Oscillation Theorems for Second-Order Half-Linear Neutral Delay Dynamic Equations with Damping on Time Scales

Abstract

We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.

1. Introduction

In reality, it is known that the movement in the vacuum or ideal state is rare, while the movement with damping and disturbance is extensive. In recent years, the study of the oscillation of the second-order dynamic equations with damping on time scales is emerging; see [1–7], for example. Besides, the study of the oscillation for the second-order linear and nonlinear or semilinear dynamic equations can be found in [8–23] and of the oscillation for the high-order dynamic equations can be found in [24–33]. Then, inspired by the above work, this paper will study the oscillatory behavior of all solutions of a more extensive second-order half-linear neutral delay dynamic equation with damping, which is given as follows: 𝑎(𝑡)Φ𝑧^Δ(𝑡)^Δ+𝑝(𝑡)Φ𝑧^Δ(𝑡)+𝑞(𝑡)𝑓(Φ(𝑥(𝜏(𝑡))))=0,𝑡∈𝕋,𝑡≥𝑡₀, (1) where Φ(𝑠)=|𝑠|^𝛾−2𝑠, 𝑧(𝑡)=𝑥(𝑡)+𝑟(𝑡)𝑥(𝜏(𝑡)), 𝛾>1.

Here, we give the following hypotheses at first.(H₁)𝕋 is a time scale (i.e., a nonempty closed subset of the real numbers ℝ) which is unbounded above and when 𝑡₀∈𝕋 with 𝑡₀>0, we define the time scale interval of the form [𝑡₀,∞)_𝕋 by [𝑡₀,∞)_𝕋=[𝑡₀,∞)∩𝕋.(H₂)𝑎,𝑟,𝑝,𝑞∶𝕋→ℝ are positive rd-continuous functions such that 0≤𝑟(𝑡)<1,−𝑝/𝑎∈ℛ⁺, where ℛ is defined as the set of all regressive and rd-continuous functions and ℛ⁺ is all positively regressive elements of ℛ.(H₃)𝜏∶𝕋→𝕋 is a strictly increasing and differentiable function such that 𝜏(𝑡)≤𝑡,lim_𝑡→∞𝜏(𝑡)=∞,𝜏(𝕋)=𝕋. (2) (H₄)𝑓∶ℝ→ℝ is a continuous function such that, for some positive constant 𝐿, 𝑓(𝑥)𝑥≥𝐿forall𝑥≠0. (3)

The solution of (1) defines a nontrivial real-valued function 𝑥 satisfying (1) for 𝑡∈𝕋. A solution 𝑥 of (1) is called oscillatory if it is neither eventually positive nor negative; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory. Here, we pay attention to those solutions of (1) which are not the eventually identical zero.

The purpose of this paper is to establish the oscillation criteria of Philos [34] for (1). The two famous results of Philos [34] about oscillation of second-order linear differential equations are extended to (1), while it satisfies ^∞_𝑡01𝑎(𝑡)𝑒_−𝑝/𝑎(𝑡,𝑡₀)^1/(𝛾−1)Δ𝑡=∞. (4) Besides, two criteria of (1) about the fact that each solution is either oscillatory or converges to zero are obtained when ^∞_𝑡01𝑒𝑎(𝑡)^−𝑝/𝑎(𝑡,𝑡₀)^1/(𝛾−1)Δ𝑡<∞. (5)

The paper is organized as follows. In Section 2, we present some basic definitions and results about the theory of calculus on time scales. In Section 3, we give some lemmas. Section 4 introduces the main results of this paper. We established four new oscillatory criteria when conditions (4) and (5) hold, respectively, for the solutions of (1) in this paper.

2. Some Preliminaries

On the time scale 𝕋 we define the forward and backward jump operators by 𝜎(𝑡){},𝜌(𝑡){}.=inf𝑠∈𝕋∶𝑠>𝑡=sup𝑠∈𝕋∶𝑠<𝑡 (6) A point 𝑡∈𝕋 is said to be left-dense if it satisfies 𝜌(𝑡)=𝑡, right-dense if it satisfies 𝜎(𝑡)=𝑡, left-scattered if it satisfies 𝜌(𝑡)<𝑡, and right-scattered if it satisfies 𝜎(𝑡)>𝑡. The graininess function 𝜇∶𝕋→[𝑡₀,∞) of the time scale is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. For a function 𝑓∶𝕋→ℝ, the (delta) derivative is defined by 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜎(𝑡),−𝑡 (7) if 𝑓 is continuous at 𝑡 and 𝑡 is right-scattered. If 𝑡 is right-dense, then the derivative is defined by 𝑓^Δ(𝑡)=lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠),𝑡−𝑠 (8) provided this limit exists. A function 𝑓∶𝕋→ℝ is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit at every left-dense point. Denote by 𝐶_rd(𝕋,ℝ) the set of rd-continuous functions 𝑓∶𝕋→ℝ, and denote by 𝐶¹_rd(𝕋,ℝ) the set of functions 𝑓 which is Δ-differentiable and the derivative 𝑓^Δ is rd-continuous. The derivative 𝑓^Δ of 𝑓, the shift 𝑓^𝜎 of 𝑓, and the graininess function 𝜇 are related by the following formula: 𝑓^𝜎=𝑓+𝜇𝑓^Δwhere𝑓^𝜎=𝑓∘𝜎. (9)

We will make use of the following product and quotient rules for the derivative of the product 𝑓𝑔 and the quotient 𝑓/𝑔 of two differentiable functions 𝑓 and 𝑔: 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔(𝜎(𝑡)),𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)−𝑓(𝑡)𝑔^Δ(𝑡)𝑔(𝑡)𝑔(𝜎(𝑡))if𝑔𝑔^𝜎≠0. (10)(11) For 𝑏,𝑐∈𝕋, the Cauchy integral of 𝑓^Δ is defined by ^𝑐_𝑏𝑓^Δ(𝑡)Δ𝑡=𝑓(𝑐)−𝑓(𝑏). (12) The integration by parts formula reads ^𝑐_𝑏𝑓^Δ(𝑡)𝑔(𝑡)(𝑐)𝑔(𝑐)(𝑏)𝑔(𝑏)−Δ𝑡=𝑓−𝑓_𝑐𝑏𝑓^𝜎(𝑡)𝑔^Δ(𝑡)Δ𝑡, (13) and the infinite integral is defined by ^∞_𝑏𝑓(𝑠)Δ𝑠=lim_𝑡→∞^𝑡_𝑏𝑓(𝑠)Δ𝑠. (14) For more details, see [8, 9].

3. Several Lemmas

In this section, we present six lemmas that are needed in Section 4. The first lemma is well known, and it can be found in Chapter 2 of [8]. Lemma 2 is Theorem 1.93 of [8]; Lemma 3 is the simple corollary of Theorem 1.90 in [8]; Lemma 4 is Theorem 41 in [35]; and Lemma 5 is Theorem 3 in [36].

Lemma 1. If 𝑔∈ℛ⁺, that is, 𝑔∶𝕋→ℝ is rd-continuous, such that 1+𝜇(𝑡)𝑔(𝑡)>0 for all 𝑡∈[𝑡₀,∞)_𝕋, then the initial value problem 𝑦^Δ=𝑔(𝑡)𝑦,𝑦(𝑡₀)=𝑦₀∈ℝ has a unique and positive solution on [𝑡₀,∞)_𝕋, denoted by 𝑒_𝑔(⋅,𝑡₀). This “exponential function” satisfies the semigroup property 𝑒_𝑔(𝑎,𝑏)𝑒_𝑔(𝑏,𝑐)=𝑒_𝑔(𝑎,𝑐).

Lemma 2. Assume that 𝑣∶𝕋→ℝ is strictly increasing and 𝕋∶=𝑣(𝕋) is a time scale. Let 𝑤∶𝕋→ℝ. If 𝑣^Δ(𝑡) and 𝑤^Δ(𝑣(𝑡)) exist on 𝕋^𝑘, where 𝕋^𝑘=𝕋⧵𝜌sup𝕋,sup𝕋,𝑖𝑓sup𝕋<∞,𝕋,𝑖𝑓sup𝕋=∞, (15) then (𝑤∘𝑣)^Δ=𝑤^Δ∘𝑣𝑣^Δ. (16)

Lemma 3. If 𝑥 is differentiable, then 𝑥^𝛾Δ=𝛾𝑥^Δ1₀ℎ𝑥^𝜎+(1−ℎ)𝑥^𝛾−1dℎ. (17)

Lemma 4. Assume that 𝑋 and 𝑌 are nonnegative real numbers; then 𝜆𝑋𝑌^𝜆−1−𝑋^𝜆≤(𝜆−1)𝑌^𝜆𝑓𝑜𝑟𝑎𝑙𝑙𝜆>1, (18) where the equality holds if and only if 𝑋=𝑌.

Lemma 5. Let 𝑎,𝑏∈𝕋 and 𝑎<𝑏. Then for positive rd-continuous functions 𝑓,𝑔∶[𝑎,𝑏]→ℝ we have ^𝑏_𝑎𝑓(𝑠)𝑔(𝑠)Δ𝑠≤^𝑏_𝑎𝑓(𝑠)^𝑝Δ𝑠^1/𝑝𝑏_𝑎𝑔(𝑠)^𝑞Δ𝑠^1/𝑞, (19) where 𝑝>1 and (1/𝑝)+(1/𝑞)=1.

Lemma 6. Assume that (H₁)–(H₄) and (4) hold. Let 𝑥(𝑡) be an eventually positive solution of (1). Then there exists 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑧^Δ(𝑡)>0,𝑎(𝑡)𝑧^Δ(𝑡)^𝛾−2𝑧^Δ(𝑡)^Δ<0. (20)

Proof. Suppose that 𝑥(𝑡) is an eventually positive solution of (1). There exists 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑥(𝑡)>0 and 𝑥(𝜏(𝑡))>0 for 𝑡∈[𝑡₁,∞)_𝕋. From the definition of 𝑧(𝑡), we get 𝑧(𝑡)>0 for 𝑡∈[𝑡₁,∞)_𝕋, and at the same time for 𝑡∈[𝑡₁,∞)_𝕋, from (1), we get 𝑎(𝑡)𝑧^Δ(𝑡)^𝛾−2𝑧^Δ(𝑡)^Δ+𝑝(𝑡)𝑧^Δ(𝑡)^𝛾−2𝑧^Δ(𝑡)<0. (21) Hence, from Lemma 1 and (11) we obtain 𝑎𝑧^Δ𝛾−2𝑧^Δ𝑒_−𝑝/𝑎(⋅,𝑡₀)^Δ=𝑎𝑧^Δ𝛾−2𝑧^ΔΔ𝑒_−𝑝/𝑎⋅,𝑡₀−𝑒^Δ_−𝑝/𝑎⋅,𝑡₀𝑎𝑧^Δ𝛾−2𝑧^Δ𝑒_−𝑝/𝑎⋅,𝑡₀𝑒^𝜎_−𝑝/𝑎⋅,𝑡₀=𝑎𝑧^Δ𝛾−2𝑧^ΔΔ+𝑝𝑧^Δ𝛾−2𝑧^Δ𝑒^𝜎_−𝑝/𝑎⋅,𝑡₀<0 (22) for [𝑡₁,∞)_𝕋. So 𝑎𝑧^Δ𝛾−2𝑧^Δ𝑒_−𝑝/𝑎⋅,𝑡₀ (23) is decreasing. By Lemma 1, 𝑧^Δ(𝑡) is either eventually positive or eventually negative. Therefore, for arbitrary 𝑡∈[𝑡₁,∞)_𝕋, we have 𝑧^Δ(𝑡)>0. (24) Otherwise, we assume that (24) is not satisfied; then there exits 𝑡₂∈[𝑡₁,∞)_𝕋 such that 𝑧^Δ(𝑡)<0 for all 𝑡∈[𝑡₂,∞)_𝕋. Because (23) is decreasing, from Lemma 1 we have 𝑎(𝑡)𝑧^Δ(𝑡)^𝛾−2𝑧^Δ(𝑡)𝑒_−𝑝/𝑎𝑡,𝑡₀≤𝑎𝑡₂𝑧^Δ𝑡₂^𝛾−2𝑧^Δ𝑡₂𝑒_−𝑝/𝑎𝑡₂,𝑡₀𝑀=−^𝛾−1𝑒_−𝑝/𝑎𝑡₂,𝑡₀ (25) for 𝑡∈[𝑡₂,∞)_𝕋, where 𝑀=𝑎(𝑡₂)^1/(𝛾−1)|𝑧^Δ(𝑡₂)|>0. By (25) and Lemma 1, we get −𝑧^Δ(𝑡)^𝛾−1≥𝑀^𝛾−1𝑎(𝑡)𝑒_−𝑝/𝑎𝑡,𝑡₂,𝑡∈𝑡₂,∞_𝕋; (26) that is, 𝑧^Δ(𝑡)1≤−𝑀𝑎(𝑡)𝑒_−𝑝/𝑎(𝑡,𝑡₂)^1/(𝛾−1),𝑡∈𝑡₂,∞_𝕋. (27) After integrating the two sides of inequality (27) from 𝑡₂ to 𝑡∈[𝑡₂,∞)_𝕋, we have 𝑧(𝑡)≤𝑧𝑡₂−𝑀^𝑡_𝑡21𝑎(𝑠)𝑒_−𝑝/𝑎𝑠,𝑡₂^1/(𝛾−1)𝑡Δ𝑠,𝑡∈₂,∞_𝕋. (28) Next, we find the limits of the two sides of (28) when 𝑡→∞. From (4), we get lim_𝑡→∞𝑧(𝑡)=−∞. Therefore, 𝑧(𝑡) is eventually negative, which is contradictory to 𝑧(𝑡)>0. So the inequality (24) holds.
From (24) and (21), it is obvious that the second inequality of (20) holds. This completes the proof.

4. Main Results

Firstly, the two famous results of Philos [24] about oscillation of second-order linear differential equations are extended to (1) when condition (4) is satisfied.

Theorem 7. Assume that (H₁)–(H₄) and (4) hold. Let ∶𝐷_𝕋≡{(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀, 𝑡,𝑠∈[𝑡₀,∞)_𝕋}→ℝ be rd-continuous function, such that 𝐻(𝑡,𝑡)=0,𝑡≥𝑡₀;𝐻(𝑡,𝑠)>0,𝑡>𝑠≥𝑡₀,𝑡,𝑠∈𝑡₀,∞_𝕋, (29) and 𝐻 has a nonpositive continuous Δ-partial derivative 𝐻^Δ𝑠(𝑡,𝑠) with respect to the second variable and satisfies (31). Let ℎ∶𝐷_𝕋→ℝ be a rd-continuous function and satisfies −𝐻^Δ𝑠(𝑡,𝑠)=ℎ(𝑡,𝑠)(𝐻(𝑡,𝑠))^{(𝛾−1)/𝛾},(𝑡,𝑠)∈𝐷_𝕋,0<inf_𝑠≥𝑇0liminf_𝑡→∞𝐻(𝑡,𝑠)𝐻𝑡,𝑇₀≤∞,𝑇₀∈𝑡₀,∞_𝕋. (30)(31) If there exist a positive and differentiable function 𝛿∶𝕋→ℝ such that 𝛿^Δ(𝑡)≥0 for 𝑡∈[𝑡₀,∞)_𝕋 and a real rd-continuous function Ψ∶[𝑡₀,∞)_𝕋→ℝ such that limsup_𝑡→∞1𝐻𝑡,𝑇₀^𝑡_𝑇0𝑎(𝜏(𝑠))𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠<∞,_∞𝑇0𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^1/(𝛾−1)Ψ₊(𝜎(𝑠))𝛿(𝜎(𝑠))^{𝛾/(𝛾−1)}Δ𝑠=∞,limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1−𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊((𝑇),𝑡,𝑠)Δ𝑠≥Ψ (32)(33)(34) where 𝐺(𝑡,𝑠)=(𝛿^Δ(𝑠)−(𝑝(𝑠)/𝑎(𝑠))𝛿(𝑠))(𝐻(𝑡,𝑠))^1/𝛾−𝛿(𝑠)ℎ(𝑡,𝑠),𝐺₊(𝑡,𝑠)=max{0,𝐺(𝑡,𝑠)}, Ψ₊(𝑡)=max{0,Ψ(𝑡)}, and 𝑇∈[𝑇₀,∞)_𝕋, then (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that (1) has a nonoscillatory solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Without loss of generality we may assume that there exists 𝑡₁∈[𝑡₀,∞)_𝕋, such that 𝑥(𝑡)>0 and 𝑥[𝜏(𝑡)]>0 for all 𝑡∈[𝑡₁,∞)_𝕋. By the definition of 𝑧(𝑡), it follows 𝑥(𝑡)(𝑡)(𝑡)𝑥(𝜏(𝑡(𝑡)(𝑡)𝑧(𝜏(𝑡≥((𝑡𝑧(𝑡)𝑡=𝑧−𝑟))≥𝑧−𝑟))1−𝑟)),𝑡∈₁,∞_𝕋. (35) Since it satisfies lim_𝑡→∞𝜏(𝑡)=∞, there exists 𝑇₀∈[𝑡₀,∞)_𝕋 such that 𝜏(𝑡)≥𝑡₁ for all 𝑡∈[𝑇₀,∞)_𝕋. Then if it satisfies 𝑡∈[𝑇₀,∞)_𝕋, we have 𝑥(𝜏(𝑡))≥(1−𝑟(𝜏(𝑡)))𝑧(𝜏(𝑡)). (36) By Lemma 6 and (H₃), we obtain that 1≥1𝑧∘𝜏𝑧∘𝜏^𝜎,𝑎𝑧^Δ𝛾−1≥𝑎^𝜎𝑧^{Δ𝜎𝛾−1} (37) on [𝑇₀,∞)_𝕋 (where (𝑧^Δ)^𝜎 is short hand for 𝑧^Δ𝜎), and 𝑧^Δ𝑎∘𝜏≥^{𝜎1/(𝛾−1)}(𝑎∘𝜏)^1/(𝛾−1)𝑧^Δ𝜎 (38) holds. Moreover, using Lemmas 3 and 6, it follows that (𝑧∘𝜏)^𝛾−1Δ=𝛾−1(𝑧∘𝜏)^Δ×¹₀ℎ𝑧∘𝜏^𝜎+(1−ℎ)(𝑧∘𝜏)^𝛾−2≥()dℎ𝛾−1𝑧∘𝜏^Δ×¹₀[ℎ(𝑧∘𝜏)+(1−ℎ)(𝑧∘𝜏)]^𝛾−2=()dℎ𝛾−1𝑧∘𝜏^𝛾−2(𝑧∘𝜏)^Δ. (39) In Lemma 2, let 𝑣=𝜏,𝑤=𝑧, and 𝕋 is unbounded above by (H₁), so 𝕋^𝑘=𝕋, and 𝕋=𝑣(𝕋)=𝜏(𝕋)=𝕋 by (H₃); using Lemma 2, we get (𝑧∘𝜏)^Δ=𝑧^Δ∘𝜏𝜏^Δ. (40) Thus (𝑧∘𝜏)^𝛾−1Δ≥𝛾−1(𝑧∘𝜏)^𝛾−2𝑧^Δ∘𝜏𝜏^Δ. (41) By the above inequality and the first inequality in (37), we obtain that (𝑧∘𝜏)^𝛾−1Δ(𝑧∘𝜏)^𝛾−1≥𝛾−1𝑧^Δ∘𝜏𝜏^Δ𝑧∘𝜏^𝜎 (42) holds on [𝑇₀,∞)_𝕋. Now we define the function 𝑊 by 𝑎𝑧𝑊=𝛿^Δ𝛾−1(𝑧∘𝜏)^𝛾−1. (43) Then we have 𝑊>0 on [𝑇₀,∞)_𝕋, and 𝑊=^Δ(10)𝛿(𝑧∘𝜏)^𝛾−1𝑎𝑧^Δ𝛾−1Δ+𝑎^𝜎𝑧^{Δ𝜎𝛾−1}(𝑧∘𝜏)^𝛾−1𝛿^Δ−𝛿(𝑧∘𝜏)^𝛾−1Δ(𝑧∘𝜏)^𝛾−1(𝑧∘𝜏^𝜎)₍₁^𝛾−1_)H4()≤−𝐿𝑞𝛿𝑥∘𝜏^𝛾−1(𝑧∘𝜏)^𝛾−1−𝑝𝛿(𝑧∘𝜏)^𝛾−1𝑧^Δ𝛾−1+𝑎^𝜎𝑧^{Δ𝜎𝛾−1}(𝑧∘𝜏)^𝛾−1𝛿^Δ−𝛿(𝑧∘𝜏)^𝛾−1Δ(𝑧∘𝜏)^𝛾−1(𝑧∘𝜏^𝜎)^𝛾−1₍₃₆₎≤−𝐿𝑞𝛿(1−𝑟∘𝜏)^𝛾−1−𝑝𝛿(𝑧∘𝜏)^𝛾−1𝑧^Δ𝛾−1+𝑎^𝜎𝑧^{Δ𝜎𝛾−1}(𝑧∘𝜏)^𝛾−1𝛿^Δ−𝛿(𝑧∘𝜏)^𝛾−1Δ(𝑧∘𝜏)^𝛾−1(𝑧∘𝜏^𝜎)^𝛾−1₍₄₃₎≤−𝐿𝑞𝛿(1−𝑟∘𝜏)^𝛾−1−𝑝𝑎𝛿𝑊+^Δ𝛿^𝜎𝑊^𝜎−𝛿𝑎^𝜎𝑧^{Δ𝜎𝛾−1}(𝑧∘𝜏)^𝛾−1Δ(𝑧∘𝜏)^𝛾−1(𝑧∘𝜏^𝜎)^𝛾−1₍₃₇₎≤−𝐿𝑞𝛿(1−𝑟∘𝜏)^𝛾−1−𝑝𝛿𝑎𝛿^𝜎𝑊^𝜎+𝛿^Δ𝛿^𝜎𝑊^𝜎−𝛿𝑎^𝜎𝑧^{Δ𝜎𝛾−1}(𝑧∘𝜏)^𝛾−1Δ(𝑧∘𝜏)^𝛾−1(𝑧∘𝜏^𝜎)^𝛾−1₍₄₂₎≤−𝐿𝑞𝛿(1−𝑟∘𝜏)^𝛾−1+𝛿^Δ−𝑝𝑎𝛿𝑊^𝜎𝛿^𝜎−𝛾−1𝛿𝑎^𝜎𝑧^{Δ𝜎𝛾−1}𝑧^Δ∘𝜏𝜏^Δ(𝑧∘𝜏^𝜎)^𝛾₍₃₈₎≤−𝐿𝑞𝛿(1−𝑟∘𝜏)^𝛾−1+𝛿^Δ−𝑝𝑎𝛿𝑊^𝜎𝛿^𝜎−𝛾−1𝛿𝜏^Δ𝑎^{𝜎𝛾/(𝛾−1)}𝑧^Δ𝜎𝛾(𝑎∘𝜏)^1/(𝛾−1)(𝑧∘𝜏^𝜎)^𝛾₍₄₃₎≤−𝐿𝑞𝛿(1−𝑟∘𝜏)^𝛾−1+𝛿^Δ−𝑝𝑎𝛿𝑊^𝜎𝛿^𝜎−𝛾−1𝛿𝜏^Δ(𝑎∘𝜏)^1/(𝛾−1)𝛿^{𝜎𝛾/(𝛾−1)}𝑊^{𝜎𝛾/(𝛾−1)}; (44) then we obtain 𝑊^Δ(𝑡)≤−𝐿𝑞(𝑡)𝛿(𝑡)(1−𝑟(𝜏(𝑡)))^𝛾−1+𝛿(𝑡)𝛿(𝜎(𝑡𝑊(𝜎(𝑡−𝛿(𝑡)𝜏))))𝛾−1^Δ(𝑡)(𝑎(𝜏(𝑡)))^𝜆−1(𝛿(𝜎(𝑡)))^𝜆(𝑊(𝜎(𝑡)))^𝜆 (45) on [𝑇₀,∞)_𝕋, where 𝜆=𝛾/(𝛾−1),𝛿(𝑡)=𝛿^Δ(𝑡)−(𝑝(𝑡)/𝑎(𝑡))𝛿(𝑡). Thus, for every 𝑡,𝑇∈[𝑇₀,∞)_𝕋 with 𝑡≥𝑇≥𝑇₀, by (13), we get 𝐿𝐻(^𝑡_𝑇𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1()𝑊(𝑇)−Δ𝑠≤𝐻𝑡,𝑇_𝑡𝑇−𝐻^Δ𝑠()𝑊(𝜎(𝑠+𝑡,𝑠))Δ𝑠_𝑡𝑇𝐻(𝑡,𝑠)𝛿(𝑠)𝛿(𝜎(𝑠𝑊(𝜎(𝑠))))Δ𝑠−^𝑡_𝑇𝐻()𝛿(𝑠)𝜏𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆Δ𝑠₍₃₀₎=𝐻(𝑡,𝑇)𝑊(𝑇)+^𝑡_𝑇𝛿(𝑠)𝐻(𝑡,𝑠)−𝛿(𝜎(𝑠))ℎ(𝑡,𝑠)𝐻^1/𝜆(𝑡,𝑠)𝛿(𝜎(𝑠𝑊(𝜎(𝑠−))))Δ𝑠_𝑡𝑇𝐻()𝛿(𝑠)𝜏𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆()𝑊(𝑇)+Δ𝑠≤𝐻𝑡,𝑇_𝑡𝑇𝛿(𝑠)𝐻^{(𝜆−1)/𝜆}(𝑡,𝑠)−𝛿(𝑠)ℎ(𝑡,𝑠)𝛿(𝜎(𝑠))×𝐻^1/𝜆(𝑡,𝑠)𝑊(𝜎(𝑠))Δ𝑠−^𝑡_𝑇𝐻()𝛿(𝑠)𝜏𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆()𝑊(𝑇)Δ𝑠≤𝐻𝑡,𝑇+^𝑡_𝑇𝐺₊(𝑡,𝑠)𝛿(𝜎(𝑠𝐻))^1/𝜆()𝑊(𝜎(𝑠−𝑡,𝑠))Δ𝑠_𝑡𝑇𝐻()𝛿(𝑠)𝜏𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆Δ𝑠, (46) where 𝐺(𝑡,𝑠)=𝛿(𝑠)𝐻^{(𝜆−1)/𝜆}(𝑡,𝑠)−𝛿(𝑠)ℎ(𝑡,𝑠)=(𝛿^Δ(𝑠)−(𝑝(𝑠)/𝑎(𝑠))𝛿(𝑠))(𝐻(𝑡,𝑠))^1/𝛾− 𝛿(𝑠)ℎ(𝑡,𝑠),𝐺₊(𝑡,𝑠)=max{0,𝐺(𝑡,𝑠)}. So using Lemma 4, let 𝑋=𝐻(𝑡,𝑠)(𝛾−1)𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆1/𝜆𝑊(𝜎(𝑠)),𝐺𝑌=⁺(𝑡,𝑠)𝜆𝛿(𝜎(𝑠))(𝛾−1)𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^{𝜆−1/𝜆1/(𝜆−1)}. (47)(48) Using the inequality (18), we have 𝐺₊(𝑡,𝑠)𝛿(𝜎(𝑠𝐻))^1/𝜆()𝑊(𝜎(𝑠()×𝛿(𝑠)𝜏𝑡,𝑠))−𝐻𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆𝐺≤𝐶⁺(𝑡,𝑠)𝛿(𝜎(𝑠))^{𝜆/(𝜆−1)}𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^{𝜆−1/(𝜆−1)}, (49) where 𝐶=(𝜆−1)𝜆^{−𝜆/(𝜆−1)}(𝛾−1)^{−1/(𝜆−1)}=1/𝛾^𝛾. Thus 𝐺₊(𝑡,𝑠)𝛿(𝜎(𝑠𝐻))^1/𝜆()𝑊(𝜎(𝑠()𝛿(𝑠)𝜏𝑡,𝑠))−𝐻𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆≤𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠). (50) From (46) and (50), we obtain 𝐿𝐻(^𝑡_𝑇𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1−𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠≤𝐻(𝑡,𝑇)𝑊(𝑇); (51) that is, 1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1−𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠≤𝑊(𝑇). (52) From condition (34), we have Ψ(𝑇)≤𝑊(𝑇),𝑇∈𝑇₀,∞_𝕋,limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1(𝑇).Δ𝑠≥Ψ (53)(54) By (46), we have 1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1(𝑇)+1Δ𝑠≤𝑊𝐻(𝑡,𝑇)^𝑡_𝑇𝐺₊(𝑡,𝑠)𝛿(𝜎(𝑠𝐻))^1/𝜆()𝑊(𝜎(𝑠−1𝑡,𝑠))Δ𝑠𝐻(𝑡,𝑇)^𝑡_𝑇𝐻()𝛿(𝑠)𝜏𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆×(𝑊(𝜎(𝑠)))^𝜆Δ𝑠, (55) and from the above inequality, let 𝑇=𝑇₀, and denote that 𝐴(𝑡)=1𝐻𝑡,𝑇₀^𝑡_𝑇0𝐺₊(𝑡,𝑠)𝛿(𝜎(𝑠𝐻))^1/𝜆()𝑊(𝜎(𝑠𝐵(𝑡)=1𝑡,𝑠))Δ𝑠,𝐻𝑡,𝑇₀^𝑡_𝑇0𝐻()𝛿(𝑠)𝜏𝑡,𝑠𝛾−1^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆×(𝑊(𝜎(𝑠)))^𝜆Δ𝑠; (56) meanwhile noting (54), we obtain liminf_𝑡→∞[𝐵(𝑡)(𝑡𝑇−𝐴)]≤𝑊₀−limsup_𝑡→∞1𝐻𝑡,𝑇₀×^𝑡_𝑇0𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1𝑇Δ𝑠≤𝑊₀−Ψ𝑇₀<∞. (57) Now we assert that ^∞_𝑇0𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆Δ𝑠<∞ (58) holds. Suppose to the contrary that ^∞_𝑇0𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆Δ𝑠=∞. (59) By (31), there exists a constant 𝜀>0 such that inf_𝑠≥𝑇0liminf_𝑡→∞𝐻(𝑡,𝑠)𝐻𝑡,𝑇₀>𝜀>0. (60) From (59), there exists a 𝑇∈[𝑇₀,∞)_𝕋 for arbitrary real number 𝑀>0 such that ^𝑡_𝑇0𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆𝑀Δ𝑠≥𝜀,𝛾−1 (61) for 𝑡∈[𝑇,∞)_𝕋. By (13), we have 𝐵(𝑡)=1𝐻𝑡,𝑇₀^𝑡_𝑇0𝐻()×𝛾−1𝑡,𝑠_𝑠𝑇0𝛿(𝑢)𝜏^Δ(𝑢)(𝑎(𝜏(𝑢)))^𝜆−1(𝛿(𝜎(𝑢)))^𝜆×(𝑊(𝜎(𝑢)))^𝜆Δ𝑢^Δ𝑠=1Δ𝑠𝐻𝑡,𝑇₀^𝑡_𝑇0−𝛾−1𝐻^Δ𝑠(×𝑡,𝑠)_{𝑇𝜎(𝑠)0}𝛿(𝑢)𝜏^Δ(𝑢)(𝑎(𝜏(𝑢)))^𝜆−1(𝛿(𝜎(𝑢)))^𝜆×(𝑊(𝜎(𝑢)))^𝜆≥1Δ𝑢Δ𝑠𝐻𝑡,𝑇₀^𝑡_𝑇−𝛾−1𝐻^Δ𝑠(×𝑡,𝑠)_𝑠𝑇0𝛿(𝑢)𝜏^Δ(𝑢)(𝑎(𝜏(𝑢)))^𝜆−1(𝛿(𝜎(𝑢)))^𝜆×(𝑊(𝜎(𝑢)))^𝜆≥1Δ𝑢Δ𝑠𝐻𝑡,𝑇₀^𝑡_𝑇−𝛾−1𝐻^Δ𝑠(𝑀𝑡,𝑠)𝜀=𝑀𝛾−1Δ𝑠𝜀𝐻(𝑡,𝑇)𝐻𝑡,𝑇₀. (62) From (60), there exists 𝑡₂∈[𝑇,∞)_𝕋 such that 𝐻(𝑡,𝑇)/𝐻(𝑡,𝑇₀)≥𝜀 for 𝑡∈[𝑡₂,∞)_𝕋, so 𝐵(𝑡)≥𝑀. Since 𝑀 is arbitrary, we have lim_𝑡→∞𝐵(𝑡)=∞. (63) Selecting a sequence {𝑇_𝑛}^∞_𝑛=1∶ 𝑇_𝑛∈[𝑇₀,∞)_𝕋 with lim_𝑛→∞𝑇_𝑛=∞ satisfying lim_𝑛→∞𝐵𝑇_𝑛−𝐴𝑇_𝑛=liminf_𝑡→∞[𝐵(𝑡)−𝐴(𝑡)]<∞, (64) then there exists a constant 𝑀₀>0 such that 𝐵𝑇_𝑛−𝐴𝑇_𝑛≤𝑀₀ (65) for sufficiently large positive integer 𝑛. From (63), we can easily see lim_𝑛→∞𝐵𝑇_𝑛=∞, (66) and (65) implies that lim_𝑛→∞𝐴𝑇_𝑛=∞. (67) From (65) and (66), we have 𝐴𝑇_𝑛𝐵𝑇_𝑛𝑀−1≥−⁰𝐵𝑇_𝑛𝑀>−⁰2𝑀₀1=−2; (68) that is, 𝐴𝑇_𝑛𝐵𝑇_𝑛>12 (69) for sufficiently large positive integer 𝑛, which together with (67) implies lim_𝑛→∞𝐴(𝑇_𝑛)^𝛾𝐵𝑇_𝑛^𝛾−1=lim_𝑛→∞𝐴(𝑇_𝑛)𝐵(𝑇_𝑛)^𝛾−1𝐴𝑇_𝑛=∞. (70) On the other hand, by Lemma 5, we obtain 𝐴𝑇_𝑛=1𝐻𝑇_𝑛,𝑇₀^𝑇𝑛_𝑇0𝐺₊𝑇_𝑛,𝑠𝛿(𝜎(𝑠𝐻))^1/𝜆𝑇_𝑛𝑊(𝜎(𝑠=,𝑠))Δ𝑠_{𝑇𝑛𝑇0}𝛾−1𝐻𝑇_𝑛,𝑠𝛿(𝑠)𝜏^Δ(𝑠)𝐻𝑇_𝑛,𝑇₀^{(𝛾−1)/𝛾}×𝑊(𝜎(𝑠))[𝑎(𝜏(𝑠))]^1/𝛾𝛿(𝜎(𝑠×[𝑎(𝜏(𝑠))))]^1/𝛾𝐺₊𝑇_𝑛,𝑠𝐻𝑇_𝑛,𝑇₀𝐻^{(𝛾−1)/𝛾}𝑇_𝑛×𝐻𝑇,𝑠𝛾−1_𝑛,𝑠𝛿(𝑠)𝜏^Δ(𝑠)𝐻𝑇_𝑛,𝑇₀^{(1−𝛾)/𝛾}≤Δ𝑠_{𝑇𝑛𝑇0}𝛾−1𝐻𝑇_𝑛,𝑠𝛿(𝑠)𝜏^Δ(𝑠)𝐻𝑇_𝑛,𝑇₀×𝑊(𝜎(𝑠))(𝑎(𝜏(𝑠)))^1/𝛾𝛿(𝜎(𝑠))^{𝛾/(𝛾−1)}Δ𝑠^{(𝛾−1)/𝛾}×^𝑇𝑛_𝑇0𝑎(𝜏(𝑠))𝐺^𝛾₊𝑇_𝑛,𝑠𝐻^𝛾𝑇_𝑛,𝑇₀𝐻^𝛾−1𝑇_𝑛×𝐻𝑇,𝑠𝛾−1_𝑛,𝑠𝛿(𝑠)𝜏^Δ(𝑠)𝐻𝑇_𝑛,𝑇₀^1−𝛾Δ𝑠^1/𝛾=𝐵𝑇_𝑛^{(𝛾−1)/𝛾}×𝛾−1^1−𝛾𝐻𝑇_𝑛,𝑇₀^𝑇𝑛_𝑇0𝑎(𝜏(𝑠))𝐺^𝛾₊𝑇_𝑛×𝛿(𝑠)𝜏,𝑠_Δ(𝑠)^1−𝛾Δ𝑠^1/𝛾. (71) The above inequality shows that 𝐴(𝑇_𝑛)^𝛾𝐵(𝑇_𝑛)^𝛾−1≤𝛾−1^1−𝛾𝐻𝑇_𝑛,𝑇₀^𝑇𝑛_𝑇0𝑎(𝜏(𝑠))𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊𝑇_𝑛,𝑠Δ𝑠. (72) Hence, (70) implies lim_𝑛→∞1𝐻𝑇_𝑛,𝑇₀^𝑇𝑛_𝑇0𝑎(𝜏(𝑠))𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊𝑇_𝑛,𝑠Δ𝑠=∞, (73) which contradicts (32). Therefore (58) holds. Noting Ψ(𝑇)≤𝑊(𝑇) for 𝑇∈[𝑇₀,∞)_𝕋, by using (58), we obtain ^∞_𝑇0𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆Ψ₊(𝜎(𝑠))^𝜆Δ𝑠≤^∞_𝑇0𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^𝜆−1(𝛿(𝜎(𝑠)))^𝜆(𝑊(𝜎(𝑠)))^𝜆Δ𝑠<∞, (74) which is contradicting with (33). This completes the proof.

Remark 8. From Theorem 7, we can obtain different conditions for oscillation of all solutions of (1) with different choices of 𝛿(𝑡) and 𝐻(𝑡,𝑠). For example, 𝐻(𝑡,𝑠)=(𝑡−𝑠)^𝑚 or 𝐻(𝑡,𝑠)=(ln((𝑡+1)/(𝑠+1)))^𝑚.

Theorem 9. Assume that (H₁)–(H₄), (4), (30)-(31), and (33) hold, where 𝐻, ℎ, 𝛿, and Ψ are defined in Theorem 7. Assume that liminf_𝑡→∞1𝐻𝑡,𝑇₀^𝑡_𝑇0𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1Δ𝑠<∞,liminf_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1−𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊((𝑇)𝑡,𝑠)Δ𝑠≥Ψ (75)(76) holds, where 𝑇∈[𝑇₀,∞)_𝕋, 𝐺(𝑡,𝑠)=(𝛿^Δ(𝑠)−(𝑝(𝑠)/𝑎(𝑠))𝛿(𝑠))(𝐻(𝑡,𝑠))^1/𝛾−𝛿(𝑠)ℎ(𝑡,𝑠),𝐺₊(𝑡,𝑠)=max{0,𝐺(𝑡,𝑠)}. Then (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that (1) has a nonoscillatory solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Without loss of generality we may assume that there exists 𝑡₁∈[𝑡₀,∞)_𝕋, such that 𝑥(𝑡)>0 and 𝑥[𝜏(𝑡)]>0 for all 𝑡∈[𝑡₁,∞)_𝕋. So 𝑧(𝑡)>0 and there exists 𝑇₀∈[𝑡₁,∞)_𝕋 such that 𝑥(𝜏(𝑡))≥(1−𝑟(𝜏(𝑡)))𝑧(𝜏(𝑡)) (77) for 𝑡∈[𝑇₀,∞)_𝕋. Define the function 𝑊 by 𝑎𝑧𝑊=𝛿^Δ𝛾−1(𝑧∘𝜏)^𝛾−1,𝑡∈𝑇₀,∞_𝕋. (78) We proceed as in the proof of Theorem 7 to obtain (46) and (50), so that 1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1−𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠≤𝑊(𝑇). (79) Hence, (76) implies Ψ(𝑇)≤𝑊(𝑇),𝑇∈𝑇₀,∞_𝕋,liminf_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1(𝑇);Δ𝑠≥Ψ (80)(81) then we have Ψ(𝑇)≤liminf_𝑡→∞1𝐻()×𝑡,𝑇_𝑡𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1−𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠≤liminf_𝑡→∞1𝐻()×𝑡,𝑇_𝑡𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1Δ𝑠−liminf_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠. (82) From the above inequality and (75), we have liminf_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝑎(𝜏(𝑠))𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠<∞. (83) Therefore, there exists a sequence {𝑇_𝑛}^∞_𝑛=1∶ 𝑇_𝑛∈[𝑇₀,∞)_𝕋 with lim_𝑛→∞𝑇_𝑛=∞ such that lim_𝑛→∞1𝐻𝑇_𝑛,𝑇^𝑇𝑛_𝑇𝑎(𝜏(𝑠))𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠<∞. (84) Definitions of 𝐴(𝑡) and 𝐵(𝑡) are as in Theorem 7; from (46), and noting (81), we have limsup_𝑡→∞[𝐵(𝑡)(𝑡𝑇−𝐴)]≤𝑊₀−liminf_𝑡→∞1𝐻𝑡,𝑇₀×^𝑡_𝑇0𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1𝑇Δ𝑠≤𝑊₀−Ψ𝑇₀<∞. (85) For the above sequence {𝑇_𝑛}^∞_𝑛=1, lim_𝑛→∞𝐵𝑇_𝑛−𝐴𝑇_𝑛≤limsup_𝑡→∞[𝐵(𝑡)−𝐴(𝑡)]<∞. (86) We obtain (58) by using reductio ad absurdum. The rest of the proof is similar to that of Theorem 7 and hence is omitted. This completes the proof.

If (4) is not satisfied, that is, if condition (5) holds, we can obtain the following results.

Theorem 10. Assume that (H₁)–(H₄), (5), and (30)–(34) hold, where 𝐻, ℎ, 𝛿, and Ψ are defined in Theorem 7. Assume that ^∞_𝑡01𝑎(𝑡)^𝑡_𝑡0𝑒_−𝑝/𝑎((𝑠𝑞(𝑠)×((𝜏(𝑠𝑡,𝜎))1−𝑟)))^𝛾−1Δ𝑠^1/(𝛾−1)Δ𝑡=∞ (87) holds. Then every solution 𝑥(𝑡) of (1) is either oscillatory or converges to zero on [𝑡₀,∞)_𝕋.

Proof. As the proof of Theorem 7, assume that (1) has a nonoscillatory solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Without loss of generality we may assume that there exists 𝑡₁∈[𝑡₀,∞)_𝕋, such that 𝑥(𝑡)>0 and 𝑥[𝜏(𝑡)]>0 for all 𝑡∈[𝑡₁,∞)_𝕋. So 𝑧(𝑡)>0 and there exists 𝑡₂∈[𝑡₁,∞)_𝕋 such that 𝑥(𝜏(𝑡))≥(1−𝑟(𝜏(𝑡)))𝑧(𝜏(𝑡)) (88) for 𝑡∈[𝑡₂,∞)_𝕋. In the proof of Lemma 6, we find that 𝑧^Δ(𝑡) is either eventually positive or eventually negative. Thus, we will distinguish the following two cases:(I)𝑧^Δ(𝑡)>0  for  𝑡∈[𝑡₂,∞)_𝕋;(II)𝑧^Δ(𝑡)<0  for  𝑡∈[𝑡₂,∞)_𝕋.
Case (I). When 𝑧^Δ(𝑡) is an eventually positive and the proof is similar to that of the proof of Theorem 7, we can obtain that (1) is oscillatory.
Case (II). When 𝑧^Δ(𝑡) is an eventually negative, 𝑧(𝑡) is decreasing and lim_𝑡→∞𝑧(𝑡)=∶𝑏≥0 exists. Therefore, there exists 𝑇₀∈[𝑡₂,∞)_𝕋 such that 𝑧(𝜏(𝑡))>𝑧(𝑡)>𝑧(𝜎(𝑡))≥𝑏≥0 (89) for 𝑡∈[𝑇₀,∞)_𝕋. Define the function 𝑢(𝑡)=𝑎(𝑡)|𝑧^Δ(𝑡)|^𝛾−2𝑧^Δ(𝑡)=−𝑎(𝑡)|𝑧^Δ(𝑡)|^𝛾−1. Equations (1) and (89) yield 𝑢^Δ(𝑡)𝑝(𝑡)=−𝑎(𝑡)𝑢(𝑡)−𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))^𝛾−1𝑝(𝑡)≤−𝑎(𝑡)𝑢(𝑡)−𝐿𝑏^𝛾−1𝑞(𝑡)(1−𝑟(𝜏(𝑡)))^𝛾−1,𝑡∈𝑇₀,∞_𝕋. (90) The inequality (90) is the assumed inequality of [8, Theorem 6.1] (see also [37, Lemma 1]). All assumptions of [8, Theorem 6.1], for example, −𝑝/𝑎∈ℛ⁺, are satisfied as well. Hence the conclusion of [8, Theorem 6.1] holds; that is, 𝑢(𝑡)≤𝑢𝑇₀𝑒_−𝑝/𝑎𝑡,𝑇₀−𝐿𝑏^𝛾−1×^𝑡_𝑇0𝑒_−𝑝/𝑎(𝑡,𝜎(𝑠))𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1Δ𝑠<−𝐿𝑏^𝛾−1𝑡_𝑇0𝑒_−𝑝/𝑎(𝑡,𝜎(𝑠))𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1Δ𝑠 (91) for all 𝑡∈[𝑇₀,∞)_𝕋, and thus ^𝑙_𝑇0𝑧^Δ(𝑡)Δ𝑡<−𝑏𝐿^{1/(𝛾−1)𝑙}_𝑇01𝑎(𝑡)^𝑡_𝑇0𝑒_−𝑝/𝑎((𝑠𝑞(𝑠)×((𝜏(𝑠𝑡,𝜎))1−𝑟)))^𝛾−1Δ𝑠^1/(𝛾−1)Δ𝑡 (92) for all 𝑙∈[𝑇₀,∞)_𝕋. Assuming 𝑏>0 and using (87) in (92), we can get lim_𝑙→∞𝑧(𝑙)=−∞, and this is a contradiction to the fact that 𝑧(𝑡)>0 for 𝑡∈[𝑡₁,∞)_𝕋. Thus 𝑏=0; that is, lim_𝑡→∞𝑧(𝑡)=0. Then, it follows from (1−𝑟(𝑡))𝑧(𝑡)≤𝑥(𝑡)≤𝑧(𝑡) that lim_𝑡→∞𝑥(𝑡)=0 holds. This completes the proof.

Using the same method as in the proofs of Theorems 9 and 10, we can easily obtain the following results.

Theorem 11. Assume that (H₁)–(H₃), (5), (30)-(31), (33), (75)-(76), and (87) hold, where 𝐻, ℎ, 𝛿, and Ψ are defined in Theorem 9. Then every solution 𝑥(𝑡) of (1) is either oscillatory or converges to zero on [𝑡₀,∞)_𝕋.

Remark 12. The theorems in this paper are new even for the cases of 𝕋=ℝ and 𝕋=ℤ.

Example 13. Consider a second-order half-linear delay 2-difference equation with damping 1𝑡²𝑧^Δ(𝑡)𝑧^Δ(𝑡)^Δ+1𝑡⁴𝑧^Δ(𝑡)𝑧^Δ(𝑡)+1𝑡³𝑥𝑡2𝑥𝑡2=0,𝑡∈2^ℤ,𝑡≥𝑡₀∶=2, (93) where 𝑧(𝑡)=𝑥(𝑡)+(1/2)𝑥(𝑡/2). Here, we have 𝑎(𝑡)=1𝑡²(𝑡)=1,𝑟2(𝑡)=1,𝑝𝑡⁴,𝑞(𝑡)=1𝑡³(𝑢)(𝑡)=𝑡,𝑓=𝑢,𝜏2,𝛾=3. (94) Then 𝕋=2^ℤ is unbounded above, 𝜎(𝑡)=2𝑡, and 𝜇(𝑡)=𝑡. Conditions (H₁) and (H₃) are clearly satisfied, (H₄) holds with 𝐿=1, and (H₂) is satisfied as (𝑡)𝑝(𝑡)1−𝜇𝑎(𝑡)=1−𝑡⋅1/𝑡⁴1/𝑡²1=1−𝑡>0∀𝑡≥2. (95) Next, by [37, Lemma 2] and (H₂), we obtain 𝑒_−𝑝/𝑎(𝑡,2)≥1−^𝑡₂𝑝(𝑠)𝑎(𝑠)Δ𝑠=1−^𝑡₂𝑠⁻²2Δ𝑠=𝑡∀𝑡≥2, (96) so ^𝑡₂1𝑎(𝑠)𝑒_−𝑝/𝑎(𝑠,2)^1/(𝛾−1)≥Δ𝑠_𝑡2𝑠²⋅2𝑠^1/2Δ𝑠=^𝑡₂√2𝑠^1/2Δ𝑠⟶∞as𝑡⟶∞. (97) Hence (4) is satisfied. Now let 𝐻(𝑡,𝑠)=(𝑡−𝑠)²; then 𝐻^Δ𝑠()=()𝑡,𝑠𝑡−2𝑠²−(𝑡−𝑠)²𝑠=(2𝑡−3𝑠)⋅(−𝑠)𝑠=−(2𝑡−3𝑠)<0,∀𝑡>𝑠≥𝑡₀∶=2. (98) Since −𝐻^Δ𝑠(𝑡,𝑠)=2𝑡−3𝑠=2𝑡−3𝑠(𝑡−𝑠)^4/3(𝑡−𝑠)^22/3=2𝑡−3𝑠(𝑡−𝑠)^4/3[𝐻(𝑡,𝑠)]^{(𝛾−1)/𝛾}, (99) let ℎ(𝑡,𝑠)=(2𝑡−3𝑠)/(𝑡−𝑠)^4/3; then condition (30) holds. We have 0<inf_𝑠≥𝑇0liminf_𝑡→∞𝐻(𝑡,𝑠)𝐻𝑡,𝑇₀=inf_𝑠≥𝑇0liminf_𝑡→∞(𝑡−𝑠)²𝑡−𝑇₀²=1<∞,∀𝑇₀∈𝑡₀,∞_𝕋, (100) so condition (31) holds. Let 𝛿(𝑡)=𝑡 as 𝑡≥2; then 𝛿^Δ(𝑡)=1 for all 𝑡∈[𝑡₀,∞)_𝕋, and 𝐺(𝑡,𝑠)=𝛿^Δ(𝑠)−𝑝(𝑠)𝑎(𝑠)𝛿(𝑠)(𝐻(𝑡,𝑠))^1/𝛾(𝑠)ℎ()=1−𝛿𝑡,𝑠1−𝑠𝐻^1/3()−𝑠()𝑡,𝑠2𝑡−3𝑠𝐻^2/3(𝑡,𝑠)=𝐻^1/3()−𝐻𝑡,𝑠^1/3(𝑡,𝑠)𝑠−𝑠(2𝑡−3𝑠)𝐻^2/3(𝑡,𝑠)<𝐻^1/3(𝑡,𝑠), (101) for all 𝑡>𝑠≥2. Hence ^𝑡_𝑇0𝑎(𝜏(𝑠))𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊()<𝑡,𝑠Δ𝑠_𝑡𝑇0(𝑠/2)⁻²(𝑠⋅(1/2))²𝐻^1/3(𝑡,𝑠)³Δ𝑠=16^𝑡_𝑇0(𝑡−𝑠)²𝑠⁴−8Δ𝑠=16+87𝑡−23𝑡𝑡−16−8𝑡²7𝑇³₀+8𝑡3𝑇²₀−2𝑇₀. (102) We get limsup_𝑡→∞1𝐻𝑡,𝑇₀^𝑡_𝑇0𝑎(𝜏(𝑠))𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠≤limsup_𝑡→∞−816+87𝑡−23𝑡𝑡−16−8𝑡²7𝑇³₀+8𝑡3𝑇²₀−2𝑇₀×𝑡−𝑇₀⁻²=12871𝑇³₀<∞; (103) thus condition (32) holds. Let Ψ(𝑡)=1/4𝑡; then ^∞_𝑇0𝛿(𝑠)𝜏^Δ(𝑠)(𝑎(𝜏(𝑠)))^1/(𝛾−1)Ψ₊(𝜎(𝑠))𝛿(𝜎(𝑠))^{𝛾/(𝛾−1)}=Δ𝑠_∞𝑇0𝑠⋅(1/2)(𝑠/2)^−21/21/8𝑠2𝑠^3/2=Δ𝑠_∞𝑇0𝑠²4116𝑠^23/2=1Δ𝑠256^∞_𝑇01𝑠1Δ𝑠=256ln𝑠ln2^∞_𝑇0=∞; (104) that is, condition (33) holds. Since 𝐿𝐻(^𝑡_𝑇𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1=1Δ𝑠4^𝑡_𝑇(𝑡−𝑠)²⋅𝑠𝑠³=1Δ𝑠4^𝑡_𝑇𝑡²𝑠²−2𝑡𝑠=1+1Δ𝑠4−2𝑡²𝑠−2𝑡ln𝑠ln2+𝑠^𝑡_𝑇=14−2𝑡−2𝑡ln𝑡−1ln2+𝑡4−2𝑡²𝑇−2𝑡ln𝑇,ln2+𝑇 (105) then limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1=1Δ𝑠.2𝑇 (106) Moreover, (103) implies limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(𝑡,𝑠)Δ𝑠≤128163𝑇³. (107) Thus, when 𝑇 is enough large, we have limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐿𝐻(𝑡,𝑠)𝛿(𝑠)𝑞(𝑠)(1−𝑟(𝜏(𝑠)))^𝛾−1−𝑎(𝜏(𝑠))𝛾^𝛾𝛿(𝑠)𝜏^Δ(𝑠)^𝛾−1𝐺^𝛾₊(≥1𝑡,𝑠)Δ𝑠−2𝑇128163𝑇³≥1(𝑇),4𝑇=Ψ (108) so (34) is satisfied. By Theorem 7, (93) is oscillatory on [𝑡₀,∞)_𝕋. Similarly, conditions (75) and (76) are satisfied as well. By Theorem 9, we can also obtain that (93) is oscillatory. But the other known results cannot be applied in (93).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This work was supported by the Natural Science Foundation of Shandong Province of China under Grant no. ZR2013AM003, the Development Program in Science and Technology of Shandong Province of China under Grant no. 2010GWZ20401, and the Science Foundation of Binzhou University under Grant no. BZXYKJ0810.