Oscillation Criteria for Certain Even Order Neutral Delay Differential Equations with Mixed Nonlinearities

Abstract

We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities: 𝑟(𝑡)𝑧^()𝑛−1(𝑡)^𝛼−1𝑧^()𝑛−1(𝑡)^′+𝑞₀(𝑡)(𝑥𝜏₀(𝑡)^𝛼−1𝑥(𝜏₀(𝑡))+𝑞₁(𝑡)(𝑥(𝜏₁(𝑡))^𝛽−1𝑥(𝜏₁𝑞(𝑡))+₂(𝑡)(𝑥(𝜏₂(𝑡))^𝛾−1𝑥(𝜏₂(𝑡))=0,𝑡≥𝑡₀, where 𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡)), 𝑛 is even integer, and 𝛾>𝛼>𝛽>0. Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities.

1. Introduction

In this paper, we are concerned with oscillation behavior of the certain even order neutral delay differential equations with mixed nonlinearities: 𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)^′+𝑞₀(𝑡)𝑥𝜏₀(𝑡)^𝛼−1𝑥𝜏₀(𝑡)+𝑞₁(𝑡)𝑥𝜏₁(𝑡)^𝛽−1𝑥𝜏₁(𝑡)+𝑞₂(𝑡)𝑥𝜏₂(𝑡)^𝛾−1𝑥𝜏₂(𝑡)=0,𝑡≥𝑡₀, (1) where 𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡)), 𝑛 is even integer, and 𝛾>𝛼>𝛽>0 are constants. 𝑟,𝑞_𝑖∈𝐶([𝑡₀,∞),𝑅⁺), 𝑟^′(𝑡)≥0, 𝑝,𝜏_𝑖∈𝐶([𝑡₀,∞),𝑅) satisfy that 𝜏_𝑖(𝑡)≤𝑡, 𝑖=0,1,2, and there exists a function 𝜎∈𝐶([𝑡₀,∞),𝑅), such that 𝜎(𝑡)≤𝑡, lim_𝑡→∞𝜎(𝑡)=∞. We assume that there exists a function 𝜏∈𝐶¹([𝑡₀,∞),𝑅), such that 𝜏(𝑡)≤𝜏_𝑖(𝑡), 𝑖=0,1,2, 𝜏(𝑡)≤𝑡, 𝜏^′(𝑡)>0, and lim_𝑡→∞𝜏(𝑡)=∞.

We will consider the two cases ^∞_𝑡01𝑟^1/𝛼(𝑡)𝑑𝑡=∞,_∞𝑡01𝑟^1/𝛼(𝑡)𝑑𝑡<∞. (2)(3)

Recently, there have been a large number of papers devoted to the oscillation of the delay differential equations; see [1–8]. Furthermore, there have been a large number of works on the oscillation of the neutral differential equations, and we refer the readers to the articles [9–25].

Agarwal and Grace [3] studied the oscillation for functional differential equations of higher order, 𝑥^(𝑛−1)(𝑡)^𝛼′+𝑞(𝑡)𝑓𝑥𝑔(𝑡)=0, (4) and established some sufficient conditions for oscillation of (4).

Sun and Meng [6] examined the oscillation of (1), where 𝑝(𝑡)=0, 𝑛=2.

Xu and Xia [7], by means of Riccati transformation technique, established some oscillation criteria for certain even order delay differential equations: (𝑥(𝑡))^{(𝑛−1)𝛼−1}(𝑥(𝑡))^{(𝑛−1)′}+𝐹𝑡,𝑥𝑔(𝑡)=0. (5)

In 2011, Zhang et al. [8] studied the oscillatory behavior of the following higher-order half-linear delay differential equation: 𝑟(𝑡)𝑥^(𝑛−1)(𝑡)^𝛼′+𝑞(𝑡)𝑥^𝛽(𝜏(𝑡))=0,𝑡≥𝑡₀, (6) where ∫^∞_𝑡0𝑟^−(1/𝛼)(𝑡)𝑑𝑡<∞.

In 2013, Zhang et al. [23] improved those reported in [8].

Han et al. [9] studied the oscillation of second-order neutral differential equations: 𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^′(𝑡)^𝛼−1𝑍^′(𝑡)^′+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0, (7) where 𝑍(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜏) and 𝛼>0, 0≤𝑝(𝑡)<1. Some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations: 𝑟(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))^′′+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0, (8) where ∫^∞_𝑡01/𝑟(𝑡)𝑑𝑡<∞, 0≤𝑝(𝑡)≤𝑝₀<+∞.

Meng and Xu [19], by using the Riccati transformation technique and inequalities, considered the oscillation for even order quasilinear neutral differential equations: 𝑟(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜎)^{(𝑛−1)𝛼−1}×𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜎)^{(𝑛−1)′}+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0, (9) where 𝑡≥𝑡₀, 0≤𝑝(𝑡)<1.

In 2012, Sun et al. [22] considered the oscillation criteria for even order nonlinear neutral differential equations: 𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^′+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0, (10) where 𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡)), 𝑛≥2 is even integer, and 0≤𝑝(𝑡)≤𝑝₀<+∞. The results are obtained when ∫^∞𝑟⁻¹(𝑡)𝑑𝑡=∞ or ∫^∞𝑟⁻¹(𝑡)𝑑𝑡<∞. These criteria obtained in this paper extended and improved some known results in the literatures.

In 2013, Agarwal et al. [24] considered the oscillation criteria for even order neutral differential equations: 𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))^𝑛+𝑞(𝑡)𝑥(𝜎(𝑡))=0. (11) Some new criteria are established that improve a number of related results reported in the literature and can be used in cases where known theorems fail to apply.

In 2014, Zhang et al. [25] study oscillation and asymptotic behavior of solutions to two classes of higher-order delay damped differential equations with 𝑝-Laplacian like operators: 𝑎(𝑡)𝑥^(𝑛−1)(𝑡)^𝛼−1𝑥^(𝑛−1)(𝑡)^′+𝑟(𝑡)|𝑥(𝑡)|^𝛼−1𝑥^(𝑛−1)(𝑡)+𝑞(𝑡)𝑥𝑔(𝑡)^𝛼−1𝑥𝑔(𝑡)=0,𝑡≥𝑡₀, (12) where 𝛼>0. Some new criteria are presented that improve the related contributions to the subject.

Clearly, the equations (4)–(12) are special cases of (1). The purpose of this paper is to extend and improve the abovementioned oscillation theorems for certain even order neutral delay differential equations with mixed nonlinearities (1).

The paper is organized as follows. In the next section, we present some lemmas which will be used in the following results. In Sections 3 and 4, by developing Riccati transformations technique and inequalities, some sufficient conditions for oscillation of all solutions of (1) are established. In Section 5, we give an example to illustrate Theorem 11.

2. Lemmas

In this section, in order to prove our main results, we need the following lemmas.

Lemma 1 (see [5]). Let 𝑢∈𝐶^𝑛([𝑡₀,∞),𝑅⁺). If 𝑢^(𝑛)(𝑡) is eventually of one sign for all large 𝑡, then there exist 𝑡_𝑥>𝑡₁, for some 𝑡₁>𝑡₀, and an integer 𝑙, 0≤𝑙≤𝑛, with 𝑛+𝑙 even for 𝑢^(𝑛)(𝑡)≥0 or 𝑛+𝑙 odd for 𝑢^(𝑛)(𝑡)≤0 such that 𝑙>0 implies that 𝑢^(𝑘)(𝑡)>0 for 𝑡>𝑡_𝑥, 𝑘=0,1,…,𝑙−1, and 𝑙≤𝑛−1 implies that (−1)^𝑙+𝑘𝑢^(𝑘)(𝑡)>0 for 𝑡>𝑡_𝑥, 𝑘=𝑙,𝑙+1,…,𝑛−1.

Lemma 2 (see [1], Lemma 2.2.2). If the function 𝑢 is as in Lemma 1 and 𝑢^(𝑛−1)(𝑡)𝑢^(𝑛)(𝑡)≤0 for 𝑡>𝑡_𝑥, then, for every 𝜆, 0<𝜆<1, there exists a constant 𝑀>0 such that 𝑢(𝜆𝑡)≥𝑀𝑡^𝑛−1𝑢^(𝑛−1)(𝑡), (13) for all large 𝑡.

Lemma 3 (see [1], Lemma 2.2.3). If the function 𝑢 is as in Lemma 1 and 𝑢^(𝑛−1)(𝑡)𝑢^(𝑛)(𝑡)≤0 for 𝑡>𝑡_𝑥, lim_𝑡→∞𝑢(𝑡)≠0, then, for every 𝜆, 0<𝜆<1, such that 𝑢(𝑡)≥𝜆()!𝑡𝑛−1^𝑛−1𝑢^(𝑛−1)(𝑡), (14) for all large 𝑡.

Lemma 4 (see [2, 5]). Consider the half-linear differential equations 𝑎(𝑡)𝑥^′(𝑡)^𝛼−1𝑥^′(𝑡)^′+𝑞(𝑡)|𝑥(𝑡)|^𝛼−1𝑥(𝑡)=0, (15) where 𝛼>0, 𝑎, 𝑞∈𝐶([𝑡₀,∞),𝑅⁺). Then every solution of (15) is nonoscillatory if and only if there exist a real number 𝑇≥𝑡₀ and a function 𝜈∈𝐶¹([𝑡₀,∞),𝑅), such that 𝜈^′(𝑡)+𝛼𝑎^−(1/𝛼)(𝑡)|𝜈(𝑡)|^{(𝛼+1)/𝛼}(𝑡)[).+𝑞≤0,𝑡∈𝑇,∞ (16)

Lemma 5. If 𝐴 and 𝐵 are nonnegative constants, then 𝐴^𝛾−𝛾𝐴𝐵^𝛾−1+𝛾−1𝐵^𝛾≥0,𝛾>1, (17) and the equality holds if and only if 𝐴=𝐵.

In the next section, by developing Riccati transformations technique and inequalities, some sufficient conditions for oscillation of all solutions of (1) are established.

3. Oscillation Criteria for an Oscillating Function 𝑝

In this section, we assume the following.

(H) 𝑝 is an oscillating function, and lim_𝑡→∞𝑝(𝑡)=0.

Lemma 6. Assume that (2) holds. Furthermore, assume that 𝑥 is an eventually positive solution of (1), which is bounded and does not converge to zero. Then there exists 𝑡₁≥𝑡₀, such that 𝑧(𝑡)>0,𝑧^′(𝑡)>0,𝑧^(𝑛−1)(𝑡)>0,𝑧^(𝑛)(𝑡)≤0,∀𝑡≥𝑡₁. (18)

Proof. Since 𝑥 is an eventually positive solution of (1), there exists a constant 𝑡₁≥𝑡₀, such that 𝑥(𝑡)>0, 𝑥(𝜎(𝑡))>0, and 𝑥(𝜏_𝑖(𝑡))>0, 𝑖=0,1,2, for all 𝑡≥𝑡₁. Then, by (1), we have (𝑟(𝑡)|𝑧^(𝑛−1)(𝑡)|^𝛼−1𝑧^(𝑛−1)(𝑡))^′≤0, 𝑡≥𝑡₁.
Furthermore, since 𝑥 is a bounded solution and lim_𝑡→∞𝑥(𝑡)≠0, by (H) we know that lim_𝑡→∞𝑝(𝑡)𝑥(𝜎(𝑡))=0; then there exists 𝑡₂≥𝑡₁, such that 𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))>0, 𝑡≥𝑡₂. So 𝑧 is eventually positive and bounded.
The rest of the proof is similar to that of Meng and Xu [19, Lemma 2.3], so it is omitted.

Theorem 7. Assume that (H) and (2) hold. Furthermore, assume that there exists a constant 𝜆, 0<𝜆<1, and, for every constant 𝑀>0, assume that there exists a positive function 𝜌∈𝐶¹([𝑡₀,∞),𝑅), for sufficiently large 𝑡₁≥𝑡₀, such that limsup_𝑡→∞^𝑡_𝑡1𝜌(𝑠)𝑄(𝑠)−𝑟(𝑠)𝜌^′(𝑠)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑠)𝜏^𝑛−2(𝑠)𝜏^′(𝑠)^𝛼𝑑𝑠=∞, (19) where 𝑄(𝑡)1∶=2^𝛼𝑞₀(𝑡)+𝑘₁𝑞₁(𝑡)^1/𝑘1𝑘₂𝑞₂(𝑡)^1/𝑘2,𝑘₁∶=𝛾−𝛽𝛾−𝛼,𝑘₂∶=𝛾−𝛽.𝛼−𝛽 (20) Then every bounded solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a bounded nonoscillatory solution 𝑥. We may assume without loss of generality that there exists a number 𝑡₁≥𝑡₀, such that 𝑥(𝑡)>0, 𝑥(𝜎(𝑡))>0, and 𝑥(𝜏(𝑡))>0, for all 𝑡≥𝑡₁. Furthermore, we assume that lim_𝑡→∞𝑥(𝑡)≠0. Using the definition of 𝑧 and Lemma 6, we have 𝑧(𝑡)>0, 𝑧^′(𝑡)>0, 𝑧^(𝑛−1)(𝑡)>0, and 𝑧^(𝑛)(𝑡)≤0, 𝑡≥𝑡₁. Hence there exists 𝑡₂≥𝑡₁, such that 𝑥(𝑡)(𝑡)(𝑡)𝑥(𝜎(𝑡≥𝑧(𝑡)=𝑧−𝑝))2,𝑡≥𝑡₂. (21) From (1) and the above inequality, we obtain 𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^𝛼′+12^𝛼𝑞₀(𝑡)𝑧^𝛼(𝜏(𝑡+1))2^𝛽𝑞₁(𝑡)𝑧^𝛽(𝜏(𝑡+1))2^𝛾𝑞₂(𝑡)𝑧^𝛾(𝜏(𝑡))≤0,𝑡≥𝑡₃≥𝑡₂. (22) Because of 𝑧^′(𝑡)>0, by Lemma 2, 𝑧^(𝑛−1)(𝑡)>0, and 𝑧^(𝑛)(𝑡)≤0, there exists 𝑡₄≥𝑡₃, and, for every 0<𝜆<1, there exists a constant 𝑀>0, we have 𝑧^′(𝜆𝜏(𝑡))≥𝑀𝜏^𝑛−2(𝑡)𝑧^(𝑛−1)𝜏(𝑡)≥𝑀𝜏^𝑛−2(𝑡)𝑧^(𝑛−1)(𝑡), (23) for 𝑡≥𝑡₄. We define the function 𝜔 by 𝜔(𝑡)(𝑡)𝑟(𝑡)𝑧=𝜌^(𝑛−1)(𝑡)𝑧(𝜆𝜏(𝑡))^𝛼,𝑡≥𝑡₄. (24) Then 𝜔(𝑡)>0, 𝑡≥𝑡₄. Next differentiating (24), we get 𝜔^′(𝑡)=𝜌^′(𝑡)𝑟(𝑡𝑧)^(𝑛−1)(𝑡)𝑧(𝜆𝜏(𝑡))^𝛼(𝑡)𝑟(𝑡)𝑧+𝜌^(𝑛−1)(𝑡)^𝛼′𝑧^𝛼((𝑡(𝑡)𝑟(𝑡)𝑧𝜆𝜏))−𝛼𝜆𝜌^(𝑛−1)(𝑡)^𝛼𝜏^′(𝑡)𝑧^′(𝜆𝜏(𝑡))𝑧^𝛼+1((𝑡.𝜆𝜏)) (25) So by (22) and (23), we obtain 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)×1−𝜌2^𝛼𝑞₀(𝑡)+12^𝛽𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡+1))2^𝛾𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡𝑧(𝜏(𝑡))))𝑧(𝜆𝜏(𝑡))^𝛼−𝛼𝜆𝑀𝜏^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼𝜔^{(𝛼+1)/𝛼}(𝑡). (26) Let 𝑎=𝑘₁12^𝛽𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡)),𝑏=𝑘₂12^𝛾𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡)), (27) where 𝑘₁ and 𝑘₂ are defined as in Theorem 7. Using the inequality |𝑎|𝑝+|𝑏|𝑞≥|𝑎|^1/𝑝|𝑏|^1/𝑞1,𝑝>1,𝑞>1,𝑝+1𝑞=1, (28) we have 12^𝛽𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡+1))2^𝛾𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡≥𝑘))₁12^𝛽𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡))^1/𝑘1𝑘₂12^𝛾𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡))^1/𝑘2=12^𝛼𝑘₁𝑞₁(𝑡)^1/𝑘1𝑘₂𝑞₂(𝑡)^1/𝑘2, (29) so we get 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)𝑄(𝑡)−−𝜌𝛼𝜆𝑀𝜏^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼𝜔^{(𝛼+1)/𝛼}(𝑡). (30) Let 𝜏𝐴=𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼1/𝛾𝜔(𝑡),𝜌𝐵=^′(𝑡)𝜏𝛾𝜌(𝑡)𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^{1/𝛼−(1/𝛾)1/(𝛾−1)}, (31) where 𝛾=(𝛼+1)/𝛼>1. Applying the inequality in Lemma 5, we obtain 𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)𝜏−𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼𝜔^𝛾(𝑡)≤𝑟(𝑡)𝜌^′(𝑡)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑡)𝜏^𝑛−2(𝑡)𝜏^′(𝑡)^𝛼. (32) Thus, by (30) and (32), we get 𝜔^′(𝑡)𝜌(𝑡)𝑄(𝑡)−𝑟(𝑡)𝜌≤−_′(𝑡)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑡)𝜏^𝑛−2(𝑡)𝜏^′(𝑡)^𝛼. (33) Integrating (33) from 𝑡₁ to 𝑡, we have 𝜔(𝑡)≤𝜔𝑡₁−^𝑡_𝑡1𝜌(𝑠)𝑄(𝑠)−𝑟(𝑠)𝜌^′(𝑠)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑠)𝜏^𝑛−2(𝑠)𝜏^′(𝑠)^𝛼𝑑𝑠. (34) Let 𝑡→∞ in (34), which leads to a contradiction with (19). The proof is complete.

Theorem 8. Assume that (H) and (2) hold, and there exists a constant 𝜆, 0<𝜆<1, and, for every constant 𝑀>0, such that (𝑟(𝑡))^1/𝛼𝜆𝑀𝜏^𝑛−2(𝑡)𝜏^′(𝑡)^𝛼𝑥^′(𝑡)^𝛼−1𝑥^′(𝑡)^′+𝑄(𝑡)|𝑥(𝑡)|^𝛼−1𝑥(𝑡)=0, (35) is oscillatory, where 𝑄 is defined as in Theorem 7. Then every bounded solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a bounded nonoscillatory solution 𝑥. We may assume without loss of generality that there exists 𝑡₁≥𝑡₀, such that 𝑥(𝑡)>0, 𝑥(𝜎(𝑡))>0, and 𝑥(𝜏(𝑡))>0, for all 𝑡≥𝑡₁. Furthermore, we assume that lim_𝑡→∞𝑥(𝑡)≠0. We define 𝜈 by 𝜈(𝑡)(𝑡)𝑧=𝑟^(𝑛−1)(𝑡)𝑧(𝜆𝜏(𝑡))^𝛼. (36) Proceeding as in the proof of Theorem 7, for every 0<𝜆<1, there exists 𝑀>0, and we have 𝜈^′(𝑡)+𝑄(t)+𝛼𝜆𝑀𝜏^𝑛−2(𝑡)𝜏^′(𝑡)(𝑟(𝑡))^1/𝛼𝜈^{(𝛼+1)/𝛼}(𝑡)≤0. (37) That is, 𝜈^′(𝑡)(𝑡)(𝑟(𝑡+𝑄+𝛼))^1/𝛼𝜆𝑀𝜏^𝑛−2(𝑡)𝜏^′(𝑡)^{𝛼−(1/𝛼)}𝜈^{(𝛼+1)/𝛼}(𝑡)≤0. (38) Based on Lemma 4, we obtain that (35) is nonoscillatory, which leads to a contradiction. The proof is complete.

Theorem 9. Assume that (H) and (3) hold. Furthermore, assume that there exists a constant 𝜆, 0<𝜆<1, and, for every constant 𝑀>0, assume that there exists a positive function 𝜌∈𝐶¹([𝑡₀,∞),𝑅), such that (19) holds. If, for sufficiently large 𝑡₁≥𝑡₀, limsup_𝑡→∞^𝑡_𝑡1𝜆(𝑛−2)!^𝛼𝑄(𝑠)𝜏^{𝛼(𝑛−2)}(𝑠)𝛿^𝛼(𝑠)−𝛼𝛼+1^𝛼+11𝑟^1/𝛼(𝑠)𝛿(𝑠)𝑑𝑠=∞, (39) where 𝑄 and ℎ are defined as in Theorem 7, and 𝛿(𝑡)=∫^∞_𝑡(1/𝑟^1/𝛼(𝑠))𝑑𝑠, then every bounded solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a bounded nonoscillatory solution 𝑥. We may assume without loss of generality that there exists 𝑡₁≥𝑡₀, such that 𝑥(𝑡)>0, 𝑥(𝜎(𝑡))>0, and 𝑥(𝜏(𝑡))>0, for all 𝑡≥𝑡₁. Then it follows from (1) that 𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)^′≤0,𝑡≥𝑡₁. (40) Therefore, 𝑟(𝑡)|𝑧^(𝑛−1)(𝑡)|^𝛼−1𝑧^(𝑛−1)(𝑡) is a nonincreasing function on [𝑡₁,∞). Consequently, it is easy to conclude that there exist two possible cases of the sign of 𝑧^(𝑛−1)(𝑡). Furthermore, we assume that lim_𝑡→∞𝑥(𝑡)≠0.
Case I. If 𝑧^(𝑛−1)(𝑡)>0, for 𝑡≥𝑡₁, then we go back to the proof of Theorem 7, and we get a contradiction to (19), so we omit the details.
Case II. 𝑧^(𝑛−1)(𝑡)<0, for 𝑡≥𝑡₁. Applying Lemma 1, we get 𝑧^(𝑛−2)(𝑡)>0. Define the function 𝜐 by 𝜐(𝑡)=𝑟(𝑡)−𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−2)(𝑡)^𝛼,𝑡≥𝑡₁. (41) Then 𝜐(𝑡)<0 for 𝑡≥𝑡₁. Noting that 𝑟(𝑡)|𝑧^(𝑛−1)(𝑡)|^𝛼−1𝑧^(𝑛−1)(𝑡) is nonincreasing, we obtain 𝑟^1/𝛼(𝑠)𝑧^(𝑛−1)(𝑠)≤𝑟^1/𝛼(𝑡)𝑧^(𝑛−1)(𝑡),𝑠≥𝑡. (42) Dividing (42) by 𝑟^1/𝛼(𝑠) and integrating it from 𝑡 to 𝑙(𝑙≥𝑡), we have 𝑧^(𝑛−2)(𝑙)≤𝑧^(𝑛−2)(𝑡)+𝑟^1/𝛼(𝑡)𝑧^(𝑛−1)(𝑡)^𝑙_𝑡1𝑟^1/𝛼(𝑠)𝑑𝑠. (43) Letting 𝑙→∞ in the above inequality, we get 0≤𝑧^(𝑛−2)(𝑡)+𝑟^1/𝛼(𝑡)𝑧^(𝑛−1)(𝑡)𝛿(𝑡), (44) which implies that 𝑟−1≤^1/𝛼(𝑡)𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−2)(𝑡)𝛿(𝑡),𝑡≥𝑡₁, (45) where 𝛿 is defined as in Theorem 9. Hence, by (41), we obtain −1≤𝜐(𝑡)𝛿^𝛼(𝑡)≤0,𝑡≥𝑡₁. (46) Differentiating (41), we have 𝜐^′(𝑡)=𝑟(𝑡)−𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)^′𝑧^(𝑛−2)(𝑡)^𝛼−𝛼𝑟(𝑡)−𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−2)(𝑡)^𝛼+1. (47) From (22), we get 𝜐^′(𝑡)≤−12^𝛼𝑞₀(𝑡)𝑧^𝛼(𝜏(𝑡−1))2^𝛽𝑞₁(𝑡)𝑧^𝛽(𝜏(𝑡−1))2^𝛾𝑞₂(𝑡)𝑧^𝛾(𝜏(𝑡1))𝑧^(𝑛−2)(𝑡)^𝛼−𝛼𝑟(𝑡)−𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−2)(𝑡)^𝛼+1. (48) On the other hand, by lim_𝑡→∞𝑧(𝑡)≠0 and Lemma 3, we obtain 𝑧(𝑡)≥𝜆()!𝑡𝑛−2^𝑛−2𝑧^(𝑛−2)(𝑡); (49) that is, because of 𝑧^(𝑛−1)<0, 𝑧(𝜏(𝑡≥𝜆))()!𝜏𝑛−2^𝑛−2(𝑡)𝑧^(𝑛−2)(𝜏(𝑡≥𝜆))()!𝜏𝑛−2^𝑛−2(𝑡)𝑧^(𝑛−2)(𝑡), (50) for every 0<𝜆<1 and 𝑡≥𝑡₁. Then from (41), (48), and (50), we have 𝜐^′(𝑡)≤−12^𝛼𝑞₀(𝑡)−12^𝛽𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡−1))2^𝛾𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡𝑧))^𝛼(𝜏(𝑡))𝑧^(𝑛−2)(𝑡)^𝛼−𝛼𝑟(𝑡)−𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−1)(𝑡)𝑧^(𝑛−2)(𝑡)^𝛼+1𝜆≤−(𝑛−1)!^𝛼𝑄(𝑡)𝜏^{𝛼(𝑛−2)}(𝑡)−𝛼𝑟^1/𝛼(𝑡)(−𝜐(𝑡))^{(𝛼+1)/𝛼}, (51) where 𝑄 is defined as in Theorem 7. Multiplying (51) by 𝛿^𝛼(𝑡) and integrating it from 𝑡₁ to 𝑡, we get 𝛿^𝛼(𝑡)𝜐(𝑡)−𝛿^𝛼𝑡₁𝜐𝑡₁+𝛼^𝑡_𝑡1𝛿^𝛼−1(𝑠)𝑟^1/𝛼(𝑠)𝜐(𝑠)+𝜆𝑑𝑠(𝑛−2)!^𝛼𝑡_𝑡1𝑄(𝑠)𝜏^{𝛼(𝑛−2)}(𝑠)𝛿^𝛼(𝑠)𝑑𝑠+𝛼^𝑡_𝑡1𝛿^𝛼(𝑠)𝑟^1/𝛼(𝑠)(−𝜐(𝑠))^{(𝛼+1)/𝛼}𝑑𝑠≤0. (52) Let 𝐴=𝛼𝛿^𝛼(𝑠)𝑟^1/𝛼(𝑠)^1/𝛾((𝑠,𝛼−𝜐))𝐵=²𝛿𝛼+1^𝛼−1(𝑠)𝑟^1/𝛼(𝑠)𝛼𝛿^𝛼(𝑠)𝑟^1/𝛼(𝑠)^{−(1/𝛾)1/(𝛾−1)}, (53) where 𝛾=(𝛼+1)/𝛼>1. Applying the inequality in Lemma 5, we obtain 𝛼𝛿^𝛼−1(𝑠)𝑟^1/𝛼(𝑠)(−𝜐(𝑠))−𝛼𝛿^𝛼(𝑠)𝑟^1/𝛼(𝑠)(−𝜐(𝑠))^{(𝛼+1)/𝛼}≤𝛼𝛼+1^𝛼+11r^1/𝛼(𝑠)𝛿(𝑠). (54) Therefore, it follows from (52) that 𝛿^𝛼(𝑡)𝜐(𝑡)+^𝑡_𝑡1𝜆(𝑛−2)!^𝛼𝑄(𝑠)𝜏^{𝛼(𝑛−2)}(𝑠)𝛿^𝛼(𝑠)−𝛼𝛼+1^𝛼+11𝑟^1/𝛼(𝑠)𝛿(𝑠)𝑑𝑠≤𝛿^𝛼𝑡₁𝜐𝑡₁. (55) From (39) and the above inequality, we get a contradiction to (46). The proof is complete.

Theorem 10. Assume that (H) and (3) hold. Furthermore, assume that there exists a constant 𝜆, 0<𝜆<1, and for every constant 𝑀>0 such that (35) is oscillatory. If, for sufficiently large 𝑡₁≥𝑡₀, one has (39), where 𝑄 and ℎ are defined as in Theorem 7 and 𝛿 is defined as in Theorem 9, then every bounded solution of (1) is oscillatory or converges to zero.

4. Oscillation Criteria for 0≤𝑝(𝑡)<1

In this section, we assume that 0≤𝑝(𝑡)<1.

Theorem 11. Assume that (2) holds, there exists a constant 𝜆, 0<𝜆<1, and, for every constant 𝑀>0, there exists a positive function 𝜌∈𝐶¹([𝑡₀,∞),𝑅), such that, for sufficiently large 𝑡₁≥𝑡₀, limsup_𝑡→∞^𝑡_𝑡1𝜌(𝑠)−𝑟(𝑠)𝜌𝑄(𝑠)_′(𝑠)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑠)𝜏^𝑛−2(𝑠)𝜏^′(𝑠)^𝛼𝑑𝑠=∞, (56) where 𝑄(𝑡)=𝜉^𝛼(𝑡)𝑞₀(𝑡)+𝑘₁𝑞₁(𝑡)^1/𝑘1𝑘₂𝑞₂(𝑡)^1/𝑘2,𝜉(𝑡)=min1−𝑝𝜏₀(𝑡),1−𝑝𝜏₁(𝑡),1−𝑝𝜏₂(𝑡), (57) and 𝑘₁ and 𝑘₂ are defined as in Theorem 7. Then every solution of (1) is oscillatory.

Proof. Suppose that (1) has a nonoscillatory solution 𝑥. Without loss of generality, we may assume that there exists 𝑡₁≥𝑡₀, such that 𝑥(𝑡)>0, 𝑥(𝜎(𝑡))>0, and 𝑥(𝜏(𝑡))>0, for all 𝑡≥𝑡₁. Similar to the proof of Lemma 2.3 in [19], there exists 𝑡₂≥𝑡₁, such that 𝑧(𝑡)>0,𝑧^′(𝑡)>0,𝑧^(𝑛−1)(𝑡)>0,𝑧^(𝑛)(𝑡)≤0,𝑡≥𝑡₂. (58) From the definition of 𝑧, we have 𝑥(𝑡)(𝑡)(𝑡)𝑥(𝜎(𝑡(𝑡)(𝑡)𝑧(𝜎(𝑡≥(𝑡)𝑧(𝑡)=𝑧−𝑝))≥𝑧−𝑝))1−𝑝,𝑡≥𝑡₃≥𝑡₂. (59) Since lim_𝑡→∞𝜏(𝑡)=∞, there exists 𝑡₄≥𝑡₃, such that 𝜏(𝑡)≥𝑡₄, 𝑡≥𝑡₄, so 𝑥(𝜏(𝑡))≥1−𝑝(𝜏(𝑡))𝑧(𝜏(𝑡)),𝑡≥𝑡₄. (60) From (1), (59), and (60), we get 𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^𝛼′+1−𝑝𝜏₀(𝑡)^𝛼𝑞₀(𝑡)𝑧^𝛼(𝜏(𝑡𝜏))+1−𝑝₁(𝑡)^𝛽𝑞₁(𝑡)𝑧^𝛽(𝜏(𝑡𝜏))+1−𝑝₂(𝑡)^𝛾𝑞₂(𝑡)𝑧^𝛾(𝜏(𝑡))≤0,𝑡≥𝑡₄. (61) For every 0<𝜆<1, we define the function 𝜔(𝑡)(𝑡)𝑟(𝑡)𝑧=𝜌^(𝑛−1)(𝑡)𝑧(𝜆𝜏(𝑡))^𝛼,𝑡≥𝑡₄. (62) Then 𝜔(𝑡)>0, 𝑡≥𝑡₄. Next differentiating (62), we obtain 𝜔^′(𝑡)=𝜌^′(𝑡)𝑟(𝑡𝑧)^(𝑛−1)(𝑡)𝑧(𝜆𝜏(𝑡))^𝛼(𝑡)𝑟(𝑡)𝑧+𝜌^(𝑛−1)(𝑡)^𝛼′𝑧^𝛼((𝑡(𝑡)𝑟(𝑡)𝑧𝜆𝜏))−𝛼𝜆𝜌^(𝑛−1)(𝑡)^𝛼𝑧^′(𝜆𝜏(𝑡))𝑧^𝛼+1((𝑡.𝜆𝜏)) (63) From (23), (61), and (62), we have 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)×𝜉−𝜌_𝛼(𝑡)𝑞₀(𝑡)+𝜉^𝛽(𝑡)𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡))+𝜉^𝛾(𝑡)𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡𝑧(𝜏(𝑡))))𝑧(𝜆𝜏(𝑡))^𝛼𝜏−𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼𝜔^{(𝛼+1)/𝛼}(𝑡), (64) where 𝜉 is defined as in Theorem 11. Setting 𝑎=𝑘₁𝜉^𝛽(𝑡)𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡)),𝑏=𝑘₂𝜉^𝛾(𝑡)𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡)), (65) by inequality (28), we get 𝜉^𝛽(𝑡)𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡))+𝜉^𝛾(𝑡)𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡≥𝑘))₁𝜉^𝛽(𝑡)𝑞₁(𝑡)𝑧^𝛽−𝛼(𝜏(𝑡))^1/𝑘1×𝑘₂𝜉^𝛾(𝑡)𝑞₂(𝑡)𝑧^𝛾−𝛼(𝜏(𝑡))^1/𝑘2=𝜉^𝛼(𝑡)𝑘₁𝑞₁(𝑡)^1/𝑘1𝑘₂𝑞₂(𝑡)^1/𝑘2; (66) hence, 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)−𝜌(𝑡)𝜏𝑄(𝑡)−𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼𝜔^{(𝛼+1)/𝛼}(𝑡). (67) Let 𝜏𝐴=𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼1/𝛾𝜔(𝑡),𝜌𝐵=^′(𝑡)𝜏𝛾𝜌(𝑡)𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^{1/𝛼−(1/𝛾)1/(𝛾−1)}, (68) where 𝛾=(𝛼+1)/𝛼>1. Applying the inequality in Lemma 5, we obtain 𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)𝜏−𝛼𝜆𝑀^𝑛−2(𝑡)𝜏^′(𝑡)𝜌(𝑡)𝑟(𝑡)^1/𝛼𝜔^𝛾(𝑡)≤𝑟(𝑡)𝜌^′(𝑡)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑡)𝜏^𝑛−2(𝑡)𝜏^′(𝑡)^𝛼. (69) Thus, by (67) and (69), we get 𝜔^′(𝑡)≤−𝜌(𝑡)−𝑟(𝑡)𝜌𝑄(𝑡)_′(𝑡)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑡)𝜏^𝑛−2(𝑡)𝜏^′(𝑡)^𝛼. (70) Integrating (70) from 𝑡₁ to 𝑡, we have 𝜔(𝑡)≤𝜔𝑡₁−^𝑡_𝑡1𝜌(𝑠)𝑄(𝑠)−𝑟(𝑠)𝜌^′(𝑠)^𝛼+1(𝛼+1)^𝛼+1𝜆𝑀𝜌(𝑠)𝜏^𝑛−2(𝑠)𝜏^′(𝑠)^𝛼𝑑𝑠. (71) Letting 𝑡→∞ in (71), we get a contradiction with (56). This completes the proof of Theorem 11.

Remark 12. From Theorem 11, we can obtain different conditions for oscillation of all solutions of (1) with different choices of 𝜌.

Theorem 13. Assume that (3) holds, assume that there exists a constant 𝜆, 0<𝜆<1, and, for every constant 𝑀>0, there exists a positive function 𝜌∈𝐶¹([𝑡₀,∞),𝑅), such that, for sufficiently large 𝑡₁≥𝑡₀, (56) holds. If there exists a positive function 𝜂∈𝐶¹([𝑡₀,∞),𝑅), 𝜂^′(𝑡)≥0, such that ^∞_𝑡11𝜂(𝑡)𝑟(𝑡)^𝑡_𝑡1𝜆𝑀()!𝜏𝑛−2^𝑛−2(𝑠)𝛿(𝜏(𝑠))^𝛼×𝜂(𝑠)𝑄(𝑠)𝑑𝑠^1/𝛼𝑑𝑡=∞,_∞𝑡11𝜂(𝑡)𝑟(𝑡)^𝑡_𝑡1𝜆𝑀()!𝑠𝑛−2^𝑛−2𝛿(𝑠)^𝛼×𝜂(𝑠)𝑄(𝑠)𝑑𝑠^1/𝛼𝑑𝑡=∞, (72)(73) where 𝛿 is defined as in Theorem 9 and 𝑄 is defined as in Theorem 11, then every solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a nonoscillatory solution 𝑥. We may assume without loss of generality that there exists 𝑡₁≥𝑡₀, such that 𝑥(𝑡)>0, 𝑥(𝜎(𝑡))>0, 𝑥(𝜏(𝑡))>0, and 𝑧(𝑡)>0, for all 𝑡≥𝑡₁. Furthermore, we assume that lim_𝑡→∞𝑥(𝑡)≠0. Similar to the proof of Theorem 9, we find that 𝑟(𝑡)|𝑧^(𝑛−1)(𝑡)|^𝛼−1𝑧^(𝑛−1)(𝑡) is a nonincreasing function on [𝑡₁,∞) and there exist two possible cases of the sign of 𝑧^(𝑛−1)(𝑡).
Case I. If 𝑧^(𝑛−1)(𝑡)>0, for 𝑡≥𝑡₁, then we go back to the proof of Theorem 11, and we get a contradiction to (56), so we omit the details.
Case II. 𝑧^(𝑛−1)(𝑡)<0, for 𝑡≥𝑡₁. Applying Lemma 1, we get 𝑧^(𝑛−2)(𝑡)>0, 𝑧^′(𝑡)>0 or 𝑧^(𝑛−2)(𝑡)>0, 𝑧^′(𝑡)<0.
If 𝑧^(𝑛−2)(𝑡)>0, 𝑧^′(𝑡)>0. Since 𝑟(𝑡)|𝑧^(𝑛−1)(𝑡)|^𝛼−1𝑧^(𝑛−1)(𝑡) is nonincreasing, we obtain 𝑟(𝑠)𝑧^(𝑛−1)(𝑠)^𝛼−1𝑧^(𝑛−1)(𝑠)≤𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)≤𝑟𝑡₁𝑧^(𝑛−1)𝑡₁^𝛼−1𝑧^(𝑛−1)𝑡₁, (74) for 𝑠≥𝑡≥𝑡₁; that is, −𝑧^(𝑛−1)(𝑠)≥𝑟(𝑡)𝑟(𝑠)^1/𝛼−𝑧^(𝑛−1)(𝑡). (75) Integrating (75) from 𝑡 to ∞, we get 𝑧^(𝑛−2)(𝑡)≥𝑟^1/𝛼(𝑡)−𝑧^(𝑛−1)(𝑡)^∞_𝑡1𝑟^1/𝛼(𝑠)𝑑𝑠≥𝑀𝛿(𝑡),𝑡≥𝑡₁, (76) where 𝑀=𝑟^1/𝛼(𝑡₁)(−𝑧^(𝑛−1)(𝑡₁)). From (49) and the above inequality, we obtain 𝑧(𝑡)𝜆≥𝑀()!𝑡𝑛−2^𝑛−2𝛿(𝑡). (77) Using (59) and (77) in (1) and noting that 𝑧^′(𝑡)>0, we have −𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)^′=𝑞₀(𝑡)𝑥^𝛼𝜏₀(𝑡)+𝑞₁(𝑡)𝑥^𝛽𝜏₁(𝑡)+𝑞₂(𝑡)𝑥^𝛾𝜏₂(𝑡)≥𝑞₀(𝑡)1−𝑝𝜏₀(𝑡)^𝛼𝑧^𝛼𝜏₀(𝑡)+𝑞₁(𝑡)1−𝑝𝜏₁(𝑡)^𝛽𝑧^𝛽𝜏₁(𝑡)+𝑞₂(𝑡)1−𝑝𝜏₂(𝑡)^𝛾𝑧^𝛾𝜏₂(𝑡)≥𝜉^𝛼(𝑡)𝑞₀(𝑡)𝑧^𝛼(𝜏(𝑡))+𝜉^𝛽(𝑡)𝑞₁(𝑡)𝑧^𝛽(𝜏(𝑡))+𝜉^𝛾(𝑡)𝑞₂(𝑡)𝑧^𝛾(𝜏(𝑡)). (78) Setting 𝑎=𝑘₁𝜉^𝛽(𝑡)𝑞₁(𝑡)𝑧^𝛽(𝜏(𝑡)),𝑏=𝑘₂𝜉^𝛾(𝑡)𝑞₂(𝑡)𝑧^𝛾(𝜏(𝑡)), (79) by inequality (28), we get 𝜉^𝛽(𝑡)𝑞₁(𝑡)𝑧^𝛽(𝜏(𝑡))+𝜉^𝛾(𝑡)𝑞₂(𝑡)𝑧^𝛾(𝜏(𝑡≥𝑘))₁𝜉^𝛽𝑞₁(𝑡)𝑧^𝛽(𝜏(𝑡))^1/𝑘1𝑘₂𝜉^𝛾𝑞₂(𝑡)𝑧^𝛾(𝜏(𝑡))^1/𝑘2≥𝜉^𝛼(𝑡)𝑘₁𝑞₁(𝑡)^1/𝑘1𝑘₂𝑞₂(𝑡)^1/𝑘2𝑧^𝛼(𝜏(𝑡)). (80) Therefore, combining (77), (78), and (80), we obtain −𝑟(𝑡)𝑧^(𝑛−1)(𝑡)^𝛼−1𝑧^(𝑛−1)(𝑡)^′≥𝑄(𝑡)𝑧^𝛼(𝜏(𝑡))≥𝜆𝑀()!𝜏𝑛−2^𝑛−2(𝑡)𝛿(𝜏(𝑡))^𝛼𝑄(𝑡)