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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 653689, 26 pages
http://dx.doi.org/10.1155/2011/653689
Research Article

Oscillation Criteria for a Class of Second-Order Neutral Delay Dynamic Equations of Emden-Fowler Type

1School of Science, University of Jinan, Jinan, Shandong 250022, China
2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
3Department of Mathematics and Statistics, Missouri University of Science and Technology Rolla, Missouri 65409-0020, USA
4Department of Mathematics, University of Science and Technology, Hefei 230026, China

Received 31 August 2010; Accepted 30 September 2010

Academic Editor: Elena Braverman

Copyright © 2011 Zhenlai Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish some new oscillation criteria for the second-order neutral delay dynamic equations of Emden-Fowler type, [𝑎(𝑡)(𝑥(𝑡)+𝑟(𝑡)𝑥(𝜏(𝑡)))Δ]Δ+𝑝(𝑡)𝑥𝛾(𝛿(𝑡))=0, on a time scale unbounded above. Here 𝛾>0 is a quotient of odd positive integers with a and p being real-valued positive functions defined on 𝕋. Our results in this paper not only extend and improve the results in the literature but also correct an error in one of the references.

1. Introduction

The study of dynamic equations on time scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. It was partly created in order to unify the study of differential and difference equations. Many results concerning differential equations are carried over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies and helps avoid proving results twice|once for differential equations and once again for difference equations.

The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [2]), that is, when 𝕋=,𝕋=, and 𝕋=𝑞0={𝑞𝑡𝑡0}, 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, for example, winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population (see [3]). There are applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. A recent cover story article in New Scientist [4] discusses several possible applications. Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [5] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [3], summarizes and organizes much of time scale calculus; see also the book by Bohner and Peterson [6] for advances results of dynamic equations on time scales.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales unbounded above and neutral differential equations; we refer the reader to the papers [719]. Some authors are especially interested in obtaining sufficient conditions for the oscillation or nonoscillation of solutions of first and second-order linear and nonlinear neutral functional dynamic equations on time scales; we refer to the articles [2028].

Agarwal et al. [7] considered the second-order delay dynamic equations𝑥ΔΔ(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))=0,𝑡𝕋(1.1) and established some sufficient conditions for oscillation of (1.1). Şahiner [11] studied the second-order nonlinear delay dynamic equations 𝑥ΔΔ(𝑡)+𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡𝕋(1.2) and obtained some sufficient conditions for oscillation by employing Riccati transformation technique. Zhang and Zhu [13] examined the second-order dynamic equations𝑥ΔΔ(𝑡)+𝑝(𝑡)𝑓(𝑥(𝑡𝜏))=0,𝑡𝕋,(1.3) and by using comparison theorems, they proved that oscillation of (1.3) is equivalent to the oscillation of the nonlinear dynamic equations 𝑥ΔΔ(𝑡)+𝑝(𝑡)𝑓(𝑥(𝜎(𝑡)))=0,𝑡𝕋(1.4) and established some sufficient conditions for oscillation by applying the results established in [15]. Erbe et al. [16] investigated the oscillation of the second-order nonlinear delay dynamic equations𝑟(𝑡)𝑥Δ(𝑡)Δ+𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡𝕋(1.5) and by employing the generalized Riccati technique, they established some new sufficient conditions which ensure that every solution of (1.5) oscillates or converges to zero. Mathsen et al. [20] investigated the first-order neutral delay dynamic equations [𝑦](𝑡)𝑟(𝑡)𝑦(𝜏(𝑡))Δ+𝑝(𝑡)𝑦(𝛿(𝑡))=0,𝑡𝕋(1.6) and established some new oscillation criteria which as a special case involve some well-known oscillation results for first-order neutral delay differential equations. Zhu and Wang [21] studied the nonoscillatory solutions to neutral dynamic equations[𝑦](𝑡)+𝑝(𝑡)𝑦(𝑔(𝑡))Δ+𝑓(𝑡,𝑥((𝑡)))=0,𝑡𝕋(1.7) and gave a classification scheme for the eventually positive solutions of (1.7). Agarwal et al. [22], Şahíner [23], Saker et al. [2426], Wu et al. [27], and Zhang and Wang [28] considered the second-order nonlinear neutral delay dynamic equations𝑟(𝑡)(𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡)))Δ𝛾Δ+𝑓(𝑡,𝑦(𝛿(𝑡)))=0,𝑡𝕋,(1.8) where 𝛾>0 is a quotient of odd positive integers, the delay function 𝜏 and 𝛿 satisfy 𝜏𝕋𝕋 and 𝛿𝕋𝕋 for all 𝑡𝕋, and 𝑟 and 𝑝 are real-valued positive functions defined on 𝕋, and(1)𝑟(𝑡)>0,𝑡0(1/𝑟(𝑡))1/𝛾Δ𝑡=, and 0𝑝(𝑡)<1;(2)𝑓𝕋× is continuous function such that 𝑢𝑓(𝑢)>0 for all 𝑢0, and there exists a nonnegative function 𝑞 defined on 𝕋 such that |𝑓(𝑡,𝑢)|𝑞(𝑡)|𝑢|𝛾.

By employing different Riccati transformation technique, the authors established some oscillation criteria for all solutions of (1.8).

Recently, some authors have been interested in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of Emden-Fowler type dynamic equations on time scales, differential equations, and difference equations; see, for example, [2947].

Han et al. [32] studied the second-order Emden-Fowler delay dynamic equations𝑥ΔΔ(𝑡)+𝑝(𝑡)𝑥𝛾(𝜏(𝑡))=0,𝑡𝕋(1.9) and established some sufficient conditions for oscillation of (1.9) and extended the results given in [7].

Saker [34] studied the second-order superlinear neutral delay dynamic equation of Emden-Fowler type𝑎(𝑡)(𝑦(𝑡)+𝑟(𝑡)𝑦(𝜏(𝑡)))ΔΔ||||+𝑝(𝑡)𝑦(𝛿(𝑡))𝛾sign𝑦(𝛿(𝑡))=0(1.10) on a time scale 𝕋.

The author assumes that(𝐴1)𝛾>1;(𝐴2) the delay functions 𝜏 and 𝛿 satisfy 𝜏𝕋𝕋,𝛿𝕋𝕋,𝜏(𝑡)𝑡,𝛿(𝑡)𝑡 for all 𝑡𝕋, and lim𝑡𝜏(𝑡)=lim𝑡𝛿(𝑡)=;(𝐴3)𝑎,𝑟 and 𝑝 are positive rd-continuous functions defined on 𝕋 such that 𝑎Δ(𝑡)0,𝑡0(Δ𝑡/𝑎(𝑡))=, and 0𝑟(𝑡)<1.

The main result for the oscillation of (1.10) in [34] is the following.

Theorem 1.1 (see, [34, Theorem 3.1]). Assume that (𝐴1)-(𝐴3) hold. Furthermore, assume that 𝑡0𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝛿𝛾(𝑡)Δ𝑡=,(1.11) and there exists a Δ-differentiable function 𝜂 such that for all constants 𝑀>0,limsup𝑡𝑡𝑡0𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾𝜂𝑎(𝑠)Δ(𝑠)24𝛾𝑀𝛾1𝜂(𝑠)Δ𝑠=.(1.12) Then every solution of (1.10) is oscillatory.

We note that in [34], the author gave an open problem, that is, how to establish oscillation criteria for (1.10) when 𝛾<1.

In [35], the author examined the oscillation of the second-order neutral delay dynamic equations(𝑥(𝑡)𝑟𝑥(𝜏(𝑡)))ΔΔ+𝐻𝑡,𝑥1(𝑡)=0,𝑡𝕋.(1.13)

The author assumes that(𝐻1)𝜏 and 1𝐶𝑟𝑑(𝕋,𝕋),𝜏(𝑡)<𝑡,𝜏(𝑡) as 𝑡,1(𝑡)<𝑡,1(𝑡) as 𝑡, and 0𝑟<1;(𝐻2)𝐻𝐶(𝕋×,) for each 𝑡𝕋 which are nondecreasing in 𝑢, and 𝐻(𝑡,𝑢)>0, for 𝑢>0;(𝐻3)|𝐻(𝑡,𝑢)|𝛼(𝑡)|𝑢|𝜆, where 𝛼(𝑡)0, and 0𝜆=𝑝/𝑞<1 with 𝑝,𝑞 being odd integers.

The main result for the oscillation of (1.13) in [35] is the following.

Theorem 1.2 (see, [35, Theorem 3.4]). Assume that (𝐻1)–(𝐻3) hold. If for all sufficiently large 𝑡1𝑡0,𝑡1𝜏𝛼(𝑠)1(𝑠)𝜆Δ𝑠=,(1.14) then (1.13) oscillates.

We find that the conclusion of this theorem is wrong. The following is a counter example of this theorem.

Counter Example
Consider the second-order differential equation 1𝑥(𝑡)3𝑥𝑡3+1𝑒271/3𝑒1/3𝑒2𝑡/3𝑥1/3(𝑡1)=0,𝑡𝑡0.(1.15)
Let 𝛼(𝑡)=𝑒1/3/27𝑒1/3𝑒2𝑡/3,𝑟(𝑡)=1/3,𝜏(𝑡)=𝑡/3,and1(𝑡)=𝑡1,𝜆=1/3. For all sufficiently large 𝑡1𝑡0, we find that 𝑡1𝜏𝛼(𝑠)1(𝑠)𝜆Δ𝑠=𝑡1𝜏𝛼(𝑠)1(𝑠)𝜆d𝑠=𝑡11𝑒271/3𝑒1/3𝑒2𝑠/3𝑠131/3d𝑠.(1.16) It is easy to see that 𝑡11𝑒271/3𝑠131/3d𝑠=,𝑡1𝑒2𝑠/3𝑠131/3d𝑠𝑡1𝑒2𝑠/3𝑠1/3d𝑠.(1.17) Integrating by parts, we obtain 𝑡1𝑒2𝑠/3𝑠1/3d𝑠=𝑡11/332𝑒2𝑡1/3+12𝑡1𝑒2𝑠/3𝑠2/3d𝑠<.(1.18) Hence 𝑡1𝜏𝛼(𝑠)1(𝑠)𝜆d𝑠=.(1.19) Therefore, by the above theorem, (1.15) is oscillatory. However, 𝑥(𝑡)=e𝑡 is a positive solution of (1.15). Therefore, the above theorem is wrong. Tracing the error to its source, we find that the following false assertion was used in the proof of the aforementioned theorem.

Assertion A
If 𝑥 is an eventually positive solution of (1.13), then 𝑧(𝑡)=𝑥(𝑡)𝑟(𝑡)𝑥(𝜏(𝑡)) is eventually positive.

Abdalla [37] studied the second-order superlinear neutral delay differential equations𝑎(𝑡)(𝑦(𝑡)+𝑟(𝑡)𝑦(𝜏(𝑡)))||||+𝑝(𝑡)𝑦(𝛿(𝑡))𝛾𝑡sign𝑦(𝛿(𝑡))=0,𝑡0,.(1.20) Most of the oscillation criteria are unsatisfactory since additional assumptions have to be imposed on the unknown solutions. Also, the author proved that if 𝑡0d𝑡=𝑎(𝑡)𝑡0𝑝(𝑡)d𝑡=,(1.21) then every solution of (1.20) oscillates for every 𝑟(𝑡)>0, but one can easily see that this result cannot be applied when 𝑝(𝑡)=𝑡𝛼 for 𝛼>1.

Lin [38] considered the second-order nonlinear neutral differential equations[]𝑥(𝑡)𝑝(𝑡)𝑥(𝑡𝜏)+𝑞(𝑡)𝑓(𝑥(𝑡𝜎))=0,𝑡0,(1.22) where 0𝑝(𝑡)1,𝑞(𝑡)0,𝜏,𝜎>0. The author investigated the oscillation for (1.22) when 𝑓 is superlinear.

Wong [46, 47] studied the second-order neutral differential equations[]𝑦(𝑡)𝑝𝑦(𝑡𝜏)+𝑞(𝑡)𝑓(𝑦(𝑡𝜎))=0,𝑡0,(1.23)𝑞𝐶[0,),𝑞(𝑡)0,𝑓𝐶1(,),𝑦𝑓(𝑦)>0 whenever 𝑦0,𝑓(𝑦)0 for all 𝑦, and 0<𝑝<1,𝜏>0,𝜎>0 are constants.

The main results for the oscillation of (1.23) in [46, 47] are the following.

Theorem 1.3 (see, [46, 47]). Suppose that 𝑓 is superlinear. Then a solution of (1.23) is either oscillatory or tends to zero if and only if 𝑡𝑞(𝑡)d𝑡=.(1.24)

Theorem 1.4 (see, [46, 47]). Suppose that 𝑓 is sublinear and in addition satisfies 𝑓(𝑢𝑣)𝑓(𝑢)𝑓(𝑣),𝑢𝑣0.(1.25) Then a solution of (1.23) is either oscillatory or tends to zero if and only if 𝑓(𝑡)𝑞(𝑡)d𝑡=.(1.26)

Li and Saker [40] investigated the second-order sublinear neutral delay difference equationsΔ𝑎𝑛Δ𝑥𝑛+𝑝𝑛𝑥𝑛𝜏+𝑞𝑛𝑥𝛾𝑛𝜎=0,(1.27) where 0<𝛾<1 is a quotient of odd positive integers, 𝑎𝑛>0,Δ𝑎𝑛0,𝑛=01/𝑎𝑛=,0𝑝𝑛<1, for all 𝑛0 and 𝑞𝑛0.

The main result for the oscillation of (1.27) in [40] is the following.

Theorem 1.5 (see, [40, Theorem 2.1]). Assume that there exists a positive sequence {𝜌𝑛} such that for every 𝛼1,limsup𝑛𝑛𝑙=0𝜌𝑙𝑄𝑙𝑎𝑙𝜎(𝛼(𝑙+1𝜎))1𝛾Δ𝜌𝑙24𝛾𝜌𝑙=,(1.28) where 𝑄𝑛=𝑞𝑛(1𝑝𝑛𝜎)𝛾. Then every solution of (1.27) oscillates.

Yildiz and Öcalan [41] studied the higher-order sublinear neutral delay difference equations of the typeΔ𝑚𝑦𝑛+𝑝𝑛𝑦𝑛𝑙+𝑞𝑛𝑦𝛼𝑛𝑘=0,𝑛,(1.29) where 0<𝛼<1 is a ratio of odd positive integers. The authors established some oscillation criteria of (1.29).

The main results for the oscillation of (1.29) when 𝑚=2 in [41] are the following.

Theorem 1.6 (see, [41, Theorem 2.1(𝑎),𝑚=2]). Assume that 0𝑝𝑛<1, and 𝑛=0𝑞𝑛1𝑝𝑛𝑘𝑛𝛼=.(1.30) Then all solutions of (1.29) are oscillatory.

Theorem 1.7 (see, [41, Theorem 2.2,𝑚=2]). Assume that 1<𝑝2𝑝𝑛0, where 𝑝2>0 is a constant, and 𝑛=0𝑞𝑛𝑛𝛼=.(1.31) Then every solution of (1.29) either oscillates or tends to zero as 𝑛.

Cheng [42] considered the oscillation of the second-order nonlinear neutral difference equationsΔ𝑝𝑛Δ𝑥𝑛+𝑐𝑛𝑥𝑛𝜏𝛾+𝑞𝑛𝑥𝛽𝑛𝜎=0(1.32) and established some oscillation criteria of (1.32) by means of Riccati transformation techniques.

Following this trend, in this paper, we are concerned with oscillation of the second-order neutral delay dynamic equations of Emden-Fowler type𝑎(𝑡)(𝑥(𝑡)+𝑟(𝑡)𝑥(𝜏(𝑡)))ΔΔ+𝑝(𝑡)𝑥𝛾(𝛿(𝑡))=0,𝑡𝕋.(1.33)

As we are interested in oscillatory behavior, we assume throughout this paper that the given time scales 𝕋 are unbounded above; that is, it is a time scale interval of the form [𝑡0,) with 𝑡0𝕋.

We assume that 𝛾>0 is a quotient of odd positive integers, the delay functions 𝜏 and 𝛿 satisfy 𝜏𝕋𝕋,𝛿𝕋𝕋,𝜏(𝑡)𝑡,𝛿(𝑡)𝑡 for all 𝑡𝕋, and lim𝑡𝜏(𝑡)=lim𝑡𝛿(𝑡)=;𝑎,𝑟 and 𝑝 are real-valued rd-continuous functions defined on 𝕋,𝑎(𝑡)>0,𝑝(𝑡)>0,𝑡0Δ𝑡/𝑎(𝑡)=.

We note that if 𝕋=, then 𝜎(𝑡)=𝑡,𝜇(𝑡)=0,𝑥Δ(𝑡)=𝑥(𝑡), and (1.33) becomes the second-order nonlinear delay differential equation𝑎(𝑡)(𝑥(𝑡)+𝑟(𝑡)𝑥(𝜏(𝑡)))+𝑝(𝑡)𝑥𝛾(𝛿(𝑡))=0,𝑡.(1.34)

If 𝕋=, then 𝜎(𝑡)=𝑡+1,𝜇(𝑡)=1,𝑥Δ(𝑡)=Δ𝑥(𝑡)=𝑥(𝑡+1)𝑥(𝑡), and (1.33) becomes the second-order nonlinear delay differential equationΔ[]𝑎(𝑡)Δ(𝑥(𝑡)+𝑟(𝑡)𝑥(𝜏(𝑡)))+𝑝(𝑡)𝑥𝛾(𝛿(𝑡))=0,𝑡.(1.35)

In the case of 𝛾>1, (1.33) is the prototype of a wide class of nonlinear dynamic equations called Emden-Fowler sublinear dynamic equations, and if 𝛾<1, (1.33) is the prototype of dynamic equations called Emden-Fowler sublinear dynamic equations. It is interesting to study (1.33) because the continuous version, that is, (1.34), has several physical applications; see, for example, [1, 39], and when 𝑡 is a discrete variable, it is (1.35), and it is also important in applications.

2. Main Results

In this section, we give some new oscillation criteria of (1.33). In order to prove our main results, we will use the formula ((𝑥(𝑡))𝛾)Δ=𝛾10[𝑥𝜎](𝑡)+(1)𝑥(𝑡)𝛾1𝑥Δ(𝑡)d,(2.1) which is a simple consequence of Keller's chain rule [3, Theorem 1.90]. Also, we need the following auxiliary results.

For the sake of convenience, we assume that 𝑧(𝑡)=𝑥(𝑡)+𝑟(𝑡)𝑥(𝜏(𝑡)),𝑅𝑡,𝑡=𝑎(𝑡)𝑡𝑡Δ𝑠𝑎(𝑠),𝛼𝑡,𝑡=𝑡𝛿(𝑡)Δ𝑠/𝑎(𝑠)𝑡𝑡Δ𝑠/𝑎(𝑠),𝑡𝑡0.(2.2)

Lemma 2.1. Assume that (1.11) holds, 𝑎Δ(𝑡)0, and 0𝑟(𝑡)<1. Then an eventually positive solution 𝑥 of (1.33) eventually satisfies that 𝑧(𝑡)𝑡𝑧Δ(𝑡)>0,𝑧ΔΔ(𝑡)<0,𝑎(𝑡)𝑧Δ(𝑡)Δ<0,𝑧(𝑡)𝑡isnonincreasing.(2.3)

Proof. From (1.11), the proof is similar to that of Saker et al. [24, Lemma 2.1], so it is omitted.

Lemma 2.2. Assume that 𝑡0𝑝(𝑡)𝛿𝛾(𝑡)Δt=,(2.4)𝑎Δ(𝑡)0,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟1>1. Then an eventually positive solution 𝑥 of (1.33) eventually satisfies that 𝑧(𝑡)𝑡𝑧Δ(𝑡)>0,𝑧ΔΔ(𝑡)<0,𝑎(𝑡)𝑧Δ(𝑡)Δ<0,𝑧(𝑡)𝑡isnonincreasing,(2.5) or lim𝑡𝑥(𝑡)=0.

Proof. Let 𝑥 be an eventually positive solution of (1.33). Then there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1. Assume that lim𝑡𝑥(𝑡)0, that is, limsup𝑡𝑥(𝑡)>0. Then, we have to show that (2.5) holds. It follows from (1.33) that 𝑎(𝑡)𝑧Δ(𝑡)Δ=𝑝(𝑡)𝑥𝛾(𝛿(𝑡))<0,𝑡𝑡1,(2.6) which implies that 𝑎𝑧Δ is nonincreasing on [𝑡1,)𝕋. Since the function 𝑎 is nondecreasing, 𝑧Δ must be nonincreasing on [𝑡1,)𝕋, that is, 𝑧Δ is eventually either positive or negative. In both cases, 𝑧 is eventually monotonic, so that 𝑧 has a limit at infinity (finite or infinite). This implies that lim𝑡𝑧(𝑡)0; that is, 𝑧 is eventually positive (see [19, Lemma 3]). Then we proceed as in the proof of [24, Lemma 2.1] to obtain (2.5). The proof is complete.

Lemma 2.3. Assume that 0𝑟(𝑡)<1. Further, 𝑥 is an eventually positive solution of (1.33). Then there exists a 𝑡𝑡0 such that for 𝑡𝑡,𝑧Δ(𝑡)>0,𝑎(𝑡)𝑧Δ(𝑡)Δ<0,𝑧(𝑡)𝑅𝑡,𝑡𝑧Δ(𝑡),𝑧(𝛿(𝑡))𝛼𝑡,𝑡𝑧(𝑡).(2.7)

Proof. Let 𝑥 be an eventually positive solution of (1.33). Then there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1. It follows from (1.33) that (2.6) holds. From (2.6), we know that 𝑎(𝑡)𝑧Δ(𝑡) is an eventually decreasing function. We claim that 𝑧Δ(𝑡)>0 eventually. Otherwise, if there exists a 𝑡2𝑡1 such that 𝑧Δ(𝑡)<0, by (2.6), we have 𝑎(𝑡)𝑧Δ(𝑡𝑡)𝑎2𝑧Δ𝑡2=𝑏<0,𝑡𝑡2.(2.8) Thus 𝑧Δ1(𝑡)𝑏𝑎(𝑡).(2.9) Integrating the above inequality from 𝑡2 to 𝑡 leads to lim𝑡𝑧(𝑡)=, which contradicts 𝑧(𝑡)>0. Hence, 𝑧Δ(𝑡)>0 on [𝑡2,)𝕋. Therefore, 𝑡𝑧(𝑡)>𝑧(𝑡)𝑧2=𝑡𝑡2𝑎(𝑠)𝑧Δ(𝑠)𝑎(𝑠)Δ𝑠𝑎(𝑡)𝑧Δ(𝑡)𝑡𝑡2Δ𝑠𝑎(𝑠),(2.10) which yields 𝑧(𝑡)𝑎(𝑡)𝑡𝑡2Δ𝑠𝑧𝑎(𝑠)Δ(𝑡).(2.11) Since 𝑎(𝑡)𝑧Δ(𝑡) is strictly decreasing, we have 𝑧(𝑡)𝑧(𝛿(𝑡))=𝑡𝛿(𝑡)𝑎(𝑠)𝑧Δ(𝑠)𝑎(𝑠)Δ𝑠𝑎(𝛿(𝑡))𝑧Δ(𝛿(𝑡))𝑡𝛿(𝑡)Δ𝑠𝑎(𝑠),(2.12) and so 𝑧(𝑡)𝑧(𝛿(𝑡))1+𝑎(𝛿(𝑡))𝑧Δ(𝛿(𝑡))𝑧(𝛿(𝑡))𝑡𝛿(𝑡)Δ𝑠.𝑎(𝑠)(2.13) Also, we have that for large 𝑡,𝑡𝑧(𝛿(𝑡))𝑧(𝛿(𝑡))𝑧2=𝑡𝛿(𝑡)2𝑎(𝑠)𝑧Δ(𝑠)𝑎(𝑠)Δ𝑠𝑎(𝛿(𝑡))𝑧Δ(𝛿(𝑡))𝑡𝛿(𝑡)2Δ𝑠𝑎(𝑠),(2.14) so we obtain 𝑎(𝛿(𝑡))𝑧Δ(𝛿(𝑡))𝑧(𝛿(𝑡))𝑡𝛿(𝑡)2Δ𝑠𝑎(𝑠)1.(2.15) Therefore, from (2.13), we have 𝑧(𝛿(𝑡))𝛼𝑡,𝑡2𝑧(𝑡).(2.16) This completes the proof.

Lemma 2.4. Assume that 1<𝑟0𝑟(𝑡)0,lim𝑡𝑟(𝑡)=𝑟1>1. Then an eventually positive solution 𝑥 of (1.33) satisfies that, for sufficiently large 𝑡𝑡0,𝑧Δ(𝑡)>0,𝑎(𝑡)𝑧Δ(𝑡)Δ<0,𝑧(𝑡)𝑅𝑡,𝑡𝑧Δ(𝑡),𝑧(𝛿(𝑡))𝛼𝑡,𝑡𝑧(𝑡),𝑡𝑡,(2.17) or lim𝑡𝑥(𝑡)=0.

Proof. The proof is similar to that of the proof Lemmas 2.2 and 2.3, so we omit the details.

Theorem 2.5. Assume that (1.11) holds, 𝑎Δ(𝑡)0, and 0𝑟(𝑡)<1. Then every solution of (1.33) oscillates if the inequality 𝑦Δ(𝑡)+𝐴(𝑡)𝑦𝛾(𝛿(𝑡))0,(2.18) where 𝐴(𝑡)=𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾(𝛿(𝑡))𝛾(𝑎(𝛿(𝑡)))𝛾,(2.19) has no eventually positive solution.

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution 𝑥. We may assume without loss of generality that there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0 and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1. From Lemma 2.1, there is some 𝑡2𝑡1 such that 𝑥(𝑡)=𝑧(𝑡)𝑟(𝑡)𝑥(𝜏(𝑡))𝑧(𝑡)𝑟(𝑡)𝑧(𝜏(𝑡))(1𝑟(𝑡))𝑧(𝑡),𝑡𝑡2.(2.20) From (1.33), there exists a 𝑡3𝑡2 such that 𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾(𝑧(𝛿(𝑡)))𝛾0,𝑡𝑡3.(2.21) By Lemma 2.1, there exists a 𝑡4𝑡3 such that 𝑧(𝛿(𝑡))𝛿(𝑡)𝑧Δ(𝛿(𝑡)).(2.22) Substituting the last inequality in (2.21) we obtain for 𝑡𝑡4 that 𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾(𝛿(𝑡))𝛾𝑧Δ(𝛿(𝑡))𝛾0.(2.23) Set 𝑦(𝑡)=𝑎(𝑡)𝑧Δ(𝑡). Then from (2.23), 𝑦 is positive and satisfies the inequality (2.18), and this contradicts the assumption of our theorem. Thus every solution of (1.33) oscillates. This completes the proof.

By [41, Lemma 1.1] and Theorem 2.5 in this paper, we have the following result.

Corollary 2.6. If 𝕋=,𝑎(𝑡)=1,𝛿(𝑡)=𝑡𝑙,𝑙 is a positive integer, and 0𝑟(𝑡)<1, then every solution of (1.33) oscillates if 𝑡=𝑛0𝑡𝛾𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾=.(2.24)

Theorem 2.7. Assume that (2.4) holds, and 𝑎Δ(𝑡)0,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟1>1. Then every solution of (1.33) either oscillates or tends to zero as 𝑡 if the inequality 𝑦Δ(𝑡)+𝐵(𝑡)𝑦𝛾(𝛿(𝑡))0,(2.25) where 𝐵(𝑡)=𝑝(𝑡)(𝛿(𝑡))𝛾(𝑎(𝛿(𝑡)))𝛾,(2.26) has no eventually positive solution.

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution 𝑥. We may assume without loss of generality that there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1.
From Lemma 2.2, if (i) holds, there is some 𝑡2𝑡1 such that 𝑥(𝑡)=𝑧(𝑡)𝑟(𝑡)𝑥(𝜏(𝑡))𝑧(𝑡)>0,𝑡𝑡2.(2.27) From (1.33), there exists a 𝑡3𝑡2 such that 𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑝(𝑡)(𝑧(𝛿(𝑡)))𝛾0,𝑡𝑡3.(2.28) By Lemma 2.2, there exists a 𝑡3𝑡2 such that 𝑧(𝛿(𝑡))𝛿(𝑡)𝑧Δ(𝛿(𝑡)).(2.29) Substituting the last inequality in (2.28), we obtain for 𝑡𝑡3 that 𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑝(𝑡)(𝛿(𝑡))𝛾𝑧Δ(𝛿(𝑡))𝛾0.(2.30) Set 𝑦(𝑡)=𝑎(𝑡)𝑧Δ(𝑡). Then from (2.30), 𝑦 is positive and satisfies the inequality (2.25), and this contradicts the assumption of our theorem.
If (ii) holds, by Lemma 2.2, we have lim𝑡𝑥(𝑡)=0. This completes the proof.

By [41, Lemma 1.1] and Theorem 2.7 in this paper, we have the following result.

Corollary 2.8. Assume that 𝕋=,𝑎(𝑡)=1,𝛿(𝑡)=𝑡𝑙,𝑙 is a positive integer, 1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟>1. Then every solution of (1.33) either oscillates or tends to zero as 𝑡 if 𝑡=𝑛0𝑡𝛾𝑝(𝑡)=.(2.31)

Remark 2.9. Theorems 2.5 and 2.7 reduce the question of (1.33) to the absence of eventually positive solution (the oscillatory) of the differential inequalities (2.18) and (2.25).

Remark 2.10. From Theorem 2.5, Theorem 2.7, and the results given in [79, 12, 14], we can obtain some oscillation criteria for (1.33) in the case when 𝛾=1,𝑎Δ(𝑡)0.

Theorem 2.11. Assume that (1.11) holds, 𝛾<1,𝑎Δ(𝑡)0, and 0𝑟(𝑡)<1. Then every solution of (1.33) oscillates if t0𝑝(𝑠)(𝑎(𝛿(𝑠)))𝛾(1𝑟(𝛿(𝑠)))𝛾(𝛿(𝑠))𝛾Δ𝑠=.(2.32)

Proof. We assume that (1.33) has a nonoscillatory solution such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1𝑡0. By proceeding as in the proof of Theorem 2.5, we get (2.21). By Lemma 2.1, note that (𝑎(𝑡)𝑧Δ(𝑡))Δ<0, and from Keller's chain rule, we obtain 𝑎(𝑡)𝑧Δ(𝑡)1𝛾Δ=(1𝛾)10𝑎(𝑡)𝑧Δ(𝑡)𝜎+(1)𝑎(𝑡)𝑧Δ(𝑡)𝛾𝑎(𝑡)𝑧Δ(𝑡)Δd(1𝛾)10𝑎(𝑡)𝑧Δ(𝑡)+(1)𝑎(𝑡)𝑧Δ(𝑡)𝛾𝑎(𝑡)𝑧Δ(𝑡)Δd=(1𝛾)𝑎(𝑡)𝑧Δ(𝑡)𝛾𝑎(𝑡)𝑧Δ(𝑡)Δ<0,(2.33) so 𝑎(𝑡)𝑧Δ(𝑡)𝛾𝑎(𝑡)𝑧Δ(𝑡)Δ𝑎(𝑡)𝑧Δ(𝑡)1𝛾Δ1𝛾.(2.34) Using (2.21), we have 0𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾(𝑧(𝛿(𝑡)))𝛾𝑎(𝑡)𝑧Δ(𝑡)𝛾=𝑎(𝑡)𝑧Δ(𝑡)𝛾𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝑧(𝛿(𝑡))𝑎(𝑡)𝑧Δ(𝑡)𝛾𝑎(𝑡)𝑧Δ(𝑡)1𝛾Δ+1𝛾𝑝(𝑡)(𝑎(𝛿(𝑡)))𝛾(1𝑟(𝛿(𝑡)))𝛾(𝛿(𝑡))𝛾.(2.35) Hence, 𝑝(𝑡)(𝑎(𝛿(𝑡)))𝛾(1𝑟(𝛿(𝑡)))𝛾(𝛿(𝑡))𝛾𝑎(𝑡)𝑧Δ(𝑡)1𝛾Δ𝛾1.(2.36) Upon integration we arrive at 𝑡𝑡1𝑝(𝑠)(𝑎(𝛿(𝑠)))𝛾(1𝑟(𝛿(𝑠)))𝛾(𝛿(𝑠))𝛾Δ𝑠𝑡𝑡1𝑎(𝑠)𝑧Δ(𝑠)1𝛾Δ𝑎𝑡𝛾1Δ𝑠1𝑧Δ𝑡11𝛾1𝛾.(2.37) This contradicts (2.32) and finishes the proof.

Theorem 2.12. Assume that (2.4) holds, and 𝛾<1,𝑎Δ(𝑡)0,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟1>1. Then every solution of (1.33) either oscillates or tends to zero as 𝑡 if 𝑡0𝑝(𝑠)(𝑎(𝛿(𝑠)))𝛾(𝛿(𝑠))𝛾Δ𝑠=.(2.38)

Proof. By Lemma 2.2, the proof is similar to that of the proof of Theorem 2.11, so we omit the details.

Theorem 2.13. Assume that 𝛾<1 and 0𝑟(𝑡)<1. Then every solution of (1.33) oscillates if 𝑡0𝑝(𝑠)(𝑎(𝛿(𝑠)))𝛾(1𝑟(𝛿(𝑠)))𝛾𝑅𝛿(𝑠),𝑡𝛾Δ𝑠=(2.39) holds for all sufficiently large 𝑡.

Proof. By Lemma 2.3, the proof is similar to that of the proof Theorem 2.11, so we omit the details.

Theorem 2.14. Assume that 𝛾<1,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟1>1. Then every solution of (1.33) either oscillates or tends to zero as 𝑡 if 𝑡0𝑝(𝑠)(𝑎(𝛿(𝑠)))𝛾𝑅𝛿(𝑠),𝑡𝛾Δ𝑠=(2.40) holds for all sufficiently large 𝑡.

Proof. By using Lemma 2.4 and (2.28), the proof is similar to that of the proof of Theorem 2.11, so we omit the details.

Theorem 2.15. Assume that (1.11) holds, 𝛾1,𝑎Δ(𝑡)0, and 0𝑟(𝑡)<1. Then every solution of (1.33) oscillates if limsup𝑡𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾Δ𝑠=.(2.41)

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution 𝑥. We may assume without loss of generality that there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1. By proceeding as in the proof of Theorem 2.5, we get (2.21). Thus from Lemma 2.1, we have for 𝑇𝑡𝑡1,𝑇𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾(𝑧(𝛿(𝑠)))𝛾Δ𝑠𝑇𝑡𝑎(𝑠)𝑧Δ(𝑠)ΔΔ𝑠=𝑎(𝑡)𝑧Δ(𝑡)𝑎(𝑇)𝑧Δ(𝑇),(2.42) and hence 𝑇𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾(𝑧(𝛿(𝑠)))𝛾Δ𝑠𝑎(𝑡)𝑧Δ(𝑡).(2.43) This and Lemma 2.1 provide, for sufficiently large 𝑡𝕋,𝑧(𝑡)𝑡𝑧Δ𝑡(𝑡)𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾(𝑧(𝛿(𝑠)))𝛾𝑡Δ𝑠𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾𝑧𝛾(𝑠)Δ𝑠𝑧𝛾𝑡(𝑡)𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾.Δ𝑠(2.44) So 𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾1Δ𝑠𝑧(𝑡)𝛾1.(2.45) We note that 𝛾1 and 𝑧Δ(𝑡)>0 imply 𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾1Δ𝑠𝑧𝑡1𝛾1.(2.46) This contradicts (2.41) and completes the proof.

Theorem 2.16. Assume that (2.4) holds, and 𝛾1,𝑎Δ(𝑡)0,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟1>1. Then every solution of (1.33) either oscillates or tends to zero as 𝑡 if limsup𝑡t𝑎(𝑡)𝑡𝑝(𝑠)𝛿(𝑠)𝑠𝛾Δ𝑠=.(2.47)

Proof. By using Lemma 2.2 and (2.28), the proof is similar to that of the proof of Theorem 2.15, so we omit the details.

Theorem 2.17. Assume that 𝛾1,0𝑟(𝑡)<1. Then every solution of (1.33) oscillates if limsup𝑡𝑅𝑡,𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾Δ𝑠=(2.48) holds for all sufficiently large 𝑡.

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution 𝑥. We may assume without loss of generality that there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1. By proceeding as in the proof of Theorem 2.5, we obtain (2.21). Thus from Lemma 2.3, we have, for 𝑇𝑡𝑡1,𝑇𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾(𝑧(𝛿(𝑠)))𝛾Δ𝑠𝑇𝑡𝑎(𝑠)𝑧Δ(𝑠)ΔΔ𝑠=𝑎(𝑡)𝑧Δ(𝑡)𝑎(𝑇)𝑧Δ(𝑇),(2.49) and hence 𝑇𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾(𝑧(𝛿(𝑠)))𝛾Δ𝑠𝑎(𝑡)𝑧Δ(𝑡).(2.50) This and Lemma 2.3 provide, for sufficiently large 𝑡𝕋,𝑧(𝑡)𝑅𝑡,𝑡𝑧Δ𝑅(𝑡)𝑡,𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾(𝑧(𝛿(𝑠)))𝛾𝑅Δ𝑠𝑡,𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾𝑧𝛾(𝑠)Δ𝑠𝑧𝛾𝑅(𝑡)𝑡,𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾.Δ𝑠(2.51) So 𝑅𝑡,𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾1Δ𝑠𝑧(𝑡)𝛾1.(2.52) We note that 𝛾1 and 𝑧Δ(𝑡)>0 imply 𝑅𝑡,𝑡𝑎(𝑡)𝑡𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾1Δ𝑠𝑧𝑡1𝛾1.(2.53) This contradicts (2.48) and completes the proof.

Theorem 2.18. Assume that (2.4) holds, and 𝛾1,𝑎Δ(𝑡)0,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟>1. Then every solution of (1.33) either oscillates or tends to zero as 𝑡 if limsup𝑡𝑅𝑡,𝑡𝑎(𝑡)𝑡𝛼𝑝(𝑠)𝑠,𝑡𝛾Δ𝑠=(2.54) holds for all sufficiently large 𝑡.

Proof. By using Lemma 2.4 and (2.28), the proof is similar to that of the proof of Theorem 2.17, so we omit the details.

Theorem 2.19. Assume that (1.11) holds, 𝛾>1,𝑎Δ(𝑡)0, and 0𝑟(𝑡)<1. Then every solution of (1.33) oscillates if 𝑡0𝜎(𝑠)𝑝(𝑠)𝑎(s)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝜎(𝑠)𝛾Δ𝑠=.(2.55)

Proof. We assume that (1.33) has a nonoscillatory solution such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1𝑡0. By proceeding as in the proof of Theorem 2.5, we get (2.21). Define the function 𝜔(𝑡)=𝑡𝑎(𝑡)𝑧Δ(𝑡)𝑧𝛾(𝑡),𝑡𝑡1.(2.56) By Lemma 2.1, 𝜔(𝑡)>0. We calculate 𝜔Δ(𝑡)=𝑎(𝑡)𝑧Δ(𝑡)+𝜎(𝑡)𝑎(𝑡)𝑧Δ(𝑡)Δ(𝑧𝛾(𝑡))𝜎+𝑡𝑎(𝑡)𝑧Δ(𝑡)(𝑧𝛾(𝑡))Δ.(2.57) From (2.21), we have 𝜔Δ(𝑡)𝑎(𝑡)𝑧Δ(𝑡)(𝑧𝛾(𝑡))𝜎𝜎(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝑧(𝛿(𝑡))𝑧(𝜎(𝑡))𝛾+𝑡𝑎(𝑡)𝑧Δ(𝑡)(𝑧𝛾(𝑡))Δ,(2.58) and by Lemma 2.1, we have 𝜔Δ(𝑡)𝑎(𝑡)𝑧Δ(𝑡)(𝑧𝛾(𝑡))𝜎𝜎(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝛿(𝑡)𝜎(𝑡)𝛾,(2.59) because (𝑧𝛾(𝑡))Δ0 due to Keller's chain rule. Since (𝑧(𝑡))1𝛾Δ=(1𝛾)10[𝑧𝜎](𝑡)+(1)𝑧(𝑡)𝛾𝑧Δ(𝑡)d(1𝛾)10[𝑧𝜎(𝑡)+(1)𝑧𝜎](𝑡)𝛾𝑧Δ(𝑡)d=(1𝛾)(𝑧𝜎(𝑡))𝛾𝑧Δ(𝑡),(2.60) thus 𝜔Δ(𝑡)𝑎(𝑡)(𝑧(𝑡))1𝛾Δ1𝛾𝜎(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝛿(𝑡)𝜎(𝑡)𝛾.(2.61) Upon integration we arrive at 𝑡𝑡1𝜎(𝑠)𝑝(𝑠)𝑎(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝜎(𝑠)𝛾Δ𝑠𝑡𝑡1(𝑧(𝑠))1𝛾Δ𝜔1𝛾Δ(𝑠)=(𝑎(𝑠)Δ𝑠𝑧(𝑡))1𝛾𝑧𝑡1𝛾11𝛾1𝛾𝑡𝑡1𝜔Δ(𝑠)=𝑎(𝑠)Δ𝑠(𝑧(𝑡))1𝛾𝑧𝑡1𝛾11𝛾1𝛾𝜔(𝑡)+𝜔𝑡𝑎(𝑡)1𝑎𝑡1+𝑡𝑡1𝜔𝜎1(𝑠)𝑎(𝑠)ΔΔ𝑠.(2.62) Noting that (1/𝑎(𝑡))Δ0, we have 𝑡𝑡1𝜎(𝑠)𝑝(𝑠)𝑎(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝜎(𝑠)𝛾𝑧𝑡Δ𝑠11𝛾+𝜔𝑡𝛾11𝑎𝑡1.(2.63) This contradicts (2.55) and finishes the proof.

Theorem 2.20. Assume that (2.4) holds, and 𝛾>1,𝑎Δ(𝑡)0,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟1>1. Then every solution of (1.33) either oscillates or tends to zero as 𝑡 if 𝑡0𝜎(𝑠)𝑝(𝑠)𝛿(𝑠)𝜎(𝑠)𝛾Δ𝑠=.(2.64)

Proof. By using Lemma 2.2 and (2.28), the proof is similar to that of the proof of Theorem 2.19, so we omit the details.

In the following, we use a Riccati transformation technique to establish new oscillation criteria for (1.33).

Theorem 2.21. Assume that 𝛾1, and 0𝑟(𝑡)<1. Furthermore, suppose that there exists a positive Δ-differentiable function 𝜂 such that for all sufficiently large 𝑡, and for all constants 𝑀>0, for 𝑡1𝑡,limsup𝑡𝑡𝑡1𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾𝜂𝑎(𝑠)Δ(𝑠)24𝛾𝑀𝛾1𝜂(𝑠)Δ𝑠=.(2.65) Then every solution of (1.33) oscillates.

Proof. We assume that (1.33) has a nonoscillatory solution such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1𝑡0. By proceeding as in the proof of Theorem 2.5, we get (2.21). Define the function 𝜔 by the Riccati substitution 𝜔(𝑡)=𝜂(𝑡)𝑎(𝑡)𝑧Δ(𝑡)𝑧𝛾(𝑡),𝑡𝑡1.(2.66) Then 𝜔(𝑡)>0. By the product rule and then the quotient rule 𝜔Δ(𝑡)=𝑎(𝑡)𝑧Δ(𝑡)𝜎𝜂(𝑡)𝑧𝛾(𝑡)Δ+𝜂(𝑡)𝑧𝛾(𝑡)𝑎(𝑡)𝑧Δ(𝑡)Δ=𝜂(𝑡)𝑧𝛾(𝑡)𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑎(𝑡)𝑧Δ(𝑡)𝜎𝑧𝛾(𝑡)𝜂Δ(𝑡)𝜂(𝑡)(𝑧𝛾(𝑡))Δ𝑧𝛾(𝑡)(𝑧𝜎(𝑡))𝛾.(2.67) In view of (2.21) and (2.66), we have 𝜔Δ(𝑡)𝜂(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝑧(𝛿(𝑡))𝑧(𝑡)𝛾+𝜂Δ(𝑡)𝜂𝜎𝜔(𝑡)𝜎(𝑡)𝜂(𝑡)𝑎(𝑡)𝑧Δ(𝑡)𝜎(𝑧𝛾(𝑡))Δ𝑧𝛾(𝑡)(𝑧𝜎(𝑡))𝛾.(2.68) By the chain rule and 𝛾1, we obtain (𝑧𝛾(𝑡))Δ𝛾𝑧𝛾1(𝑡)𝑧Δ(𝑡)𝛾𝑀𝛾1𝑧Δ(𝑡),(2.69) where 𝑀=𝑧(𝑡1)>0. In view of (𝑎(𝑡)𝑧Δ(𝑡))Δ<0, we have 𝑎(𝑡)𝑧Δ(𝑡)𝑎(𝑡)𝑧Δ(𝑡)𝜎,(2.70) and by Lemma 2.3, we see that 𝜔Δ(𝑡)𝜂(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝛼𝑡,𝑡𝛾+𝜂Δ(𝑡)𝜂𝜎𝜔(𝑡)𝜎(𝑡)𝛾𝑀𝛾1𝜂(𝑡)𝑎(𝑡)(𝜂𝜎(𝑡))2(𝜔𝜎(𝑡))2.(2.71) Integrating (2.71) from 𝑡1 to 𝑡, we obtain 𝑡𝑡1𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾Δ𝑠𝑡𝑡1𝜔Δ+(𝑠)Δ𝑠𝑡𝑡1𝜂Δ(𝑠)𝜂𝜎𝜔(𝑠)𝜎(𝑠)Δ𝑠𝑡𝑡1𝛾𝑀𝛾1𝜂(𝑠)𝑎(𝑠)(𝜂𝜎(𝑠))2(𝜔𝜎(𝑠))2Δ𝑠.(2.72) Hence 𝑡𝑡1𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾𝜂𝑎(𝑠)Δ(𝑠)24𝛾𝑀𝛾1𝑡𝜂(𝑠)Δ𝑠𝜔1,(2.73) which contradicts condition (2.65). The proof is complete.

Theorem 2.22. Assume that 𝛾1,1<𝑟0𝑟(𝑡)0, and lim𝑡𝑟(𝑡)=𝑟1>1. If there exists a positive Δ-differentiable function 𝜂 such that for all sufficiently large 𝑡, and for all constants 𝑀>0, for 𝑡1𝑡,limsup𝑡𝑡𝑡1𝛼𝜂(𝑠)𝑝(𝑠)𝑠,𝑡𝛾𝜂𝑎(𝑠)Δ(𝑠)24𝛾𝑀𝛾1𝜂(𝑠)Δ𝑠=,(2.74) then every solution of (1.33) either oscillates or tends to zero as 𝑡.

Proof. By Lemma 2.4 and (2.28), the proof is similar to that of the proof of Theorem 2.21, so we omit the details.

Theorem 2.23. Assume that (1.11) holds, 𝛾1,𝑎Δ(𝑡)0, and 0𝑟(𝑡)<1. Furthermore, suppose that there exists a positive Δ-differentiable function 𝜂 such that for all sufficiently large 𝑡1, and for all constants 𝑀>0,limsup𝑡𝑡𝑡1𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾𝜂𝑎(𝑠)Δ(𝑠)24𝛾𝑀𝛾1(𝜎(𝑠))𝛾1𝜂(𝑠)Δ𝑠=.(2.75) Then every solution of (1.33) oscillates.

Proof. We assume that (1.33) has a nonoscillatory solution such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1𝑡0. By proceeding as in the proof of Theorem 2.5, we obtain (2.21). Define the function 𝜔 by the Riccati substitution as (2.66). Then 𝜔(𝑡)>0. By the product rule and then the quotient rule 𝜔Δ(𝑡)=𝑎(𝑡)𝑧Δ(𝑡)𝜎𝜂(𝑡)𝑧𝛾(𝑡)Δ+𝜂(𝑡)𝑧𝛾(𝑡)𝑎(𝑡)𝑧Δ(𝑡)Δ=𝜂(𝑡)𝑧𝛾(𝑡)𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑎(𝑡)𝑧Δ(𝑡)𝜎𝑧𝛾(𝑡)𝜂Δ(𝑡)𝜂(𝑡)(𝑧𝛾(𝑡))Δ𝑧𝛾(𝑡)(𝑧𝜎(𝑡))𝛾.(2.76) In view of (2.21) and (2.66), we have 𝜔Δ(𝑡)𝜂(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝑧(𝛿(𝑡))𝑧(𝑡)𝛾+𝜂Δ(𝑡)𝜂𝜎𝜔(𝑡)𝜎(𝑡)𝜂(𝑡)𝑎(𝑡)𝑧Δ(𝑡)𝜎(𝑧𝛾(𝑡))Δ𝑧𝛾(𝑡)(𝑧𝜎(𝑡))𝛾.(2.77) From the chain rule and 𝛾1, we get (𝑧𝛾(𝑡))Δ𝛾𝑧𝛾1(𝜎(𝑡))𝑧Δ(𝑡).(2.78) Noting that 𝑧(𝑡)/𝑡 is nonincreasing, and there exists a constant 𝑀>0, such that 𝑧(𝑡)𝑀𝑡, hence we have (𝑧𝛾(𝑡))Δ𝛾𝑧𝛾1(𝜎(𝑡))𝑧Δ(𝑡)𝛾𝑀𝛾1(𝜎(𝑡))𝛾1𝑧Δ(𝑡).(2.79) In view of (𝑎(𝑡)𝑧Δ(𝑡))Δ<0, we have 𝑎(𝑡)𝑧Δ(𝑡)𝑎(𝑡)𝑧Δ(𝑡)𝜎,(2.80) and by Lemma 2.1, we see that 𝜔Δ(𝑡)𝜂(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝛿(𝑡)𝑡𝛾+𝜂Δ(𝑡)𝜂𝜎𝜔(𝑡)𝜎(𝑡)𝛾𝑀𝛾1(𝜎(𝑡))𝛾1𝜂(𝑡)𝑎(𝑡)(𝜂𝜎(𝑡))2(𝜔𝜎(𝑡))2.(2.81) Integrating (2.81) from 𝑡1 to 𝑡, we obtain 𝑡𝑡1𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾Δ𝑠𝑡𝑡1𝜔Δ(𝑠)Δ𝑠+𝑡𝑡1𝜂Δ(𝑠)𝜂𝜎(𝜔𝑠)𝜎(𝑠)Δ𝑠𝑡𝑡1𝛾𝑀𝛾1(𝜎(𝑠))𝛾1𝜂(𝑠)𝑎(𝑠)(𝜂𝜎(𝑠))2(𝜔𝜎(𝑠))2Δ𝑠.(2.82) Hence 𝑡𝑡1𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛿(𝑠)𝑠𝛾𝜂𝑎(𝑠)Δ(𝑠)24𝛾𝑀𝛾1(𝜎(𝑠))𝛾1𝑡𝜂(𝑠)Δ𝑠𝜔1,(2.83) which contradicts condition (2.75). The proof is complete.

Theorem 2.24. Assume that (2.4) holds, 𝛾1,𝑎Δ(𝑡)0,1<𝑟0𝑟(𝑡)0,andlim𝑡𝑟(𝑡)=𝑟1>1. If there exists a positive Δ-differentiable function 𝜂 such that for all sufficiently large 𝑡1, and for all constants 𝑀>0,limsup𝑡𝑡𝑡1𝜂(𝑠)𝑝(𝑠)𝛿(𝑠)𝑠𝛾𝜂𝑎(𝑠)Δ(𝑠)24𝛾𝑀𝛾1(𝜎(𝑠))𝛾1𝜂(𝑠)Δ𝑠=,(2.84) then every solution of (1.33) either oscillates or tends to zero as 𝑡.

Proof. By Lemma 2.2 and (2.28), the proof is similar to that of the proof of Theorem 2.23, so we omit the details.

Theorem 2.25. Assume that 𝛾1,𝑎Δ(𝑡)0, and 0𝑟(𝑡)<1. Furthermore, suppose that there exists a positive Δ-differentiable function 𝜂 such that for all sufficiently large 𝑡, and for all constants 𝑀>0, for 𝑡1𝑡,limsup𝑡𝑡𝑡1𝜂(s)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾𝑎(𝑠)(𝜎(𝑠))1𝛾𝜂Δ(𝑠)24𝛾𝑀𝛾1(𝑎(𝜎(𝑠)))1𝛾𝜂(𝑠)Δ𝑠=.(2.85) Then every solution of (1.33) oscillates.

Proof. We assume that (1.33) has a nonoscillatory solution such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, and 𝑥(𝛿(𝑡))>0 for all 𝑡𝑡1𝑡0. By proceeding as in the proof of Theorem 2.5, we have (2.21). Define the function 𝜔 by the Riccati substitution as (2.66). Then 𝜔(𝑡)>0. By the product rule and then the quotient rule 𝜔Δ(𝑡)=𝑎(𝑡)𝑧Δ(𝑡)𝜎𝜂(𝑡)𝑧𝛾(𝑡)Δ+𝜂(𝑡)𝑧𝛾(𝑡)𝑎(𝑡)𝑧Δ(𝑡)Δ=𝜂(𝑡)𝑧𝛾(𝑡)𝑎(𝑡)𝑧Δ(𝑡)Δ+𝑎(𝑡)𝑧Δ(𝑡)𝜎𝑧𝛾(𝑡)𝜂Δ(𝑡)𝜂(𝑡)(𝑧𝛾(𝑡))Δ𝑧𝛾(𝑡)(𝑧𝜎(𝑡))𝛾.(2.86) In view of (2.21) and (2.66), we have 𝜔Δ(𝑡)𝜂(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝑧(𝛿(𝑡))𝑧(𝑡)𝛾+𝜂Δ(𝑡)𝜂𝜎𝜔(𝑡)𝜎(𝑡)𝜂(𝑡)𝑎(𝑡)𝑧Δ(𝑡)𝜎(𝑧𝛾(𝑡))Δ𝑧𝛾(𝑡)(𝑧𝜎(𝑡))𝛾.(2.87) By the chain rule and 𝛾1, we obtain (𝑧𝛾(𝑡))Δ𝛾𝑧𝛾1(𝜎(𝑡))𝑧Δ(𝑡),(2.88) and noting that (𝑎(𝑡)𝑧Δ(𝑡))Δ<0 and there exists a constant 𝐿>0 such that 𝑎(𝑡)𝑧Δ(𝑡)𝐿, so 𝑡𝑧(𝑡)=𝑧1+𝑡𝑡1𝑧Δ(𝑡𝑠)Δ𝑠𝑧1+𝑡𝑡1𝐿𝑎(𝑠)Δ𝑠.(2.89) From 𝑎Δ(𝑡)0, there exists a positive constant 𝑀 such that 𝑡𝑧(𝑡)𝑧1+𝐿𝑎(𝑡)𝑡𝑡1=𝑧𝑡1𝑎(𝑡)+𝐿𝑡𝑡1𝑎(𝑡)𝑀𝑡𝑎(𝑡).(2.90) Hence (𝑧𝛾(𝑡))Δ𝛾𝑧𝛾1(𝜎(𝑡))𝑧Δ(𝑡)𝛾𝑀𝛾1𝜎(𝑡)𝑎(𝜎(𝑡))𝛾1𝑧Δ(𝑡).(2.91) In view of (𝑎(𝑡)𝑧Δ(𝑡))Δ<0, we have 𝑎(𝑡)𝑧Δ(𝑡)𝑎(𝑡)𝑧Δ(𝑡)𝜎,(2.92) and by Lemma 2.3, we see that 𝜔Δ(𝑡)𝜂(𝑡)𝑝(𝑡)(1𝑟(𝛿(𝑡)))𝛾𝛼𝑡,𝑡𝛾+𝜂Δ(𝑡)𝜂𝜎𝜔(𝑡)𝜎(𝑡)𝛾𝑀𝛾1𝜂(𝑡)𝑎(𝑡)(𝜂𝜎(𝑡))2𝜎(𝑡)𝑎(𝜎(𝑡))𝛾1(𝜔𝜎(𝑡))2.(2.93) Integrating (2.93) from 𝑡1 to 𝑡, we obtain 𝑡𝑡1𝜂(𝑠)𝑝(𝑠)(1𝑟(𝛿(𝑠)))𝛾𝛼𝑠,𝑡𝛾Δ𝑠𝑡𝑡1𝜔Δ(𝑠)Δ𝑠+𝑡𝑡1𝜂Δ(𝑠)𝜂𝜎𝜔(𝑠)𝜎(𝑠)Δ𝑠𝑡𝑡