Oscillatory and Asymptotic Properties on a Class of Third Nonlinear Dynamic Equations with Damping Term on Time Scales

Abstract

We establish some new oscillatory and asymptotic criteria for a class of third-order nonlinear dynamic equations with damping term on time scales. The established results on one hand extend some known results in the literature on the other hand unify continuous and discrete analysis. For illustrating the validity of the established results, we also present some applications for them.

1. Introduction

The theory of time scale, which was initiated by Hilger [1], trying to treat continuous and discrete analysis in a consistent way, has received a lot of attention in recent years. Various investigations have been done by many authors. Among these investigations, some authors have taken research in the oscillatory and asymptotic properties of dynamic equations on time scales, and there has been increasing interest in obtaining sufficient conditions for the oscillation and asymptotic behavior of solutions of various dynamic equations on time scales (e.g., we refer the reader to [2–20]). But we notice that most of the investigations are concerned with oscillatory and asymptotic properties of solutions of first- or second-order dynamic equations on time scales, while relatively less attention has been paid to oscillatory and asymptotic properties of third-order dynamic equations on time scales. For recent results about the oscillation and asymptotic behavior of solutions of third-order dynamic equations on time scales, we refer the reader to [21–33]. In [34, 35], Saker researched oscillation of the following third-order dynamic equations: 𝑝(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0. (1) Based on the Riccati substitution and the analysis of the associated Riccati dynamic inequality, some new sufficient oscillatory conditions were presented.

Moreover, to our best knowledge, none of the existing results deal with oscillatory and asymptotic behavior of solutions of third-order nonlinear dynamic equations with damping term on time scales, in which the damping term brings new difficulty in obtaining oscillatory and asymptotic criteria. We now list some important results.

In this paper, we are concerned with oscillatory and asymptotic behavior of solutions of the third-order nonlinear dynamic equation with damping term on time scales of the following form: 𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ+𝑝(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡∈𝕋₀, (2) where 𝕋 is an arbitrary time scale, 𝕋₀=[𝑡₀,∞)⋂𝕋, 𝑎,𝑟,𝑝,𝑞∈𝐶_rd(𝕋₀,ℝ₊),𝑓∈𝐶(ℝ,ℝ) satisfying 𝑥𝑓(𝑥)>0,𝑓(𝑥)/𝑥^𝛾≥𝐿>0 for 𝑥≠0, and 𝛾≥1 is a quotient of two odd positive integers.

A solution of (2) is said to be oscillatory if it is neither eventually positive nor eventually negative otherwise it is nonoscillatory. Equation (2) is said to be oscillatory in case all its solutions are oscillatory.

We will establish some new criteria of oscillatory and asymptotic behavior for (2) by a generalized Riccati transformation technique in Section 2 and present some applications for our results in Section 3. Throughout this paper, ℝ denotes the set of real numbers and ℝ₊=(0,∞), while ℤ denotes the set of integers. 𝕋 denotes an arbitrary time scale and 𝑡_𝑖∈𝕋,𝑖=1,2,…,5. On 𝕋 we define the forward and backward jump operators 𝜎∈(𝕋,𝕋) and 𝜌∈(𝕋,𝕋) such that 𝜎(𝑡)=inf{𝑠∈𝕋,𝑠>𝑡}, 𝜌(𝑡)=sup{𝑠∈𝕋,𝑠<𝑡}. A point 𝑡∈𝕋 with 𝑡>inf𝕋 is said to be left-dense if 𝜌(𝑡)=𝑡, right-dense if 𝜎(𝑡)=𝑡, left-scattered if 𝜌(𝑡)<𝑡, and right-scattered if 𝜎(𝑡)>𝑡. A function 𝑓∈(𝕋,ℝ) is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while 𝑓 is called regressive if 1+𝜇(𝑡)𝑓(𝑡)≠0, where 𝜇(𝑡)=𝜎(𝑡)−𝑡. 𝐶_rd denotes the set of rd-continuous functions, while ℜ denotes the set of all regressive and rd-continuous functions, and ℜ⁺={𝑓∣𝑓∈ℜ,1+𝜇(𝑡)𝑓(𝑡)>0,∀𝑡∈𝕋}.

Definition 1. For 𝑝∈ℜ, the exponential function is defined by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏for𝑠,𝑡∈𝕋. (3)

Remark 2. If 𝕋=ℝ, then 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝑝(𝜏)𝑑𝜏,for𝑠,𝑡∈ℝ. (4) If 𝕋=ℤ, then 𝑒_𝑝(𝑡,𝑠)=^𝑡−1_𝜏=𝑠1+𝑝(𝜏),for𝑠,𝑡∈ℤ,𝑠<𝑡. (5)
The following two theorems include some known properties on the exponential function.

Theorem 3 (see [36, Theorem 5.1]). If 𝑝∈ℜ and fix 𝑡₀∈𝕋, then the exponential function 𝑒_𝑝(𝑡,𝑡₀) is the unique solution of the following initial value problem 𝑦^Δ(𝑡)(𝑡)𝑦(𝑡),𝑦𝑡=𝑝₀=1. (6)

Theorem 4 (see [36, Theorem 5.2]). If 𝑝∈ℜ⁺, then 𝑒_𝑝(𝑡,𝑠)>0 for  ∀𝑠,𝑡∈𝕋.

For more details about the calculus of time scales, we refer to [37].

2. Main Results

For the sake of convenience, in the rest of this paper, we set 𝛿₁(𝑡,𝑎)=∫^𝑡_𝑎([𝑒_−𝑝/𝑎(𝑠,𝑡₀)]^1/𝛾/𝑎^1/𝛾(𝑠))Δ𝑠,𝛿₂(𝑡,𝑎)=∫^𝑡_𝑎(𝛿₁(𝑠,𝑎)/𝑟(𝑠))Δ𝑠.

Lemma 5. Suppose −𝑝/𝑎∈ℜ₊, and assume that ^∞_𝑡0𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)Δ𝑠=∞,_∞𝑡01𝑟(𝑠)Δ𝑠=∞, (7)(8) and (2) has an eventually positive solution 𝑥. Then there exists a sufficiently large 𝑇^∗₁∈𝕋 such that 𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾𝑒_−𝑝/𝑎𝑡,𝑡₀^Δ𝑟(𝑡)𝑥<0,_Δ(𝑡)^Δ>0𝑜𝑛𝑇^∗₁,∞_𝕋. (9)

Proof. By −𝑝/𝑎∈ℜ₊, we have 𝑒_−𝑝/𝑎(𝑡,𝑡₀)>0. Since 𝑥 is eventually a positive solution of (2), there exists a sufficiently large 𝑡₁ such that 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋, and for 𝑡∈[𝑡₁,∞)_𝕋, we obtain that 𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾𝑒_−𝑝/𝑎𝑡,𝑡₀^Δ=𝑒_−𝑝/𝑎𝑡,𝑡₀𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ−𝑒_−𝑝/𝑎𝑡,𝑡₀^Δ𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾×𝑒_−𝑝/𝑎𝑡,𝑡₀𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀⁻¹=𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ+𝑝(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀=−𝑞(𝑡)𝑓(𝑥(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀<0. (10) Then 𝑎(𝑡)([𝑟(𝑡)𝑥^Δ(𝑡)]^Δ)^𝛾/𝑒_−𝑝/𝑎(𝑡,𝑡₀) is strictly decreasing on [𝑡₁,∞)_𝕋, and together with 𝑎(𝑡)>0, 𝑒_−𝑝/𝑎(𝑡,𝑡₀)>0 we deduce that [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ is eventually of one sign. We claim [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ>0 on [𝑡₂,∞)_𝕋. Otherwise, assume there exists a sufficiently large 𝑡₃ such that [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ<0 on [𝑡₃,∞)_𝕋. Then 𝑟(𝑡)𝑥^Δ(𝑡)−𝑟𝑡₃𝑥^Δ𝑡₃=^𝑡_𝑡3𝑒_−𝑝/𝑎𝑠,𝑡₀𝑎(𝑠)^1/𝛾𝑟(𝑠)𝑥^Δ(𝑠)^Δ𝑒_−𝑝/𝑎𝑠,𝑡₀𝑎(𝑠)^1/𝛾≤𝑎Δ𝑠_1/𝛾𝑡₃𝑟𝑡₃𝑥^Δ𝑡₃^Δ𝑒_−𝑝/𝑎𝑡₃,𝑡₀^{1/𝛾𝑡𝑡3}𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)Δ𝑠. (11) By (7), we have lim_𝑡→∞𝑟(𝑡)𝑥^Δ(𝑡)=−∞, and thus there exists a sufficiently large 𝑡₄∈[𝑡₃,∞)_𝕋 such that 𝑟(𝑡)𝑥^Δ(𝑡)<0 on [𝑡₄,∞)_𝕋. By the assumption [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ<0 one can see 𝑟(𝑡)𝑥^Δ(𝑡) is strictly decreasing on [𝑡₄,∞)_𝕋, and then 𝑥(𝑡)−𝑥𝑡₄=^𝑡_𝑡4𝑟(𝑠)𝑥^Δ(𝑠)𝑟(𝑠)Δ𝑠≤𝑟𝑡₄𝑥^Δ𝑡₄^𝑡_𝑡41𝑟(𝑠)Δ𝑠. (12) Using (8), we have lim_𝑡→∞𝑥(𝑡)=−∞, which leads to a contradiction. So [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ>0 on [𝑡₂,∞)_𝕋, and the proof is complete with taking 𝑇^∗₁=𝑡₂.

Lemma 6. Under the conditions of Lemma 5, furthermore, assume that lim_𝑡→∞sup^𝑡_𝑡01𝑟(𝜉)×^∞_𝜉𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)×^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠^1/𝛾Δ𝜏Δ𝜉=∞. (13) Then either there exists a sufficiently large 𝑇^∗₂∈𝕋 such that 𝑥^Δ(𝑡)>0 on [𝑇^∗₂,∞)_𝕋 or lim_𝑡→∞𝑥(𝑡)=0.

Proof. By Lemma 5, we deduce that 𝑥^Δ(𝑡) is eventually of one sign. So there exists a sufficiently large 𝑡₅>𝑡₄ such that either 𝑥^Δ(𝑡)>0 or 𝑥^Δ(𝑡)<0 on [𝑡₅,∞)_𝕋, where 𝑡₄ is defined as in Lemma 5. If 𝑥^Δ(𝑡)<0, together with 𝑥(𝑡) is an eventually positive solution of (2), we obtain lim_𝑡→∞𝑥(𝑡)=𝛼≥0 and lim_𝑡→∞𝑟(𝑡)𝑥^Δ(𝑡)=𝛽≤0. We claim 𝛼=0. Otherwise, assume 𝛼>0. Then 𝑥(𝑡)≥𝛼 on [𝑡₅,∞)_𝕋, and for 𝑡∈[𝑡₅,∞)⋂𝕋, an integration for (10) from 𝑡 to ∞ yields −𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾𝑒_−𝑝/𝑎𝑡,𝑡₀=−lim_𝑡→∞𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾𝑒_−𝑝/𝑎𝑡,𝑡₀+^∞_𝑡−𝑞(𝑠)𝑓(𝑥(𝑠))𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠≤−lim_𝑡→∞𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾𝑒_−𝑝/𝑎𝑡,𝑡₀+^∞_𝑡−𝐿𝑞(𝑠)𝑥^𝛾(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠≤−𝐿^∞_𝑡𝑞(𝑠)𝑥^𝛾(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠≤−𝐿𝛼^𝛾∞_𝑡𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠, (14) which is followed by −𝑟(𝑡)𝑥^Δ(𝑡)^Δ≤−𝐿𝛼^𝛾𝑒_−𝑝/𝑎𝑡,𝑡₀𝑎(𝑡)^∞_𝑡𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠^1/𝛾. (15) Substituting 𝑡 with 𝜏 in (15), an integration for (15) with respect to 𝜏 from 𝑡 to ∞ yields 𝑟(𝑡)𝑥^Δ(𝑡)=lim_𝑡→∞𝑟(𝑡)𝑥^Δ(𝑡)−𝛼𝐿^1/𝛾×^∞_𝑡𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠^1/𝛾Δ𝜏=𝛽−𝛼𝐿^1/𝛾∞_𝑡𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠^1/𝛾Δ𝜏≤−𝛼𝐿^1/𝛾∞_𝑡𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠^1/𝛾Δ𝜏, (16) which implies 𝑥^Δ(𝑡)≤−𝛼𝐿^1/𝛾1𝑟(𝑡)^∞_𝑡𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)×^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠^1/𝛾Δ𝜏. (17) Substituting 𝑡 with 𝜉 in (17), an integration for (17) with respect to 𝜉 from 𝑡₅ to 𝑡 yields 𝑥(𝑡)−𝑥𝑡₅≤−𝛼𝐿^1/𝛾𝑡_𝑡51𝑟(𝜉)×^∞_𝜉𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)×^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀Δ𝑠^1/𝛾Δ𝜏Δ𝜉. (18) By (18) and (13) we have lim_𝑡→∞𝑥(𝑡)=−∞, which leads to a contradiction. So one has 𝛼=0, and the proof is complete.

Lemma 7. Suppose −𝑝/𝑎∈ℜ₊, and assume that 𝑥 is a positive solution of (2) such that 𝑟(𝑡)𝑥^Δ(𝑡)^Δ>0,𝑥^Δ(𝑡)>0𝑜𝑛𝑇^∗₃,∞_𝕋, (19) where 𝑇^∗₃∈𝕋 is sufficiently large. Then for 𝑡∈[𝑇^∗₃,∞)_𝕋, we have 𝑥^Δ(𝑡)≥𝛿₁𝑡,𝑇^∗₃𝑟(𝑡)𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾,𝑥(𝑡)≥𝛿₂𝑡,𝑇^∗₃𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾. (20)

Proof. Take 𝑇^∗₃>max(𝑇^∗₁,𝑇^∗₂), where 𝑇^∗₁,𝑇^∗₂ are defined as in Lemmas 5 and 6, respectively. By Lemma 5 we have 𝑎(𝑡)([𝑟(𝑡)𝑥^Δ(𝑡)]^Δ)^𝛾/𝑒_−𝑝/𝑎(𝑡,𝑡₀)strictly decreasing on [𝑇^∗₃,∞). So 𝑟(𝑡)𝑥^Δ(𝑡)≥𝑟(𝑡)𝑥^Δ(𝑡)−𝑟𝑇^∗₃𝑥^Δ𝑇^∗₃=^𝑡_𝑇∗3𝑒_−𝑝/𝑎𝑠,𝑡₀𝑎(𝑠)^1/𝛾𝑟(𝑠)𝑥^Δ(𝑠)^Δ𝑒_−𝑝/𝑎𝑠,𝑡₀𝑎(𝑠)^1/𝛾≥𝑎Δ𝑠_1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾×^𝑡_𝑇∗3𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)Δ𝑠=𝛿₁𝑡,𝑇^∗₃𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾, (21) and then 𝑥^Δ(𝑡)≥𝛿₁𝑡,𝑇^∗₃𝑟(𝑡)𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾. (22) Furthermore, 𝑥(𝑡)≥𝑥(𝑡)−𝑥𝑇^∗₃=^𝑡_𝑇∗3𝑥^Δ(𝑠)≥Δ𝑠_𝑡𝑇∗3𝛿₁𝑠,𝑇^∗₃𝑟(𝑠)𝑎^1/𝛾(𝑠)𝑟(𝑠)𝑥^Δ(𝑠)^Δ𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾≥𝑎Δ𝑠_1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾×^𝑡_𝑇∗3𝛿₁𝑠,𝑇^∗₃𝑟(𝑠)Δ𝑠=𝛿₂𝑡,𝑇^∗₃𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾, (23) which is the desired result.

Lemma 8 (see [38, Theorem 41]). Assume that 𝑋 and 𝑌 are nonnegative real numbers. Then 𝜆𝑋𝑌^𝜆−1−𝑋^𝜆≤(𝜆−1)𝑌^𝜆∀𝜆>1. (24)

Theorem 9. Suppose −𝑝/𝑎∈ℜ₊, and assume that (7), (8), and (13) hold, and for all sufficiently large 𝑇,lim_𝑡→∞sup^𝑡_𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+𝛾+1𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1Δ𝑠=∞, (25)where 𝜌,𝜂 are two given nonnegative functions on 𝕋 with 𝜌(𝑡)>0. Then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution 𝑥 on 𝕋₀. Without loss of generality, we may assume 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋, where 𝑡₁ is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large 𝑡₂ such that [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ>0 on [𝑡₂,∞)_𝕋, and either 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋 or lim_𝑡→∞𝑥(𝑡)=0. Now we assume 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋. Define the generalized Riccati function: 𝜔(𝑡)(𝑡)𝑎(𝑡)𝑟(𝑡)𝑥=𝜌^Δ(𝑡)^Δ𝛾𝑥^𝛾(𝑡)𝑒_−𝑝/𝑎𝑡,𝑡₀+𝜂(𝑡). (26) Then for 𝑡∈[𝑡₂,∞)_𝕋, we have 𝜔^Δ(𝑡)=𝜌(𝑡)𝑥^𝛾(𝑡)𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾𝑒_−𝑝/𝑎𝑡,𝑡₀^Δ+𝜌(𝑡)𝑥^𝛾(𝑡)^Δ×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ+𝜌^Δ(𝑡)𝑎(𝜎(𝑡𝜂(𝜎(𝑡=𝜌(𝑡)))))𝑥^𝛾(𝑡)×𝑒_−𝑝/𝑎𝑡,𝑡₀𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ−𝑒_−𝑝/𝑎𝑡,𝑡₀^Δ𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾×𝑒_−𝑝/𝑎𝑡,𝑡₀𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀⁻¹+𝑥^𝛾(𝑡)𝜌^Δ(𝑡)−𝑥^𝛾(𝑡)^Δ𝜌(𝑡)𝑥^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡×𝑎(𝜎(𝑡𝑟(𝜎(𝑡𝑥))))))^Δ(𝜎(𝑡))^Δ𝛾𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ+𝜌^Δ(𝑡)𝑎(𝜎(𝑡𝜂(𝜎(𝑡=𝜌(𝑡)))))𝑥^𝛾(𝑡)×𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ+𝑝(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾×𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀⁻¹+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡−𝜌(𝑡)𝑥))))^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑥^𝛾(𝜎(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ𝜌(𝑡)=−𝑥^𝛾(𝑡)𝑞(𝑡)𝑓(𝑥(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡−𝜌(𝑡)𝑥))))_𝛾(𝑡)^Δ𝑥^𝛾(𝑡)𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑥^𝛾(𝜎(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ𝑞(𝑡)𝜌(𝑡)≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡−𝜌(𝑡)𝑥))))_𝛾(𝑡)^Δ𝑥^𝛾(𝑡)×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑥^𝛾(𝜎(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ. (27) By [37, Theorem 1.93], we have (𝑥^𝛾(𝑡))^Δ≥𝛾𝑥^𝛾−1(𝑡)𝑥^Δ(𝑡). Then 𝜔^Δ(𝑡)𝑞(𝑡)𝜌(𝑡)≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡(𝑡))))−𝜌)𝛾𝑥^𝛾−1(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑥^𝛾(𝜎(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ𝑞(𝑡)𝜌(𝑡)≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡−(𝑡)))))𝛾𝜌𝑥(𝜎(𝑡×𝛿))₁𝑡,𝑡₂𝑟(𝑡)𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑥^𝛾(𝜎(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ𝑞(𝑡)𝜌(𝑡)=−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡𝜌(𝑡)𝛿))))−𝛾¹𝑡,𝑡₂𝑟(𝑡)×𝜔(𝜎(𝑡))𝜌(𝜎(𝑡(𝜎(𝑡𝜂(𝜎(𝑡))−𝑎))))^1+1/𝛾+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ. (28) Using the following inequality (see [25, (2.17)]): (𝑢−𝑣)^1+1/𝛾≥𝑢^1+1/𝛾+1𝛾𝑣^1+1/𝛾−11+𝛾𝑣^1/𝛾𝑢, (29) where 𝑢,𝑣 are constants and 𝛾≥1 is a quotient of two odd positive integers, we obtain 𝜔(𝜎(𝑡))𝜌(𝜎(𝑡(𝜎(𝑡𝜂(𝜎(𝑡))−𝑎))))^1+1/𝛾≥𝜔^1+1/𝛾(𝜎(𝑡))𝜌^1+1/𝛾(𝜎(𝑡+1))𝛾𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1+1/𝛾−11+𝛾𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1/𝛾𝜔(𝜎(𝑡))𝜌(𝜎(𝑡.)) (30) A combination of (28) and (30) yields 𝜔^Δ(𝑡)𝑞(𝑡)𝜌(𝑡)≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ−𝜌(𝑡)𝛿₁𝑡,𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1+1/𝛾𝑟(𝑡)+𝑟(𝑡)𝜌^Δ(𝑡)+𝛾+1𝜌(𝑡)𝛿₁𝑡,𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1/𝛾𝑟(𝑡)𝜌(𝜎(𝑡(𝜎(𝑡𝜌(𝑡)𝛿))×𝜔))−𝛾¹𝑡,𝑡₂𝑟(𝑡)𝜔^1+1/𝛾(𝜎(𝑡))𝜌^1+1/𝛾(𝜎(𝑡.)) (31) Setting 1𝜆=1+𝛾,𝑋^𝜆𝜌(𝑡)𝛿=𝛾¹𝑡,𝑡₂𝑟(𝑡)𝜔^1+1/𝛾(𝜎(𝑡))𝜌^1+1/𝛾(𝜎(𝑡,𝑌))_𝜆−1=𝛾^1/(𝛾+1)×𝑟(𝑡)𝜌^Δ(𝑡)+𝛾+1𝜌(𝑡)𝛿₁𝑡,𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑡)𝜌^{𝛾/(𝛾+1)}(𝑡)𝛿₁^{𝛾/(𝛾+1)}𝑡,𝑡₂. (32) Using Lemma 8 in (31) we get that 𝜔^Δ(𝑡)𝑞(𝑡)𝜌(𝑡)≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ−𝜌(𝑡)𝛿₁𝑡,𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1+1/𝛾𝑟(𝑡)+𝑟(𝑡)𝜌^Δ(𝑡)+(𝛾+1)𝜌(𝑡)𝛿₁(𝑡,𝑡₂)𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑡)𝜌^{𝛾/(𝛾+1)}(𝑡)𝛿₁^{𝛾/(𝛾+1)}𝑡,𝑡₂^𝛾+1. (33) Substituting 𝑡 with 𝑠 in (33), an integration for (33) with respect to 𝑠 from 𝑡₂ to 𝑡 yields^𝑡_𝑡2𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+𝛾+1𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1×Δ𝑠≤𝜔𝑡₂−𝜔(𝑡)≤𝜔𝑡₂<∞, (34)which contradicts (25). So the proof is complete.

In Theorem 9, if we take 𝕋 for some special cases, then we can obtain the following corollaries.

Corollary 10. Let 𝕋=ℝ. Assume that ^∞_𝑡0𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞,_∞𝑡01𝑟(𝜉)^∞_𝜉𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)×^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₀𝑑𝑠^1/𝛾𝑑𝜏𝑑𝜉=∞, (35) and for all sufficiently large 𝑇,lim_𝑡→∞sup∫^𝑡_𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^′+𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝑠)𝜂(𝑠)^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^′(𝑠)+𝛾+1𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝑠)𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1𝑑𝑠=∞. (36) Then every solution of (2) is oscillatory or tends to zero.

Corollary 11. Let 𝕋=ℤ and −𝑝/𝑎∈ℜ₊. Assume that ^∞_𝑠=𝑡0𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)=∞,_{∞𝑠=𝑡0}1𝑟(𝑠)=∞,_{∞𝜉=𝑡0}1𝑟(𝜉)^∞_𝜏=𝜉𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)^∞_𝑠=𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝑠+1,𝑡₀^1/𝛾=∞, (37) and for all sufficiently large 𝑇, lim_𝑡→∞sup∑^𝑡−1_𝑠=𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠+1,𝑡₀(𝑡𝑎()𝜂()(𝑠)𝜂(𝑠+𝜌(𝑠)𝛿−𝜌)𝑠+1𝑠+1−𝑎)¹(𝑠,𝑇)𝑎(𝑠+1)𝜂(𝑠+1)^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)(𝜌(𝑠+1)−𝜌(𝑠))+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝑠+1)𝜂(𝑠+1)^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1=∞. (38)Then every solution of (2) is oscillatory or tends to zero.

Theorem 12. Suppose −𝑝/𝑎∈ℜ₊, and assume that (7), (8), and (13) hold, and for all sufficiently large 𝑇, lim_𝑡→∞sup^𝑡_𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)Δ𝑠=∞, (39)where 𝜌,𝜂 are defined as in Theorem 9, then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution 𝑥 on 𝕋₀. Similar to Theorem 9, we may assume 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋, where 𝑡₁ is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large 𝑡₂ such that [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ>0 on [𝑡₂,∞)_𝕋, and either 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋 or lim_𝑡→∞𝑥(𝑡)=0. Now we assume 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋. Let 𝜔(𝑡) be defined as in Theorem 9. By Lemma 7, we have the following observation: 𝑥^Δ(𝑡)𝑥(𝑡)≥𝛿₁𝑡,𝑡₂𝑟(𝑡)𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾1𝑥(𝜎(𝑡=𝛿))₁𝑡,𝑡₂𝑟(𝑡)𝑎^1/𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑎𝑡,𝑡₀^1/𝛾𝑥^𝛾(𝜎(𝑡𝑥))^𝛾−1(𝜎(𝑡≥𝛿))₁𝑡,𝑡₂𝑟(𝑡)𝑎^1/𝛾(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀^1/𝛾𝑥^𝛾(𝜎(𝑡𝑥))^𝛾−1(𝜎(𝑡≥𝛿))₁𝑡,𝑡₂𝑟(𝑡)𝑎^1/𝛾(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀^1/𝛾𝑥^𝛾(𝜎(𝑡𝛿))^2𝛾−1×𝜎(𝑡),𝑡₂𝑎^1/𝛾(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀^{1/𝛾𝛾−1}=𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑟(𝑡)×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀𝑥^𝛾(𝜎(𝑡.)) (40) Using (40) in (28) we get that𝜔^Δ(𝑡)𝑞(𝑡)𝜌(𝑡)≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡(𝑡))))−𝜌)𝛾𝑥^Δ(𝑡)𝑥(𝑡)×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑥^𝛾(𝜎(𝑡))𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ𝑞(𝑡)𝜌(𝑡)≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡(𝑡)𝛿))))−𝛾𝜌¹𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑟(𝑡)×𝑎(𝜎(𝑡))𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^Δ𝛾𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀𝑥^𝛾(𝜎(𝑡))²+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ𝑞(𝑡)𝜌(𝑡)=−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌^Δ(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡−(𝑡)𝛿))))𝛾𝜌₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑟(𝑡)×𝜔(𝜎(𝑡))𝜌(𝜎(𝑡(𝜎(𝑡𝜂(𝜎(𝑡))−𝑎))))²+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ𝑞(𝑡)𝜌(𝑡)=−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ−𝛾𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑎²(𝜎(𝑡))𝜂²(𝜎(𝑡))𝑟(𝑡)+𝑟(𝑡)𝜌^Δ(𝑡)+2𝛾𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))𝑟(𝑡)𝜌(𝜎(𝑡𝜔(𝜎(𝑡−(𝑡)𝛿))))𝛾𝜌₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑟(𝑡)𝜌²(𝜎(𝑡𝜔))²(𝜎(𝑡𝑞(𝑡)𝜌(𝑡)))≤−𝐿𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀+𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ−𝛾𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑎²(𝜎(𝑡))𝜂²(𝜎(𝑡))𝑟(𝑡)+𝑟(𝑡)𝜌^Δ(𝑡)+2𝛾𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))²4𝛾𝑟(𝑡)𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂. (41)Substituting 𝑡 with 𝑠 in (41), an integration for (41) with respect to 𝑠 from 𝑡₂ to 𝑡 yields∫^𝑡_𝑡2𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂Δ𝑠≤𝜔𝑡₂−𝜔(𝑡)≤𝜔𝑡₂<∞, (42)which contradicts (39). So the proof is complete.

Based on Theorems 9 and 12, we will establish some Philos-type oscillation criteria for (2).

Theorem 13. Suppose −𝑝/𝑎∈ℜ₊, and assume that (7), (8), and (13) hold, and define 𝔻={(𝑡,𝑠)∣𝑡≥𝑠≥𝑡₀}. If there exists a function 𝐻∈𝐶_rd(𝔻,ℝ) such that 𝐻(𝑡,𝑡)=0,𝑓𝑜𝑟𝑡≥𝑡₀,𝐻(𝑡,𝑠)>0,𝑓𝑜𝑟𝑡>𝑠≥𝑡₀, (43) and 𝐻 has a nonpositive continuous Δ− partial derivative 𝐻^Δ𝑠(𝑡,𝑠) with respect to the second variable, and for all sufficiently large 𝑇,lim_𝑡→∞1sup𝐻𝑡,𝑡₀∫^𝑡_𝑡0𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+𝛾+1𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1Δ𝑠=∞, (44)where 𝜌,𝜂 are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution 𝑥 on 𝕋₀. Without loss of generality, we may assume 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋, where 𝑡₁ is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large 𝑡₂ such that [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ>0 on [𝑡₂,∞)_𝕋, and either 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋 or lim_𝑡→∞𝑥(𝑡)=0. Now we assume 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋. Let 𝜔(𝑡) be defined as in Theorem 9. By (33) we have 𝐿𝑞(𝑡)𝜌(𝑡)𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀−𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ+𝜌(𝑡)𝛿₁𝑡,𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1+1/𝛾𝑟(𝑡)−𝑟(𝑡)𝜌^Δ(𝑡)+𝛾+1𝜌(𝑡)𝛿₁𝑡,𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑡)𝜌^{𝛾/(𝛾+1)}(𝑡)𝛿₁^{𝛾/(𝛾+1)}𝑡,𝑡₂^𝛾+1≤−𝜔^Δ(𝑡). (45) Substituting 𝑡 with 𝑠 in (45) and multiplying both sides by 𝐻(𝑡,𝑠) and then integrating with respect to 𝑠 from 𝑡₂ to 𝑡 yield∫^𝑡_𝑡2𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑡₂)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1Δ𝑠≤−∫^𝑡_𝑡2𝐻(𝑡,𝑠)𝜔^Δ(𝑠)Δ𝑠=𝐻𝑡,𝑡₂𝜔𝑡₂+∫^𝑡_𝑡2𝐻^Δ𝑠(𝑡,𝑠)𝜔(𝜎(𝑠))Δ𝑠≤𝐻𝑡,𝑡₂𝜔𝑡₂≤𝐻𝑡,𝑡₀𝜔𝑡₂. (46) Then ^𝑡_𝑡0𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑡₂)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1=Δ𝑠_𝑡2𝑡0𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑡₂)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1Δ𝑠+^𝑡_𝑡2𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑡₂)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1Δ𝑠≤𝐻𝑡,𝑡₀𝜔𝑡₂+𝐻𝑡,𝑡₀×^𝑡2_𝑡0𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑡₂)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1Δ𝑠. (47) So lim_𝑡→∞1sup𝐻𝑡,𝑡₀∫^𝑡_𝑡0𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑡₂)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1𝑡Δ𝑠≤𝜔₂+∫^𝑡2_𝑡0𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁𝑠,𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑡₂)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}𝑠,𝑡₂^𝛾+1Δ𝑠<∞, (48)which contradicts (44). So the proof is complete.

Theorem 14. Suppose −𝑝/𝑎∈ℜ₊, and assume that (7), (8), and (13) hold. Let 𝐻 be defined as in Theorem 13, and for all sufficiently large 𝑇,lim_𝑡→∞1sup𝐻𝑡,𝑡₀∫^𝑡_𝑡0𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)Δ𝑠=∞, (49)where 𝜌,𝜂 are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Proof. Assume (2) has a nonoscillatory solution 𝑥 on 𝕋₀. Without loss of generality, we may assume 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋, where 𝑡₁ is sufficiently large. By Lemmas 5 and 6, there exists sufficiently large 𝑡₂ such that [𝑟(𝑡)𝑥^Δ(𝑡)]^Δ>0 on [𝑡₂,∞)_𝕋, and either 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋 or lim_𝑡→∞𝑥(𝑡)=0. Now we assume 𝑥^Δ(𝑡)>0 on [𝑡₂,∞)_𝕋. Let 𝜔(𝑡) be defined as in Theorem 9. By (41) we have𝐿𝑞(𝑡)𝜌(𝑡)𝑒_−𝑝/𝑎𝜎(𝑡),𝑡₀−𝜌(𝑡)𝑎(𝑡)𝜂(𝑡)^Δ+𝛾𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑎²(𝜎(𝑡))𝜂²(𝜎(𝑡))𝑟(𝑡)−𝑟(𝑡)𝜌^Δ(𝑡)+2𝛾𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂𝑎(𝜎(𝑡))𝜂(𝜎(𝑡))²4𝛾𝑟(𝑡)𝜌(𝑡)𝛿₁𝑡,𝑡₂𝛿₂^𝛾−1𝜎(𝑡),𝑡₂≤−𝜔^Δ(𝑡). (50)Substituting 𝑡 with 𝑠 in (45), multiplying both sides by 𝐻(𝑡,𝑠), and then integrating with respect to 𝑠 from 𝑡₂ to 𝑡 yield^𝑡_𝑡2𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂Δ𝑠≤−^𝑡_𝑡2𝐻(𝑡,𝑠)𝜔^Δ(𝑠)Δ𝑠=𝐻𝑡,𝑡₂𝜔𝑡₂+^𝑡_𝑡2𝐻^Δ𝑠(𝑡,𝑠)𝜔(𝜎(𝑠))Δ𝑠≤𝐻𝑡,𝑡₂𝜔𝑡₂≤𝐻𝑡,𝑡₀𝜔𝑡₂. (51)Then similar to Theorem 13, we obtainlim_𝑡→∞1sup𝐻𝑡,𝑡₀×∫^𝑡_𝑡0𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑡Δ𝑠≤𝜔₂+∫^𝑡2_𝑡0𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁𝑠,𝑡₂𝛿₂^𝛾−1𝜎(𝑠),𝑡₂Δ𝑠<∞, (52)which contradicts (44). So the proof is complete.

In Theorems 13 and 14, if we take 𝐻(𝑡,𝑠) for some special functions such as (𝑡−𝑠)^𝑚 or ln(𝑡/𝑠), then we can obtain some corollaries. For example, if we take 𝐻(𝑡,𝑠)=(𝑡−𝑠)^𝑚,𝑚≥1, then we have the following corollaries.

Corollary 15. Suppose −𝑝/𝑎∈ℜ₊, and assume that (7), (8), and (13) hold, and for all sufficiently large 𝑇,lim_𝑡→∞1sup𝑡−𝑡₀^𝑚×^𝑡_𝑡0(𝑡−𝑠)^𝑚𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1Δ𝑠=∞. (53)Then every solution of (2) is oscillatory or tends to zero.

Corollary 16. Suppose −𝑝/𝑎∈ℜ₊, and assume that (7), (8), and (13) hold, and for all sufficiently large 𝑇,lim_𝑡→∞1sup𝑡−𝑡₀^𝑚×∫^𝑡_𝑡0(𝑡−𝑠)^𝑚𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)Δ𝑠=∞. (54)Then every solution of (2) is oscillatory or tends to zero.

Remark 17. The established results above extend the main results in [25, Theorems 2.1--2.4] except that the latter is related to time delay.

Remark 18. In Theorems 12–14, if we take 𝕋 for some special time scales, we can obtain similar results as in Corollaries 10 and 11, which are omitted here.
In Theorems 9, 12, 13, and 14, if we let 𝑝=0, then 𝑒_−𝑝/𝑎(𝑡,𝑡₀)≡1, and subsequently we obtain the following four corollaries concerning oscillatory criteria of the following equation: 𝑎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡∈𝕋₀. (55)

Corollary 19. Assume (8) holds. If ^∞_𝑡01𝑎^1/𝛾(𝑠)Δ𝑠=∞,lim_𝑡→∞sup^𝑡_𝑡01𝑟(𝜉)^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)Δ𝑠^1/𝛾Δ𝜏Δ𝜉=∞. (56)(57) and for all sufficiently large 𝑇,lim_𝑡→∞sup^𝑡_𝑇𝐿𝑞(𝑠)𝜌(𝑠)−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1Δ𝑠=∞, (58)where 𝜌,𝜂 are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Corollary 20. Assume that (8), (56), and (57) hold, and for all sufficiently large 𝑇,lim_𝑡→∞sup^𝑡_𝑇𝐿𝑞(𝑠)𝜌(𝑠)−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)Δ𝑠=∞, (59)where 𝜌,𝜂 are defined as in Theorem 9. Then every solution of (2) is oscillatory or tends to zero.

Corollary 21. Assume that (8), (56), and (57) hold. If for all sufficiently large 𝑇,lim_𝑡→∞1sup𝐻𝑡,𝑡₀∫^𝑡_𝑡0𝐻(𝑡,𝑠)𝐿𝑞(𝑠)𝜌(𝑠)−𝜌(𝑡)𝑎(𝑠)𝜂(𝑠)^Δ+𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1+1/𝛾𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+(𝛾+1)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))^1/𝛾𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1Δ𝑠=∞, (60)where 𝜌,𝜂,𝐻 are defined as in Theorems 9 and 13, respectively, then every solution of (2) is oscillatory or tends to zero.

Corollary 22. Assume that (8), (56), and (57) hold. If for all sufficiently large 𝑇,lim_𝑡→∞1sup𝐻𝑡,𝑡₀∫^𝑡_𝑡0𝐻()𝐿𝑞(𝑠)𝜌(𝑠)𝑡,𝑠𝑒_−𝑝/𝑎𝜎(𝑠),𝑡₀−𝜌(𝑠)𝑎(𝑠)𝜂(𝑠)^Δ+𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎²(𝜎(𝑠))𝜂²(𝜎(𝑠))𝑟(𝑠)−𝑟(𝑠)𝜌^Δ(𝑠)+2𝛾𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)𝑎(𝜎(𝑠))𝜂(𝜎(𝑠))²4𝛾𝑟(𝑠)𝜌(𝑠)𝛿₁(𝑠,𝑇)𝛿₂^𝛾−1(𝜎(𝑠),𝑇)Δ𝑠=∞, (61)where 𝜌,𝜂,𝐻 are defined as in Theorems 9 and 13, respectively, then every solution of (2) is oscillatory or tends to zero.

Remark 23. In [34, Theorems 3.3-3.4] and [35, Theorems 2.7--2.9], Saker established some new oscillatory criteria for the equation 𝑝(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝛾Δ+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0 (62) under the condition 𝜏(𝑡)≤𝑡. We note that the conditions (8) and (56) in Corollaries 19–22 are consistent with those in [34, (24)] and [35, (1.12)], which were used in [34, Theorems 3.3-3.4] and [35, Theorems 2.7--2.9], respectively, while 𝑟(𝑡)≡1 is assumed in [34, 35]. Moreover, in the results established above, the Riccati substitution function is defined by 𝜔(𝑡)=𝜌(𝑡)𝑎(𝑡)[([𝑟(𝑡)𝑥^Δ(𝑡)]^Δ)^𝛾/𝑥^𝛾(𝑡)𝑒_−𝑝/𝑎(𝑡,𝑡₀)+𝜂(𝑡)] (see Theorem 9), which is different form that in [34, Theorem 3.3] and [35, Theorem 2.7], where the Riccati substitution function is defined by 𝑢(𝑡)=𝑦^[2]/(𝑦^[1])^𝛾=𝑥^ΔΔ/(𝑥^Δ)^𝛾. Since the Riccati substitution function is the most important fact in establishing sufficient oscillatory conditions, so our results in Corollaries 19–22 are essentially different from Saker's results in [34, 35].

3. Applications

In this section, we will present some applications for the established results above. First we consider the following third-order nonlinear differential equation with damping term.

Example 24. Consider 𝑡𝑥^′′(𝑡)^𝛾′+1𝑡^𝛾+1𝑥^′′(𝑡)^𝛾+1𝑡^𝛾+1𝑥^𝛾(𝑡)𝑒^𝑥(𝑡)+1=0,𝑡∈[2,∞), (63) where 𝛾≥1 is a quotient of two odd positive integers.
We have in (2) 𝕋=ℝ,𝑎(𝑡)=𝑡^𝛾,𝑝(𝑡)=𝑞(𝑡)=1/𝑡^𝛾+1,𝑓(𝑥)=𝑥^𝛾[𝑒^𝑥+1],𝑟(𝑡)=1,𝑡₀=2. Then 𝑓(𝑥)/𝑥^𝛾≥1=𝐿,𝜇(𝑡)=𝜎(𝑡)−𝑡=0, and −𝑝/𝑎∈ℜ₊. So 𝑒_−𝑝/𝑎(𝑡,𝑡₀)=𝑒_−𝑝/𝑎(𝑡,2)=exp(−∫^𝑡₂(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠). Moreover, we have 1>exp−^𝑡₂𝑝(𝑠)𝑎(𝑠)𝑑𝑠≥1−^𝑡₂𝑝(𝑠)𝑎(𝑠)𝑑𝑠=1−^𝑡₂1𝑠^2𝛾+11𝑑𝑠=1+𝑡2𝛾^−2𝛾−2^−2𝛾>12. (64) Then we have ^∞_𝑡0𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)1𝑑𝑠>2^1/𝛾∞₂1𝑠𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞. (65) Furthermore, ^∞_𝑡01𝑟(𝜉)^∞_𝜉𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₀𝑑𝑠^1/𝛾=𝑑𝜏𝑑𝜉_∞2∞𝜉𝑒_−𝑝/𝑎(𝜏,2)𝜏^𝛾∞_𝜏1𝑠^𝛾+1𝑒_−𝑝/𝑎(𝑠,2)𝑑𝑠^1/𝛾>1𝑑𝜏𝑑𝜉2^1/𝛾∞₂^∞_𝜉1𝜏^𝛾∞_𝜏1𝑠^𝛾+1𝑑𝑠^1/𝛾=1𝑑𝜏𝑑𝜉2𝛾^{1/𝛾∞2∞𝜉}1𝜏²1𝑑𝜏𝑑𝜉=2𝛾^1/𝛾∞21𝜉𝑑𝜉=∞. (66) On the other hand, for a sufficiently large 𝑇, we have 𝛿₁(𝑡,𝑇)=^𝑡_𝑇𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)1𝑑𝑠>2^1/𝛾𝑡_𝑇1𝑠𝑑𝑠⟶∞. (67) So there exists a sufficiently large 𝑇^∗>𝑇 such that 𝛿₁(𝑡,𝑇)>1 for 𝑡∈[𝑇^∗,∞). Taking 𝜌(𝑡)=𝑡^𝛾,𝜂(𝑡)=0 in (36), we get thatlim_𝑡→∞sup^𝑡_𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₀−𝑟(𝑠)𝜌^′(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1𝑑𝑠=lim_𝑡→∞sup^𝑇∗_𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₀−𝑟(𝑠)𝜌^′(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1+𝑑𝑠_𝑡𝑇∗𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₀−𝑟(𝑠)𝜌^′(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1𝑑𝑠>lim_𝑡→∞sup^𝑇∗_𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₀−𝑟(𝑠)𝜌^′(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1𝑑𝑠+^𝑡_𝑇∗𝛾1−𝛾+1^𝛾+11𝑠𝑑𝑠⟶∞. (68)So (35)–(36) all hold, and by Corollary 10 we deduce that every solution of (63) is oscillatory or tends to zero.

Next we consider the following third-order difference equation:

Example 25. Consider Δ𝑡Δ²𝑥(𝑡)^𝛾+1𝑡^𝛾+1Δ²𝑥(𝑡)^𝛾+𝑀𝑡^𝛾+1𝑥^𝛾(𝑡)[)=0,𝑡∈2,∞_ℤ, (69) where Δ denotes the difference operator, 𝑀>0 is a constant, and 𝛾≥1 is a quotient of two odd positive integers.
We have in (2) 𝕋=ℤ,𝑎(𝑡)=𝑡^𝛾,𝑝(𝑡)=𝑞(𝑡)=1/𝑡^𝛾+1, 𝑓(𝑥)=𝑀𝑥^𝛾,𝑟(𝑡)=1,𝑡₀=2. Then 𝑓(𝑥)/𝑥^𝛾≥𝑀=𝐿,𝜇(𝑡)=𝜎(𝑡)−𝑡=1, and (𝑡)𝑝(𝑡)1−𝜇𝑎(𝑡)1=1−𝑡^2𝛾+11≥1−2>0, (70) which implies −𝑝/𝑎∈ℜ₊. So by [2, Lemma 2] we obtain 𝑒_−𝑝/𝑎𝑡,𝑡₀=𝑒_−𝑝/𝑎(𝑡,2)≥1−^𝑡₂𝑝(𝑠)𝑎(𝑠)Δ𝑠=1−^𝑡₂1𝑠^2𝛾+1Δ𝑠=1−^𝑡−1_𝑠=21𝑠^2𝛾+1≥1−₁^𝑡−11𝑠^2𝛾+11𝑑𝑠=1+)2𝛾(𝑡−1^−2𝛾>1−12,𝑒_−𝑝/𝑎𝑡,𝑡₀≤exp−^𝑡₂𝑝(𝑠)𝑎(𝑠)Δ𝑠<1. (71) Then we have ^∞_𝑠=𝑡0𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)=^∞_𝑠=2𝑒_−𝑝/𝑎(𝑠,2)^1/𝛾𝑎^1/𝛾(𝑠)=^∞_𝑠=2𝑒_−𝑝/𝑎(𝑠,2)^1/𝛾𝑠>12^∞1/𝛾_𝑠=21𝑠=∞,_{∞𝑠=𝑡0}1𝑟(𝑠)=∞. (72) Furthermore, ^∞_𝜉=𝑡01𝑟(𝜉)^∞_𝜏=𝜉𝑒_−𝑝/𝑎𝜏,𝑡₀𝑎(𝜏)^∞_𝑠=𝜏𝑞(𝑠)𝑒_−𝑝/𝑎𝑠+1,𝑡₀^1/𝛾=^∞_𝜉=21𝑟(𝜉)^∞_𝜏=𝜉𝑒_−𝑝/𝑎(𝜏,2)𝑎(𝜏)×^∞_𝑠=𝜏𝑞(𝑠)𝑒_−𝑝/𝑎(𝑠+1,2)^1/𝛾>12^∞1/𝛾_𝜉=2^∞_𝜏=𝜉1𝜏^𝛾∞_𝑠=𝜏1𝑠^𝛾+11/𝛾>12^∞1/𝛾_𝜉=2^∞_𝜏=𝜉1𝜏^𝛾∞_𝑠=𝜏1𝑠^𝛾+1𝑑𝑠^1/𝛾=12𝛾^∞1/𝛾_𝜉=2^∞_𝜏=𝜉1𝜏²>12𝛾^∞1/𝛾∞_{𝜉=2𝜏=𝜉}1𝜏()=1𝜏+12𝛾^∞1/𝛾_𝜉=21𝜉=∞. (73) On the other hand, for a sufficiently large 𝑇>1, we have 𝛿₁(𝑡,𝑇)=^𝑡−1_𝑠=𝑇𝑒_−𝑝/𝑎𝑠,𝑡₀^1/𝛾𝑎^1/𝛾(𝑠)>12^{1/𝛾𝑡−1}_𝑠=𝑇1𝑠⟶∞. (74) So there exists 𝑇^∗>𝑇 such that 𝛿₁(𝑠,𝑇)>1 for 𝑡∈[𝑇^∗,∞)_ℤ. Let 𝜌(𝑡)=𝑡^𝛾,𝜂(𝑡)=0 in (38). Then by the inequality (𝑡+1)^𝛾−𝑡^𝛾≤𝛾(𝑡+1)^𝛾−1<𝛾2^𝛾−1𝑡^𝛾−1,𝑡≥𝑇^∗, we obtainlim_𝑡→∞sup∑^𝑡−1_𝑠=𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠+1,𝑡₀−𝑟(𝑠)𝜌(𝑠+1)−𝜌(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1=lim_𝑡→∞sup∑^𝑇∗_𝑠=𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠+1,𝑡₀−𝑟(𝑠)𝜌(𝑠+1)−𝜌(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1+∑^𝑡−1_𝑠=𝑇∗𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠+1,𝑡₀−𝑟(𝑠)𝜌(𝑠+1)−𝜌(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1>lim_𝑡→∞sup∑^𝑇∗_𝑠=𝑇𝐿𝑞(𝑠)𝜌(𝑠)𝑒_−𝑝/𝑎𝑠+1,𝑡₀−𝑟(𝑠)𝜌(𝑠+1)−𝜌(𝑠)𝛾+1𝑟^1/(𝛾+1)(𝑠)𝜌^{𝛾/(𝛾+1)}(𝑠)𝛿₁^{𝛾/(𝛾+1)}(𝑠,𝑇)^𝛾+1+∑^𝑡−1_𝑠=𝑇∗𝛾𝑀−𝛾+1^𝛾+12^𝛾2−11𝑠⟶∞, (75)provided that 𝑀>(𝛾/(𝛾+1))^𝛾+12^𝛾2−1. So (37) and (38) all hold, and by Corollary 11 we obtain that every solution of (69) is oscillatory or tends to zero under the condition 𝑀>(𝛾/(𝛾+1))^𝛾+12^𝛾2−1.

Acknowledgments

The authors would like to thank the reviewers very much for their valuable suggestions on this paper.