t))1/(γ+1)1(Rσ(t))1/(γ+1).
Then, using the inequality
(3.28)App+Bqq≥AB,1p+1q=1,
we have
(3.29)γRγσ(t)r-1/γ(t)ω(γ+1)/γ(t)+(γγ+1)γ+11Rσ(t)r1/γ(t)≥-γ(1r(t))1/γ(Rσ(t))γ-1ω(t).
Thus, by (3.26) and (3.29), we get (3.30)Rγ(t)ω(t)-Rγ(t1)ω(t1)+∫t1t[p(s)(m(g(s))m(σ(s)))γRγσ(s)-(γγ+1)γ+11Rσ(s)r1/γ(s)]Δs≤0.
which contradicts (3.5) due to (3.12). The proof is complete.Ifg(t)=t, similar as in the proof of Theorem 3.2, we obtain the following result.Theorem 3.3.
Assume that (1.10) holds, g(t)=t and γ≤1. Furthermore, assume that there exists a positive real-valued Δ-differentiable function δ such that (3.4) holds for all sufficiently large T. If
(3.31)limsup⁡t→∞∫Tt[p(s)Rγσ(s)-(γγ+1)γ+11Rσ(s)r1/γ(s)]Δs=∞,
then (1.1) is oscillatory.Theorem 3.4.
Assume that (1.10) holds, g(t)≥σ(t) and γ≥1.  Suppose further that there exists a positive real-valued Δ-differentiable function δ such that (3.4) holds for all sufficiently large T. If
(3.32)limsup⁡t→∞∫Tt[p(s)(m(g(s))m(σ(s)))γRγσ(s)-(γγ+1)γ+1Rγ2-1(s)(Rσ(s))γ2r1/γ(s)]Δs=∞,
then (1.1) is oscillatory.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x(t)>0 and x(g(t))>0 for t∈[t0,∞)𝕋. In view of (1.1), we obtain (3.6). Therefore, r(t)(xΔ(t))γ is an eventually strictly decreasing function, and so there exists a t1∈[t0,∞)𝕋 such that xΔ(t)>0,t∈[t1,∞)𝕋 or xΔ(t)<0,t∈[t1,∞)𝕋.
Case 1.
Assume thatxΔ(t)>0,t∈[t1,∞)𝕋. From Theorem A, we can obtain a contradiction to (3.4).
Case 2.
Assume thatxΔ(t)<0,t∈[t1,∞)𝕋. Define ω as in (3.7). We have (3.12). Δ-differentiating (3.7) and using (3.6), we obtain (3.16). In view of Keller’s chain rule [3, Theorem 1.90], we see that
(3.33)(xγ(t))Δ≤γ(xσ(t))γ-1xΔ(t),sinceγ≥1.
Thus, we get
(3.34)ωΔ(t)≤-p(t)(m(g(t))m(σ(t)))γ-γr(t)(xΔ(t))γ+1xγ(t)x(σ(t)).
On the other hand, from xΔ(t)<0, we have x(t)≥xσ(t) and (3.35)-γr(t)(xΔ(t))γ+1xγ(t)x(σ(t))≤-γ(1r(t))1/γω(γ+1)/γ(t).
Hence by (3.34), we get (3.20). Then we obtain that (3.22) and (3.23) hold. By Keller’s chain rule [3, Theorem 1.90], we have (3.24). From (3.24), γ≥1 and RΔ(t)=-(1/r(t))1/γ<0, we see that (3.36)-∫t1t(Rγ(s))Δω(s)Δs≥γ∫t1t(1r(s))1/γRγ-1(s)ω(s)Δs.
It follows from (3.22), (3.23), and (3.36) that (3.37)Rγ(t)ω(t)-Rγ(t1)ω(t1)+∫t1tp(s)(m(g(s))m(σ(s)))γRγσ(s)Δs+γ∫t1t(1r(s))1/γRγ-1(s)ω(s)Δs+γ∫t1tRγσ(s)r-1/γ(s)ω(γ+1)/γ(s)Δs≤0.
Set p∶=(γ+1)/γ,q∶=γ+1, (3.38)A∶=-(γ+1)γ/(γ+1)(Rγσ(t)r1/γ(t))γ/(γ+1)ω(t),B∶=γγ+1(γ+1)1/(γ+1)(1r1/γ(t))1/(γ+1)Rγ-1(t)(Rσ(t))γ2/(γ+1).
By the inequality (3.28), we have(3.39)γRγσ(t)r-1/γ(t)ω(γ+1)/γ(t)+(γγ+1)γ+1Rγ2-1(t)(Rσ(t))γ2r1/γ(t)≥-γ(1r(t))1/γRγ-1(t)ω(t).
Thus, (3.37) and (3.39) implies
(3.40)Rγ(t)ω(t)-Rγ(t1)ω(t1)+∫t1t[p(s)(m(g(s))m(σ(s)))γRγσ(s)-(γγ+1)γ+1Rγ2-1(s)(Rσ(s))γ2r1/γ(s)]Δs≤0.
which contradicts (3.32) when using (3.12). This completes the proof.Ifg(t)=t, similar as in the proof of Theorem 3.4, we establish the following result.Theorem 3.5.
Assume that (1.10) holds, g(t)=t and γ≥1. Moreover, assume that there exists a positive real-valued Δ-differentiable function δ such that (3.4) holds for all sufficiently large T. If
(3.41)limsup⁡t→∞∫Tt[p(s)Rγσ(s)-(γγ+1)γ+1Rγ2-1(s)(Rσ(s))γ2r1/γ(s)]Δs=∞,
then (1.1) is oscillatory.Theorem 3.6.
Assume that (1.10) holds, g(t)≥σ(t) and γ>0. Assume further that there exists a positive real-valued Δ-differentiable function δ such that (3.4) holds for all sufficiently large T. If
(3.42)∫Ttp(s)(m(g(s))m(σ(s)))γRγ+1(σ(s))Δs=∞,
then (1.1) is oscillatory.Proof.
Letx be a nonoscillatory solution of (1.1). Without loss of generality, we assume that x(t)>0 and x(g(t))>0 for t∈[t0,∞)𝕋. Similar as in the proof of Theorem 3.2 or Theorem 3.4, we consider two cases.
Case 1.
AssumexΔ(t)>0,t∈[t1,∞)𝕋. By (3.4), this case is not true.
Case 2.
SupposexΔ(t)<0,t∈[t1,∞)𝕋. If γ≤1, proceeding as in the proof of Case 2 of Theorem 3.2, we obtain (3.12) and (3.20). Multiplying (3.20) by Rγ+1(σ(t)), and integrating it from t1 to t, we get
(3.43)∫t1tRγ+1(σ(s))ωΔ(s)Δs+∫t1tp(s)(m(g(s))m(σ(s)))γRγ+1(σ(s))Δs+γ∫t1tRγ+1(σ(s))r-1/γ(s)ω(γ+1)/γ(s)Δs≤0.
Integrating by parts, we see that
(3.44)∫t1tRγ+1(σ(s))ωΔ(s)Δs=Rγ+1(t)ω(t)-Rγ+1(t1)ω(t1)-∫t1t(Rγ+1(s))Δω(s)Δs.
By Keller’s chain rule [3, Theorem 1.90], we obtain
(3.45)(Rγ+1(t))Δ=(γ+1)∫01[hRσ(t)+(1-h)R(t)]γdhRΔ(t).
Note that RΔ(t)=-(1/r(t))1/γ<0, we have
(3.46)-∫t1t(Rγ+1(s))Δω(s)Δs≥(γ+1)∫t1t(1r(s))1/γRγ(s)ω(s)Δs.
Thus, from (3.44), we get
(3.47)Rγ+1(t)ω(t)-Rγ+1(t1)ω(t1)+∫t1tp(s)(m(g(s))m(σ(s)))γRγ+1(σ(s))Δs+(γ+1)∫t1t(1r(s))1/γRγ(s)ω(s)Δs+γ∫t1tRγ+1(σ(s))r-1/γ(s)ω(γ+1)/γ(s)Δs≤0.
It follows from (3.12) that
(3.48)-Rγ+1(t)ω(t)≤R(t)<∞,t⟶∞,-∫t1∞(1r(s))1/γRγ(s)ω(s)Δs≤∫t1∞(1r(s))1/γΔs<∞.
Noting that Rσ(t)/R(t)≤1, we obtain
(3.49)∫t1∞Rγ+1(σ(s))r-1/γ(s)ω(γ+1)/γ(s)Δs=∫t1∞r-1/γ(s)(Rσ(s)R(s))γ+1(Rγ(s)ω(s))(γ+1)/γΔs≤∫t1∞r-1/γ(s)Δs<∞.
Thus, from (3.47), we get
(3.50)∫t1∞p(s)(m(g(s))m(σ(s)))γRγ+1(σ(s))Δs<∞,
which contradicts (3.42).
Whenγ≥1, the proof is similar to that of the case where γ≤1, so we omit the details. The proof is complete.Ifg(t)=t, similar as in the proof of Theorem 3.6, we have the following result.Theorem 3.7.
Assume that (1.10) holds, g(t)=t and γ>0. Furthermore, assume that there exists a positive real-valued Δ-differentiable function δ such that (3.4) holds for all sufficiently large T. If
(3.51)∫T∞p(s)Rγ+1(σ(s))Δs=∞,
then (1.1) is oscillatory.


## 4. Examples

For some applications, we give the following examples.Example 4.1.
Consider the second-order linear dynamic equation(4.1)(tσ(t)xΔ(t))Δ+λσ(t)tx(σ(t))=0,t∈[1,∞)T,
where λ>0 is a constant.Let(4.2)r(t)=tσ(t),p(t)=λσ(t)t,g(t)=σ(t).
Then R(t)=1/t. Set m(t)=R(t)=1/t and δ(t)=1. Using Theorem 3.2, it is easy to see that every solution of (4.1) is oscillatory if λ>1/4. This result extends that of [20]. But recent results on the oscillation of second-order dynamic equations on time scales cannot be applied in (4.1).Example 4.2.
Consider the second-order advanced differential equation(4.3)(t2x′(t))′+p0x(2t)=0,t≥1,
where p0>0 is a constant.Letm(t)=t-1. Applying Theorem 3.2, we see that (4.3) is oscillatory if p0>1/2. However, results of [15] cannot be applied to (4.3), since (4.4)∫1∞[1r(t)∫1tp(s)(∫g(s)∞dur1/γ(u))γds]1/γdt<∞.Example 4.3.
Consider the second-order superlinear advanced dynamic equation(4.5)((tσ(t))γ(xΔ(t))γ)Δ+λgγ(t)txγ(g(t))=0,t∈[1,∞)T,
where γ≥1 is the ratio of odd positive integers, λ>0 is a constant, k1t≤σ(t)≤k2t, and g(t)≥σ(t).Let(4.6)r(t)=(tσ(t))γ,p(t)=λgγ(t)t.
Then R(t)=1/t. Set δ(t)=1. Clearly, (3.4) holds. Put m(t)=R(t)=1/t. Then (4.7)limsup⁡t→∞∫Tt[p(s)(m(g(s))m(σ(s)))γRγσ(s)-(γγ+1)γ+1Rγ2-1(s)(Rσ(s))γ2r1/γ(s)]Δs=limsup⁡t→∞∫Tt[λgγ(s)sσγ(s)gγ(s)1σγ(s)-(γγ+1)γ+1(σ(s)s)γ21σ(s)]Δs≥(λ-(γγ+1)γ+1k2γ2k1)limsup⁡t→∞∫Tt1sΔs=∞if(4.8)λ>(γγ+1)γ+1k2γ2k1.Therefore, by Theorem3.4, every solution of (4.5) is oscillatory when the above inequality holds.


## 5. Summary

This paper is concerned with the oscillatory behavior of advanced equation (1.1). By using the generalized Riccati substitution, we establish some new oscillation criteria for (1.1). On one hand, these criteria can be extended to the equation of the form (5.1)(r(t)(xΔ(t))γ)Δ+p(t)xβ(g(t))=0.
On the other hand, the main results supplement and improve those results of [9] and extend those results of [20]. The established results are easily applicable and are illustrated on three suitable examples.



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