Oscillation Theorems for Second-Order Quasilinear Neutral Functional Differential Equations

Abstract

New oscillation criteria are established for the second-order nonlinear neutral functional differential equations of the form (𝑟(𝑡)|𝑧^′(𝑡)|^𝛼−1𝑧^′(𝑡))^′+𝑓(𝑡,𝑥[𝜎(𝑡)])=0, 𝑡≥𝑡₀, where 𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡)), 𝑝∈𝐶¹([𝑡₀,∞),[0,∞)), and 𝛼≥1. Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results.

1. Introduction

This paper is concerned with the oscillation problem of the second-order nonlinear functional differential equation of the following form: 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′+𝑓(𝑡,𝑥[𝜎(𝑡)])=0,𝑡≥𝑡₀, (1.1) where 𝛼≥1 is a constant, 𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥[𝜏(𝑡)].

Throughout this paper, we will assume the following hypotheses:(A₁)𝑟∈𝐶¹([𝑡₀,∞),ℝ),𝑟(𝑡)>0 for 𝑡≥𝑡₀,(A₂)𝑝∈𝐶¹([𝑡₀,∞),[0,∞)), (A₃)𝜏∈𝐶²([𝑡₀,∞),ℝ),𝜏^′(𝑡)>0, lim_𝑡→∞𝜏(𝑡)=∞,(A₄)𝜎∈𝐶([𝑡₀,∞),ℝ),lim_𝑡→∞𝜎(𝑡)=∞,𝜏∘𝜎=𝜎∘𝜏;(A₅)𝑓(𝑡,𝑢)∈𝐶([𝑡₀,∞)×ℝ,ℝ), and there exists a function 𝑞∈𝐶([𝑡₀,∞),[0,∞)) such that 𝑓(𝑡,𝑢)sgn𝑢≥𝑞(𝑡)|𝑢|^𝛼,for𝑢≠0,𝑡≥𝑡₀. (1.2)

By a solution of (1.1), we mean a function 𝑥∈𝐶([𝑇_𝑥,∞),ℝ) for some 𝑇_𝑥≥𝑡₀ which has the property that 𝑟(𝑡)|𝑧′(𝑡)|^𝛼−1𝑧′(𝑡)∈𝐶¹([𝑇_𝑥,∞),ℝ) and satisfies (1.1) on [𝑇_𝑥,∞). As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [𝑡₀,∞); otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all of its nonconstant solutions are oscillatory.

We note that neutral delay differential equations find numerous applications in electric networks. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines which rise in high-speed computers where the lossless transmission lines are used to interconnect switching circuits; see [1]. Therefore, there is constant interest in obtaining new sufficient conditions for the oscillation or nonoscillation of the solutions of varietal types of the second-order equations, see, e.g., papers [2–17].

Known oscillation criteria require various restrictions on the coefficients of the studied neutral differential equations.

Agarwal et al. [2], Chern et al. [3], Džurina and Stavroulakis [4], Kusano et al. [5, 6], Mirzov [7], and Sun and Meng [8] observed some similar properties between 𝑟(𝑡)𝑥^′(𝑡)^𝛼−1𝑥^′(𝑡)^′+𝑞(𝑡)|𝑥[𝜎(𝑡)]|^𝛼−1𝑥[𝜎(𝑡)]=0 (1.3) and the corresponding linear equation 𝑟(𝑡)𝑥^′(𝑡)′+𝑞(𝑡)𝑥(𝑡)=0. (1.4) Liu and Bai [10], Xu and Meng [11, 12], and Dong [13] established some oscillation criteria for (1.3) with neutral term under the assumption that ^∞_𝑡01𝑟^1/𝛼(𝑡)d𝑡=∞. (1.5)

Han et al. [14] examined the oscillation of second-order linear neutral differential equation 𝑟(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))^′′+𝑞(𝑡)𝑥[𝜎(𝑡)]=0,𝑡≥𝑡₀, (1.6) where 𝜏^′(𝑡)=𝜏₀>0,0≤𝑝(𝑡)≤𝑝₀<∞, and obtained some oscillation criteria for (1.6) when ^∞_𝑡01𝑟(𝑡)d𝑡=∞. (1.7) Han et al. [15] studied the oscillation of (1.6) under the case 0≤𝑝(𝑡)≤1 and ^∞_𝑡01𝑟(𝑡)d𝑡<∞. (1.8)

Tripathy [16] considered the nonlinear dynamic equation of the form 𝑟(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜏)^Δ𝛾Δ+𝑞(𝑡)|𝑥(𝑡−𝛿)|^𝛾sgn𝑥(𝑡−𝛿)=0, (1.9) where 0≤𝑝(𝑡)≤𝑝₀<∞,𝛾 is a the ratios of two positive odd integers, and obtained some oscillation criteria under the following conditions: ^∞_𝑡01𝑟(𝑡)^𝛾d𝑡=∞. (1.10)

Džurina [17] was concerned with the oscillation behavior of the solutions of the second-order neutral differential equations as follows 𝑎(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))^′𝛾′+𝑞(𝑡)𝑥^𝛽(𝜎(𝑡))=0, (1.11) where 0≤𝑝(𝑡)≤𝑝₀<∞,𝛾 is a the ratios of two positive odd integers, and obtained some new results under the following conditions ^∞_𝑡01𝑎(𝑡)^𝛾d𝑡=∞. (1.12)

Our purpose of this paper is to establish some new oscillation criteria for (1.1), and we will also consider the cases (1.5) and ^∞_𝑡01𝑟^1/𝛼(𝑡)d𝑡<∞. (1.13)

To the best of my knowledge, there is no result for the oscillation of (1.1) under the conditions both 0≤𝑝(𝑡)≤𝑝₀<∞ and (1.13).

In this paper, we will use a new inequality to establish some oscillation criteria for (1.1) for the first time. Some examples will be given to show the importance of these results. In Sections 3 and 4, for the sake of convenience, we denote that 𝑄(𝑡)∶=min𝑞(𝑡),𝑞[𝜏(𝑡)],𝑑₊(𝑡)∶=max{0,𝑑(𝑡)},𝜉(𝑡)∶=𝛼𝑝^′[𝜎(𝑡)]𝜎^′(𝑡)𝑝[𝜎(𝑡−𝜏)]^′′(𝑡)𝜏^′(𝑡),𝜁(𝑡)𝜌∶=^′(𝑡)₊𝜌(𝑡)(𝑡)(𝑡)𝜌+𝜉,𝜑∶=^′+(𝑡)𝜌(𝑡)^𝛼+1+𝑝^𝛼[𝜎(𝑡)]𝜁₊(𝑡)^𝛼+1𝜏^′(𝑡),𝛿(𝑡)∶=^∞_𝜂(𝑡)d𝑠𝑟^1/𝛼(𝑠). (1.14)

2. Lemma

In this section, we give the following lemma, which we will use in the proofs of our main results.

Lemma 2.1. Assume that 𝛼≥1,𝑎,𝑏∈ℝ. If 𝑎≥0,𝑏≥0, then one has 𝑎^𝛼+𝑏^𝛼≥12^𝛼−1(𝑎+𝑏)^𝛼. (2.1)

Proof. (i) Suppose that 𝑎=0 or 𝑏=0. Then we have (2.1). (ii) Suppose that 𝑎>0,𝑏>0. Define the function 𝑔 by 𝑔(𝑥)=𝑥^𝛼,𝑥∈(0,∞). Then 𝑔^′′(𝑥)=𝛼(𝛼−1)𝑥^𝛼−2≥0 for 𝑥>0. Thus, 𝑔 is a convex function. By the definition of convex function, for 𝜆=1/2,𝑎,𝑏∈(0,∞), we have 𝑔𝑎+𝑏2≤𝑔(𝑎)+𝑔(𝑏)2, (2.2) that is, 𝑎^𝛼+𝑏^𝛼≥12^𝛼−1(𝑎+𝑏)^𝛼. (2.3) This completes the proof.

3. Oscillation Criteria for the Case (1.5)

In this section, we will establish some oscillation criteria for (1.1) under the case (1.5).

Theorem 3.1. Suppose that (1.5) holds, 𝜎∈𝐶([𝑡₀,∞),ℝ),𝜎^′(𝑡)>0,𝜎(𝑡)≤𝑡,and𝜎(𝑡)≤𝜏(𝑡) for 𝑡≥𝑡₀. Furthermore, assume that there exists a function 𝜌∈𝐶([𝑡₀,∞),(0,∞)) such that limsup_𝑡→∞^𝑡_𝑡0𝜌(𝑠𝑄(𝑠))2^𝛼−1−𝑟[𝜎(𝑠)]𝜑(𝑠)(𝛼+1)^𝛼+1𝜎^′(𝑠)^𝛼d𝑠=∞. (3.1) Then (1.1) is oscillatory.

Proof. Let 𝑥 be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>0,𝑥[𝜏(𝑡)]>0 and 𝑥[𝜎(𝑡)]>0 for all 𝑡≥𝑡₁. By applying (1.1), for all sufficiently large 𝑡, we obtain that 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′+𝑞(𝑡)𝑥^𝛼[𝜎(𝑡)]+𝑞[𝜏(𝑡)]𝑝^𝛼[𝜎(𝑡)]𝑥^𝛼(𝜎[𝜏(𝑡+𝑝)])_𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼−1𝑧^′[𝜏(𝑡)]^′≤0. (3.2) Using (2.1) and the definition of 𝑧, we conclude that 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′+12^𝛼−1𝑄(𝑡)𝑧^𝛼[𝜎(𝑡+𝑝)]^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼−1𝑧^′[𝜏(𝑡)]^′≤0. (3.3) In view of (1.1), we obtain that 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′≤−𝑞(𝑡)𝑥^𝛼[𝜎(𝑡)]≤0,𝑡≥𝑡₁. (3.4) Thus, 𝑟(𝑡)|𝑧′(𝑡)|^𝛼−1𝑧′(𝑡) is decreasing function. Now we have two possible cases for 𝑧^′(𝑡)∶(i)𝑧′(𝑡)<0 eventually and (ii)𝑧′(𝑡)>0 eventually.
(i) Suppose that 𝑧′(𝑡)<0 for 𝑡≥𝑡₂≥𝑡₁. Then, from (3.4), we get 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)≤𝑟𝑡₂𝑧^′𝑡₂^𝛼−1𝑧^′𝑡₂,𝑡≥𝑡₂, (3.5) which implies that 𝑧(𝑡)≤𝑧𝑡₂−𝑟^1/𝛼𝑡₂𝑧^′𝑡₂^𝑡_𝑡2𝑟^−1/𝛼(𝑠)d𝑠. (3.6) Letting 𝑡→∞, by (1.5), we find 𝑧(𝑡)→−∞, which is a contradiction.
(ii) Suppose that 𝑧′(𝑡)>0 for 𝑡≥𝑡₂≥𝑡₁. We define a Riccati substitution 𝜔(𝑡)(𝑡)𝑟(𝑡)𝑧=𝜌^′(𝑡)^𝛼(𝑧[𝜎(𝑡)])^𝛼,𝑡≥𝑡₂. (3.7) Then 𝜔(𝑡)>0. From (3.4), we have 𝑧^′[𝜎(𝑡≥𝑟(𝑡))]𝑟[𝜎(𝑡)]^1/𝛼𝑧^′(𝑡). (3.8) Differentiating (3.7), we find that 𝜔^′(𝑡)=𝜌^′(𝑡)𝑟(𝑡)𝑧^′(𝑡)^𝛼(𝑧[𝜎(𝑡)])^𝛼(𝑡)𝑟(𝑡)𝑧+𝜌^′(𝑡)^𝛼′(𝑧[𝜎(𝑡)])^𝛼(𝑡)𝑟(𝑡)𝑧−𝛼𝜌^′(𝑡)^𝛼𝑧^𝛼−1[𝜎(𝑡)]𝑧^′[𝜎(𝑡)]𝜎^′(𝑡)(𝑧[𝜎(𝑡)])^2𝛼. (3.9) Therefore, by (3.7), (3.8), and (3.9), we see that 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)𝑟(𝑡)𝑧+𝜌^′(𝑡)^𝛼′(𝑧[𝜎(𝑡)])^𝛼−𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡). (3.10) Similarly, we introduce a Riccati substitution 𝜐(𝑡)(𝑡)𝑟[𝜏(𝑡𝑧=𝜌)]^′[𝜏(𝑡)]^𝛼(𝑧[𝜎(𝑡)])^𝛼,𝑡≥𝑡₂. (3.11) Then 𝜐(𝑡)>0. From (3.4), we have 𝑧^′[𝜎(𝑡≥𝑟[𝜏(𝑡)])]𝑟[𝜎(𝑡)]^1/𝛼𝑧^′[𝜏(𝑡)]. (3.12) Differentiating (3.11), we find that 𝜐^′(𝑡)=𝜌^′(𝑡)𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼(𝑧[𝜎(𝑡)])^𝛼(𝑡)𝑟[𝜏(𝑡𝑧+𝜌)]^′[𝜏(𝑡)]^𝛼′(𝑧[𝜎(𝑡)])^𝛼(𝑡)𝑟[𝜏(𝑡𝑧−𝛼𝜌)]^′[𝜏(𝑡)]^𝛼𝑧^𝛼−1[𝜎(𝑡)]𝑧^′[𝜎(𝑡)]𝜎^′(𝑡)(𝑧[𝜎(𝑡)])^2𝛼. (3.13) Therefore, by (3.11), (3.12), and (3.13), we see that 𝜐^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜐(𝑡)(𝑡)𝑟[𝜏(𝑡𝑧+𝜌)]^′[𝜏(𝑡)]^𝛼′(𝑧[𝜎(𝑡)])^𝛼−𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡). (3.14) Thus, from (3.10) and (3.14), we have 𝜔^′(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐^′(𝑡)(𝑡)𝑟(𝑡)𝑧≤𝜌^′(𝑡)^𝛼′(𝑧[𝜎(𝑡)])^𝛼+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼′(𝑧[𝜎(𝑡)])^𝛼+𝜌′(𝑡)𝜌(𝑡)𝜔(𝑡)−𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜌^′(𝑡)𝜌(𝑡)𝜐(𝑡)−𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡). (3.15) It is follows from (3.3) that 𝜔^′(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐^′(𝑡)1≤−2^𝛼−1𝜌(𝑡)𝑄(𝑡)+𝜌^′₊(𝑡)𝜌(𝑡)𝜔(𝑡)−𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜌^′₊(𝑡)𝜌(𝑡)𝜐(𝑡)−𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡). (3.16) Integrating the above inequality from 𝑡₂ to 𝑡, we obtain that 𝜔(𝑡)−𝜔𝑡₂+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐(𝑡)−𝑝^𝛼𝜎𝑡₂𝜏^′𝑡₂𝜐𝑡₂≤−^𝑡_𝑡212^𝛼−1𝜌(𝑠)𝑄(𝑠)+d𝑠_𝑡𝑡2𝜌^′₊(𝑠)𝜌(𝑠)𝜔(𝑠)−𝛼𝜎^′(𝑠)𝜌^1/𝛼(𝑠)𝑟^1/𝛼[𝜎(𝑠𝜔)]^{(𝛼+1)/𝛼}(𝑠+)d𝑠_𝑡𝑡2𝑝^𝛼[𝜎(𝑠)]𝜏^′(𝑠)𝜌^′₊(𝑠)𝜌(𝑠)+𝜉(𝑠)₊𝜐(𝑠)−𝛼𝜎^′(𝑠)𝜌^1/𝛼(𝑠)𝑟^1/𝛼[𝜎(𝑠𝜐)]^{(𝛼+1)/𝛼}(𝑠)d𝑠. (3.17) Define 𝐴∶=𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡)]^{𝛼/(𝛼+1)}𝜔(𝑡)𝜌,𝐵∶=^′+(𝑡)𝜌(𝑡)𝛼𝛼+1𝛼𝜎′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡)]^{−𝛼/(𝛼+1)𝛼}. (3.18) Using the following inequality: 𝛼+1𝛼𝐴𝐵^1/𝛼−𝐴^{(𝛼+1)/𝛼}≤1𝛼𝐵^{(𝛼+1)/𝛼},for𝐴≥0,𝐵≥0areconstants, (3.19) we have 𝜌^′₊(𝑡)𝜌(𝑡)𝜔(𝑡)−𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡)≤1(𝛼+1)^𝛼+1𝑟[𝜎(𝑡)]𝜌^′₊(𝑡)^𝛼+1𝜌(𝑡)𝜎^′(𝑡)^𝛼. (3.20) On the other hand, define 𝐴∶=𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡)]^{𝛼/(𝛼+1)}𝜐(𝑡),𝐵∶=𝜁₊(𝑡)𝛼𝛼+1𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡)]^{−𝛼/(𝛼+1)𝛼}. (3.21) So we have 𝜁₊(𝑡)𝜐(𝑡)−𝛼𝜎^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜎(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡)≤1(𝛼+1)^𝛼+1𝑟[𝜎(𝑡)]𝜁₊(𝑡)^𝛼+1𝜌(𝑡)𝜎^′(𝑡)^𝛼. (3.22) Thus, from (3.17), we get 𝜔(𝑡)−𝜔𝑡₂+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐(𝑡)−𝑝^𝛼𝜎𝑡₂𝜏^′𝑡₂𝜐𝑡₂≤−^𝑡_𝑡2𝜌(𝑠)𝑄(𝑠)2^𝛼−1−𝑟[𝜎(𝑠)](𝛼+1)^𝛼+1𝜎^′(𝑠)^𝛼𝜌^′₊(𝑠)𝜌(𝑠)^𝛼+1+𝑝^𝛼[𝜎(𝑠)]𝜁₊(𝑠)^𝛼+1𝜏^′(𝑠)d𝑠, (3.23) which contradicts (3.1). This completes the proof.

When 𝑝(𝑡)≤𝑝₀<∞,𝜏′(𝑡)≥𝜏₀>0, where 𝑝₀,𝜏₀ are constants, we obtain the following result.

Theorem 3.2. Suppose that (1.5) holds, 𝑝(𝑡)≤𝑝₀<∞,𝜏′(𝑡)≥𝜏₀>0,𝜎∈𝐶([𝑡₀,∞),ℝ),𝜎′(𝑡)>0, 𝜎(𝑡)≤𝑡,𝜎(𝑡)≤𝜏(𝑡) for 𝑡≥𝑡₀. Further, assume that there exists a function 𝜌∈𝐶([𝑡₀,∞),(0,∞)) such that limsup_𝑡→∞^𝑡_𝑡0𝜌(𝑠)𝑄(𝑠)2^𝛼−1−1(𝛼+1)^𝛼+1𝑝1+^𝛼0𝜏₀𝑟[𝜎(𝑠)]𝜌^′₊(𝑠)^𝛼+1𝜌(𝑠)𝜎^′(𝑠)^𝛼d𝑠=∞. (3.24) Then (1.1) is oscillatory.

Proof. Let 𝑥 be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>0,𝑥[𝜏(𝑡)]>0 and 𝑥[𝜎(𝑡)]>0 for all 𝑡≥𝑡₁. Using (1.1), for all sufficiently large 𝑡, we obtain that 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′+𝑞(𝑡)𝑥^𝛼[𝜎(𝑡)]+𝑝^𝛼₀𝑞[𝜏(𝑡)]𝑥^𝛼(𝜎[𝜏(𝑡+𝑝)])_𝛼0𝜏₀𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼−1𝑧^′[𝜏(𝑡)]^′≤0. (3.25) By applying (2.1) and the definition of 𝑧, we conclude that 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′+12^𝛼−1𝑄(𝑡)𝑧^𝛼[𝜎(𝑡+𝑝)]_𝛼0𝜏₀𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼−1𝑧^′[𝜏(𝑡)]^′≤0. (3.26) The remainder of the proof is similar to that of Theorem 3.1 and hence is omitted.

Theorem 3.3. Suppose that (1.5) holds, 𝜏(𝑡)≤𝑡,𝜎(𝑡)≥𝜏(𝑡) for 𝑡≥𝑡₀. Furthermore, assume that there exists a function 𝜌∈𝐶([𝑡₀,∞),(0,∞)) such that limsup_𝑡→∞^𝑡_𝑡0𝜌(𝑠𝑄(𝑠))2^𝛼−1−𝑟[𝜏(𝑠)]𝜑(𝑠)(𝛼+1)^𝛼+1𝜏^′(𝑠)^𝛼d𝑠=∞. (3.27) Then (1.1) is oscillatory.

Proof. Let 𝑥 be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>0,𝑥[𝜏(𝑡)]>0 and 𝑥[𝜎(𝑡)]>0 for all 𝑡≥𝑡₁. Proceeding as in the proof of Theorem 3.1, we get (3.3) and (3.4). In view of (3.4), 𝑟(𝑡)|𝑧′(𝑡)|^𝛼−1𝑧′(𝑡) is decreasing function. Now we have two possible cases for 𝑧′(𝑡)∶(i)𝑧′(𝑡)<0 eventually and (ii)𝑧′(𝑡)>0 eventually.
(i) Suppose that 𝑧′(𝑡)<0 for 𝑡≥𝑡₂≥𝑡₁. Then, similar to the proof of case (i) of Theorem 3.1, we obtain a contradiction.
(ii) Suppose that 𝑧′(𝑡)>0 for 𝑡≥𝑡₂≥𝑡₁. We define a Riccati substitution 𝜔(𝑡)(𝑡)𝑟(𝑡)𝑧=𝜌^′(𝑡)^𝛼(𝑧[𝜏(𝑡)])^𝛼,𝑡≥𝑡₂. (3.28) Then 𝜔(𝑡)>0. From (3.4), we have 𝑧′[𝜏(𝑡)]≥𝑟(𝑡)𝑟[𝜏(𝑡)]^1/𝛼𝑧′(𝑡). (3.29) Differentiating (3.28), we find that 𝜔^′(𝑡)=𝜌^′(𝑡)𝑟(𝑡)𝑧^′(𝑡)^𝛼(𝑧[𝜏(𝑡)])^𝛼(𝑡)𝑟(𝑡)𝑧+𝜌^′(𝑡)^𝛼′(𝑧[𝜏(𝑡)])^𝛼(𝑡)𝑟(𝑡)𝑧−𝛼𝜌^′(𝑡)^𝛼𝑧^𝛼−1[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]𝜏^′(𝑡)(𝑧[𝜏(𝑡)])^2𝛼. (3.30) Therefore, by (3.28), (3.29), and (3.30), we see that 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)𝑟(𝑡)𝑧+𝜌^′(𝑡)^𝛼′(𝑧[𝜏(𝑡)])^𝛼−𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡). (3.31)
Similarly, we introduce a Riccati substitution 𝜐(𝑡)(𝑡)𝑟[𝜏(𝑡𝑧=𝜌)]^′[𝜏(𝑡)]^𝛼(𝑧[𝜏(𝑡)])^𝛼,𝑡≥𝑡₂. (3.32) Then 𝜐(𝑡)>0. Differentiating (3.32), we find that 𝜐^′(𝑡)=𝜌^′(𝑡)𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼(𝑧[𝜏(𝑡)])^𝛼(𝑡)𝑟[𝜏(𝑡𝑧+𝜌)]^′[𝜏(𝑡)]^𝛼′(𝑧[𝜏(𝑡)])^𝛼(𝑡)𝑟[𝜏(𝑡𝑧−𝛼𝜌)]^′[𝜏(𝑡)]^𝛼𝑧^𝛼−1[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]𝜏^′(𝑡)(𝑧[𝜏(𝑡)])^2𝛼. (3.33) Therefore, by (3.32) and (3.33), we see that 𝜐^′(𝑡)=𝜌^′(𝑡)𝜌(𝑡)𝜐(𝑡)(𝑡)𝑟[𝜏(𝑡𝑧+𝜌)]^′[𝜏(𝑡)]^𝛼′(𝑧[𝜏(𝑡)])^𝛼−𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡). (3.34) Thus, from (3.31) and (3.33), we have 𝜔^′(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐^′(𝑡)(𝑡)𝑟(𝑡)𝑧≤𝜌^′(𝑡)^𝛼′(𝑧[𝜏(𝑡)])^𝛼+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝑟[𝜏(𝑡)]𝑧^′[𝜏(𝑡)]^𝛼′(𝑧[𝜏(𝑡)])^𝛼+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)−𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜌^′(𝑡)𝜌(𝑡)𝜐(𝑡)−𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡). (3.35) It follows from (3.3) that 𝜔^′(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐^′(𝑡)1≤−2^𝛼−1𝜌(𝑡)𝑄(𝑡)+𝜌^′₊(𝑡)𝜌(𝑡)𝜔(𝑡)−𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡)+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜌^′₊(𝑡)𝜌(𝑡)𝜐(𝑡)−𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡). (3.36) Integrating the above inequality from 𝑡₂ to 𝑡, we obtain that 𝜔(𝑡)−𝜔𝑡₂+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐(𝑡)−𝑝^𝛼𝜎𝑡₂𝜏^′𝑡₂𝜐𝑡₂≤−^𝑡_𝑡212^𝛼−1𝜌(𝑠)𝑄(𝑠)+d𝑠_𝑡𝑡2𝜌^′₊(𝑠)𝜌(𝑠)𝜔(𝑠)−𝛼𝜏^′(𝑠)𝜌^1/𝛼(𝑠)𝑟^1/𝛼[𝜏(𝑠𝜔)]^{(𝛼+1)/𝛼}(𝑠+)d𝑠_𝑡𝑡2𝑝^𝛼[𝜎(𝑠)]𝜏^′(𝑠)𝜌^′₊(𝑠)𝜌(𝑠)+𝜉(𝑠)₊𝜐(𝑠)−𝛼𝜏^′(𝑠)𝜌^1/𝛼(𝑠)𝑟^1/𝛼[𝜏(𝑠𝜐)]^{(𝛼+1)/𝛼}(𝑠)d𝑠. (3.37) Define 𝐴∶=𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡)]^{𝛼/(𝛼+1)}𝜔(𝑡)𝜌,𝐵∶=^′+(𝑡)𝜌(𝑡)𝛼𝛼+1𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡)]^{−𝛼/(𝛼+1)𝛼}. (3.38) Using (3.19), we have 𝜌^′₊(𝑡)𝜌(𝑡)𝜔(𝑡)−𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜔)]^{(𝛼+1)/𝛼}(𝑡)≤1(𝛼+1)^𝛼+1𝑟[𝜏(𝑡)]𝜌^′₊(𝑡)^𝛼+1𝜌(𝑡)𝜏^′(𝑡)^𝛼. (3.39) On the other hand, define 𝐴∶=𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡)]^{𝛼/(𝛼+1)}𝜐(𝑡),𝐵∶=𝜁₊(𝑡)𝛼𝛼+1𝛼𝜏′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡)]^{−𝛼/(𝛼+1)𝛼}. (3.40) So we have 𝜁₊(𝑡)𝜐(𝑡)−𝛼𝜏^′(𝑡)𝜌^1/𝛼(𝑡)𝑟^1/𝛼[𝜏(𝑡𝜐)]^{(𝛼+1)/𝛼}(𝑡)≤1(𝛼+1)^𝛼+1𝑟[𝜏(𝑡)]𝜁₊(𝑡)^𝛼+1𝜌(𝑡)𝜏^′(𝑡)^𝛼. (3.41) Thus, from (3.37), we get 𝜔(𝑡)−𝜔𝑡₂+𝑝^𝛼[𝜎(𝑡)]𝜏^′(𝑡)𝜐(𝑡)−𝑝^𝛼𝜎𝑡₂𝜏^′𝑡₂𝜐𝑡₂≤−^𝑡_𝑡2𝜌(𝑠)𝑄(𝑠)2^𝛼−1−𝑟[𝜏(𝑠)](𝛼+1)^𝛼+1𝜏^′(𝑠)^𝛼𝜌^′₊(𝑠)𝜌(𝑠)^𝛼+1+𝑝^𝛼[𝜎(𝑠)]𝜁₊(𝑠)^𝛼+1𝜏^′(𝑠)d𝑠, (3.42) which contradicts (3.27). This completes the proof.

When 𝑝(𝑡)≤𝑝₀<∞,𝜏′(𝑡)≥𝜏₀>0, where 𝑝₀,𝜏₀ are constants, we obtain the following result.

Theorem 3.4. Suppose that (1.5) holds, 𝑝(𝑡)≤𝑝₀<∞,𝜏′(𝑡)≥𝜏₀>0,𝜏(𝑡)≤𝑡,𝜎(𝑡)≥𝜏(𝑡) for 𝑡≥𝑡₀. Furthermore, assume that there exists a function 𝜌∈𝐶([𝑡₀,∞),(0,∞)) such that limsup_𝑡→∞^𝑡_𝑡0𝜌(𝑠)𝑄(𝑠)2^𝛼−1−1(𝛼+1)^𝛼+1𝑝1+^𝛼0𝜏₀𝑟[𝜏(𝑠)]𝜌^′₊(𝑠)^𝛼+1𝜏₀𝜌(𝑠)^𝛼d𝑠=∞. (3.43) Then (1.1) is oscillatory.

Proof. Let 𝑥 be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>0,𝑥[𝜏(𝑡)]>0 and 𝑥[𝜎(𝑡)]>0 for all 𝑡≥𝑡₁. Using (1.1) and the definition of 𝑧, we obtain (3.26) for all sufficiently large 𝑡. The remainder of the proof is similar to that of Theorem 3.3 and hence is omitted.

4. Oscillation Criteria for the Case (1.13)

In this section, we will establish some oscillation criteria for (1.1) under the case (1.13).

In the following, we assume that 𝑝₀,𝜏₀ are constants.

Theorem 4.1. Suppose that (1.13) holds, 𝑝(𝑡)≤𝑝₀<∞,𝜏′(𝑡)≥𝜏₀>0,𝜎(𝑡)≤𝑡,𝜎∈𝐶([𝑡₀,∞),ℝ),𝜎′(𝑡)>0,𝜎(𝑡)≤𝜏(𝑡)? ?for 𝑡≥𝑡₀. Further, assume that there exists a function 𝜌∈𝐶([𝑡₀,∞),(0,∞)) such that (3.24) holds. If there exists a function 𝜂∈𝐶¹([𝑡₀,∞),ℝ),𝜂(𝑡)≥𝑡,𝜂′(𝑡)>0 for 𝑡≥𝑡₀ such that limsup_{𝑡′→∞𝑡}^𝑡′₀𝑄(𝑠)2^𝛼−1𝛿^𝛼(𝑠)−𝑝1+^𝛼0𝜏₀𝛼𝛼+1^𝛼+1𝜂^′(𝑠)𝛿(𝑠)𝑟^1/𝛼𝜂(𝑠)d𝑠=∞, (4.1) then (1.1) is oscillatory.

Proof. Let 𝑥 be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>0,𝑥[𝜏(𝑡)]>0 and 𝑥[𝜎(𝑡)]>0 for all 𝑡≥𝑡₁. Proceeding as in the proof of Theorem 3.2, we get (3.26). In view of (1.1), we have (3.4). Thus, 𝑟(𝑡)|𝑧′(𝑡)|^𝛼−1𝑧′(𝑡) is decreasing function. Now we have two possible cases for 𝑧′(𝑡)∶(i)𝑧′(𝑡)<0 eventually and (ii)𝑧′(𝑡)>0 eventually.
(i) Suppose that 𝑧′(𝑡)>0 for 𝑡≥𝑡₂≥𝑡₁. Then, by Theorem 3.2, we obtain a contradiction with (3.24).
(ii) Suppose that 𝑧′(𝑡)<0 for 𝑡≥𝑡₂≥𝑡₁. We define the function 𝑢 by 𝑢(𝑡)𝑟(𝑡)=−−𝑧^′(𝑡)^𝛼𝑧^𝛼𝜂(𝑡),𝑡≥𝑡₂. (4.2) Then 𝑢(𝑡)<0. Noting that 𝑟(𝑡)(−𝑧′(𝑡))^𝛼 is increasing, we get 𝑟^1/𝛼(𝑠)𝑧^′(𝑠)≤𝑟^1/𝛼(𝑡)𝑧^′(𝑡),𝑠≥𝑡≥𝑡₂. (4.3) Dividing the above inequality by 𝑟^1/𝛼(𝑠), and integrating it from 𝜂(𝑡) to 𝑡′, we obtain that 𝑧𝑡^′≤𝑧𝜂(𝑡)+𝑟^1/𝛼(𝑡)𝑧^′(𝑡)_𝜂(𝑡)^𝑡′d𝑠𝑟^1/𝛼(𝑠). (4.4) Letting 𝑡′→∞, we have 0≤𝑧𝜂(𝑡)+𝑟^1/𝛼(𝑡)𝑧^′(𝑡)𝛿(𝑡), (4.5) that is, (𝑡)𝑟−𝛿^1/𝛼(𝑡)𝑧^′(𝑡)𝑧𝜂(𝑡)≤1. (4.6) Hence, by (4.2), we get −𝛿^𝛼(𝑡)𝑢(𝑡)≤1. (4.7)
Similarly, we define the function 𝑣 by 𝑣(𝑡)𝑟[𝜏(𝑡=−)]−𝑧^′[𝜏(𝑡)]^𝛼𝑧^𝛼𝜂(𝑡),𝑡≥𝑡₂. (4.8) Then 𝑣(𝑡)<0. Noting that 𝑟(𝑡)(−𝑧′(𝑡))^𝛼 is increasing, we get the following: 𝑟(𝑡)−𝑧^′(𝑡)^𝛼≥𝑟[𝜏(𝑡)]−𝑧^′[𝜏(𝑡)]^𝛼. (4.9) Thus 0<−𝑣(𝑡)≤−𝑢(𝑡). So by (4.7), we see that −𝛿^𝛼(𝑡)𝑣(𝑡)≤1. (4.10) Differentiating (4.2), we obtain that 𝑢^′(𝑡)=−𝑟(𝑡)−𝑧^′(𝑡)^𝛼′𝑧^𝛼𝜂(𝑡)+𝛼𝑟(𝑡)−𝑧^′(𝑡)^𝛼𝑧^𝛼−1𝜂(𝑡)𝑧^′𝜂(𝑡)𝜂^′(𝑡)𝑧^2𝛼𝜂(𝑡,) (4.11) by (3.4), and we have 𝑧′[𝜂(𝑡)]≤(𝑟(𝑡)/𝑟[𝜂(𝑡)])^1/𝛼𝑧′(𝑡), so 𝑢′(𝑡)≤−𝑟(𝑡)−𝑧^′(𝑡)^𝛼′𝑧^𝛼𝜂(𝑡𝜂)−𝛼^′(𝑡)𝑟^1/𝛼𝜂(𝑡((𝑡)−𝑢))^{(𝛼+1)/𝛼}. (4.12)
Similarly, we see that 𝑣′(𝑡)≤−𝑟[𝜏(𝑡)]−𝑧^′[𝜏(𝑡)]^𝛼′𝑧^𝛼𝜂(𝑡𝜂)−𝛼^′(𝑡)𝑟^1/𝛼𝜂(𝑡((𝑡)−𝑣))^{(𝛼+1)/𝛼}. (4.13) Therefore, by (4.12) and (4.13), we get the following: 𝑢^′(𝑡)+𝑝^𝛼₀𝜏₀𝑣^′(𝑡)≤−𝑟(𝑡)−𝑧^′(𝑡)^𝛼′𝑧^𝛼𝜂(𝑡+𝑝)𝜏^𝛼0₀−𝑟[𝜏(𝑡)]−𝑧^′[𝜏(𝑡)]^𝛼′𝑧^𝛼𝜂(𝑡𝜂)−𝛼^′(𝑡)𝑟^1/𝛼𝜂(𝑡((𝑡)−𝑢))^{(𝛼+1)/𝛼}−𝛼𝑝^𝛼₀𝜏₀𝜂^′(𝑡)𝑟^1/𝛼𝜂(𝑡((𝑡)−𝑣))^{(𝛼+1)/𝛼}. (4.14) Using (3.26) and (4.14), we obtain that 𝑢^′(𝑡)+𝑝^𝛼₀𝜏₀𝑣^′(𝑡)𝑄(𝑡)≤−2^𝛼−1𝜂−𝛼^′(𝑡)𝑟^1/𝛼𝜂(𝑡((𝑡)−𝑢))^{(𝛼+1)/𝛼}−𝛼𝑝^𝛼₀𝜏₀𝜂^′(𝑡)𝑟^1/𝛼𝜂(𝑡((𝑡)−𝑣))^{(𝛼+1)/𝛼}. (4.15) Multiplying (4.15) by 𝛿^𝛼(𝑡), and integrating it from 𝑡₂ to 𝑡′, we have 𝑢𝑡^′𝛿^𝛼𝑡^′−𝑢𝑡₂𝛿^𝛼𝑡₂+𝛼_𝑡^𝑡′₂𝛿^𝛼−1(𝑠)𝜂^′(𝑠)𝑢(𝑠)𝑟^1/𝛼𝜂(𝑠)d𝑠+𝛼_𝑡^𝑡′₂𝜂^′(𝑠)𝛿^𝛼(𝑠)𝑟^1/𝛼𝜂(𝑠((𝑠)−𝑢))^{(𝛼+1)/𝛼}+𝑝d𝑠_𝛼0𝜏₀𝑣𝑡^′𝛿^𝛼𝑡^′−𝑝^𝛼₀𝜏₀𝑣𝑡₂𝛿^𝛼𝑡₂+𝛼𝑝^𝛼₀𝜏_0𝑡^𝑡′₂𝛿^𝛼−1(𝑠)𝜂^′(𝑠)𝑣(𝑠)𝑟^1/𝛼𝜂(𝑠+)d𝑠𝛼𝑝^𝛼₀𝜏_0𝑡^𝑡′₂𝜂^′(𝑠)𝛿^𝛼(𝑠)𝑟^1/𝛼𝜂(𝑠((𝑠)−𝑣))^{(𝛼+1)/𝛼}d𝑠+^𝑡′_𝑡2𝑄(𝑠)2^𝛼−1𝛿^𝛼(𝑠)d𝑠≤0. (4.16) Using (3.19), (4.7), and (4.10), we find that _𝑡^𝑡′₂𝑄(𝑠)2^𝛼−1𝛿^𝛼(𝑠)−𝑝1+^𝛼0𝜏₀𝛼𝛼+1^𝛼+1𝜂′(𝑠)𝛿(𝑠)𝑟^1/𝛼𝜂(𝑠𝑡)d𝑠≤𝑢₂𝛿^𝛼𝑡₂+𝑝^𝛼₀𝜏₀𝑣𝑡₂𝛿^𝛼𝑡₂𝑝+1+^𝛼0𝜏₀. (4.17) Letting 𝑡′→∞, we obtain a contradiction with (4.1). This completes the proof.

From Theorems 3.4 and 4.1, we have the following result.

Theorem 4.2. Suppose that (1.13) holds, 𝑝(𝑡)≤𝑝₀<∞,𝜏′(𝑡)≥𝜏₀>0,𝜏(𝑡)≤𝑡,𝜎(𝑡)≥𝜏(𝑡) for 𝑡≥𝑡₀. Further, assume that there exists a function 𝜌∈𝐶([𝑡₀,∞),(0,∞)) such that (3.43) holds. If there exists a function 𝜂∈𝐶¹([𝑡₀,∞),ℝ),𝜂(𝑡)≥𝑡,𝜂′(𝑡)>0,𝜎(𝑡)≤𝜂(𝑡) for 𝑡≥𝑡₀ such that (4.1) holds, then (1.1) is oscillatory.

5. Examples

In this section, we will give some examples to illustrate the main results.

Example 5.1. Study the second-order neutral differential equation 𝑥(𝑡)+𝑡𝑥𝑡−𝜆₁^′𝛼−1𝑥(𝑡)+𝑡𝑥𝑡−𝜆₁^′′+𝛽𝑥𝑡−𝜆₂^𝛼−1𝑥𝑡−𝜆₂=0,𝑡≥𝑡₀, (5.1) where 𝛼≥1,0<𝜆₁≤𝜆₂<1,𝛽>0 are constants.

Let 𝑟(𝑡)=1,𝑝(𝑡)=𝑡,𝜌(𝑡)=1. It is easy to see that all the conditions of Theorem 3.1 hold. Hence, (5.1) is oscillatory.

Example 5.2. Consider the second-order quasilinear neutral differential equation 𝑥(𝑡)+𝑝(𝑡)𝑥𝜆₁𝑡^′𝛼−1𝑥(𝑡)+𝑝(𝑡)𝑥𝜆₁𝑡^′′+𝛽𝑡^𝛼+1𝑥𝜆₂𝑡^𝛼−1𝑥𝜆₂𝑡=0,𝑡≥𝑡₀, (5.2) where 𝛼≥1,𝛽>0 are constants, 𝜆₁,𝜆₂∈(0,1),𝜆₂≤𝜆₁.

Let 𝑟(𝑡)=1,0≤𝑝(𝑡)≤𝑝₀<∞,𝑞(𝑡)=𝛽/𝑡^𝛼+1,𝜏(𝑡)=𝜆₁𝑡,𝜎(𝑡)=𝜆₂𝑡,and𝜌(𝑡)=𝑡^𝛼. Then, we have limsup_𝑡→∞^𝑡_𝑡0𝜌(𝑠)𝑄(𝑠)2^𝛼−1−1(𝛼+1)^𝛼+1𝑝1+^𝛼0𝜏₀𝑟[𝜎(𝑠)](𝜌′(𝑠)₊^𝛼+1𝜌(𝑠)𝜎^′(𝑠)^𝛼=𝛽d𝑠2^𝛼−1−𝛼^𝛼+1(𝛼+1)^𝛼+1𝜆^𝛼₂𝑝1+^𝛼0𝜆₁limsup_𝑡→∞^𝑡_𝑡0d𝑠𝑠=∞, (5.3) if 𝛽>2^𝛼−1𝛼^𝛼+1(1+𝑝^𝛼₀/𝜆₁)/[(𝛼+1)^𝛼+1𝜆^𝛼₂]. Hence, by Theorem 3.2, (5.2) is oscillatory if 2𝛽>^𝛼−1𝛼^𝛼+1(𝛼+1)^𝛼+1𝜆^𝛼₂𝑝1+^𝛼0𝜆₁. (5.4)

Example 5.3. Investigate the second-order neutral differential equation 𝑥(𝑡)+(𝑡−1)(𝑡−4𝜋)𝑡𝑥(𝑡−4𝜋)^′′+(𝑡−2𝜋)𝑥(𝑡−2𝜋)=0,𝑡≥𝑡₀. (5.5)

Let 𝑟(𝑡)=1,𝑝(𝑡)=(𝑡−1)(𝑡−4𝜋)/𝑡,𝑞(𝑡)=𝑡−2𝜋,and𝜌(𝑡)=1. It is easy to see that all the conditions of Theorem 3.3 hold. Hence, (5.5) is oscillatory, for example, 𝑥(𝑡)=sin𝑡/𝑡 is a solution of (5.5).

Example 5.4. Discuss the second-order quasilinear neutral differential equation 𝑥(𝑡)+𝑝(𝑡)𝑥𝜆₁𝑡^′𝛼−1𝑥(𝑡)+𝑝(𝑡)𝑥𝜆₁𝑡^′′+𝛽𝑡^𝛼+1𝑥𝜆₂𝑡^𝛼−1𝑥𝜆₂𝑡=0,𝑡≥𝑡₀, (5.6) where 𝛼≥1,𝛽>0 are constants, 𝜆₁∈(0,1),𝜆₂∈[𝜆₁,∞).

Let 𝑟(𝑡)=1,0≤𝑝(𝑡)≤𝑝₀<∞,𝑞(𝑡)=𝛽/𝑡^𝛼+1,𝜏(𝑡)=𝜆₁𝑡,𝜎(𝑡)=𝜆₂𝑡,𝜌(𝑡)=𝑡^𝛼. Then, we have limsup_𝑡→∞^𝑡_𝑡0𝜌(𝑠)𝑄(𝑠)2^𝛼−1−1(𝛼+1)^𝛼+1𝑝1+^𝛼0𝜏₀𝑟[𝜏(𝑠)]𝜌′(𝑠)₊^𝛼+1𝜏₀𝜌(𝑠)^𝛼=𝛽d𝑠2^𝛼−1−𝛼^𝛼+1(𝛼+1)^𝛼+1𝜆^𝛼₁𝑝1+^𝛼0𝜆₁limsup_𝑡→∞^𝑡_𝑡0d𝑠𝑠=∞ (5.7) if 𝛽>2^𝛼−1𝛼^𝛼+1(1+𝑝^𝛼₀/𝜆₁)/[(𝛼+1)^𝛼+1𝜆^𝛼₁]. Hence, by Theorem 3.4, (5.6) is oscillatory if 2𝛽>^𝛼−1𝛼^𝛼+1(𝛼+1)^𝛼+1𝜆^𝛼₁𝑝1+^𝛼0𝜆₁. (5.8)

Example 5.5. Examine the second-order quasilinear neutral differential equation 𝑡^2𝛼𝑥(𝑡)+𝑝(𝑡)𝑥𝜆₁𝑡^′𝛼−1𝑥(𝑡)+𝑝(𝑡)𝑥𝜆₁𝑡^′′+𝛽𝑡^𝛼−1𝑥𝜆₂𝑡^𝛼−1𝑥𝜆₂𝑡=0,𝑡≥𝑡₀, (5.9) where 𝛼≥1,𝛽>0 are constants, 𝜆₁,𝜆₂∈(0,1),𝜆₂≤𝜆₁.

Let 𝑟(𝑡)=𝑡^2𝛼,0≤𝑝(𝑡)≤𝑝₀<∞,𝑞(𝑡)=𝛽𝑡^𝛼−1,𝜏(𝑡)=𝜆₁𝑡,𝜎(𝑡)=𝜆₂𝑡,and𝜌(𝑡)=1. Then, 𝑄(𝑡)=𝛽𝜆₁^𝛼−1𝑡^𝛼−1. It is easy to see that (3.24) holds. On the other hand, taking 𝜂(𝑡)=𝑡, then 𝛿(𝑡)=1/𝑡. Therefore, one has limsup_{𝑡′→∞𝑡}^𝑡′₀𝑄(𝑠)2^𝛼−1𝛿^𝛼(𝑠)−𝑝1+^𝛼0𝜏₀𝛼𝛼+1^𝛼+1𝜂^′(𝑠)𝛿(𝑠)𝑟^1/𝛼𝜂(𝑠=𝛽)d𝑠2^𝛼−1𝜆₁^𝛼−1−12𝑝1+^𝛼0𝜆₁𝛼𝛼+1^𝛼+1limsup_{𝑡′→∞𝑡}^𝑡′₀d𝑠𝑠=∞ (5.10) if 𝛽>2^𝛼−2(1+𝑝^𝛼₀/𝜆₁)(𝛼/𝛼+1)^𝛼+1/𝜆₁^𝛼−1. Thus, by Theorem 4.1, (5.9) oscillates if 2𝛽>^𝛼−2𝜆₁^𝛼−1𝑝1+^𝛼0𝜆₁𝛼𝛼+1^𝛼+1. (5.11)

6. Conclusions

Inequality technique plays an important role in studying the oscillatory behavior of differential equations; in this paper, we establish a new inequality (2.1); by using (2.1) and Riccati substitution, we establish some new oscillation criteria for (1.1). Theorem 3.1 can be applied to the case 𝜏(𝑡)≥𝑡. Specially, taking 𝛼=1, our results include and improve the results in [15]; for example, and Theorem 4.1 includes [15, Theorem 3.1], Theorem 4.2 includes [15, Theorem 3.2]. The method can be applied on the second-order Emden-Fowler neutral differential equations 𝑟(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))^′′+𝑞(𝑡)|𝑥(𝛿(𝑡))|^𝛼−1𝑥(𝛿(𝑡))=0,𝑡≥𝑡₀, (6.1) where 𝛼≥1. It would be interesting to find another method to investigate (1.1) when 𝜏∘𝜎≢𝜎∘𝜏.

Acknowledgment

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This paper is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003), and by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).