Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations

Abstract

We present new oscillation criteria for the differential equation of the form [𝑟(𝑡)𝑈(𝑡)]^′+𝑝(𝑡)𝑘₂(𝑥(𝑡),𝑥^′(𝑡))|𝑥(𝑡)|^𝜈𝑈(𝑡)+𝑞(𝑡)𝜙(𝑥(𝑔₁(𝑡)),𝑥^′(𝑔₂(𝑡)))𝑓(𝑥(𝑡))=0, where 𝑈(𝑡)=𝑘₁(𝑥(𝑡),𝑥^′(𝑡))|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡), 𝛼≤𝛽,𝜈=(𝛽−𝛼)/(𝛼+1). Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking 𝛽=𝛼, the above-mentioned equation can be reduced into the various types of equations concerned by people currently.

1. Introduction

The existence of the oscillatory solutions of the nonlinear differential equation with damping,

[𝑟(𝑡)𝑥^′(𝑡)]^′+𝑝(𝑡)𝑥^′(𝑡)+𝑞(𝑡)𝑔(𝑥(𝑡))=0,𝑡≥𝑡₀, (1.1) has received considerable attention from researchers for a long time.

People previously focused on the cases 𝑟(𝑡)>0,𝑝(𝑡)≥0,𝑞(𝑡)>0. In recent years, people concerned that 𝑟(𝑡)>0,𝑝(𝑡),𝑞(𝑡) may change sign for 𝑡∈[𝑡₀,∞), regarding work in this area can be seen in literature [1–4].

Recently, Li [5] has extended (1.1) to more general equations of the form

(𝑟(𝑡)𝑥^′(𝑡))^′+𝑝(𝑡)𝑥^′(𝑡)+𝑞(𝑡)𝑔(𝑥(𝑡))𝑓(𝑥^′(𝑡))=0,𝑡≥𝑡₀. (1.2) Yamaoka [6] has studied the following class of particular equation:

(|𝑥^′(𝑡)|^𝑝−2𝑥^′(𝑡))^′+2𝑝−1𝑡|𝑥^′(𝑡)|^𝑝−2𝑥^′(𝑡)+𝑞(𝑡)𝑔(𝑥(𝑡))=0,𝑡≥𝑡₀,𝑝>1. (1.3) Tiryaki and Zafer [1] and other authors [2, 7] have considered the following equation of the form

(𝑟(𝑡)ℎ(𝑥(𝑡))𝑥^′(𝑡))^′+𝑝(𝑡)𝑥^′(𝑡)+𝑞(𝑡)𝑔(𝑥(𝑡))=0. (1.4) Zheng [8] has discussed the oscillation problem for the following equation:

(𝑟(𝑡)ℎ(𝑥(𝑡))𝜓(𝑥^′(𝑡)))^′+𝑝(𝑡)𝜓(𝑥^′(𝑡))+𝑞(𝑡)𝑔(𝑥(𝑡))=0. (1.5)

It is worth noting that (1.1), (1.2), and (1.3) can transform into an undamping equation. For example, the equation

(𝑟(𝑡)|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡))^′+𝑝(𝑡)|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡)+𝑞(𝑡)𝑔(𝑥(𝑡))=0 (1.6) can transform into the undamping equation

(𝑟(̃𝑡)|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡))^′+𝑞(𝑡)𝑔(𝑥(𝑡))𝑓(𝑥^′(𝑡))=0,𝑡≥𝑡₀, (1.7) where 𝑟(̃𝑡)=𝑟(𝑡)𝑒^{∫𝑡𝑇(𝑝(𝑠)/𝑟(𝑠))𝑑𝑠},𝑞(𝑡)=𝑞(𝑡)𝑒^{∫𝑡𝑇(𝑝(𝑠)/𝑟(𝑠))𝑑𝑠}. Although (1.4) and (1.5) can not be transformed into the undamping equation, but from the conditions 0<𝑐≤ℎ(𝑥)≤𝑐₁<∞ given by [1, 8, 9], if ℎ(𝑡) is changed into 𝑐 or 𝑐₁, the above-mentioned equations consistent with (1.6). This shows that under the above conditions, there is no essential difference between (1.4), (1.5), and the undamping equation. We note that the condition ∫^∞_𝑡0(1/𝑟(𝑠))^1/𝛼𝑑𝑠=∞ must be used for (1.5); however, at this point the condition ∫^∞_𝑡0(1/𝑟(̃𝑠))^1/𝛼𝑑𝑠=∞ cannot be guaranteed.

We have removed the condition ∫^∞_𝑡0(1/𝑟(𝑠))^1/𝛼𝑑𝑠=∞, considered the oscillation problem for the following equation: [𝑟(𝑡)𝜒(𝑥^′(𝑡))]^′+𝑞(𝑡)𝜙(𝑥(𝑔₁(𝑡)),𝑥^′(𝑔₂(𝑡)))𝑓(𝑥(𝑡))=0, (1.8) applied the results to the above-mentioned equation, and obtained a very good result.

In this paper, we consider the oscillatory behavior of the following differential equation of the form:

[𝑟(𝑡)𝑈(𝑡)]^′+𝑝(𝑡)𝑘₂(𝑥(𝑡),𝑥^′(𝑡))|𝑥(𝑡)|^{(𝛽−𝛼)/(𝛼+1)}𝑈(𝑡)+𝑞(𝑡)𝜙(𝑥(𝑔₁(𝑡)),𝑥^′(𝑔₂(𝑡)))𝑓(𝑥(𝑡))=0, (1.9) where 𝑈(𝑡)=𝑘₁(𝑥(𝑡),𝑥^′(𝑡))|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡).

Today, Riccati transformation, and its generalized forms are one of the most effective method in the oscillatory theory of nonlinear differential equations. Most obvious merits of Riccati's approach is that 𝑞(𝑡) may change sign in (1.8). For getting the more general results [10, 11], a lot of authors have introduced to a class of Y function

𝑌={Φ∈𝐶¹(𝐸,𝑅)∣Φ(𝑡,𝑡,𝑙)=Φ(𝑡,𝑙,𝑙)=0,Φ(𝑡,𝑠,𝑙)≠0,𝑙<𝑠<𝑡},𝐸={(𝑡,𝑠,𝑙)∣𝑡₀≤𝑙≤𝑠≤𝑡<∞}, (1.10) where 𝜕Φ/𝜕𝑠 exists on E and is integral with respect to s. By using this method, peoples have obtained some general results, but its shortcoming is that the property of 𝑞(𝑡) can be weakened as 𝑡→∞. We use the method similar to [4], that is, replace the above-mentioned function Φ(𝑡,𝑠,𝑙) with 𝜌∈𝐶¹([𝑡₀,∞),𝑅⁺). Perhaps the reason that people like to use this method is that integrating by parts with respect to s on [𝑙,𝑡] can employ Φ(𝑡,𝑡,𝑙)=Φ(𝑡,𝑙,𝑙)=0.

For (1.9), we make the following assumptions:

(A)𝑥𝑓(𝑥)>0,𝑥≠0,𝑓^′(𝑥)|𝑓(𝑥)|^{(1−𝛽)/𝛽}≥𝐶₁,𝛼,𝛽>0,𝜙∈𝐶(𝑅²,𝑅⁺),0<𝑛_𝜙≤𝜙(𝑥,𝑦)≤𝑁_𝜙;(B)0<𝐶₀≤𝑘₁(𝑢,𝑣)≤𝐶₂,0≤𝐶₃≤𝑘₂(𝑢,𝑣)≤𝐶₄;(C)𝑟∈𝐶¹(𝐼,𝑅⁺),𝑔_𝑖∈𝐶(𝐼,𝑅⁺),0≤𝑔^′_𝑖(𝑡),𝑖=1,2,𝐼=[𝑡₀,∞),𝑝,𝑞∈𝐶([𝑡₀,∞),𝑅).

In the paper, a solution of (1.9) is called oscillatory if it has zeros unbounded set. If the solutions are oscillatory, (1.9) is called to be oscillatory equation.

2. Main Theorem

We establish some lemmas which are useful in our discussions.

Lemma 2.1. Let 𝑎,𝑏,𝛾,𝜆>0, then 𝑎𝑡−𝑏𝑡^1+1/𝛾≤𝛾^𝛾(𝛾+1)^𝛾+1𝑎^𝛾+1𝑏^𝛾,𝑎𝑡^−𝜆+𝑏𝑡^𝛾≥(𝜆+𝛾)𝜆^{−𝜆/(𝜆+𝛾)}𝛾^{−𝛾/(𝜆+𝛾)}𝑎^{𝛾/(𝜆+𝛾)}𝑏^{𝜆/(𝜆+𝛾)}. (2.1)(2.2)

Lemma 2.1 can easily be proved by using the extremum of one variable function. For the sake of convenience, we denote

𝑞₋(𝑡)=min{𝑞(𝑡),0},𝑞₊(𝑡)=max{𝑞(𝑡),0},𝑞(𝑡)=𝑛_𝜙𝑞₊(𝑠)+𝑁_𝜙𝑞₋𝑝(𝑠),₁(𝑡)=𝐶₄𝑝₊(𝑡)+𝐶₃𝑝₋(𝑡),𝑝₂(𝑡)=𝐶₃𝑝₊(𝑡)+𝐶₄𝑝₋(𝑡). (2.3)

Theorem 2.2. Assume that 𝛼=𝛽 holds and there exists 𝜌∈𝐶¹([𝑡₀,∞),𝑅⁺),∀𝑇∈[𝑡₀,∞), such that ^𝑡_𝑇(𝐶𝑞(𝑠)−²𝛼^𝛼𝑟(𝑠)𝐶^𝛼₁(𝛼+1)^𝛼+1|𝜌^′(𝑠)|𝜌(𝑠)^𝛼+1)𝑒^{∫𝑠𝑡0(𝑝𝑖(𝜏)/𝑟(𝜏))𝑑𝜏}𝜌(𝑠)𝑑𝑠→∞,𝑡→∞,𝑖=1,2,lim_𝑡→∞𝑄(𝑡)=^𝑡_𝑇𝑞(𝑠)𝑒^{∫𝑠𝑡0(𝑝1(𝜏)/𝑟(𝜏))𝑑𝜏}𝑑𝑠>0. (2.4)(2.5) If any one of the following two conditions holds, then the solution 𝑥=𝑥(𝑡) of (1.9) is oscillatory.
(1) 𝑅(𝑡)=^𝑡_𝑡0𝑟^−1/𝛼(𝑠)𝑒^{−1/𝛼∫𝑠𝑡0(𝑝1(𝜏)/𝑟(𝜏))𝑑𝜏}𝑑𝑠→∞,𝑡→∞. (2.6)
(2)𝑅(∞)<∞, and ^𝑡_𝑇(𝑄(𝑠)𝑟⁻¹(𝑠)𝑒^{−∫𝑠𝑡0(𝑝1(𝜏)/𝑟(𝜏))𝑑𝜏})^1/𝛼=𝑑𝑠_𝑡𝑇(𝑟⁻¹(𝑠)^𝑠_𝑇𝑞(𝜍)𝑒^{−∫𝑠𝜍(𝑝1(𝜏)/𝑟(𝜏))𝑑𝜏}𝑑𝜍)^1/𝛼𝑑𝑠→∞,𝑡→∞,lim_𝑡→∞𝑄^1/𝛼𝛼(𝑡)(𝑅(∞)−𝑅(𝑡))≥𝐶₁𝐶₂^1/𝛼. (2.7)(2.8)

Proof. Let 𝑥=𝑥(𝑡) be a nonoscillatory solution of (1.9). Then, there exists 𝑇≥𝑡₀ such that 𝑥=𝑥(𝑡)≠0,𝑡∈(𝑇,∞). Without loss of generality, we may assume that 𝑥=𝑥(𝑡)≥0,𝑡∈(𝑇,∞).
Define the Riccati Transformation by 𝑊(𝑡)=𝜌(𝑡)𝑟(𝑡)𝑘₁(𝑥(𝑡),𝑥^′(𝑡))|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡)𝑓(𝑥(𝑡)),𝑡≥𝑇. (2.9) From conditions (A) and (B), we have |𝑊(𝑡)|^1/𝛼|𝑥(𝑡)|𝐶₀^1/𝛼(𝜌(𝑡)𝑟(𝑡))^1/𝛼≥|𝑥^′(𝑡)|≥|𝑊(𝑡)|^1/𝛼|𝑥(𝑡)|𝐶₂^1/𝛼(𝜌(𝑡)𝑟(𝑡))^1/𝛼𝐶,|𝑓(𝑥)|≥(¹𝛼)^𝛼|𝑥|^𝛼,𝑡≥𝑇. (2.10) Differentiating 𝑊(𝑡), and applying (1.9) and (2.10), we have 𝑊^′𝜌(𝑡)=^′(𝑡)𝜌(𝑡)𝑊(𝑡)−𝑝(𝑡)𝑘₂(𝑥(𝑡),𝑥^′(𝑡))−𝑟(𝑡)𝑊(𝑡)𝜌(𝑡)𝑞(𝑡)|𝑓(𝑥(𝑡))||𝑥(𝑡)|^𝛼𝜙(𝑥(𝑔₁(𝑡)),𝑥^′(𝑔₂(𝑡)))−𝛼𝑊(𝑡)𝑥^′(𝑡)≤(𝜌𝑥(𝑡)_′(𝑡)−𝑝𝜌(𝑡)¹(𝑡)𝐶𝑟(𝑡))𝑊(𝑡)−(¹𝛼)^𝛼𝜌(𝑡)𝑞(𝑡)−𝛼|𝑊(𝑡)|^{(𝛼+1)/𝛼}𝐶₂^1/𝛼(𝜌(𝑡)𝑟(𝑡))^1/𝛼(𝜌,𝑊(𝑡)<0,_′(𝑡)−𝑝𝜌(𝑡)²(𝑡)𝐶𝑟(𝑡))𝑊(𝑡)−(¹𝛼)^𝛼𝜌(𝑡)𝑞(𝑡)−𝛼|𝑊(𝑡)|^{(𝛼+1)/𝛼}𝐶₂^1/𝛼(𝜌(𝑡)𝑟(𝑡))^1/𝛼,𝑊(𝑡)>0. (2.11)
By (2.1), we have 𝑊^′𝐶(𝑡)≤−((¹𝛼)^𝛼𝐶𝜌(𝑡)𝑞(𝑡)−²𝑟(𝑡)(𝛼+1)^𝛼+1(𝜌^′(𝑡)−𝑝𝜌(𝑡)¹(𝑡))𝑟(𝑡)^𝛼+1𝐶)𝜌(𝑡),𝑊(𝑡)<0,−((¹𝛼)^𝛼𝐶𝜌(𝑡)𝑞(𝑡)−²𝑟(𝑡)(𝛼+1)^𝛼+1(𝜌^′(𝑡)−𝑝𝜌(𝑡)²(𝑡))𝑟(𝑡)^𝛼+1)𝜌(𝑡),𝑊(𝑡)>0. (2.12) Integrating the above inequality from 𝑇 to 𝑡≥𝑇, we have 𝑊(𝑡)≤𝑊(𝑇)−^𝑡_𝑇𝐶((¹𝛼)^𝛼𝐶𝑞(𝑠)−²𝑟(𝑠)(𝛼+1)^𝛼+1(𝜌^′(𝑠)−𝑝𝜌(𝑠)^𝑖(𝑠))𝑟(𝑠)^𝛼+1)𝜌(𝑠),𝑖=1,2. (2.13)
Condition (2.4) shows that lim_𝑡→∞𝑊(𝑡)=−∞. Without loss of generality, we may assume that 𝑊(𝑡)<0,𝑡≥𝑇, by applying (2.11), we have 𝑞(𝑡)𝑒^{∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}+𝐶₁𝐶₂^−1/𝛼𝑟^−1/𝛼(𝑡)𝑒^{−(1/𝛼)∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}(𝑡)≤𝑍^′(𝑡),𝑍(𝑡)=−𝑊(𝑡)𝑒𝜌(𝑡)^{∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}>0,𝑡≥𝑇. (2.14)
Integrating the above inequality from 𝑇 to 𝑡≥𝑇, we obtain 𝑍(𝑇)+𝑄(𝑡)+𝐶₁𝐶₂^{−1/𝛼𝑡}_𝑇𝑟^−1/𝛼(𝜏)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}(𝜏)𝑑𝜏≤𝑍(𝑡). (2.15) Let 𝐹(𝑡)=^𝑡_𝑇𝑟^−1/𝛼(𝜏)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}𝐹(𝑠)𝑑𝑠>0,_′(𝑡)=𝑟^−1/𝛼(𝑡)𝑒^{−(1/𝛼)∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}(𝑡),𝑡>𝑇. (2.16) We will discuss in the following two cases.
(1) By (2.1) and (2.5), we see that 𝐶₁^{(𝛼+1)/𝛼}𝐶^{−(𝛼+1)/𝛼2}₂𝑟^−1/𝛼(𝑡)𝑒^{−(1/𝛼)∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}≤𝐹^′(𝑡)𝐹^{−(𝛼+1)/𝛼}(𝑡),𝑡≥𝑇₁>𝑇. (2.17) Integrating the above inequality from 𝑇₁ to 𝑡≥𝑇₁, we have 𝛼𝐹^−1/𝛼(𝑇₁)>𝛼𝐹^−1/𝛼(𝑇₁)−𝛼𝐹^−1/𝛼(𝑡)>𝐶₁^{(𝛼+1)/𝛼}𝐶^{−(𝛼+1)/𝛼2}₂^𝑡_𝑇1𝑟^−1/𝛼(𝑠)𝑒^{−(1/𝛼)∫𝑠𝑡0(𝑝1(𝜏)/𝑟(𝜏))𝑑𝜏}𝑑𝑠→∞,𝑡→∞. (2.18) But, it is impossible that the above inequality holds.
(2) Observe that 𝑊(𝑡)<0 and by (2.8), we have 𝑥^′(𝑡)<0, so that 𝑥=𝑥(𝑡) is monotonic decreasing function for 𝑡≥𝑇, and if lim_𝑡→∞𝑥(𝑡)=𝑐, then 𝑐=0. Otherwise, if 𝑐>0, by (2.15) and (2.9), we have 𝑄(𝑡)𝑓(𝑐)≤𝑍(𝑡)𝑓(𝑐)≤𝑍(𝑡)𝑓(𝑥(𝑡))=−𝑒^{∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑟(𝑡)𝑘₁(𝑥(𝑡),𝑥^′(𝑡))|𝑥^′(𝑡)|^𝛼,𝑡≥𝑇. (2.19) By condition (B), we have −𝑥^′(𝑡)≥𝐶₂^−1/𝛼𝑓^1/𝛼(𝑐)𝑄^1/𝛼(𝑡)𝑟^−1/𝛼(𝑡)𝑒^{−(1/𝛼)∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠},𝑡≥𝑇. (2.20) Integrating the above inequality from 𝑇₁ to 𝑡≥𝑇₁ leads to 𝑥(𝑇)−𝑐≥𝑥(𝑇)−𝑥(𝑡)≥𝐶₂^−1/𝛼𝑓^1/𝛼(𝑐)^𝑡_𝑇𝑄^1/𝛼(𝑠)𝑟^−1/𝛼(𝑠)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑑𝑠→∞,𝑡→∞. (2.21) But, this is impossible. We choose 𝜌(𝑡)=1, thus (2.15) has the following form: 𝑍(𝑇)+𝑄(𝑡)+𝐶₁𝐶₂^{−1/𝛼𝑡}_𝑇𝑟^−1/𝛼(𝜏)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}(𝜏)𝑑𝜏≤𝑍(𝑡),𝑍(𝑡)=−𝑒^{∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑊(𝑡). (2.22)
When ∫^𝑡_𝑇𝑟^−1/𝛼(𝜏)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}(𝜏)𝑑𝜏<𝑀^𝛼+1, by considering Hölder^′s inequality and (2.10), we have −ln𝑥(𝑡)=𝑥(𝑇)^𝑡_𝑇−𝑥^′(𝑠)𝑥⁻¹(𝑠)𝑑𝑠≤(^𝑡_𝑇𝑟(𝑠)𝑒^{−∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}|𝑥^′(𝑠)|^𝛼+1𝑥^𝛼+1(𝑠)𝑑𝑠)^1/(𝛼+1)(^𝑡_𝑇𝑟^−1/𝛼(𝑠)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑑𝑠)^{𝛼/(𝛼+1)}≤𝐶₀^{−(𝛼+1)/𝛼}(^𝑡_𝑡𝑁𝑟^−1/𝛼(𝑠)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}(𝑠)𝑑𝑠)^1/(𝛼+1)(𝑅(𝑡)−𝑅(𝑇))^{𝛼/(𝛼+1)}≤𝑀𝐶₀^{−(𝛼+1)/𝛼}𝑅^{𝛼/(𝛼+1)}(𝑡), (2.23) such that 𝑀𝐶₀^{−(𝛼+1)/𝛼}𝑅^{𝛼/(𝛼+1)}(∞)≥𝑀𝐶₀^{−(𝛼+1)/𝛼}𝑅^{𝛼/(𝛼+1)}(𝑡)≥−ln(𝑥(𝑡)/𝑥(𝑇))→∞,𝑡→∞. However, this is also impossible.
If ∫^𝑡_𝑇𝑟^−1/𝛼(𝜏)𝑒^{−(1/𝛼)∫𝜏𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠}𝑍^{(𝛼+1)/𝛼}(𝜏)𝑑𝜏→∞,𝑡→∞, then by (2.8), we see that lim_𝑡→∞𝑄(𝑡)=∞. From (2.15), we obtain 𝑍(𝑡)≥𝑍(𝑇)+𝑄(𝑇₂)+𝐶₁𝐶₂^−1/𝛼𝐹(𝑡),𝑡≥𝑇₂𝐶≥𝑇,₁𝐶₂^−1/𝛼𝐹^′(𝑡)(𝑍(𝑇)+𝑄(𝑇₂)+𝐶₁𝐶₂^−1/𝛼𝐹(𝑡))^{−(𝛼+1)/𝛼}≥𝐶₁𝐶₂^−1/𝛼𝑟^−1/𝛼(𝑡)𝑒^{−(1/𝛼)∫𝑡𝑡0(𝑝1(𝑠)/𝑟(𝑠))𝑑𝑠},𝑡≥𝑇₂. (2.24) Integrating the above inequality from 𝑇₂ to 𝑡≥𝑇₂, we have 𝛼𝑄^−1/𝛼(𝑇₂)>𝛼(𝑍(𝑇)+𝑄(𝑇₂)+𝐶₁𝐶₂^−1/𝛼𝐹(𝑇₂))^−1/𝛼≥𝐶₁𝐶₂^−1/𝛼(𝑅(∞)−𝑅(𝑇₂)),𝑇₂≥𝑇. (2.25) Let 𝑇₂→∞; the above inequality contradicts (2.8); this completes the proof.

Theorem 2.3. Suppose that 𝛼≤𝛽 and ^∞_𝑡0𝐶((¹𝛽)^𝛽𝐶𝑞(𝑠)−²𝛼^𝛼|𝑝₁(𝑠)|^𝛼+1(𝛼+1)^𝛼+1𝛽^𝛼𝑟^𝛼(𝑠))𝑑𝑠=∞,𝑖=1,2. (2.26) If there exists 0<𝜀<((𝛼+1)𝛽/𝛼)𝐶₂^−1/𝛼,𝑡₀≤𝑡₁<𝑡₂<⋯<𝑡_𝑛<⋯→∞, such that ^𝑡_𝑡𝑛𝜃(𝑠)𝑑𝑠>0,𝑡>𝑡_𝑛,lim_𝑡→∞(^𝑡_𝑡𝑛𝜃(𝑠)𝑑𝑠)^1/𝛼(𝑅(∞)−𝑅(𝑡))^{(𝛽−𝛼)/𝜆(𝛼+1)+1}=∞,𝑛=1,2,…, (2.27) where 𝐶𝜃(𝑡)=(¹𝛽)^𝛽1𝑞(𝑡)−𝛼+1|𝑝₁(𝑡)|^𝛼+1𝜀^𝛼𝑟^𝛼(𝑡),𝑅(𝑡)=^𝑡_𝑡0𝑟^−1/𝛼𝐶(𝑠)𝑑𝑠,_𝜀=(𝛽𝐶₂^−1/𝛼−𝜀𝛼𝐶𝛼+1),𝛿=(⁰𝐶₂)^{(𝛼+1)/𝛼},𝜆=𝛽(𝛿−1)𝛼+1−𝜀𝛿,𝛼+1 (2.28) then every solution of (1.9) is oscillatory.

Note
From (2.27), it is easy to obtain the following equation: lim_𝑡→∞(^𝑡_𝑡𝑛𝜃(𝑠𝑑𝑠))^1/𝛼(𝑅(∞)−𝑅(𝑡))=∞,lim_𝑡→∞(^𝑡_𝑡𝑛𝜃(𝑠)𝑑𝑠)^1/𝛼ln𝑅(∞)𝑅(𝑡)=∞,𝑛=1,2,…. (2.29)

Proof. Let 𝑥=𝑥(𝑡) be a nonoscillatory solution of (1.9). Then, there exists 𝑇≥𝑡₀ such that 𝑥=𝑥(𝑡)≠0,𝑡>𝑇. We may assume that 𝑥=𝑥(𝑡)>0,𝑡>𝑇.
Introduce the Riccati transformation 𝑊(𝑡)=𝑟(𝑡)𝑈(𝑡)/𝑥(𝑡)^𝛽,𝑡≥𝑇. From conditions (A) and (B), we have (𝑊(𝑡)𝑥^𝛽(𝑡)𝐶₀)𝑟(𝑡)^1/𝛼≥|𝑥^′(𝑡)|≥(𝑊(𝑡)𝑥^𝛽(𝑡)𝐶₂)𝑟(𝑡)^1/𝛼𝐶,|𝑓(𝑥)|≥(¹𝛽)^𝛽|𝑥|^𝛽,𝑡≥𝑇. (2.30) Differentiating 𝑊(𝑡), and applying (1.9) and the above inequality, leads to 𝑊^′(𝑡)=−𝑝(𝑡)𝑘₂(𝑥(𝑡),𝑥^′(𝑡))𝑥^{(𝛽−𝛼)/(𝛼+1)}(𝑡)𝑊(𝑡)−𝑟(𝑡)𝑞(𝑡)𝜙(𝑥(𝑔₁(𝑡)),𝑥^′(𝑔₂(𝑡)))𝑥^𝛽−(𝑡)𝛽𝑊(𝑡)𝑥^′(𝑡)≤−𝑝𝑥(𝑡)₁(𝑡)𝑥𝑟(𝑡)^{(𝛽−𝛼)/(𝛼+1)}𝐶(𝑡)𝑊(𝑡)−(¹𝛽)^𝛽𝑞(𝑡)−𝛽𝐶₂^−1/𝛼𝑟^−1/𝛼(𝑡)𝑥^{(𝛽−𝛼)/𝛼}(𝑡)|𝑊(𝑡)|^{(𝛼+1)/𝛼}−𝑝,𝑊(𝑡)<0₂(𝑡)𝑥𝑟(𝑡)^{(𝛽−𝛼)/(𝛼+1)}𝐶(𝑡)𝑊(𝑡)−(¹𝛽)^𝛽𝑞(𝑡)−𝛽𝐶₂^−1/𝛼𝑟^−1/𝛼(𝑡)𝑥^{(𝛽−𝛼)/𝛼}(𝑡)𝑊^{(𝛼+1)/𝛼}(𝑡),𝑊(𝑡)>0. (2.31) By (2.1), we see that 𝑊^′𝐶(𝑡)≤−(¹𝛽)^𝛽𝐶𝑞(𝑡)+²𝛼^𝛼|𝑝₁(𝑡)|^𝛼+1(𝛼+1)^𝛼+1𝛽^𝛼𝑟^𝛼(𝑡),𝑍^′𝐶(𝑡)<0−(¹𝛽)^𝛽𝐶𝑞(𝑡)+²𝛼^𝛼|𝑝₂(𝑡)|^𝛼+1(𝛼+1)^𝛼+1𝛽^𝛼𝑟^𝛼(𝑡),𝑍^′(𝑡)>0. (2.32) The following proof is similar to that in Theorem 2.2, using (2.26), we find that lim_𝑡→∞𝑊(𝑡)=−∞. Thus there exists 𝑇₁≥𝑇, such that 𝑊(𝑡)<0,𝑥^′(𝑡)<0,𝑡≥𝑇₁. Because 𝑥=𝑥(𝑡) is monotonic decreasing function on [𝑇₁,∞); hence lim_𝑡→∞𝑥(𝑡)=𝑐≥0. By (2.31), we have 𝑊^′(𝑡)≤|𝑝₁(𝑡)|𝑥𝑟(𝑡)^{(𝛽−𝛼)/(𝛼+1)}𝐶(𝑡)|𝑊(𝑡)|−(¹𝛽)^𝛽𝑞(𝑡)−𝛽𝐶₂^−1/𝛼𝑟^−1/𝛼(𝑡)𝑥^{(𝛽−𝛼)/(𝛼+1)}(𝑡)|𝑊(𝑡)|^{(𝛼+1)/𝛼},𝑊(𝑡)<0,𝑡≥𝑇₁. (2.33)
By using of weighted mean inequality, we can transform the above inequality into 𝑊^′(𝑡)≤−𝜃(𝑡)−𝐶_𝜀𝑟^−1/𝛼(𝑡)|𝑥^{(𝛽−𝛼)/(𝛼+1)}(𝑡)𝑊(𝑡)|^{(𝛼+1)/𝛼},𝑊(𝑡)<0. (2.34) We need to show that 𝑐=0. Otherwise, if 𝑐>0, by the above inequality, we have 𝑊^′(𝑡)≤−𝜃(𝑡)−𝐶_𝜀𝑐^{(𝛽−𝛼)/(𝛼+1)}𝑟^−1/𝛼(𝑡)|𝑊(𝑡)|^{(𝛼+1)/𝛼},𝑊(𝑡)<0. (2.35) By choosing 𝑁≥1, such that 𝑡_𝑁≥𝑇₁, integrating the above inequality from 𝑇_𝑁 to 𝑡≥𝑇_𝑁 and by (2.27), we can get 𝑊(𝑡)≤𝑊(𝑡_𝑁)−^𝑡𝑛_𝑡𝑁𝜃(𝑠)𝑑𝑠−𝐶_𝜀𝑐^{(𝛽−𝛼)/(𝛼+1)𝑡}_𝑡𝑁𝑟^1/𝛼(𝑠)|𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑑𝑠,𝑡≥𝑡_𝑛,𝑛>𝑁,𝑊(𝑡)<0, (2.36) or 𝐶_𝜀𝑐^{(𝛽−𝛼)/(𝛼+1)}𝑟^−1/𝛼(𝑡)|𝑊(𝑡)|^{(𝛼+1)/𝛼}(|𝑊(𝑡_𝑁)|+∫^𝑡𝑛_𝑡𝑁𝜃(𝑠)𝑑𝑠+𝐶_𝜀𝑐^{(𝛽−𝛼)/(𝛼+1)}∫^𝑡_𝑡𝑁𝑟^−1/𝛼(𝑠)|𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑑𝑠)^{(𝛼+1)/𝛼}≥𝐶_𝜀𝑐^{(𝛽−𝛼)/(𝛼+1)}𝑟^−1/𝛼(𝑡),𝑡≥𝑡_𝑛,𝑛>𝑁. (2.37) Differentiating the above inequality on the interval [𝑡_𝑛,∞], we have 𝛼(|𝑊(𝑡_𝑁)|+^𝑡𝑛_𝑡𝑁𝜃(𝑠)𝑑𝑠+𝐶_𝜀𝑐^{(𝛽−𝛼)/(𝛼+1)𝑡𝑛}_𝑡𝑁𝑟^−1/𝛼(𝑠)|𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑑𝑠)^−1/𝛼≥𝐶_𝜀𝑐^{(𝛽−𝛼)/(𝛼+1)}(𝑅(∞)−𝑅(𝑡_𝑛)),𝑡>𝑡_𝑛,𝑛>𝑁. (2.38) This is a contradiction to (2.29); hence, we have lim_𝑡→∞𝑥(𝑡)=0. According to the above discussion, we have |𝑊(𝑡)|≥|𝑊(𝑡_𝑁)|+^𝑡𝑛_𝑡𝑁𝜃(𝑠)𝑑𝑠+𝐶_𝜀^𝑡_𝑡𝑁𝑟^−1/𝛼(𝑠)|𝑥^{(𝛽−𝛼)/(𝛼+1)}(𝑡)𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑑𝑠,𝑡≥𝑡_𝑛,𝑛>𝑁,𝑊(𝑡)<0,(|𝑊(𝑡_𝑁)|+^𝑡𝑛_𝑡𝑁𝜃(𝑠)𝑑𝑠+𝐶_𝜀^𝑡𝑛_𝑡𝑁𝑟^−1/𝛼(𝑠)|𝑥^{(𝛽−𝛼)/(𝛼+1)}(𝑡)𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑑𝑠)^−1/𝛼≥𝐶_𝜀^∞_𝑡𝑛𝑟^−1/𝛼(𝑠)𝑥^{(𝛽−𝛼)/(𝛼+1)}(𝑠)𝑑𝑠,𝑛>𝑁. (2.39)(2.40) We will discuss in the following two cases.
If there exists 𝑀>0, such that ∫^𝑡_𝑡𝑁𝑟^−1/𝛼(𝑠)𝑥^{(𝛽−𝛼)/𝛼}(𝑠)|𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑑𝑠≤𝑀^𝛼+1,𝑡≥𝑡_𝑁, by considering Hölder's inequality and (2.39), we have 𝛼+1𝛽−𝛼(𝑥^{(𝛼−𝛽)/(𝛼+1)}(𝑡)−𝑥^{(𝛼−𝛽)/(𝛼+1)}(𝑡_𝑁=))_𝑡𝑡𝑁−𝑥^′(𝑠)𝑥^{−(𝛽+1)/(𝛼+1)}(𝑠)𝑑𝑠≤(^𝑡_𝑡𝑁𝑟(𝑠)|𝑥^′(𝑠)|^𝛼+1𝑥^𝛽+1(𝑠)𝑑𝑠)^1/(𝛼+1)(^𝑡_𝑡𝑁𝑟^−1/𝛼(𝑠)𝑑𝑠)^{𝛼/(𝛼+1)}≤𝐶₀^{−(𝛼+1)/𝛼}(^𝑡_𝑡𝑁𝑟^−1/𝛼(𝑠)𝑥^{(𝛽−𝛼)/𝛼}(𝑠)|𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑑𝑠)^1/(𝛼+1)(𝑅(𝑡)−𝑅(𝑡_𝑁))^{𝛼/(𝛼+1)}≤𝑀𝐶₀^{−(𝛼+1)/𝛼}𝑅^{𝛼/(𝛼+1)}(𝑡). (2.41) By choosing 𝑛>𝑁, such that 2^{(𝛼+1)/(𝛽−𝛼)}𝑥(𝑡)<𝑥(𝑡_𝑁),𝑡≥𝑡_𝑛,𝑛>𝑁, from the above inequality, we have 𝑥^{(𝛽−𝛼)/(𝛼+1)}(𝑡)≥((𝛼+1)𝐶₀^{(𝛼+1)/𝛼}/2(𝛽−𝛼)𝑀)𝑅^{−𝛼/(𝛼+1)}(𝑡),𝑡≥𝑡_𝑛,𝑛>𝑁. Inserting it in (2.40), we can get (^𝑡𝑛_𝑡𝑁𝜃(𝑠)𝑑𝑠)^−1/𝛼≥(|𝑊(𝑡_𝑁)|+^𝑡𝑛_𝑡𝑁𝜃(𝑠)𝑑𝑠+^𝑡𝑛_𝑡𝑁𝐶_𝜀|𝑊(𝑠)|^{(𝛼+1)/𝛼}𝑅(𝑠)𝑟^1/𝛼(𝑠)𝑑𝑠)^−1/𝛼≥𝐶_𝜀𝛼ln𝑅(∞)𝑅(𝑡_𝑛),𝐶_𝜀=𝐶_𝜀((𝛼+1)𝐶₀^{(𝛼+1)/𝛼})2(𝛽−𝛼)𝑀^{(𝛼+1)/𝛼}. (2.42) This is contradiction to (2.29).
If ∫^∞_𝑡𝑁(𝑥^{(𝛽−𝛼)/𝛼}(𝑠)|𝑊(𝑠)|^{(𝛼+1)/𝛼}/(𝜌(𝑠)𝑟(𝑠))^1/𝛼)𝑑𝑠=∞ for 𝑡≥𝑡_𝑛,𝑛>𝑁, along with (2.39) and (2.30), we have 𝐶₂𝑟(𝑡)|𝑥^′(𝑡)|^𝛼|𝑥(𝑡)|^𝛽≥|𝑊(𝑡)|≥|𝑊(𝑡_𝑁)|+𝐶_𝜀𝐶₀^{(𝛼+1)/𝛼𝑡}_𝑡𝑁𝑟(𝑠)|𝑥^′(𝑠)|^𝛼+1|𝑥(𝑠)|^𝛽+1𝑑𝑠,𝑡≥𝑡_𝑛,𝑛>𝑁, (2.43) leading to 𝐶_𝜀𝐶₀^{(𝛼+1)/𝛼}𝑟(𝑡)|𝑥^′(𝑡)|^𝛼+1|𝑥(𝑡)|^𝛽+1(|𝑊(𝑡_𝑁)|+𝐶_𝜀𝐶₀^{(𝛼+1)/𝛼𝑡}_𝑡𝑁𝑟(𝑠)|𝑥^′(𝑠)|^𝛼+1|𝑥(𝑠)|^𝛽+1𝑑𝑠)⁻¹𝐶≥−^𝜀𝐶₂𝐶₀^{(𝛼+1)/𝛼}𝑥^′(𝑡)𝑥(𝑡). (2.44) Integrating the above inequality from 𝑡_𝑛 to 𝑡>𝑡_𝑛 and by (2.40), we can get 𝐶ln²𝑟(𝑡)|𝑥^′(𝑡)|^𝛼−𝑊(𝑡_𝑁)|𝑥(𝑡)|^𝛽≥𝐶_𝜀𝐶₂𝐶₀^{(𝛼+1)/𝛼}ln𝑥(𝑡_𝑛)𝑥(𝑡), (2.45) or −𝑥^′(𝑡)𝑥^{(𝛽/𝛼−𝜀/(𝛼+1))𝛿−𝛽/𝛼}(𝑡)≥|𝑊(𝑡_𝑁)|^1/𝛼𝐶₂^−1/𝛼𝑥^{(𝛽/𝛼−𝜀/(𝛼+1))𝛿}(𝑡_𝑛)(𝑟(𝑡))^−1/𝛼. (2.46) Integrating the above inequality on the interval [𝑡,∞), we have 𝑥(𝑡)≥𝐶^1/𝜆(𝑅(∞)−𝑅(𝑡))^1/𝜆,𝐶=𝜆|𝑊(𝑡_𝑁)|^1/𝛼𝐶₂^−1/𝛼𝑥^{(𝛽/𝛼−𝜀/(𝛼+1))𝛿}(𝑡_𝑛). (2.47)
If 𝑅_𝜌(∞)=∞, the above inequality cannot be satisfied; hence, 𝑅_𝜌(∞)<∞. Inserting it in (2.40), we can get (𝛽−𝛼𝜆(𝛼+1)+1)≥𝐶_𝜀𝐶^{(𝛽−𝛼)/𝜆(𝛼+1)}(^𝑡𝑛_𝑡𝑁𝜃(𝑠𝑑𝑠))^1/𝛼(𝑅(∞)−𝑅(𝑡_𝑛))^{(𝛽−𝛼)/𝜆(𝛼+1)+1}. (2.48) This is contradiction to (2.26).
Hence, we complete the proof of Theorem 2.3.

3. Some Examples

Example 3.1. Let us consider the oscillatory behavior of the following differential equation: (𝑡^𝜆|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡))^′+𝑝𝑡^𝜆1|𝑥^′(𝑡)|^𝛼−1𝑥^′(𝑡)+𝑞𝑡^𝜆2|𝑥(𝑡)|^𝛼−1𝑥(𝑡)𝜙(𝑥(𝑡),𝑥^′(𝑡))=0,𝑡≥𝑡₀. (3.1)
Comparing (3.1) with (1.9), we can find that 𝑟(𝑡)=𝑡^𝜆,𝐶₂=𝐶₀=1,𝑝(𝑡)=𝑝𝑡^𝜆1,𝑝>0,𝐶₁=1,𝐶₃=𝐶₄=1,𝑞(𝑡)=𝑞𝑡^𝜆2,𝛽=𝛼,𝑝₁(𝑡)=𝑝𝑡^𝜆1,𝑝₂(𝑡)=0,𝑞(𝑡)=𝑞𝑛_𝜙𝑡^𝜆2. (3.2)
Let 𝜌(𝑡)=𝑡^𝜇, (2.4)–(2.7) are transformed into the equations ^𝑡_𝑡0(𝑞𝑛_𝜙−𝜇^𝛼+1(𝛼+1)^𝛼+1𝑠^{𝜆−𝜆2−𝛼−1})𝑒^{𝑝∫𝑠𝑡0𝜏𝜆1−𝜆𝑑𝜏}𝑠^𝜆2+𝜇𝑑𝑠→∞,𝑡→∞,_𝑡𝑇𝑠^𝜆2𝑒^{𝑝∫𝑠𝑡0𝜏𝜆1−𝜆𝑑𝜏}𝑑𝑠>0,𝑅(𝑡)=^𝑡_𝑡0𝑠^−𝜆/𝛼𝑒^{−(𝑝/𝛼)∫𝑠𝑡0𝜏𝜆1−𝜆𝑑𝜏}𝑑𝑠→∞,𝑡→∞,_𝑡𝑇(𝑠^−𝜆𝑠_𝑇𝜍^𝜆2𝑒^{−𝑝∫𝑠𝜍𝜏𝜆1−𝜆𝑑𝜏}𝑑𝜍)^1/𝛼𝑑𝑠→∞,𝑡→∞,lim_𝑡→∞(^𝑡_𝑇𝑠^𝜆2𝑒^{𝑝∫𝑠𝑡0𝜏𝜆1−𝜆𝑑𝜏}𝑑𝑠)^1/𝛼(𝑅(∞)−𝑅(𝑡))>𝛼𝐶₂^1/𝛼𝑞𝑛_𝜙𝐶₁. (3.3)(3.4)(3.5)(3.6)(3.7) We will discuss inthe following cases.
(1) 𝜆₁+1<𝜆, choosing 𝜇=−min{𝜆₂+1,0}, provided that 𝜆≤𝛼,𝜆−𝜆₂<𝛼+1, or 𝑞𝑛_𝜙>𝜇^𝛼+1/(𝛼+1)^𝛼+1 is satisfied for 𝛼≥𝜆=𝜆₂+𝛼+1, then (3.3)–(3.5) hold, and the solution of (3.1) is oscillatory.
(2) 𝜆=𝜆₁+1,𝑝+𝜆+𝛼≥0, choosing 𝜇=−min{𝜆₂+1,0}, provided that 𝜆≤𝛼, and 𝜆−𝜆₂<𝛼+1 or 𝑞𝑛_𝜙>|𝜇−𝑝|^𝛼+1/(𝛼+1)^𝛼+1,𝛼≥𝜆=𝜆₂+𝛼+1, then (3.3)–(3.5) hold, the solution of (3.1) is oscillatory.
(3) 𝜆=𝜆₁+1,𝑝+𝜆+𝛼<0, we can see that 𝑅(∞)<∞, choosing 𝜇=−min{𝜆₂+1,0}, and therefore, (3.6) and (3.7) are transformed into the following equation: ^𝑡_𝑇(𝑞𝑛_𝜙𝜆₂+𝑝+1(𝑠^{−𝜆+𝜆2+1}−𝑠^{−𝜆−𝑝}𝑇^𝜆2+𝑝+1))^1/𝛼𝑑𝑠→∞,𝑡→∞,𝜆₂+𝑝+1>0,lim_𝑡→∞𝑡^{(𝜆2−𝜆+𝛼+1)/𝛼}>(𝜆+𝑝−𝛼)𝐶₂^1/𝛼𝑞𝑛_𝜙𝐶₁(𝜆₂+𝑝+1)^1/𝛼,𝜆₂+𝑝+1>0, (3.8) provided that 𝜆₂+𝑝+1>0 and 𝜆<𝜆₂+𝛼+1, or 𝜆=𝜆₂+𝛼+1,((𝜆+𝑝−𝛼)𝐶₂^1/𝛼/𝑞𝑛_𝜙𝐶₁)(𝜆₂+𝑝+1)^1/𝛼<1, then (3.3)-(3.4) and (3.6)-(3.7) hold, the solution of (3.1) is oscillatory.
In particular, we chose 𝛼=1,𝑝=0,𝜆=0,𝜆₂=−2,𝜙(𝑥(𝑡),𝑥^′(𝑡))=1=𝑛_𝜙,𝜇=1,𝑞>1/4, thus the conditions of the case (1) can be satisfied. This is the sufficient condition for all solutions of 𝑥^′′(𝑡)+(𝑞/𝑡²)𝑥(𝑡)=0 to be oscillatory.
If we choose 𝛼=𝑃−1,𝑝=2(𝑃−1),𝜆=0,𝜆₁=−1,𝜆₂=−𝑃,𝜙(𝑥,𝑦)≥((𝑃−1)/𝑃)^𝑃+𝜆/ln²|𝑥|>((𝑃−1)/𝑃)^𝑃=𝑛_𝜙,𝜆>0,𝜇=𝑃−1,𝑞≥1, the conditions of the case (2) can be satisfied. Compared with the conditions: 𝑞=1,𝜆>(1/2)((𝑃−1)/𝑃)^𝑃+1 in [6], our results are more general.

Example 3.2. Let us consider the oscillatory behavior of the following differential equation: 𝑥^′′(𝑡)−sin𝑡𝑥^′(𝑡)+1+cos𝑡1+sin²𝑡𝑥(𝑡)(1+|𝑥(𝑡)|²),𝑡≥0. (3.9)
Comparing (3.9) with (1.9), we can see that 𝑞(𝑡)=(1+cos𝑡)/(1+sin²𝑡),𝛼=1,𝑟(𝑡)=1,𝑓(𝑥)=𝑥(1+𝑥²),𝐶₁=3/√³2,𝐶₀=𝐶₂=𝐶₃=𝐶₄=1,𝑝₁(𝑡)=𝑝₂(𝑡)=−sin𝑡,𝛽=3. Choosing 𝜀=1, clearly, the conditions (2.26) and (2.27) of Theorem 6 can be satisfied. Therefore, we may conclude that (3.9) is oscillatory. Example 3.2 is Example 2 of [5]. It is easy to verify that Example 1 of  [5] also satisfies with Theorem 6.

Example 3.3. Let us consider the oscillatory behavior of the following differential equation: [11+𝑡²𝑥^′(𝑡)]^′−𝑎+𝑏sin𝑡𝑡𝑥^′(𝑡)+𝑞(𝑡)|𝑥(𝑡)|^𝛽−1𝑥(𝑡),𝑡≥0. (3.10)
Comparing (3.10) with (1.9), we can see that 𝑞(𝑡)=𝑞(𝑡),𝛼=1,𝑟(𝑡)=1/(1+𝑡²),𝑓(𝑥)=|𝑥|^𝛽−1𝑥,𝐶₁=𝛽,𝐶₀=𝐶₂=𝐶₃=𝐶₄=1,and𝑝₁(𝑡)=𝑝₂(𝑡)=(𝑎+𝑏sin𝑡)/𝑡. Thus, we have 𝑅(𝑡)=𝑡−𝑡₀+(1/3)(𝑡³−𝑡³₀)→∞,𝑡→∞, so the second condition of Theorem 2.3 in (2.27) is satisfied. Therefore, another condition is ^∞_𝑡0(𝑞(𝑠)−(𝑎+𝑏sin𝑠)²14𝛽(1+𝑠²))𝑑𝑠=∞, (3.11) and there exists 0<𝜀<2𝛽,𝑡₀≤𝑡₁<𝑡₂<⋯<𝑡_𝑛<⋯→∞ such that ^𝑡_𝑡𝑛(𝑞(𝑠)−(𝑎+𝑏sin𝑠)²12𝜀(1+𝑠²))𝑑𝑠>0,𝑡>𝑡_𝑛,𝑛=1,2,…. (3.12) If only ∫^∞_𝑡0(𝑞(𝑠)−(2𝑎²+𝑏²)/8𝛽)𝑑𝑠=∞,∫^∞_𝑡0(𝑞(𝑠)−(2𝑎²+𝑏²)/4𝜀)𝑑𝑠>𝑀, where M is a sufficiently large constant, then the conditions (3.11) and (3.12) can be satisfied, the solution of (3.10) is oscillatory.
By taking 𝑞(𝑡)=sin²𝑡, provided that 2𝑎²+𝑏²<4𝛽,𝜀=(1/2𝛽+1/(2𝑎²+𝑏²))⁻¹, (3.11) and (3.12) hold. By the way, we also note that for (3.10), the example given in [3], 𝑞(𝑡)=3+𝑡²4[2+𝑡−(6𝑛−4)𝜋1+𝑡²𝑡]²7,(6𝑛−4)𝜋≤𝑡≤(6𝑛−2)𝜋,3+𝑡²4[2−(6𝑛−3)𝜋−𝑡1+𝑡²𝑡]²7,(6𝑛−2)𝜋≤𝑡≤(6𝑛−3)𝜋,𝑛={1,2,…} (3.13) are not continuous at 𝑡=(6𝑛−4)𝜋,(6𝑛−3)𝜋,𝑛=1,2,…, and ^[𝑡/6𝜋]_𝑛=1^{(6𝑛−7/2)𝜋}_{(6𝑛−4)𝜋}3+𝑡²4[2+𝑡−(6𝑛−4)𝜋1+𝑡²𝑡]²𝑑𝑡=∞,_{(6𝑛−3)𝜋(6𝑛−7/3)𝜋}3+𝑡²