Oscillation Criteria for a Class of Second-Order Nonlinear Differential Equations with Damping Term

Abstract

A class of second-order nonlinear differential equations with damping term (𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡))^′+𝑝(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)+𝑞(𝑡)𝑓(𝑥(𝑡))=0 are investigated in this paper. By using a new method, we obtain some new sufficient conditions for the oscillation of the above equation, and some references are extended in this paper. Examples are inserted to illustrate this result.

1. Introduction

Consider the following second-order nonlinear differential equations with damping term: (𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡))^′+𝑝(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡≥𝑡₀, (1.1) where 𝑟(𝑡)∈𝐶¹([𝑡₀,∞);𝑅⁺), 𝑝(𝑡),𝑞(𝑡)∈𝐶([𝑡₀,∞);𝑅), 𝜎 is a positive constant, and 𝑓 is a continuous real-valued function on the real line 𝑅 and satisfies 𝑥𝑓(𝑥)≥0 for 𝑥≠0. We restrict our attention to those solutions 𝑥(𝑡) of (1.1) which exist on some half line [𝑡_𝑥,∞) and satisfy sup{|𝑥(𝑡)|∶𝑡≥𝑇}>0 for any 𝑇≥𝑡_𝑥.

Recently, there are many authors who have investigated the oscillation for second-order differential equations [1–9], Li [10] and Zhao investigated oscillation criteria for the following equation: (𝑟(𝑡)(𝑥^′(𝑡))^𝜎)^′+𝑝(𝑡)(𝑥^′(𝑡))^𝜎+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡≥𝑡₀, (1.2) where 𝜎 is a quotient of odd positive integer. It is obvious that (1.2) is a special case of (1.1). In fact, the conditions of Theorem 3.2 in [10] are too complex.

More recently, Rogovchenko and Tuncay [11] have obtained oscillation criteria of the following: (𝑟(𝑡)𝑥^′(𝑡))^′+𝑝(𝑡)𝑥^′(𝑡)+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡≥𝑡₀. (1.3)

Motivated by the above discussions, we investigate the oscillation of (1.1) in this paper; our oscillatory conditions and the proof of the main results are more simple than those of Theorem 3.2 in [10].

A solution 𝑥(𝑡) of (1.1) is oscillatory if and only if it has arbitrarily large zeros; otherwise, it is nonoscillatory. Equation (1.1) is oscillatory if and only if every solution of (1.1) is oscillatory.

The paper is arranged as follows. In Section 2, we will establish our main results. Finally, examples are given to illustrate our results.

2. Main Results

To obtain our results, we introduce a lemma as follows.

Lemma 2.1 (see [2, 3]). Let the function 𝐾(𝑡,𝑠,𝑥)∶𝑅×𝑅×𝑅⁺→𝑅 be such that for each fixed 𝑡, 𝑠, the function 𝐾(𝑡,𝑠,⋅) is nondecreasing. Further, let ℎ(𝑡) be a given function and 𝑢(𝑡) satisfies that 𝑢(𝑡)≥(≤)ℎ(𝑡)+^𝑡_𝑡0𝐾(𝑡,𝑠,𝑢(𝑠))𝑑𝑠,𝑡≥𝑡₀, (2.1) and 𝑣^∗(𝑡) is the minimal (maximal) solution of 𝑣(𝑡)=ℎ(𝑡)+^𝑡_𝑡0𝐾(𝑡,𝑠,𝑣(𝑠))𝑑𝑠,𝑡≥𝑡₀. (2.2) Then 𝑢(𝑡)≥(≤)𝑣^∗(𝑡) for all 𝑡≥𝑡₀.

Now, we give our main results.

Theorem 2.2. Assume that 𝑓^′(𝑥)≥0, 𝑝(𝑡)≤0, 𝑞(𝑡)>0, and ∫^∞_𝑡0(1/𝑟^1/𝜎(𝑡))𝑑𝑡=∞ hold. Suppose that there exists a positive function 𝜌(𝑡) such that ^∞_𝑡0𝑞(𝑡)𝜌(𝑡)𝑑𝑡=∞,𝑝(𝑡)𝜌(𝑡)≥𝑟(𝑡)𝜌^′(𝑡). (2.3)(2.4) Then every solution of (1.1) is oscillatory.

Proof. Assume that (1.1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, suppose that it is an eventually positive solution (if it is an eventually negative solution, the proof is similar), that is, 𝑥(𝑡)>0 for all 𝑡≥𝑡₀.
We consider the following three cases.
Case 1. Suppose that 𝑥^′(𝑡) is oscillatory. Then there exists 𝑡₁≥𝑡₀ such that 𝑥^′(𝑡₁)=0. From (1.1), we have (𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)exp(^𝑡_𝑡0𝑝(𝑠)𝑟(𝑠)𝑑𝑠))^′=(𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡))^′exp(^𝑡_𝑡0𝑝(𝑠)𝑟(𝑠)𝑑𝑠)+𝑝(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)exp(^𝑡_𝑡0𝑝(𝑠)𝑟(𝑠)𝑑𝑠)=−𝑞(𝑡)𝑓(𝑥(𝑡))exp(^𝑡_𝑡0𝑝(𝑠)𝑟(𝑠)𝑑𝑠)<0, (2.5) which means that 𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)exp(^𝑡_𝑡0𝑝(𝑠)𝑟(𝑠)𝑑𝑠)<𝑟(𝑡₁)|𝑥^′(𝑡₁)|^𝜎−1𝑥^′(𝑡₁)exp(^𝑡1_𝑡0𝑝(𝑠)𝑟(𝑠)𝑑𝑠)=0,𝑡>𝑡₁, (2.6) it follows that 𝑥^′(𝑡)<0 for all 𝑡>𝑡₁, which contradicts to the assumption that 𝑥^′(𝑡) is oscillatory.Case 2. Suppose that 𝑥^′(𝑡)<0. From (1.1), we obtain −(𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡))^′=(𝑟(𝑡)(−𝑥^′(𝑡))^𝜎)^′=−𝑝(𝑡)(−𝑥^′(𝑡))^𝜎+𝑞(𝑡)𝑓(𝑥(𝑡))≥0, (2.7) then there exists an 𝑀>0 and a 𝑡₁≥𝑡₀, such that 𝑟(𝑡)(−𝑥^′(𝑡))^𝜎≥𝑀,𝑡≥𝑡₁, (2.8) it follows 𝑥(𝑡)≤−^∞_𝑡11𝑟^1/𝜎𝑀(𝑡)^1/𝜎𝑑𝑡+𝑥(𝑡₁),𝑡≥𝑡₁, (2.9) which means that lim_𝑡→∞𝑥(𝑡)=−∞, this contradicts the assumption that 𝑥(𝑡)>0.Case 3. Suppose that 𝑥^′(𝑡)>0. Let 𝑤(𝑡)=𝜌(𝑡)𝑟(𝑡)(𝑥^′(𝑡))^𝜎, then 𝑤^′(𝑡)=(𝑟(𝑡)(𝑥^′(𝑡))^𝜎)^′𝜌(𝑡)+𝑟(𝑡)(𝑥^′(𝑡))^𝜎𝜌^′(𝑡),𝑡≥𝑡₀, (2.10) in view of (1.1), we obtain 𝑤^′(𝑡)𝑓(𝑥(𝑡))=−𝑞(𝑡)𝜌(𝑡)−𝜌(𝑡)𝑝(𝑡)(𝑥^′(𝑡))^𝜎+𝜌𝑓(𝑥(𝑡))^′(𝑡)𝑟(𝑡)(𝑥^′(𝑡))^𝜎𝑓(𝑥(𝑡)),𝑡≥𝑡₀, (2.11) noticing that (𝑤(𝑡))𝑓(𝑥(𝑡))^′=𝑤^′(𝑡)𝑓(𝑥(𝑡))−𝑤(𝑡)𝑓^′(𝑥(𝑡))𝑥^′(𝑡)𝑓²(𝑥(𝑡))=−𝑞(𝑡)𝜌(𝑡)−𝜌(𝑡)𝑝(𝑡)(𝑥^′(𝑡))^𝜎+𝜌𝑓(𝑥(𝑡))^′(𝑡)𝑟(𝑡)(𝑥^′(𝑡))^𝜎−𝑓(𝑥(𝑡))𝑤(𝑡)𝑓^′(𝑥(𝑡))𝑥^′(𝑡)𝑓²(𝑥(𝑡)),𝑡≥𝑡₀, (2.12) integrating the above from 𝑡₀ to 𝑡, we get 𝑤(𝑡)=𝑓(𝑥(𝑡))𝑤(𝑡₀)𝑓(𝑥(𝑡₀−))^𝑡_𝑡0(𝑞(𝑠)𝜌(𝑠)+(𝜌(𝑠)𝑝(𝑠)−𝜌^′(𝑠)𝑟(𝑠))(𝑥^′(𝑠))^𝜎+𝑓(𝑥(𝑠))𝑤(𝑠)𝑥^′(𝑠)𝑓^′(𝑥(𝑠))𝑓²(𝑥(𝑠)))𝑑𝑠. (2.13) Using (2.3), (2.4), and 𝑥^′(𝑡)>0, we have 0≤lim_𝑡→∞𝑤(𝑡)𝑓(𝑥(𝑡))=−∞, (2.14) this is a contradiction, the proof is complete.

Remark 2.3. If we replace 𝑝(𝑡)≤0, 𝑞(𝑡)>0 by 𝑝(𝑡)≤0, 𝑞(𝑡)≤0, lim_𝑡→∞(𝑝(𝑡)/𝑞(𝑡))=𝑀>0, Theorem 2.2 holds also.

Theorem 2.4. Assume that 𝑓^′(𝑥)≥0 holds. Suppose also that 𝜌₀(𝑡)=exp(^𝑡_𝑡0𝑝(𝑠)𝑟(𝑠)𝑑𝑠),_∞𝑡0𝑑𝑡(𝜌₀(𝑡)𝑟(𝑡))^1/𝜎=∞, (2.15)(2.16) and 𝜌₀(𝑡) such that (2.3) holds. Then every solution of (1.1) is oscillatory.

Proof. To the contrary, (1.1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we assume that 𝑥(𝑡) is an eventually positive solution. Let 𝑤(𝑡)=𝜌₀(𝑡)𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡), then 𝑤(𝑡)𝑥^′(𝑡)=𝜌₀(𝑡)𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1(𝑥^′(𝑡))²≥0 for 𝑡≥𝑡₀ and 𝑤^′(𝑡)=(𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡))^′𝜌₀(𝑡)+𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)𝜌^′₀(𝑡),𝑡≥𝑡₀, (2.17) in view of (1.1) and (2.15), we obtain 𝑤^′(𝑡)𝑓(𝑥(𝑡))=−𝑞(𝑡)𝜌₀(𝑡),𝑡≥𝑡₀, (2.18) since (𝑤(𝑡))𝑓(𝑥(𝑡))^′=𝑤^′(𝑡)𝑓(𝑥(𝑡))−𝑤(𝑡)𝑓^′(𝑥(𝑡))𝑥^′(𝑡)𝑓²(𝑥(𝑡))=−𝑞(𝑡)𝜌₀(𝑡)−𝑤(𝑡)𝑓^′(𝑥(𝑡))𝑥^′(𝑡)𝑓²(𝑥(𝑡)),𝑡≥𝑡₀, (2.19) integrating the above from 𝑡₀ to 𝑡, we have −𝑤(𝑡)𝑓(𝑥(𝑡))=−𝑤(𝑡₀)𝑓(𝑥(𝑡₀+))^𝑡_𝑡0𝑞(𝑠)𝜌₀(𝑠)𝑑𝑠+^𝑡_𝑡0𝑤(𝑠)𝑥^′(𝑠)𝑓^′(𝑥(𝑠))𝑓²(𝑥(𝑠))𝑑𝑠,𝑡≥𝑡₀. (2.20) In view of (2.3), there exists a constant 𝑚>0 and 𝑡₁≥𝑡₀ such that −𝑤(𝑡₀)𝑓(𝑥(𝑡₀+))^𝑡_𝑡0𝑞(𝑠)𝜌₀(𝑠)𝑑𝑠+^𝑡1_𝑡0𝑤(𝑠)𝑥^′(𝑠)𝑓^′(𝑥(𝑠))𝑓²(𝑥(𝑠))𝑑𝑠≥𝑚, (2.21) which means that −𝑤(𝑡)𝑓(𝑥(𝑡))≥𝑚+^𝑡_𝑡1𝑤(𝑠)𝑥^′(𝑠)𝑓^′(𝑥(𝑠))𝑓²(𝑥(𝑠))𝑑𝑠. (2.22) Because that 𝑥(𝑡) is positive, then (2.22) implies −𝑤(𝑡)>0, or equivalently 𝑥^′(𝑡)<0. Let 𝑢(𝑡)=−𝑤(𝑡)=−𝜌₀(𝑡)𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)=𝜌₀(𝑡)𝑟(𝑡)(−𝑥^′(𝑡))^𝜎, (2.23) thus (2.22) can be changed as 𝑢(𝑡)≥𝑚𝑓(𝑥(𝑡))+^𝑡_𝑡1𝑓(𝑥(𝑡))𝑓^′(𝑥(𝑠))(−𝑥^′(𝑠))𝑓²(𝑥(𝑠))𝑢(𝑠)𝑑𝑠. (2.24) Define 𝐾(𝑡,𝑠,𝑢)=𝑓(𝑥(𝑡))𝑓^′(𝑥(𝑠))(−𝑥^′(𝑠))𝑓²(𝑥(𝑠))𝑢. (2.25) Then, for any fixed 𝑡 and 𝑠, 𝐾(𝑡,𝑠,𝑢) is nondecreasing in 𝑢. Let 𝑣(𝑡) be the minimal solution of the equation 𝑣(𝑡)=𝑚𝑓(𝑥(𝑡))+^𝑡_𝑡1𝑓(𝑥(𝑡))𝑓^′(𝑥(𝑠))(−𝑥^′(𝑠))𝑓²(𝑥(𝑠))𝑣(𝑠)𝑑𝑠. (2.26) Applying Lemma 2.1, we obtain 𝑢(𝑡)≥𝑣(𝑡),𝑡≥𝑡₀. (2.27) Dividing both sides of (2.26) by 𝑓(𝑥(𝑡)) and deriving both sides of (2.26), it follows (𝑣(𝑡))𝑓(𝑥(𝑡))^′=(𝑚+^𝑡_𝑡1𝑓^′(𝑥(𝑠))(−𝑥^′(𝑠))𝑓²(𝑥(𝑠))𝑣(𝑠)𝑑𝑠)^′=𝑓^′(𝑥(𝑡))(−𝑥^′(𝑡))𝑓²(𝑥(𝑡))𝑣(𝑡). (2.28) On the other hand, (𝑣(𝑡))𝑓(𝑥(𝑡))^′=𝑣^′(𝑡)−𝑓𝑓(𝑥(𝑡))^′(𝑥(𝑡))𝑥^′(𝑡)𝑓²(𝑥(𝑡))𝑣(𝑡). (2.29) Combining (2.28) and (2.29), it means 𝑣^′(𝑡)≡0. (2.30) So 𝑣(𝑡)=𝑣(𝑡₁)=𝑚𝑓(𝑥(𝑡₁)), 𝑡≥𝑡₀. From (2.27), we obtain −𝑥^′(𝑡)≥(𝑚𝑓(𝑥(𝑡₁)))^1/𝜎1(𝜌₀(𝑡)𝑟(𝑡))^1/𝜎,𝑡≥𝑡₁. (2.31) Integrating both sides of the above from 𝑡₁ to 𝑡, we have −𝑥(𝑡)+𝑥(𝑡₁)≥(𝑚𝑓(𝑥(𝑡₁)))^1/𝜎𝑡_𝑡1𝑑𝑠(𝜌₀(𝑠)𝑟(𝑠))^1/𝜎. (2.32) Letting 𝑡→∞ in (2.32), and using (2.16), it follows that lim_𝑡→∞𝑥(𝑡)≤−∞, which contradicts to that 𝑥(𝑡) is eventually positive. The proof is complete.

In the following, we always suppose that 𝐻(𝑡)∈𝐶²(𝑅;𝑅) and it satisfies the following two conditions:

(𝐻₁)𝐻(𝑡)>0 for 𝑡≥𝑡₀, 𝐻(𝑡) is a bounded function;(𝐻₂)𝐻^′(𝑡)=ℎ(𝑡), ℎ(𝑡) is a bounded function.

Theorem 2.5. Assume that 𝑓^′(𝑥)≥0, ∫^∞_𝑡0(1/𝑟^1/𝜎(𝑡))𝑑𝑡=∞ hold, and 𝑝(𝑡)≤0,𝑞(𝑡)>0, (2.33) or 𝑝(𝑡)≤0,𝑞(𝑡)≤0,lim_𝑡→∞𝑝(𝑡)𝑞(𝑡)=𝑀>0. (2.34) Suppose further that there exists a function 𝐻(𝑡) that satisfies (𝐻₁), (𝐻₂), and such that ^∞_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡=∞,limsup_𝑡→∞𝑣(𝑡)𝑟(𝑡)<∞, (2.35)(2.36) where 𝜑(𝑡)=𝑣(𝑡)(𝑞(𝑡)−𝑝(𝑡)ℎ(𝑡)−(𝑟(𝑡)ℎ(𝑡))^′),𝑣(𝑡)=exp(^𝑡_𝑡0(𝑝(𝑠)−𝑟(𝑠)ℎ(𝑠)𝐻(𝑠))𝑑𝑠). (2.37)(2.38) Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may assume that 𝑥(𝑡)>0 for all 𝑡≥𝑡₀.
Define 𝑢(𝑡)=𝑣(𝑡)𝑟(𝑡)(|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)𝑓(𝑥(𝑡))+ℎ(𝑡)). (2.39) Deriving (2.39), we get 𝑢^′(𝑡)=(𝑝(𝑡)−𝑟(𝑡)ℎ(𝑡)𝐻(𝑡))𝑢(𝑡)+𝑣(𝑡)[−𝑝(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)𝑓(𝑥(𝑡))−𝑞(𝑡)−𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1(𝑥^′(𝑡))²𝑓^′(𝑥(𝑡))𝑓²(𝑥(𝑡))+(𝑟(𝑡)ℎ(𝑡))^′]≤−ℎ(𝑡)𝐻(𝑡)𝑢(𝑡)+𝑝(𝑡)𝑣(𝑡)ℎ(𝑡)−𝑣(𝑡)𝑞(𝑡)+𝑣(𝑡)(𝑟(𝑡)ℎ(𝑡))^′=−ℎ(𝑡)𝐻(𝑡)𝑢(𝑡)−𝜑(𝑡). (2.40) Multiplying (2.40) by 𝐻(𝑡), it follows 𝜑(𝑡)𝐻(𝑡)≤−𝐻(𝑡)𝑢^′(𝑡)−ℎ(𝑡)𝑢(𝑡). (2.41) We consider the following three cases.
Case 1 (𝑢(𝑡) is oscillatory). Then there exists a sequence {𝑡_𝑛}, 𝑛=1,2,…,𝑡_𝑛→∞ as 𝑛→∞ and such that 𝑢(𝑡_𝑛)=0, 𝑛=1,2,…. Integrating both sides of (2.41) from 𝑡₀ to 𝑡_𝑛, we obtain ^𝑡𝑛_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡≤−^𝑡𝑛_𝑡0𝐻(𝑡)𝑢^′(𝑡)𝑑𝑡−^𝑡𝑛_𝑡0ℎ(𝑡)𝑢(𝑡)𝑑𝑡=−𝐻(𝑡)𝑢(𝑡)|^𝑡𝑛_𝑡0−^𝑡𝑛_𝑡0(−𝐻^′(𝑡)𝑢(𝑡)+ℎ(𝑡)𝑢(𝑡))𝑑𝑡=𝐻(𝑡₀)𝑢(𝑡₀)−𝐻(𝑡_𝑛)𝑢(𝑡_𝑛)=𝐻(𝑡₀)𝑢(𝑡₀), (2.42) that is lim_{𝑡𝑛→∞}^𝑡𝑛_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡≤𝐻(𝑡₀)𝑢(𝑡₀), (2.43) which contradicts (2.35).Case 2 (𝑢(𝑡) is eventually positive). Integrating both sides of (2.41) from 𝑡₀ to ∞, we obtain ^∞_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡≤𝐻(𝑡₀)𝑢(𝑡₀)−lim_𝑡→∞𝐻(𝑡)𝑢(𝑡)≤𝐻(𝑡₀)𝑢(𝑡₀), (2.44) which also contradicts to (2.35).Case 3 (𝑢(𝑡) is eventually negative). If limsup_𝑡→∞𝑢(𝑡)>−∞, then there exists a sequence {𝑡_𝑛}, 𝑛=1,2,…, that satisfies {𝑡_𝑛}→∞ as 𝑛→∞ and such that lim_{𝑡𝑛→∞}𝑢(𝑡_𝑛)=limsup_𝑡→∞𝑢(𝑡)=𝑀₁>−∞. Because 𝐻(𝑡) is a bounded function, then there exists a 𝑀₂>0 such that 𝐻(𝑡_𝑛)≤𝑀₂, 𝑛=1,2,…. According to (2.41), we obtain ^𝑡𝑛_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡≤𝐻(𝑡₀)𝑢(𝑡₀)−𝐻(𝑡_𝑛)𝑢(𝑡_𝑛)≤𝐻(𝑡₀)𝑢(𝑡₀)−𝑀₂𝑢(𝑡_𝑛). (2.45) Using (2.35) and taking limit as 𝑡_𝑛→∞, it is easy to show that ∞=lim_{𝑡𝑛→∞}^𝑡𝑛_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡≤𝐻(𝑡₀)𝑢(𝑡₀)−lim_{𝑡𝑛→∞}𝐻(𝑡_𝑛)𝑢(𝑡_𝑛)≤𝐻(𝑡₀)𝑢(𝑡₀)−𝑀₁𝑀₂<∞, (2.46) which is obviously a contradiction.
If limsup_𝑡→∞𝑢(𝑡)=−∞, lim_𝑡→∞𝑢(𝑡)=−∞. From the definition of ℎ(𝑡), combining (2.36) and (2.39), it follows that 𝑥^′(𝑡)<0 and lim_𝑡→∞(|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)/𝑓(𝑥(𝑡)))=−∞, which means that lim_𝑡→∞((−𝑥^′(𝑡))^𝜎/𝑓(𝑥(𝑡)))=∞. Owing to 𝑝(𝑡)≤0, 𝑞(𝑡)≥0, or 𝑝(𝑡)≤0, 𝑞(𝑡)≤0 and lim_𝑡→∞(𝑝(𝑡)/𝑞(𝑡))=𝑀>0, using the similar method of the proof of Case 2 in Theorem 2.2, we will derive a contradiction. Then the proof is complete.

Theorem 2.6. Assume that (2.36) holds, 𝑓^′(𝑥)≥0, ∫^∞_𝑡0(1/𝑟^1/𝜎(𝑡))𝑑𝑡=∞, and 𝑝(𝑡)≤0,𝑞(𝑡)>0, (2.47) or 𝑝(𝑡)≤0,𝑞(𝑡)≤0,lim_𝑡→∞𝑝(𝑡)𝑞(𝑡)=𝑀>0. (2.48) Suppose further that there exists a function 𝐻(𝑡) that satisfies (𝐻₁), (𝐻₂) and such that ^∞_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡=∞, (2.49) where 𝜑(𝑡)=𝑣(𝑡)(𝑞(𝑡)+𝑝(𝑡)ℎ(𝑡)+(𝑟(𝑡)ℎ(𝑡))^′), (2.50) and 𝑣(𝑡) is defined in (2.38). Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution. Without loss of generality, we may assume that (1.1) has an eventually positive solution (if it has an eventually negative solution, the proof is similar), then there exists a 𝑡₁>𝑡₀ such that 𝑥(𝑡)>0 for all 𝑡≥𝑡₁. Define 𝑢(𝑡)=𝑣(𝑡)𝑟(𝑡)(|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)𝑓(𝑥(𝑡))−ℎ(𝑡)). (2.51) The rest of the proof is similar to Theorem 2.5. The proof is complete.

Theorem 2.7. Assume that (2.36) holds, 𝑓^′(𝑥)≥0,∫^∞_𝑡0(1/𝑟^1/𝜎(𝑡))𝑑𝑡=∞, and 𝑝(𝑡)≤0,𝑞(𝑡)>0, (2.52) or 𝑝(𝑡)≤0,𝑞(𝑡)≤0,lim_𝑡→∞𝑝(𝑡)𝑞(𝑡)=𝑀>0. (2.53) Suppose further that there exists a function 𝐻(𝑡) that satisfies (𝐻₁), (𝐻₂) and such that ^∞_𝑡0𝐻(𝑡)𝜙(𝑡)𝑑𝑡=∞, (2.54) where 𝜙(𝑡)=𝑣(𝑡)(−𝑝(𝑡)ℎ(𝑡)−(𝑟(𝑡)ℎ(𝑡))^′), (2.55) where 𝑣(𝑡) is defined in (2.38). Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may assume that 𝑥(𝑡)>0 for all 𝑡≥𝑡₀.
Define 𝑢(𝑡)=𝑣(𝑡)𝑟(𝑡)(|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)𝑥(𝑡)+ℎ(𝑡)). (2.56) Noting that 𝑥𝑓(𝑥)≥0 for 𝑥≠0, so 𝑓(𝑥)/𝑥≥0 for 𝑥≠0. Deriving (2.56), we obtain 𝑢^′(𝑡)=(𝑝(𝑡)−𝑟(𝑡)ℎ(𝑡)𝐻(𝑡))𝑢(𝑡)+𝑣(𝑡)[−𝑝(𝑡)|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)−𝑥(𝑡)𝑞(𝑡)𝑓(𝑥(𝑡))−𝑥(𝑡)𝑟(𝑡)|𝑥^′(𝑡)|^𝜎−1(𝑥^′(𝑡))²𝑥²(𝑡)+(𝑟(𝑡)ℎ(𝑡))^′]≤−ℎ(𝑡)𝐻(𝑡)𝑢(𝑡)+𝑝(𝑡)𝑣(𝑡)ℎ(𝑡)+𝑣(𝑡)(𝑟(𝑡)ℎ(𝑡))^′=−ℎ(𝑡)𝐻(𝑡)𝑢(𝑡)−𝜙(𝑡). (2.57) Multiplying (2.57) by 𝐻(𝑡), we get 𝐻(𝑡)𝜙(𝑡)≤−𝐻(𝑡)𝑢^′(𝑡)−ℎ(𝑡)𝑢(𝑡). (2.58) The rest of the proof is similar to Theorem 2.5; the proof is complete.

Theorem 2.8. Assume that (2.36) holds, 𝑓^′(𝑥)≥0,∫^∞_𝑡0(1/𝑟^1/𝜎(𝑡))𝑑𝑡=∞, and 𝑝(𝑡)≤0,𝑞(𝑡)>0, (2.59) or 𝑝(𝑡)≤0,𝑞(𝑡)≤0,lim_𝑡→∞𝑝(𝑡)𝑞(𝑡)=𝑀>0. (2.60) Suppose further that there exists a function 𝐻(𝑡) satisfies (𝐻₁), (𝐻₂) and such that ^∞_𝑡0𝐻(𝑡)𝜙(𝑡)𝑑𝑡=∞, (2.61) where 𝜙(𝑡)=𝑣(𝑡)(𝑝(𝑡)ℎ(𝑡)+(𝑟(𝑡)ℎ(𝑡))^′), (2.62) where 𝑣(𝑡) is defined in (2.38). Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may assume that 𝑥(𝑡)>0 for all 𝑡≥𝑡₀.
Define 𝑢(𝑡)=𝑣(𝑡)𝑟(𝑡)(|𝑥^′(𝑡)|^𝜎−1𝑥^′(𝑡)𝑥(𝑡)−ℎ(𝑡)). (2.63) The rest of the proof is similar to Theorem 2.5; the proof is complete.

3. Examples

Example 3.1. Consider the following delay differential equation: (𝑡𝑥^′(𝑡))^′−2𝑥^′3(𝑡)+(𝑡+4𝑡)𝑥(𝑡)=0. (3.1) It is obvious that 𝜎=1, 𝑞(𝑡)=(𝑡+3/4𝑡)>0, 𝑝(𝑡)=−2<0, 𝑟(𝑡)=𝑡, and ∫^∞_𝑡0(1/𝑡)𝑑𝑡=∞. It is difficult to distinguish whether every solution of (3.1) is oscillatory by Theorem  3.2 of [10].
By taking 𝜌(𝑡)=1/𝑡², then ^∞_𝑡0𝑞(𝑡)𝜌(𝑡)𝑑𝑡=^∞_𝑡03(𝑡+)14𝑡𝑡²2𝑑𝑡=∞,𝑝(𝑡)𝜌(𝑡)=−𝑡²=𝑟(𝑡)𝜌^′(𝑡). (3.2) From Theorem 2.2 or Theorem 2.4, it is easy to show that (3.1) is oscillatory.
In fact, 𝑥(𝑡)=𝑡^1/2cos𝑡 is such an oscillatory solution.

Example 3.2. Consider the following differential equation: (𝑡𝑥^′(𝑡))^′−𝑥^′(𝑡)+𝑡𝑥(𝑡)=0. (3.3) It is obvious that 𝜎=1, 𝑟(𝑡)=𝑡, 𝑝(𝑡)=−1<0, 𝑞(𝑡)=𝑡>0, and ∫^∞_𝑡0(1/𝑟(𝑡))𝑑𝑡=∫^∞_𝑡0(1/𝑡)𝑑𝑡=∞.
We are taking 𝐻(𝑡)=𝐶>0, ℎ(𝑡)=0. By a simple calculation, it is easy to show that 𝑣(𝑡)=exp(∫^𝑡_𝑡0(𝑝(𝑠)/𝑟(𝑠)−ℎ(𝑠)/𝐻(𝑠))𝑑𝑠)=exp(∫^𝑡_𝑡0(−1/𝑠−0/𝐶)𝑑𝑠)=𝑡₀/𝑡, limsup_𝑡→∞𝑣(𝑡)𝑟(𝑡)=limsup_𝑡→∞𝑡₀<∞, 𝜑(𝑡)=𝑣(𝑡)(𝑞(𝑡)−𝑝(𝑡)ℎ(𝑡)−(𝑟(𝑡)ℎ(𝑡))^′)=𝑡₀, and ∫^∞_𝑡0𝐻(𝑡)𝜑(𝑡)𝑑𝑡=∫^∞_𝑡0𝑡₀𝐶𝑑𝑡=∞.
From Theorem 2.5 or Theorem 2.6, it follows that (3.3) is oscillatory.
In fact, 𝑥(𝑡)=sin𝑡 is such an oscillatory solution.

Acknowledgments

This work was supported by the Natural Science Foundation of Hunan Province under Grant no. 07JJ3130, the Doctor Foundation of University of South China under Grant no. 5-XQD-2006-9, and the Subject Lead Foundation of University of South China under Grant no. 2007XQD13.