Oscillation of Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales

Abstract

We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale 𝕋 (𝑟(𝑡)𝑔(𝑥(𝑡), 𝑥^Δ(𝑡)))^Δ+𝑝(𝑡)𝑔(𝑥(𝑡), 𝑥^Δ(𝑡))+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0, where 𝑟(𝑡), 𝑝(𝑡), and 𝑞(𝑡) are positive right dense continuous (rd-continuous) functions on 𝕋. Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results.

1. Introduction

The purpose of this paper is to give several oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale 𝕋𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0, (1) subject to the following hypotheses.(𝐻₁)𝕋 is a time scale which is unbounded above and 𝑡₀∈𝕋 with 𝑡₀>0. The time scale interval [𝑡₀,∞)_𝕋 is defined by [𝑡₀,∞)_𝕋=[𝑡₀,∞)∩𝕋.(𝐻₂)𝑟(𝑡), 𝑝(𝑡), and 𝑞(𝑡) are positive right dense continuous functions on 𝕋 such that −𝑝/𝑟∈ℛ⁺ and ^∞_𝑡01𝑟(𝑡)𝑒_−𝑝/𝑟𝑡,𝑡₀Δ𝑡=∞, (2) or ^∞_𝑡01𝑟(𝑡)𝑒_−𝑝/𝑟𝑡,𝑡₀Δ𝑡<∞. (3)(𝐻₃)Consider 𝑓∈𝐶(ℝ,ℝ), 𝑥𝑓(𝑥)>0 for all 𝑥≠0 and there exists a positive constant 𝐿 such that 𝑓(𝑥)/𝑥≥𝐿.(𝐻₄)Consider 𝑔∈𝐶¹(ℝ²,ℝ), 𝑣𝑔(𝑢,𝑣)>0 for all 𝑣≠0 and for any fixed 𝑢∈ℝ and there exist positive constants 𝐾₁,𝐾₂ such that 𝐾₁≤𝑔(𝑢,𝑣)𝑣≤𝐾₂. (4)(𝐻₅)𝜏∶𝕋→𝕋 is a strictly increasing and differentiable function such that 𝜏(𝑡)≤𝑡,lim_𝑡→∞𝜏(𝑡)=∞. (5)

By a solution of (1), we mean that a nontrivial real valued function 𝑥 satisfies (1) for 𝑡∈𝕋. A solution 𝑥 of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory. In this work, we study the solutions of (1) which are not identically vanishing eventually.

Many results have been obtained on the oscillation and nonoscillation of dynamic equations on time scales (see e.g., the papers [1–17], the books [18, 19] and the references cited therein). It is easy to see that (1) can be transformed into the equation 𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑥^Δ(𝑡)+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡∈𝕋,𝑡≥𝑡₀, (6) where 𝑔(𝑥,𝑥^Δ)=𝑥^Δ. If 𝑟(𝑡)=1, 𝑓(𝑥)=𝑥, and 𝜏(𝑡)=𝑡, then (6) is simplified to the equation 𝑥^ΔΔ(𝑡)+𝑝(𝑡)𝑥^Δ(𝑡)+𝑞(𝑡)𝑥(𝑡)=0,𝑡∈𝕋,𝑡≥𝑡₀. (7) Also, if 𝑝(𝑡)=0, (6) is simplified to the equation 𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡∈𝕋,𝑡≥𝑡₀. (8) If 𝑟(𝑡)=1, (8) takes the form 𝑥^ΔΔ(𝑡)+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡∈𝕋,𝑡≥𝑡₀. (9) If 𝑓(𝑥)=𝑥, (9) becomes 𝑥^ΔΔ(𝑡)+𝑞(𝑡)𝑥(𝜏(𝑡))=0,𝑡∈𝕋,𝑡≥𝑡₀. (10) In 2002, Guseinov and Kaymakçalan [10] studied (7) and established some sufficient conditions for nonoscillation. They proved that if ^∞_𝑡0𝑝(𝑡)Δ𝑡<∞,^∞_𝑡0𝑡𝑞(𝑡)Δ𝑡<∞, (11) then (7) is nonoscillatory. In 2005, Agarwal et al. [2] studied the linear delay dynamic equation (10) and Şahiner [12] considered the nonlinear delay dynamic equation (9) and gave some sufficient conditions for oscillation of (10) and (9). In 2007, Erbe et al. [8] considered the general nonlinear delay dynamic equations (8). They obtained some oscillation criteria which improve the results given by Şahiner [12]. In 2011, Zhang and Gao [16] considered the oscillation of solutions of second-order nonlinear delay dynamic equation (6) with damping and establishing some new results. In this paper, we use the generalized Riccati transformation and the inequality technique to obtain some new oscillation criteria for (1). Our results generalize and improve the results in [16].

This paper is organized as follows. In Section 2, we present some preliminaries on time scales. In Section 3, we give some lemmas that we need through our work. In Section 4, we establish some new sufficient conditions for oscillation of (1). Finally, in Section 5, we present some examples to illustrate our results.

2. Some Preliminaries on Time Scales

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. On any time scale 𝕋, we define the forward and backward jump operators by 𝜎(𝑡)=inf{𝑠∈𝕋,𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋,𝑠<𝑡}. (12) A point 𝑡∈𝕋, 𝑡> inf 𝕋 is said to be left dense if 𝜌(𝑡)=𝑡, right dense if 𝑡< sup 𝕋 and 𝜎(𝑡)=𝑡, left-scattered if 𝜌(𝑡)<𝑡, and right-scattered if 𝜎(𝑡)>𝑡. The graininess function 𝜇 for a time scale 𝕋 is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. The set 𝕋^𝑘 is derived from the time scale 𝕋 as 𝕋^𝑘=𝕋−𝑚 if 𝕋 has a left-scattered maximum 𝑚. Otherwise, 𝕋^𝑘=𝕋.

A function 𝑓∶𝕋→ℝ is called rd-continuous provided that it is continuous at right dense points of 𝕋 and its left-sided limits exist at left dense points of 𝕋. The set of rd-continuous functions is denoted by 𝐶_rd(𝕋,ℝ). By 𝐶¹_rd(𝕋,ℝ), we mean the set of functions whose delta derivative belong to 𝐶_rd(𝕋,ℝ).

A function 𝑓∶𝕋→ℝ is regressive provided that 1+𝜇(𝑡)𝑓(𝑡)≠0,∀𝑡∈𝕋^𝑘, (13) holds. The set of all regressive and rd-continuous functions 𝑓∶𝕋→ℝ is denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ). (14) If 𝑞∈ℛ, then we define the exponential function 𝑒_𝑞(𝑡,𝑠) by 𝑒_𝑞(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑞(𝜏)Δ𝜏,for𝑠,𝑡∈𝕋, (15) where the cylinder function 𝜉_ℎ(𝑧) is defined by 𝜉_ℎ(𝑧)=1ℎLog(1+𝑧ℎ). (16) For a function 𝑓∶𝕋→ℝ (the range ℝ of 𝑓 may be actually replaced by any Banach space), the delta derivative 𝑓^Δ is defined by 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜎(𝑡),−𝑡 (17) provided 𝑓 is continuous at 𝑡 and 𝑡 is right-scattered. If 𝑡 is not right-scattered, then the delta derivative 𝑓^Δ(𝑡) is defined by 𝑓^Δ(𝑡)=lim_𝑠→𝑡+𝑓(𝜎(𝑡))−𝑓(𝑡)𝑡−𝑠=lim_𝑠→𝑡+𝑓(𝑡)−𝑓(𝑠)𝑡−𝑠 (18) provided this limit exists.

A function 𝑓∶[𝑎,𝑏]→ℝ is said to be differentiable if its derivative exists. The derivative 𝑓^Δ and the shift 𝑓^𝜎 of a function 𝑓 are related by the equation 𝑓^𝜎=𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡). (19) The delta derivative rules of the product 𝑓𝑔 and the quotient 𝑓/𝑔 (where 𝑔𝑔^𝜎≠0) of two differentiable functions 𝑓 and 𝑔 are given by 𝑓⋅𝑔^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). (20) An integration by parts formula reads ^𝑏_𝑎𝑓(𝑡)𝑔^Δ(𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑓^Δ(𝑡)𝑔^𝜎(𝑡)Δ𝑡 (21) or ^𝑏_𝑎𝑓^𝜎(𝑡)𝑔^Δ(𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑓^Δ(𝑡)𝑔(𝑡)Δ𝑡 (22) and the infinite integral is defined by ^∞_𝑏𝑓(𝑠)Δ𝑠=lim_𝑡→∞^𝑡_𝑏𝑓(𝑠)Δ𝑠. (23) Throughout this paper, we use 𝑑₋(𝑡){(𝑡(𝑡)𝑅(𝑡)=∶max0,−𝑑)},𝛼∶=𝑅(𝑡)(𝑡),+𝜇 (24) where 𝑅(𝑡)=𝑘𝑟(𝑡)∫^𝑡_𝑡0(Δ𝑠/𝑟(𝑠)), and positive constants 𝐾𝑘=¹𝐾₂𝐾,𝜂=²𝐿, (25) where 𝐿, 𝐾₁, and 𝐾₂ are defined in (𝐻₃), (𝐻₄).

3. Several Lemmas

In this section, we present some lemmas that we need to prove our results in the next section.

Lemma 1. (Bohner and Peterson [18, Chapter 2]). If 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).

Lemma 2. (Bohner [5, Lemma 2]). For nonnegative 𝑝 with −𝑝∈ℛ⁺, one has the inequality 1−^𝑡_𝑠𝑝(𝑢)Δ𝑢≤𝑒_−𝑝(𝑡,𝑠),∀𝑡≥𝑠. (26)

Lemma 3. If (𝐻₁)--(𝐻₅), (2) hold and (1) has a positive solution 𝑥 on [𝑡₀,∞)_𝕋, then (𝑟(𝑡)𝑔(𝑥,𝑥^Δ))^Δ<0, 𝑥^Δ(𝑡)>0, and 𝑥(𝑡)>𝛼(𝑡)𝑥(𝜎(𝑡)) for 𝑡∈[𝑡₀,∞)_𝕋.

Proof. Since 𝑥 is a positive solution of (1) on [𝑡₀,∞)_𝕋, we have 𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)=−𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))<0. (27) Therefore, 𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)𝑒_−𝑝/𝑟⋅,𝑡₀^Δ=𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)^Δ𝑒_−𝑝/𝑟⋅,𝑡₀−𝑒^Δ_−𝑝/𝑟⋅,𝑡₀𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)×𝑒_−𝑝/𝑟⋅,𝑡₀𝑒^𝜎_−𝑝/𝑟(⋅,𝑡₀)⁻¹=𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)×𝑒^𝜎_−𝑝/𝑟(⋅,𝑡₀)⁻¹<0. (28) Hence, 𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)𝑒_−𝑝/𝑟⋅,𝑡₀isdecreasing. (29) We claim that 𝑥^Δ(𝑡)>0 on [𝑡₀,∞)_𝕋. If not, then there is 𝑡≥𝑡₁ such that 𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)𝑒_−𝑝/𝑟𝑡,𝑡₀≤𝑟𝑡₁𝑔𝑥𝑡₁,𝑥^Δ𝑡₁𝑒_−𝑝/𝑟𝑡₁,𝑡₀=∶𝑐<0. (30) By (𝐻₄), we get 𝑥^Δ(𝑡)≤𝑐𝐾₁1𝑟(𝑡)𝑒_−𝑝/𝑟𝑡,𝑡₀,𝐾₁>0. (31) Integrating from 𝑡₁ to 𝑡, we get 𝑥(𝑡)≤𝑥𝑡₁+𝑐𝐾₁^𝑡_𝑡11𝑟(𝑠)𝑒_−𝑝/𝑟𝑠,𝑡₀Δ𝑠. (32) This implies that 𝑥(𝑡) is eventually negative which is a contradiction. Hence, 𝑥^Δ(𝑡)>0 on [𝑡₀,∞)_𝕋. Therefore, 𝑥(𝑡)>𝑥(𝑡)−𝑥𝑡₁=^𝑡_𝑡1𝑥^Δ(𝑠)>Δ𝑠_𝑡𝑡11𝐾₂𝑔𝑥(𝑠),𝑥^Δ(𝑠)Δ𝑠by𝐻₃>1𝐾₂^𝑡_𝑡1𝑟(𝑠)𝑔𝑥(𝑠),𝑥^Δ(𝑠)𝑟(𝑠)Δ𝑠,𝐾₂>0. (33) Using the fact that 𝑟(𝑡)𝑔(𝑥(𝑡),𝑥^Δ(𝑡)) is strictly decreasing, we get 𝑥(𝑡)>𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)𝐾₂^𝑡_𝑡1Δ𝑠𝑟(𝑠)>𝐾₁𝐾₂𝑟(𝑡)𝑥^Δ(𝑡)^𝑡_𝑡1Δ𝑠𝑟(𝑠),𝐾₁>0by𝐻₃>𝑘𝑟(𝑡)𝑥^Δ(𝑡)^𝑡_𝑡1Δ𝑠𝑟(𝑠)=𝑅(𝑡)𝑥^Δ(𝑡)𝐾,for𝑘=¹𝐾₂>0, (34) where 𝑅(𝑡)=𝑘𝑟(𝑡)∫^𝑡_𝑡1(Δ𝑠/𝑟(𝑠)).
Hence, 𝑥(𝑡)𝑥^𝜎(𝑡)=𝑥(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥^Δ(𝑡)≥𝑅(𝑡)𝑅(𝑡)(𝑡)(𝑡)𝑡+𝜇=𝛼on₀,∞_𝕋. (35)

4. Main Results

Here, we establish some new sufficient conditions for oscillation of (1).

Theorem 4. Assume that (𝐻₁)--(𝐻₅), (2) hold and ∈𝐶¹_𝑟𝑑([𝑡₀,∞)_𝕋,𝕋),𝜏([𝑡₀,∞)_𝕋)=[𝑡₀,∞)_𝕋. If there exists a positive Δ-differentiable function 𝛿(𝑡) such that limsup_𝑡→∞^𝑡_𝑡0𝛼(𝜏(𝑠𝑞(𝑠)−(𝜏(𝑠))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)𝛿^Δ(𝑠)𝛿^𝜎(𝑠)−𝑝(𝑠)𝑟(𝑠)𝛼(𝜏(𝑠))²×𝛿^𝜎(𝑠)Δ𝑠=∞, (36) then every solution of (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that (1) has a nonoscillatory solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Also, assume that 𝑥(𝑡)>0,  𝑥(𝜏(𝑡))>0 for all 𝑡∈[𝑡₁,∞)_𝕋, 𝑡₁∈[𝑡₀,∞)_𝕋. Consider the generalized Riccati substitution 𝑤(𝑡)(𝑡)𝑟(𝑡)𝑔𝑥(𝑡)=𝛿,𝑥^Δ(𝑡)𝑥(𝜏(𝑡))>0. (37) Using the delta derivative rules of product and quotient of two functions, we have 𝑤^Δ(𝑡)=𝛿^Δ(𝑡)𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)𝑥(𝜏(𝑡))+𝛿^𝜎(𝑡)𝑟(𝑡)𝑔(𝑥(𝑡),𝑥^Δ(𝑡))𝑥(𝜏(𝑡))^Δ=𝛿^Δ(𝑡)𝛿(𝑡)𝑤(𝑡)+𝛿^𝜎(𝑡)×𝑥(𝜏(𝑡))𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)^Δ−(𝑥(𝜏(𝑡)))^Δ𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)×𝑥(𝜏(𝑡))𝑥^𝜎(𝜏(𝑡))⁻¹=𝛿^Δ(𝑡)𝛿(𝑡)𝑤(𝑡)+𝛿^𝜎(𝑡)×−𝑝(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)𝑥^𝜎(𝜏(𝑡−𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))))𝑥^𝜎(𝜏(𝑡−𝑥))_Δ(𝜏(𝑡))𝜏^Δ(𝑡)𝛿(𝑡)𝑥^𝜎(𝜏(𝑡𝑤(𝑡).)) (38) From the definition of 𝑤(𝑡), we have 𝑤^Δ(𝑡)=𝛿^Δ(𝑡)𝛿(𝑡)𝑤(𝑡)+𝛿^𝜎(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡))𝛿(𝑡)𝑟(𝑡)𝑥^𝜎(𝜏(𝑡𝑤(𝑡)−𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))))𝑥^𝜎(𝜏(𝑡−𝑥))_Δ(𝜏(𝑡))𝜏^Δ(𝑡)𝛿(𝑡)𝑥^𝜎(𝜏(𝑡𝑤(𝑡).)) (39) Using the fact 𝑓(𝑥)/𝑥≥𝐿 and 𝑥(𝑡)/𝑥^𝜎(𝑡)>𝛼(𝑡), we get 𝑤^Δ(𝑡)≤𝛿^Δ(𝑡)𝛿(𝑡)𝑤(𝑡)+𝛿^𝜎(𝑡)−𝑝(𝑡)𝛼(𝜏(𝑡))𝛿(𝑡)𝑟(𝑡)𝑤(𝑡)(𝑡)𝛼(𝜏(𝑡−𝑥−𝐿𝑞))_Δ(𝜏(𝑡))𝜏^Δ(𝑡)𝛿(𝑡)𝑥^𝜎(𝜏(𝑡𝑤(𝑡).)) (40) Integrating the inequality (𝑟(𝑡)𝑔(𝑥,𝑥^Δ))^Δ<0 from 𝜏(𝑡) to 𝑡, using the definition of 𝑤(𝑡) and (𝐻₃), we get 𝑥^Δ(𝜏(𝑡>𝑥(𝜏(𝑡))))𝐾₂𝛿(𝑡)𝑟(𝜏(𝑡𝑤(𝑡).)) (41) Now, substituting (41) in (40), we have 𝑤^Δ(𝑡)≤𝛿^Δ(𝑡)𝛿(𝑡)𝑤(𝑡)+𝛿^𝜎(𝑡)−𝑝(𝑡)𝛼(𝜏(𝑡))𝛿(𝑡)𝑟(𝑡)𝑤(𝑡)(𝑡)𝛼(𝜏(𝑡−𝛼(𝜏(𝑡𝜏−𝐿𝑞))))^Δ(𝑡)𝐾₂𝛿²(𝑡)𝑟(𝜏(𝑡𝑤))²(𝑡). (42) Hence, 𝑤^Δ(𝑡)≤−𝐿𝑞(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡+𝛿))_Δ(𝑡)𝛿(𝑡)−𝑝(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))𝛿(𝑡)𝑟(𝑡)𝑤(𝑡)−𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))𝜏^Δ(𝑡)𝐾₂𝛿²(𝑡)𝑟(𝜏(𝑡𝑤))²(𝑡)=−𝐿𝑞(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡(𝑡)𝑤(𝑡)))+𝐺𝛿(𝑡)(𝑡)𝑤−𝑄²(𝑡)𝛿²(𝑡), (43) where 𝐺(𝑡)=𝛿^Δ(𝑡)−𝑝(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))/𝑟(𝑡) and 𝑄(𝑡)=𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))𝜏^Δ(𝑡)/𝐾₂𝑟(𝜏(𝑡)).
Therefore, 𝑤^Δ(𝑡)≤−𝐿𝑞(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡−))𝑄(𝑡)𝛿(𝑡)𝑤(𝑡)−𝐺(𝑡)2𝑄(𝑡)²+𝐺²(𝑡)(𝑡)(𝑡)𝛿4𝑄≤−𝐿𝑞^𝜎(𝑡)𝛼(𝜏(𝑡+𝐺))²(𝑡)(𝑡).4𝑄 (44) Hence, 𝑤^Δ(𝑡)≤−𝐿𝑞(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡+𝐾))₂𝛿^Δ(𝑡)−(𝑝(𝑡)/𝑟(𝑡))𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))²𝑟(𝜏(𝑡))4𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))𝜏^Δ(𝑡). (45) Integrating the above inequality from 𝑡₀ to 𝑡, we obtain ^𝑡_𝑡0𝛼(𝜏(𝑠𝑞(𝑠)−(𝜏(𝑠))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝛿^Δ(𝑠)𝛿^𝜎(𝑠)−𝑝(𝑠)𝑟(𝑠)𝛼(𝜏(𝑠))²𝛿^𝜎(𝑠)≤1Δ𝑠𝐿𝑤𝑡₀(𝑡)≤1−𝑤𝐿𝑤𝑡₀, (46) and taking the limit supremum as 𝑡→∞, we obtain a contradiction to condition (36). Therefore, every solution of (1) is oscillatory on [𝑡₀,∞)_𝕋.

Theorem 5. Assume that (𝐻₁)--(𝐻₅) and (2) hold. Let 𝐻 be an rd-continuous function defined as follows: 𝐻∶𝐷_𝕋≡(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀,𝑡,𝑠∈𝑡₀,∞_𝕋⟶ℝ, (47) such that 𝐻(𝑡,𝑡)=0,𝑓𝑜𝑟𝑡≥𝑡₀,𝐻(𝑡,𝑠)>0,𝑓𝑜𝑟𝑡>𝑠≥𝑡₀,𝑡,𝑠∈𝑡₀,∞_𝕋, (48) and 𝐻 has a nonpositive continuous Δ-partial derivative 𝐻^Δ𝑠(𝑡,𝑠). If there exists a positive Δ-differentiable function 𝛿(𝑡) such that limsup_𝑡→∞1𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻(𝛼(𝜏(𝑠𝑞(𝑠)−(𝜏(𝑠𝑡,𝑠)))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝛿^Δ(𝑠)𝛿^𝜎(𝑠)−𝑝(𝑠)𝑟(𝑠)𝛼(𝜏(𝑠))²×𝛿^𝜎(𝑠)Δ𝑠=∞, (49) then every solution of (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that (1) has a nonoscillatory solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Also, assume that 𝑥(𝑡)>0, 𝑥(𝜏(𝑡))>0, for all 𝑡∈[𝑡₁,∞)_𝕋,𝑡₁∈[𝑡₀,∞)_𝕋. We proceed as in the proof of Theorem 4 to get (45) 𝐿𝑞(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡−𝐾))₂𝛿^Δ(𝑡)−𝑝(𝑡)/𝑟(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))²𝑟(𝜏(𝑡))4𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))𝜏^Δ(𝑡)≤−𝑤^Δ(𝑡). (50) Multiplying the above inequality by 𝐻(𝑡,𝑠), integrating from 𝑡₀ to 𝑡 and using (48), we get ^𝑡_𝑡0𝐻(𝑡,𝑠)𝐿𝛼(𝜏(𝑠))𝑞(𝑠)𝛿^𝜎(𝑠)−𝐾₂𝑟(𝜏(𝑠))4𝛿^𝜎(𝑠)𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝛿^Δ(𝑠)−𝑝(𝑠)𝑟(𝑠)𝛿^𝜎(𝑠)𝛼(𝜏(𝑠))²Δ𝑠≤−^𝑡_𝑡0𝐻(𝑡,𝑠)𝑤^Δ(𝑠)()𝑤(𝑠Δ𝑠=−𝐻𝑡,𝑠)|^𝑡_𝑡0+^𝑡_𝑡0𝐻^Δ𝑠(𝑡,𝑠)𝑤(𝜎(𝑠))Δ𝑠≤𝐻𝑡,𝑡₀𝑤𝑡₀. (51) Thus, 1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝛼(𝜏(𝑠𝑞(𝑠)−(𝜏(𝑠𝑡,𝑠)))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝛿^Δ(𝑠)𝛿^𝜎(𝑠)−𝑝(𝑠)𝑟(𝑠)𝛼(𝜏(𝑠))²×𝛿^𝜎(𝑠)≤1Δ𝑠𝐿𝑤𝑡₀<∞, (52) which is a contradiction to (49). This completes the proof.

Now, If 𝐻(𝑡,𝑠) is a function defined by 𝐻(𝑡,𝑠)=(𝑡−𝑠)^𝑚,𝑚≥1,𝑡≥𝑠≥𝑡₀,𝑡,𝑠∈𝑡₀,∞_𝕋, (53) then, we have the following result.

Corollary 6. Assume that (𝐻₁)--(𝐻₅) and (2) hold. If there exists a positive Δ-differentiable function 𝛿(𝑡) and 𝑚≥1 such that limsup_𝑡→∞1𝑡^𝑚𝑡_𝑡0(𝑡−𝑠)^𝑚×𝛼(𝜏(𝑠𝑞(𝑠)−(𝜏(𝑠))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝛿^Δ(𝑠)𝛿^𝜎(𝑠)−𝑝(𝑠)𝑟(𝑠)𝛼(𝜏(𝑠))²×𝛿^𝜎(𝑠)Δ𝑠=∞, (54) then every solution of (1) is oscillatory on [𝑡₀,∞)_𝕋.

Theorem 7. Assume that (𝐻₁)--(𝐻₅) and (2) hold. Let 𝐻 be an rd-continuous function defined as 𝐻∶𝐷_𝕋≡(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀,𝑡,𝑠∈𝑡₀,∞_𝕋⟶ℝ, (55) such that 𝐻(𝑡,𝑡)=0,𝑓𝑜𝑟𝑡≥𝑡₀,𝐻(𝑡,𝑠)>0,𝑓𝑜𝑟𝑡>𝑠≥𝑡₀,𝑡,𝑠∈𝑡₀,∞_𝕋, (56) and 𝐻 has a nonpositive continuous Δ-partial derivative 𝐻^Δ𝑠(𝑡,𝑠). Let ℎ∶𝐷_𝕋→ℝ be an rd-continuous function satisfying −𝐻^Δ𝑠(𝑡,𝑠)=ℎ(𝑡,𝑠)𝐻(𝑡,𝑠),∀(𝑡,𝑠)∈𝐷_𝕋. (57) If there exists a positive nondecreasing Δ-differentiable function 𝛿(𝑡) such that limsup_𝑡→∞1𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻()𝑞(𝑠)𝛼(𝜏(𝑠−(𝜏(𝑠𝑡,𝑠))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝐺(𝑠)𝛿^𝜎(𝑠)𝐻()+𝛿(𝑠)𝑡,𝑠𝛿^𝜎(𝑠)ℎ₋(𝑡,𝑠)²×𝛿^𝜎(𝑠)Δ𝑠=∞, (58) where 𝐺(𝑡)=𝛿^Δ(𝑡)−𝑝(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))/𝑟(𝑡), then every solution of (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that (1) has a nonoscillatory solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Also, assume that 𝑥(𝑡)>0,  𝑥(𝜏(𝑡))>0, for all 𝑡∈[𝑡₁,∞)_𝕋, 𝑡₁∈[𝑡₀,∞)_𝕋 and there is 𝑇∈[𝑡₀,∞)_𝕋 such that 𝑥(𝑡) satisfies the conclusion of Lemma 3 on [𝑇,∞)_𝕋. Proceeding as in the proof of Theorem 4, we get (43) which has the form 𝑤^Δ(𝑡)≤−𝐿𝑞(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡(𝑡)𝑤(𝑡)))+𝐺𝛿(𝑡)(𝑡)𝑤−𝑄²(𝑡)𝛿²(𝑡), (59) where 𝐺(𝑡)=𝛿^Δ(𝑡)−𝑝(𝑡)𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))/𝑟(𝑡) and 𝑄(𝑡)=𝛿^𝜎(𝑡)𝛼(𝜏(𝑡))𝜏^Δ(𝑡)/𝐾₂𝑟(𝜏(𝑡)).
Multiplying the above inequality by 𝐻(𝑡,𝑠), integrating from 𝑡₀ to 𝑡, and using (56) and (57), we get 𝐿𝐻(^𝑡_𝑡0𝑡,𝑠)𝑞(𝑠)𝛿^𝜎(𝑠)𝛼(𝜏(𝑠))Δ𝑠≤𝐻𝑡,𝑡₀𝑤𝑡₀+^𝑡_𝑡0𝐻^Δ𝑠()𝑤(𝜎(𝑠+𝑡,𝑠))Δ𝑠_𝑡𝑡0𝐻()𝐺(𝑠)𝑤(𝑠)𝑡,𝑠𝛿(𝑠)−Δ𝑠_𝑡𝑡0𝐻()𝑄(𝑠)𝑤𝑡,𝑠²(𝑠)𝛿²(𝑠)Δ𝑠=𝐻𝑡,𝑡₀𝑤𝑡₀−^𝑡_𝑡0ℎ(𝑡,𝑠)𝐻(𝑡,𝑠)𝑤(𝜎(𝑠))Δ𝑠+^𝑡_𝑡0𝐻()𝐺(𝑠)𝑤(𝑠)𝑡,𝑠𝛿(𝑠)Δ𝑠−^𝑡_𝑡0𝐻()𝑄(𝑠)𝑤𝑡,𝑠²(𝑠)𝛿²(𝑠)Δ𝑠. (60) Using (𝐻₅) and Lemma 3, we get 1𝑥(𝜏(𝑡≥1))𝑥(𝜏(𝜎(𝑡,𝑟(𝑡)𝑔𝑥(𝑡)))),𝑥^Δ(𝑡)≥𝑟(𝑡)𝑔𝑥(𝑡),𝑥^Δ(𝑡)^𝜎. (61) Therefore, 𝐿𝐻(^𝑡_𝑡0𝑡,𝑠)𝑞(𝑠)𝛿^𝜎(𝑠)𝛼(𝜏(𝑠))Δ𝑠≤𝐻𝑡,𝑡₀𝑤𝑡₀+^𝑡_𝑡0𝐺(𝑠)𝐻(𝑡,𝑠)+𝛿(𝑠)ℎ₋(𝑡,𝑠)𝐻(𝑡,𝑠)𝛿(𝑠)𝑤(𝑠)−Δ𝑠_𝑡𝑡0𝐻()𝑄(𝑠)𝑤𝑡,𝑠²(𝑠)𝛿²(𝑠)Δ𝑠=𝐻𝑡,𝑡₀𝑤𝑡₀+^𝑡_𝑡0𝐺(𝑠)𝐻(𝑡,𝑠)+𝛿(𝑠)ℎ₋(𝑡,𝑠)𝛿(𝑠)𝐻()𝑤(𝑠)−𝑡,𝑠Δ𝑠_𝑡𝑡0𝐻()𝑄(𝑠)𝑤𝑡,𝑠²(𝑠)𝛿²(𝑠)Δ𝑠=𝐻𝑡,𝑡₀𝑤𝑡₀−^𝑡_𝑡0𝐻(𝑡,𝑠)𝛿(𝑠)𝑄(𝑠)𝑤(𝑠)−12𝑄(𝑠)×𝐺(𝑠)𝐻(𝑡,𝑠)+𝛿(𝑠)ℎ₋(𝑡,𝑠)²+Δ𝑠_𝑡𝑡01(𝑠)×𝐺(𝑠)4𝑄𝐻(𝑡,𝑠)−𝛿(𝑠)ℎ(𝑡,𝑠)²Δ𝑠<𝐻𝑡,𝑡₀𝑤𝑡₀+^𝑡_𝑡0𝐾₂𝑟(𝜏(𝑠))4𝛿^𝜎(𝑠)𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝐺(𝑠)𝐻(𝑡,𝑠)+𝛿(𝑠)ℎ₋(𝑡,𝑠)²Δ𝑠. (62) Thus 1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝑞(𝑠)𝛿^𝜎(𝑠)𝛼(𝜏(𝑠−(𝜏(𝑠))𝜂𝑟))4𝛿^𝜎(𝑠)𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝐺(𝑠)𝐻(𝑡,𝑠)+𝛿(𝑠)ℎ₋(𝑡,𝑠)²<1Δ𝑠𝐿𝑤𝑡₀<∞, (63) which is a contradiction to (58). This completes the proof.

Theorem 8. Assume that (𝐻₁)--(𝐻₅) and (3) hold. If there exists a positive Δ-differentiable function 𝛿(𝑡) such that (36) holds and ^∞_𝑡01𝑟(𝑡)^𝑡_𝑡0𝑞(𝑠)𝑒_−𝑝/𝑟(𝑡,𝜎(𝑠))Δ𝑠Δ𝑡=∞, (64) then every solution of (1) is either oscillatory or converges to zero on [𝑡₀,∞)_𝕋.

Proof. Assume that (1) has a nonoscillatory solution 𝑥(𝑡) such that 𝑥(𝑡)>0, 𝑥(𝜏(𝑡))>0 for all 𝑡∈[𝑡₁,∞)_𝕋,𝑡₁∈[𝑡₀,∞)_𝕋. As in the proof of Lemma 3, we see that there exist two possible cases for the sign of 𝑥^Δ(𝑡). When 𝑥^Δ(𝑡) is eventually positive, the proof is similar to the proof of Theorem 4. Next, suppose that 𝑥^Δ(𝑡)<0 for 𝑡∈[𝑡₁,∞)_𝕋. Then, 𝑥(𝑡) is decreasing and lim_𝑡→∞𝑥(𝑡)=𝑏≥0. Thus, 𝑥(𝜏(𝑡))>𝑥(𝑡)>𝑥(𝜎(𝑡))≥𝑏≥0,∀𝑡∈𝑡₁,∞_𝕋. (65) Hence, by (𝐻₃), we get 𝑓(𝑥(𝜏(𝑡)))≥𝐿𝑥(𝜏(𝑡))≥𝐿𝑏,∀𝑡∈𝑡₁,∞_𝕋. (66) Defining the function 𝑢(𝑡)=𝑟(𝑡)𝑔(𝑥(𝑡),𝑥^Δ(𝑡)), using (1) and (66), we get 𝑢^Δ(𝑡)𝑝(𝑡)=−𝑟(𝑡)𝑢(𝑡)(𝑡)𝑓(𝑥(𝜏(𝑡𝑝(𝑡)−𝑞)))≤−𝑟(𝑡)𝑢(𝑡)−𝐿𝑏𝑞(𝑡),for𝑡∈𝑡₁,∞_𝕋. (67) The inequality (67) is the assumed inequality of [18, Theorem 6.1]. All other assumptions of [18, Theorem 6.1]; for example, −𝑝/𝑟∈ℛ, are satisfied. Hence, the conclusion of [18, Theorem 6.1] holds; that is, 𝑢(𝑡)≤𝑢𝑡₁𝑒_−𝑝/𝑟𝑡,𝑡₁−𝐿𝑏^𝑡_𝑡1𝑞(𝑠)𝑒_−𝑝/𝑟(𝑡,𝜎(𝑠))Δ𝑠<−𝐿𝑏^𝑡_𝑡1𝑞(𝑠)𝑒_−𝑝/𝑟(𝑡,𝜎(𝑠))Δ𝑠, (68) for all 𝑡∈[𝑡₁,∞)_𝕋, and thus, for all 𝜉∈[𝑡₁,∞)_𝕋, ^𝜉_𝑡1𝑥^Δ(𝑡)Δ𝑡<−𝐿𝑏𝐾₁^𝜉_𝑡11𝑟(𝑡)^𝑡_𝑡1𝑞(𝑠)𝑒_−𝑝/𝑟((𝑠𝐾𝑡,𝜎))Δ𝑠Δ𝑡,₁>0. (69) Assuming 𝑏>0 and using (64) in (69) yield lim_𝑟→∞𝑥(𝑟)=−∞, which is a contradiction to the fact that 𝑥(𝑡)>0 for 𝑡∈[𝑡₁,∞)_𝕋. Thus, 𝑏=0 and then lim_𝑡→∞𝑥(𝑡)=0. This completes the proof.

Remark 9. Our results in this paper not only extend and improve some known results and show that some results of [2, 8–10, 12, 14] are special examples of our results but also unify the study of oscillation of second-order nonlinear delay differential equation with damping and second-order nonlinear delay difference equation with damping.

5. Examples

Example 1. Consider the second-order delay 2-difference equation with damping 𝑡^2/3𝑥²(𝑡)𝑥^Δ(𝑡)³2+𝑥²(𝑡)𝑥^Δ(𝑡)^2Δ+1𝑡²𝑥²(𝑡)𝑥^Δ(𝑡)³2+𝑥²(𝑡)𝑥^Δ(𝑡)²+1𝑡^4/3𝑥𝑡2=0,𝑡∈2^ℤ,𝑡≥𝑡₀∶=2. (70) Here, 𝑟(𝑡)=𝑡^2/3(𝑡)=1,𝑝𝑡²(𝑡)=1,𝑞𝑡^4/3,𝑔𝑥,𝑥^Δ=𝑥²(𝑡)𝑥^Δ(𝑡)³2+𝑥²(𝑡)𝑥^Δ(𝑡)²(𝑥)𝜏(𝑡)=𝑡,𝑓=𝑥,2. (71) The conditions (𝐻₁), (𝐻₅) are clearly satisfied, (𝐻₃) and (𝐻₄) hold with 𝐿=𝐾₂=1, 𝐾₁=1/2, and (𝐻₂) is satisfied as (𝑡)𝑝(𝑡)1−𝜇𝑟(𝑡)1=1−𝑡𝑡^8/31=1−𝑡^5/3>0,for𝑡≥2. (72) By Lemma 2, we get 𝑒_−𝑝/𝑟(𝑡,2)≥1−^𝑡₂𝑝(𝑠)𝑟(𝑠)Δ𝑠=1−^𝑡₂𝑠^−8/3𝑡Δ𝑠=1−^−5/3−2^−5/32^−5/3=𝑡−1_−5/3+1−2^−2/31−2^−5/3≥𝑡^−5/3+1−2^−2/3>1−2^−2/3>13, (73) for 𝑡≥2, so that ^𝑡₂1𝑟(𝑠)𝑒_−𝑝/𝑟(=𝑠,2)Δ𝑠_𝑡2𝑠^−2/3𝑒_−𝑝/𝑟()1𝑠,2Δ𝑠≥3^𝑡₂𝑠^−2/3=1Δ𝑠3𝑡^1/3−2^1/32^1/3−1⟶∞,as𝑡⟶∞. (74) Hence, (2) is satisfied. Then, 𝑅(𝑡)=𝑘𝑟(𝑡)^𝑡₂Δ𝑠𝑟(𝑠)=𝑡^2/32^𝑡₂Δ𝑠𝑠^2/3≥12()1𝑡−2,for𝑘=2. (75) Therefore, 𝛼(𝑡)=𝑅(𝑡)𝑅(𝑡)(𝑡)=𝑅(𝑡)+𝜇𝑅(𝑡)>+𝑡𝑡−2,3𝑡−2 (76) and then we can find 0≤𝜆<1 such that 𝛼(𝑡)>𝜆3. (77) If 𝛿(𝑡)=1, then ^𝑡₂𝛼(𝜏(𝑠𝑞(𝑠)−(𝜏(𝑠))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝛿^Δ(𝑠)𝛿^𝜎(𝑠)−𝑝(𝑠)𝑟(𝑠)𝛼(𝜏(𝑠))²𝛿^𝜎(𝑠)>𝜆Δ𝑠3^𝑡₂𝑠^−4/31Δ𝑠−2^5/3𝑡₂𝑠^−14/3=𝜆Δ𝑠3𝑡^−1/3−2^−1/32^−1/3−1−12^5/3𝑡^−11/3−2^−11/32^−11/3−1⟶∞as𝑡⟶∞. (78) Hence, according to Theorem 4, every solution of (70) is oscillatory on [𝑡₀,∞)_𝕋.

Example 2. Consider the second-order nonlinear delay dynamic equation with damping sin²(𝑥(𝑡))1+𝑥²(𝑡)𝑥^Δ(𝑡)^Δ+𝛽𝑡𝜎(𝑡)sin²(𝑥(𝑡))1+𝑥²(𝑡)𝑥^Δ(𝑡)^Δ+𝜆(𝑡)𝑥(𝑡𝑥𝑡𝛼)²(𝑡)+1=0,for𝑡∈𝑡₀,∞_𝕋,𝑡₀≥0, (79) where 𝜆>0 and 0<𝛽<𝑡₀≤𝑡.
Here, 𝑟(𝑡)(𝑡)=𝛽=1,𝑝(𝑡)(𝑡)=𝜆𝑡𝜎,𝑞(𝑡),𝑔𝑡𝛼𝑥,𝑥^Δ=sin²(𝑥(𝑡))1+𝑥²(𝑡)𝑥^Δ(𝑡),𝑓(𝑥)=𝑥𝑥²,𝜏(𝑡)+1=𝑡. (80) The conditions (𝐻₁), (𝐻₅) are clearly satisfied, (𝐻₃) and (𝐻₄) hold with 𝐿=𝐾₂=1, and (𝐻₂) is satisfied as (𝑡)𝑝(𝑡)1−𝜇𝑟(𝑡)(𝑡)=1−𝛽𝜇𝑡𝜎(𝑡)>0,for0<𝛽<𝑡₀≤𝑡. (81) By Lemma 2, we get ^∞_𝑡01𝑟(𝑡)𝑒_−𝑝/𝑟(≥𝑡,2)Δ𝑡_∞𝑡01−^𝑡_𝑡0𝑝(𝑠)𝑟(𝑠)=Δ𝑠Δ𝑡_∞𝑡01−𝛽^𝑡_𝑡0−1𝑠^Δ=Δ𝑠Δ𝑡_∞𝑡01−𝛽−1𝑡+1𝑡₀=Δ𝑡_∞𝑡0𝛽1−𝑡₀+𝛽𝑡Δ𝑡. (82) Hence, ^∞_𝑡01𝑟(𝑡)𝑒_−𝑝/𝑟(≥𝛽𝑡,2)Δ𝑡1−𝑡₀^∞_𝑡0Δ𝑡+𝛽^∞_𝑡01𝑡Δ𝑡=∞. (83) Therefore, (2) is satisfied. Then, 𝛼(𝑡)=𝑅(𝑡)𝑅(𝑡)(𝑡)=𝑅(𝑡)+𝜇𝑅(𝑡)(𝑡)≤𝑅(𝑡)+𝜎−𝑡𝑅(𝑡)1+𝑡−𝑡≤1,𝜎(𝑡)≤1𝑡,for𝜎(𝑡)≥𝑡. (84) If 𝐻(𝑡,𝑠)=𝑡 and 𝛿(𝑡)=1, then ℎ(𝑡,𝑠)=0, 𝐺(𝑡)=−𝛽𝛼(𝜏(𝑡))/𝑡𝜎(𝑡) and limsup_𝑡→∞1𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻()𝑞(𝑠)𝛼(𝜏(𝑠−(𝜏(𝑠𝑡,𝑠))𝜂𝑟))4𝛼(𝜏(𝑠))𝜏^Δ(𝑠)×𝐺(𝑠)𝛿^𝜎(𝑠)𝐻()+𝛿(𝑠)𝑡,𝑠𝛿^𝜎(𝑠)ℎ₋(𝑡,𝑠)²×𝛿^𝜎(𝑠)Δ𝑠=limsup_𝑡→∞1𝑡^𝑡_𝑡01𝑠𝛽𝜆𝑡−²𝛼(𝑠)4𝑠𝜎²(𝑠)𝑡Δ𝑠≥limsup_𝑡→∞1𝑡^𝑡_𝑡01𝑠𝛽𝜆𝑡−²4𝑠³𝑡Δ𝑠=limsup_𝑡→∞^𝑡_𝑡01𝑠𝛽𝜆−²4𝑠³Δ𝑠=∞,for𝜆>0. (85) Hence, according to Theorem 7, every solution of (79) is oscillatory on [𝑡₀,∞)_𝕋.

Remark 10. The results of [10] cannot be applied to equation (70) for 𝑟(𝑡)=𝑡^2/3 and 𝑔(𝑥,𝑥^Δ)=𝑥²(𝑡)(𝑥^Δ(𝑡))³/(2+𝑥²(𝑡)(𝑥^Δ(𝑡))²). Also, the results of [16] cannot be applied to (79) for 𝑔(𝑥,𝑥^Δ)=(sin²(𝑥(𝑡))/(1+𝑥²(𝑡)))𝑥^Δ(𝑡). But, according to our work, both (70) and (79) are oscillatory.

Conflict of Interests

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