Kamenev-Type Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

Abstract

Using functions from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form (𝑝(𝑡)𝜓(𝑥(𝑡))𝑘∘𝑥^Δ(𝑡))^Δ+𝑓(𝑡,𝑥(𝜎(𝑡)))=0. Two examples are included to show the significance of the results.

1. Introduction

In this paper, we study the second-order nonlinear dynamic equation 𝑝(𝑡)𝜓(𝑥(𝑡))𝑘∘𝑥^Δ(𝑡)^Δ+𝑓(𝑡,𝑥(𝜎(𝑡)))=0 (1) on a time scale 𝕋.

Throughout this paper, we will assume that(C1)𝑝∈𝐶_rd(𝕋,(0,∞)),(C2)𝜓∈𝐶(ℝ,(0,𝜂]), where 𝜂 is a fixed positive constant,(C3)𝑘∈𝐶(ℝ,ℝ), and there exist 𝛾₁≥𝛾₂>0 such that 0<𝛾₂𝑦𝑘(𝑦)≤𝑘²(𝑦)≤𝛾₁𝑦𝑘(𝑦) for all 𝑦≠0,(C4)𝑓∈𝐶(𝕋×ℝ,ℝ).

Preliminaries about time scale calculus can be found in [1–3], and hence we omit them here. Note that for some typical time scales, we have the following properties, respectively:

(1) 𝕋=ℝ₊∶=[0,∞), we have 𝜎(𝑡)=𝜌(𝑡)=𝑡,𝑓^Δ(𝑡)=𝑓^′(𝑡),_𝑏𝑎𝑓(𝑡)Δ𝑡=^𝑏_𝑎𝑓(𝑡)𝑑𝑡, (2)

(2) 𝕋=ℕ₀, we have 𝜎(𝑡)(𝑡)𝑓=𝑡+1,𝜌=𝑡−1,_Δ(𝑡)=𝑓(𝑡+1)−𝑓_𝑏𝑎(𝑡),𝑓(𝑡)Δ𝑡=^𝑏−1_𝑘=𝑎𝑓(𝑘),𝑎≤𝑏, (3)

(3) 𝕋=ℎℕ₊,ℎ∈ℝ₊⧵{0}, we have 𝜎(𝑡)(𝑡)𝑓=𝑡+ℎ,𝜌=𝑡−ℎ,_Δ(𝑡)=𝑓(𝑡+ℎ)−𝑓(𝑡)ℎ_𝑏𝑎,𝑓(𝑡)Δ𝑡=^{(𝑏/ℎ)−1}_{𝑘=𝑎/ℎ}𝑓(ℎ𝑘)ℎ,𝑎≤𝑏; (4)

(4) 𝕋={2^𝑛,𝑛∈ℕ₀}, we have 𝜎(𝑡)(𝑡)=𝑡=2𝑡,𝜌2,𝑓^Δ(𝑡)=𝑓(2𝑡)−𝑓(𝑡)𝑡_𝑏𝑎,𝑓(𝑡)Δ𝑡=^log2𝑏−1_{𝑘=log2𝑎}𝑓2^𝑘2^𝑘,𝑎≤𝑏. (5)

Without loss of generality, we assume throughout that sup𝕋=∞ since we are interested in extending oscillation criteria for the typical time scales above.

Definition 1. A solution 𝑥 of (1) is said to have a generalized zero at 𝑡^∗∈𝕋 if 𝑥(𝑡^∗)𝑥(𝜎(𝑡^∗))≤0, and it is said to be nonoscillatory on 𝕋 if there exists 𝑡₀∈𝕋 such that 𝑥(𝑡)𝑥(𝜎(𝑡))>0 for all 𝑡>𝑡₀. Otherwise, it is oscillatory. Equation (1) is said to be oscillatory if all solutions of (1) are oscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. thesis [4] in 1988 in order to unify continuous and discrete analysis; see also [5]. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales; for example, see [1–28] and the references therein. In Došlý and Hilger [10], the authors considered the second-order dynamic equation 𝑝(𝑡)𝑥^Δ(𝑡)^Δ+𝑞(𝑡)𝑥(𝜎(𝑡))=0, (6) and gave necessary and sufficient conditions for the oscillation of all solutions on unbounded time scales. In Del Medico and Kong [8, 9], the authors employed the following Riccati transformation 𝑢(𝑡)=𝑝(𝑡)𝑥^Δ(𝑡)𝑥(𝑡) (7) and gave sufficient conditions for Kamenev-type oscillation criteria of (6) on a measure chain.

In Wang [25], the author considered second-order nonlinear damped differential equation 𝑎(𝑡)𝜓(𝑥(𝑡))𝑘𝑥^′(𝑡)^′+𝑝(𝑡)𝑘𝑥^′(𝑡)+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡≥𝑡₀, (8) used the following generalized Riccati transformations 𝑣(𝑡)(𝑡)𝑎(𝑡)𝜓(𝑥(𝑡𝑘𝑥=𝜙))^′(𝑡)𝑓(𝑥(𝑡(𝑡)))+𝑅,𝑡≥𝑡₀,𝑣(𝑡)(𝑡)𝑎(𝑡)𝜓(𝑥(𝑡𝑘𝑥=𝜙))^′(𝑡)𝑥(𝑡)+𝑅(𝑡),𝑡≥𝑡₀, (9) where 𝜙∈𝐶¹([𝑡₀,∞),ℝ₊),𝑅∈𝐶([𝑡₀,∞),ℝ), and gave a new oscillation criteria of (8). In Huang and Wang [16], the authors considered second-order nonlinear dynamic equation on time scales 𝑝(𝑡)𝑥^Δ(𝑡)^Δ+𝑓(𝑡,𝑥(𝜎(𝑡)))=0. (10) By using a similar generalized Riccati transformation which is more general than (7) 𝑢(𝑡)=𝐴(𝑡)𝑝(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)+𝐵(𝑡), (11) where 𝐴∈𝐶¹_rd(𝕋,ℝ₊⧵{0}), 𝐵∈𝐶¹_𝑟𝑑(𝕋,ℝ), the authors extended the results in Del Medico and Kong [8, 9] and established some new Kamenev-type oscillation criteria.

In this paper, we will use functions in some function classes and a similar generalized Riccati transformation as (11) and was used in [25, 26] for nonlinear differential equations, and establish Kamenev-type oscillation criteria for (1) in Section 2. Finally, in Section 3, two examples are included to show the significance of the results.

For simplicity, throughout this paper, we denote (𝑎,𝑏)∩𝕋=(𝑎,𝑏)_𝕋, where 𝑎,𝑏∈ℝ, and [𝑎,𝑏]_𝕋,[𝑎,𝑏)_𝕋,(𝑎,𝑏]_𝕋 are denoted similarly.

2. Kamenev-Type Criteria

In this section we establish Kamenev-type criteria for oscillation of (1). Our approach to oscillation problems of (1) is based largely on the application of the Riccati transformation. Now, we give the first lemma.

Lemma 2. Assume that (C1)–(C4) hold and that there exists a function 𝑞∈𝐶_𝑟𝑑(𝕋,ℝ) such that 𝑢𝑓(𝑡,𝑢)≥𝑞(𝑡)𝑢². Also, suppose that 𝑥(𝑡) is a solution of (1) satisfies 𝑥(𝑡)>0 for 𝑡∈[𝑡₀,∞)_𝕋 with 𝑡₀∈𝕋. For 𝑡∈[𝑡₀,∞)_𝕋, define 𝑢(𝑡)(𝑡)𝑝(𝑡)𝜓(𝑥(𝑡=𝐴))𝑘∘𝑥^Δ(𝑡)𝑥(𝑡)+𝐵(𝑡), (12) where 𝐴∈𝐶¹_𝑟𝑑(𝕋,ℝ₊⧵{0}), 𝐵∈𝐶¹_𝑟𝑑(𝕋,ℝ), and 𝛾₁𝐴−(𝛾₁−𝛾₂)𝐴^𝜎>0 for 𝑡∈[𝑡₀,∞)_𝕋. Then, 𝑢(𝑡) satisfies 𝜇(𝑡)𝑢(𝑡)−𝜇(𝑡)𝐵(𝑡)+𝛾₁𝜂𝐴(𝑡)𝑝(𝑡)>0,𝑢^Δ(𝑡)+Φ₁(𝑡)+𝛾₁𝐴(𝑡)−𝛾₁−𝛾₂𝐴^𝜎(𝑡)𝑢²(𝑡)𝛾₁𝐴(𝑡)𝜇(𝑡)𝑢(𝑡)−𝜇(𝑡)𝐵(𝑡)+𝛾₁(𝑡)𝑝(𝑡)+𝜂𝐴−Φ₀(𝑡)𝑢(𝑡)+𝛾₂𝐴^𝜎(𝑡)𝐵²(𝑡)𝛾₁𝐴(𝑡)𝜇(𝑡)𝑢(𝑡)−𝜇(𝑡)𝐵(𝑡)+𝛾₁𝜂𝐴(𝑡)𝑝(𝑡)≤0, (13)(14) where Φ₀(𝑡)= ((2𝛾₂−𝛾₁)𝐴^𝜎(𝑡)+𝛾₁𝐴(𝑡))𝐵(𝑡) + 𝛾²₁𝜂𝐴^Δ(𝑡)𝐴(𝑡)𝑝(𝑡),  Φ₁(𝑡) = 𝐴^𝜎(𝑡)(𝑞(𝑡)−(𝐵(𝑡)/𝐴(𝑡))^Δ),𝐴^𝜎(𝑡) = 𝐴(𝜎(𝑡)).

Proof. By (C3), we see that 𝑥^Δ and 𝑘∘𝑥^Δ are both positive, both negative, or both zero. When 𝑥^Δ>0, which implies that 𝑘∘𝑥^Δ>0, it follows that 𝜇𝑢−𝜇𝐵+𝛾₁(𝑥)(𝑥)𝐴𝑝𝜓≥𝜇𝐴𝑝𝜓𝑘∘𝑥^Δ2𝑥𝑘∘𝑥^Δ+𝛾₂𝐴𝑝𝜓(𝑥)≥𝛾₂𝜇𝐴𝑝𝜓(𝑥)𝑥^Δ𝑘∘𝑥^Δ𝑥𝑘∘𝑥^Δ+𝛾₂𝐴𝑝𝜓(𝑥)=𝛾₂(𝑥)𝑥𝐴𝑝𝜓^𝜎𝑥>0. (15) When 𝑥^Δ<0, which implies that 𝑘∘𝑥^Δ<0, it follows that 𝜇𝑢−𝜇𝐵+𝛾₁(𝑥)(𝑥)𝐴𝑝𝜓≥𝜇𝐴𝑝𝜓𝑘∘𝑥^Δ2𝑥𝑘∘𝑥^Δ+𝛾₁𝐴𝑝𝜓(𝑥)≥𝛾₁𝜇𝐴𝑝𝜓(𝑥)𝑥^Δ𝑘∘𝑥^Δ𝑥𝑘∘𝑥^Δ+𝛾₁𝐴𝑝𝜓(𝑥)=𝛾₁(𝑥)𝑥𝐴𝑝𝜓^𝜎𝑥≥𝛾₂(𝑥)𝑥𝐴𝑝𝜓^𝜎𝑥>0. (16) When 𝑥^Δ=0, which implies that 𝑘∘𝑥^Δ=0 and 𝑥=𝑥^𝜎, it follows that 𝜇𝑢−𝜇𝐵+𝛾₁𝐴𝑝𝜓(𝑥)=𝛾₁𝐴𝑝𝜓(𝑥)≥𝛾₂𝐴𝑝𝜓(𝑥)=𝛾₂(𝑥)𝑥𝐴𝑝𝜓^𝜎𝑥>0. (17)
Hence, we always have 𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝≥𝜇𝑢−𝜇𝐵+𝛾₁(𝑥)𝑥𝐴𝑝𝜓>0,𝑥^𝜎≥𝛾₂𝐴𝑝𝜓(𝑥)𝜇𝑢−𝜇𝐵+𝛾₁(𝑥)≥𝛾𝐴𝑝𝜓²𝐴𝑝𝜓(𝑥)𝜇𝑢−𝜇𝐵+𝛾₁,𝜂𝐴𝑝 (18) that is, (13) holds. Then, differentiating (12) and using (1), it follows that 𝑢^Δ=𝐴^Δ𝑝𝜓(𝑥)𝑘∘𝑥^Δ𝑥+𝐴^𝜎𝑝𝜓(𝑥)𝑘∘𝑥^Δ𝑥^Δ+𝐵^Δ=𝐴^Δ𝐴(𝑢−𝐵)+𝐴^𝜎𝑝𝜓(𝑥)𝑘∘𝑥^ΔΔ𝑥−𝑝𝜓(𝑥)𝑘∘𝑥^Δ𝑥^Δ𝑥𝑥^𝜎+𝐵^Δ=𝐴^Δ𝐴𝑢+𝐵^Δ−𝐴^Δ𝐴𝐵−𝐴^𝜎𝑓𝑡,𝑥^𝜎𝑥^𝜎−𝐴^𝜎𝑝𝜓(𝑥)𝑘∘𝑥^Δ𝑥^Δ𝑥²𝑥𝑥^𝜎≤𝐴^Δ𝐴𝑢+𝐴^𝜎𝐵𝐴^Δ−𝐴^𝜎𝑞−𝐴^𝜎𝑝𝜓(𝑥)𝑘∘𝑥^Δ2𝛾₁𝑥²𝑥𝑥^𝜎≤𝐴^Δ𝐴𝑢−Φ₁−1𝛾₁𝐴^𝜎(𝑥)×()𝑝𝜓𝑢−𝐵²𝐴²𝑝²𝜓²(𝑥)𝛾₂𝐴𝑝𝜓(𝑥)𝜇𝑢−𝜇𝐵+𝛾₁=𝐴𝜂𝐴𝑝_Δ𝐴𝑢−Φ₁−𝛾₂𝛾₁𝐴^𝜎𝐴(𝑢−𝐵)²𝜇𝑢−𝜇𝐵+𝛾₁=−𝛾𝜂𝐴𝑝₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²+Φ₀𝑢−𝛾₂𝐴^𝜎𝐵²𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝−Φ₁, (19) that is, (14) holds. Lemma 2 is proved.

Remark 3. In Lemma 2, the condition 𝛾₁𝐴−(𝛾₁−𝛾₂)𝐴^𝜎>0 ensures that the coefficient of 𝑢² in (14) is always negative. The condition is obvious and easy to be fulfilled. For example, when 𝐴^Δ(𝑡)≤0 for all 𝑡∈[𝑡₀,∞)_𝕋, we have 𝐴^𝜎=𝐴+𝜇𝐴^Δ≤𝐴, by (C3), we see that 𝛾₁𝐴−(𝛾₁−𝛾₂)𝐴^𝜎>0, and when 𝛾₁=𝛾₂, the condition 𝛾₁𝐴−(𝛾₁−𝛾₂)𝐴^𝜎>0 is also fulfilled.

Let 𝐷₀={𝑠∈𝕋∶𝑠≥0} and 𝐷={(𝑡,𝑠)∈𝕋²∶𝑡≥𝑠≥0}. For any function 𝑓(𝑡,𝑠): 𝕋²→ℝ, denote by 𝑓^Δ₁ and 𝑓^Δ₂ the partial derivatives of 𝑓 with respect to 𝑡 and 𝑠, respectively. For 𝐸⊂ℝ, denote by 𝐿_loc(𝐸) the space of functions which are integrable on any compact subset of 𝐸. Define (𝒜,ℬ)=(𝐴,𝐵)∶𝐴(𝑠)∈𝐶¹_𝑟𝑑𝐷₀,ℝ₊⧵{0},𝐵(𝑠)∈𝐶¹_𝑟𝑑𝐷₀,ℝ,𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝛾>0,₁𝜂𝐴(𝑠)𝑝(𝑠)±𝜇(𝑠)𝐵(𝑠)>0,𝑠∈𝐷₀,ℋ^∗=𝐻(𝑡,𝑠)∈𝐶¹𝐷,ℝ₊()𝐻()∶𝐻𝑡,𝑡=0,𝑡,𝑠>0,𝐻^Δ₂(),ℋ𝑡,𝑠≤0,𝑡>𝑠≥0_∗=𝐻(𝑡,𝑠)∈𝐶¹𝐷,ℝ₊()𝐻()∶𝐻𝑡,𝑡=0,𝑡,𝑠>0,𝐻^Δ₁(𝑡,𝑠)≥0,𝑡>𝑠≥0, (20) These function classes will be used throughout this paper. Now, we are in a position to give our first theorem.

Theorem 4. Assume that (C1)–(C4) hold and that there exists a function 𝑞∈𝐶_𝑟𝑑(𝕋,ℝ) such that 𝑢𝑓(𝑡,𝑢)≥𝑞(𝑡)𝑢². Also, suppose that there exist (𝐴,𝐵)∈(𝒜,ℬ) and 𝐻∈ℋ^∗ such that 𝑀₁(𝑡,⋅)∈𝐿([0,𝜌(𝑡)]_𝕋) and for any 𝑡₀∈𝕋, limsup_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝜎(𝑠))Φ₁(𝑠)Δ𝑠−_𝑡^𝜌(𝑡)₀𝑀₁(𝑡,𝑠)Δ𝑠+𝐻^Δ₂𝑡,𝜌(𝑡)𝛾₁𝜌(𝑡)𝑝𝜌(𝑡)𝜌(𝑡)𝐵𝜌(𝑡)𝜂𝐴−𝜇=∞, (21) where Φ₁ is defined as before, and𝑀₁()≜𝛾𝑡,𝑠¹𝐻(𝑡,𝑠)𝐴(𝑠)𝐵(𝑠)+2𝛾₂−𝛾₁𝐻(𝑡,𝜎(𝑠))𝐴^𝜎(𝑠)𝐵(𝑠)+𝛾²₁𝜂𝐴(𝑠)𝑝(𝑠)(𝐻(𝑡,𝑠)𝐴(𝑠))^Δ𝑠24𝛾₁𝐴(𝑠)min𝛾₁𝐻(𝑡,𝑠)𝐴(𝑠)−𝛾₁−𝛾₂𝐻(𝑡,𝜎(𝑠))𝐴^𝜎(𝑠)×𝛾₁(𝑠)𝑝(𝑠)(𝑠)𝐵(𝑠,𝛾𝜂𝐴−𝜇)₂𝐻(𝑡,𝜎(𝑠))𝐴^𝜎(𝑠)𝛾₁(𝑠)𝑝(𝑠)(𝑠)𝐵(𝑠.𝜂𝐴+𝜇) (22) Then, (1) is oscillatory.

Proof. Assume that (1) is not oscillatory. Without loss of generality, we may assume there exists 𝑡₀∈[0,∞)_𝕋 such that 𝑥(𝑡)>0 for 𝑡∈[𝑡₀,∞)_𝕋. Let 𝑢(𝑡) be defined by (12). Then, by Lemma 2, (13) and (14) hold.
For simplicity in the following, we let 𝐻_𝜎=𝐻(𝑡,𝜎(𝑠)),𝐻=𝐻(𝑡,𝑠), and 𝐻^Δ₂=𝐻^Δ₂(𝑡,𝑠) and omit the arguments in the integrals. For 𝑠∈𝕋, 𝐻_𝜎−𝐻=𝐻^Δ₂𝜇. (23)
Since 𝐻^Δ₂≤0 on 𝐷, we see that 𝐻_𝜎≤𝐻. From 𝛾₁𝐴−(𝛾₁−𝛾₂)𝐴^𝜎>0 and (C3), we have 𝛾₁𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎>𝛾₁𝐻_𝜎𝐴−𝛾₁𝐻_𝜎𝐴=0. (24)
Multiplying (14), where 𝑡 is replaced by 𝑠, by 𝐻_𝜎 and integrating it with respect to 𝑠 from 𝑡₀ to 𝑡 with 𝑡∈𝕋 and 𝑡≥𝜎(𝑡₀), we obtain ^𝑡_𝑡0𝐻_𝜎Φ₁Δ𝑠≤−^𝑡_𝑡0𝐻_𝜎𝑢^Δ+𝐻_𝜎𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁+𝜂𝐴𝑝−Φ₀𝑢+𝛾₂𝐴^𝜎𝐵²𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝Δ𝑠. (25) Noting that 𝐻(𝑡,𝑡)=0, by the integration by parts formula, we have ^𝑡_𝑡0𝐻_𝜎Φ₁Δ𝑠≤𝐻𝑡,𝑡₀𝑢𝑡₀+^𝑡_𝑡0𝐻^Δ₂𝑢−𝐻_𝜎𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²−Φ₀𝑢+𝛾₂𝐴^𝜎𝐵²𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝Δ𝑠≤𝐻𝑡,𝑡₀𝑢𝑡₀+^𝑡_𝑡0𝐻^Δ₂𝑢−𝐻_𝜎𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²−Φ₀𝑢𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝Δ𝑠=𝐻𝑡,𝑡₀𝑢𝑡₀+^𝑡_𝜌(𝑡)𝐻^Δ₂+𝑢Δ𝑠_{𝑡𝜌(𝑡)0}𝐻^Δ₂𝑢−𝐻_𝜎𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²−Φ₀𝑢𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝Δ𝑠. (26)
Since 𝐻^Δ₂≤0 on 𝐷, from (13) we see that for 𝑡≥𝜎(𝑡₀), ^𝑡_𝜌(𝑡)𝐻^Δ₂𝑢Δ𝑠=𝐻^Δ₂𝑡,𝜌(𝑡)𝑢𝜌(𝑡)𝜇𝜌(𝑡)≤−𝐻^Δ₂𝑡,𝜌(𝑡)𝛾₁𝜌(𝑡)𝑝𝜌(𝑡)𝜌(𝑡)𝐵𝜌(𝑡).𝜂𝐴−𝜇 (27) For 𝑡≥𝜎(𝑡₀), 𝑠∈[𝑡₀,𝜌(𝑡))_𝕋, and 𝑢(𝑠)≤0, from (24), we have 𝐻^Δ₂𝑢−𝐻_𝜎𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²−Φ₀𝑢𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝛾𝜂𝐴𝑝=−¹𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎𝑢²𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁+𝛾𝜂𝐴𝑝₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ𝑢𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝛾𝜂𝐴𝑝=−¹𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝑢𝜂𝐴𝑝²+𝛾₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ𝛾₁𝐴𝛾₁𝑢−𝛾𝜂𝐴𝑝−𝜇𝐵₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ𝛾₁𝐴𝛾₁×𝜂𝐴𝑝−𝜇𝐵𝜇𝑢²𝜇𝑢−𝜇𝐵+𝛾₁𝛾𝜂𝐴𝑝=−²𝐻_𝜎𝐴^𝜎𝛾₁𝜂𝐴𝑝+𝜇𝐵𝛾₁𝐴𝛾₁𝜂𝐴𝑝−𝜇𝐵𝜇𝑢−𝜇𝐵+𝛾₁𝑢𝜂𝐴𝑝²+𝛾₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ𝛾₁𝐴𝛾₁𝑢𝛾𝜂𝐴𝑝−𝜇𝐵≤−²𝐻_𝜎𝐴^𝜎𝛾₁𝜂𝐴𝑝+𝜇𝐵𝛾₁𝐴𝛾₁𝜂𝐴𝑝−𝜇𝐵²𝑢²+𝛾₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ𝛾₁𝐴𝛾₁𝑢𝛾𝜂𝐴𝑝−𝜇𝐵=−²𝐻_𝜎𝐴^𝜎𝛾₁𝜂𝐴𝑝+𝜇𝐵𝛾₁𝐴𝛾₁𝜂𝐴𝑝−𝜇𝐵²×𝛾𝑢−¹𝜂𝐴𝑝−𝜇𝐵2𝛾₂𝐻_𝜎𝐴^𝜎𝛾₁×𝛾𝜂𝐴𝑝+𝜇𝐵₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ2+𝛾₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ24𝛾₁𝛾₂𝐻_𝜎𝐴^𝜎𝐴𝛾₁≤𝛾𝜂𝐴𝑝+𝜇𝐵₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ2×4𝛾₁𝐴min𝛾₁𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎×𝛾₁𝜂𝐴𝑝−𝜇𝐵,𝛾₂𝐻_𝜎𝐴^𝜎×𝛾₁𝜂𝐴𝑝+𝜇𝐵⁻¹=𝑀₁. (28)
For 𝑡≥𝜎(𝑡₀), 𝑠∈[𝑡₀,𝜌(𝑡))_𝕋, and 𝑢(𝑠)>0, from (24), we have 𝐻^Δ₂𝑢−𝐻_𝜎𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²−Φ₀𝑢𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁=−𝛾𝜂𝐴𝑝₁𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎𝑢²+𝛾₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ𝑢×𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝⁻¹𝛾=−¹𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁×𝛾𝜂𝐴𝑝𝑢−¹𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ2𝛾₁𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎2+𝛾₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ24𝛾₁𝐴𝛾₁𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎𝜇𝑢−𝜇𝐵+𝛾₁≤𝛾𝜂𝐴𝑝₁𝐻𝐴𝐵+2𝛾₂−𝛾₁𝐻_𝜎𝐴^𝜎𝐵+𝛾²₁𝜂𝐴𝑝(𝐻𝐴)^Δ2×4𝛾₁𝐴min𝛾₁𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎×𝛾₁𝜂𝐴𝑝−𝜇𝐵,𝛾₂𝐻_𝜎𝐴^𝜎×𝛾₁𝜂𝐴𝑝+𝜇𝐵⁻¹=𝑀₁. (29) Therefore, for all 𝑡≥𝜎(𝑡₀), 𝑠∈[𝑡₀,𝜌(𝑡))_𝕋, we have 𝐻^Δ₂𝑢−𝐻_𝜎𝛾₁𝐴−𝛾₁−𝛾₂𝐴^𝜎𝑢²−Φ₀𝑢𝛾₁𝐴𝜇𝑢−𝜇𝐵+𝛾₁𝜂𝐴𝑝≤𝑀₁. (30) Then, from (26), (27), and (30), we obtain that for 𝑡∈𝕋 and 𝑡>𝜎(𝑡₀), ^𝑡_𝑡0𝐻_𝜎Φ₁Δ𝑠≤𝐻𝑡,𝑡₀𝑢𝑡₀+_𝑡^𝜌(𝑡)₀𝑀₁Δ𝑠−𝐻^Δ₂(𝑡)×𝛾𝑡,𝜌₁𝜂𝐴𝜌(𝑡)𝑝𝜌(𝑡)−𝜇𝜌(𝑡)𝐵𝜌(𝑡). (31) Hence, 1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝜎(𝑠))Φ₁(𝑠)Δ𝑠−_𝑡^𝜌(𝑡)₀𝑀₁(𝑡,𝑠)Δ𝑠+𝐻^Δ₂𝑡,𝜌(𝑡)𝛾₁𝜌(𝑡)𝑝𝜌(𝑡)𝜌(𝑡)𝐵𝜌(𝑡)𝑡𝜂𝐴−𝜇≤𝑢₀<∞, (32) which contradicts (21) and completes the proof.

Remark 5. If we change the condition 𝛾₁𝐴−(𝛾₁−𝛾₂)𝐴^𝜎>0 in the definition of (𝒜,ℬ) with a stronger one 𝐴^Δ(𝑡)≤0, (24) in the proof of Theorem 4 will be changed with 𝛾₁𝐻𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴^𝜎≥𝛾₁𝐻_𝜎𝐴−𝛾₁−𝛾₂𝐻_𝜎𝐴=𝛾₂𝐻_𝜎𝐴>0, (33) Then, the definition of 𝑀₁ can be simplified as𝑀₁()≜𝛾𝑡,𝑠¹𝐻(𝑡,𝑠)𝐴(𝑠)𝐵(𝑠)+2𝛾₂−𝛾₁𝐻(𝑡,𝜎(𝑠))𝐴^𝜎(𝑠)𝐵(𝑠)+𝛾²₁𝜂𝐴(𝑠)𝑝(𝑠)(𝐻(𝑡,𝑠)𝐴(𝑠))^Δ𝑠24𝛾₁𝛾₂𝐻(𝑡,𝜎(𝑠))𝐴(𝑠)min𝐴(𝑠)𝛾₁𝜂𝐴(𝑠)𝑝(𝑠)−𝜇(𝑠)𝐵(𝑠),𝐴^𝜎(𝑠)𝛾₁(𝑠)𝑝(𝑠)(𝑠)𝐵(𝑠.𝜂𝐴+𝜇) (34)
In the sequel, we define 𝕋₁=𝑠∈𝕋∶𝑠isright-dense,𝕋₂=𝑠∈𝕋∶𝑠isright-scattered. (35)(36)
When 𝛾₁=𝛾₂=1, by (C3), we see that 𝑘(𝑦)=𝑦 and (1) is simplified as 𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^Δ(𝑡)^Δ+𝑓(𝑡,𝑥(𝜎(𝑡)))=0. (37)
Now, we have the following theorem, but we should note that this result does not apply to the case where all points in 𝕋 are right dense.

Theorem 6. Assume that (C1)–(C4) with 𝛾₁=𝛾₂=1 hold and that there exists a function 𝑞∈𝐶_𝑟𝑑(𝕋,ℝ) such that 𝑢𝑓(𝑡,𝑢)≥𝑞(𝑡)𝑢². Let (𝐴,𝐵)∈(𝒜,ℬ),𝐻∈ℋ_∗,𝑀₂(⋅,𝑡)∈𝐿_loc([𝜎(𝑡),∞)_𝕋), and 𝕋₁,𝕋₂ be defined by (35) and (36). Then, (37) is oscillatory provided there exists {𝑡_𝑛}^∞_𝑛=1⊂𝕋₂, 𝑡_𝑛→∞, such that for any 𝑡₀∈𝕋, one of the following holds(i)lim_𝑛→∞𝑁(𝑡_𝑛,𝑡₀)=∞ and limsup_𝑛→∞1𝑁𝑡_𝑛,𝑡₀×^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀Φ₁(𝑠)Δ𝑠−^𝑡𝑛_𝜎𝑡0𝑀₂𝑠,𝑡₀Δ𝑠=∞, (38)(ii)limsup_𝑛→∞𝑁(𝑡_𝑛,𝑡₀)=∞ and lim_𝑛→∞1𝑁𝑡_𝑛,𝑡₀×^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀Φ₁(𝑠)Δ𝑠−^𝑡𝑛_𝜎𝑡0𝑀₂𝑠,𝑡₀Δ𝑠=∞, (39)(iii)limsup_𝑛→∞𝑁(𝑡_𝑛,𝑡₀)<∞ and limsup_𝑛→∞^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀Φ₁(𝑠)−Δ𝑠_{𝑡𝑛𝜎𝑡0}𝑀₂𝑠,𝑡₀Δ𝑠=∞, (40) where 𝑁(𝑡,𝑠)=𝐻(𝑡,𝑠)(𝜂𝐴(𝑡)𝑝(𝑡)−𝜇(𝑡)𝐵(𝑡))/𝜇(𝑡), Φ₁ is defined as before, and𝑀₂()≜𝐻()𝐴(𝑠)𝐵(𝑠)(𝜎(𝑠))𝐴𝑠,𝑡𝑠,𝑡+𝐻,𝑡^𝜎(𝑠)𝐵(𝑠)+𝜂𝐴(𝑠)𝑝(𝑠)(𝐻(𝑠,𝑡)𝐴(𝑠))^Δ𝑠24𝐻(𝑠,𝑡)𝐴(𝑠)min𝐴(𝑠)𝜂𝐴(𝑠)𝑝(𝑠)−𝜇(𝑠)𝐵(𝑠),𝐴^𝜎(𝑠(𝑠)𝑝(𝑠)(𝑠)𝐵(𝑠.)𝜂𝐴+𝜇) (41)

Proof. Assume that (37) is not oscillatory. Without loss of generality, we may assume there exists 𝑡₀∈[0,∞)_𝕋 such that 𝑥(𝑡)>0 for 𝑡∈[𝑡₀,∞)_𝕋. Let 𝑢(𝑡) be defined by (12) with 𝑘(𝑦)=𝑦. Then, by Lemma 2, (13) and (14) hold for 𝛾₁=𝛾₂=1. So, we have 𝜇(𝑡)𝑢(𝑡)−𝜇(𝑡)𝐵(𝑡)+𝜂𝐴(𝑡)𝑝(𝑡)>0,𝑢^Δ(𝑡)+Φ₁(𝑡)+𝐴(𝑡)𝑢²(𝑡)𝐴(𝑡)𝜇(𝑡)𝑢(𝑡)(𝑡)𝐵(𝑡)(𝑡)𝑝(𝑡)−𝐴−𝜇+𝜂𝐴_𝜎(𝑡)+𝐴(𝑡)𝐵(𝑡)+𝜂𝐴^Δ(𝑡)𝐴(𝑡)𝑝(𝑡)𝑢(𝑡)𝐴(𝑡)𝜇(𝑡)𝑢(𝑡)(𝑡)𝐵(𝑡)(𝑡)𝑝(𝑡)+𝐴−𝜇+𝜂𝐴_𝜎(𝑡)𝐵²(𝑡)𝐴(𝑡)𝜇(𝑡)𝑢(𝑡)−𝜇(𝑡)𝐵(𝑡)+𝜂𝐴(𝑡)𝑝(𝑡)≤0, (13)^′(14)^′ where Φ₁(𝑡) and 𝐴^𝜎(𝑡) are defined as in Lemma 2.
For simplicity in the following, we let 𝐻^′_𝜎=𝐻(𝜎(𝑠),𝑡₀),𝐻^′=𝐻(𝑠,𝑡₀), and 𝐻^Δ₁=𝐻^Δ₁(𝑠,𝑡₀) and omit the arguments in the integrals. Multiplying (14)^′, where 𝑡 is replaced by 𝑠, by 𝐻^′_𝜎 and integrating it with respect to 𝑠 from 𝑡₀ to 𝑡 and then using the integration by parts formula, we have that ^𝑡_𝑡0𝐻^′_𝜎Φ₁Δ𝑠≤−^𝑡_𝑡0𝐻^′_𝜎𝑢^Δ+𝐻^′_𝜎×𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢+𝐴^𝜎𝐵²𝐴𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝Δ𝑠=−𝐻𝑡,𝑡₀𝑢(𝑡)+^𝑡_𝑡0𝐻^Δ₁𝑢−𝐻^′_𝜎×𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢+𝐴^𝜎𝐵²𝐴𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝Δ𝑠≤−𝐻𝑡,𝑡₀𝑢(𝑡)+^𝜎(𝑡0)_𝑡0+^𝑡_𝜎𝑡0𝐻^Δ₁𝑢−𝐻^′_𝜎𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢𝐴𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝Δ𝑠. (42) For 𝑠∈[𝑡₀,𝑡)_𝕋, 𝐻^′_𝜎−𝐻^Δ₁𝜇=𝐻^′. (43) Hence, ^𝜎(𝑡0)_𝑡0𝐻^Δ₁𝑢−𝐻^′_𝜎𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢𝐴𝑡𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝Δ𝑠=𝜇₀𝐻^Δ₁𝑢−𝐻^′_𝜎𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢𝐴𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝_𝑠=𝑡0=−𝐴𝐻^′𝑢²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝑢𝜇𝐴𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝_𝑠=𝑡0=𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝑢𝜇𝐴𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝_𝑠=𝑡0≤𝜂𝑝𝐻^′𝐴^Δ+𝐻^′_𝜎𝐴^𝜎𝐵𝐴_𝑠=𝑡0=𝜂𝑝𝑡₀𝐻^Δ₁𝑡₀,𝑡₀𝐴^𝜎𝑡₀+𝐻𝜎𝑡₀,𝑡₀𝐴^𝜎𝑡₀𝐵𝑡₀𝐴𝑡₀. (44)
Furthermore, for 𝑡≥𝑡₀, 𝑠∈[𝜎(𝑡₀),𝑡)_𝕋, and 𝑢(𝑠)≤0, 𝐻^Δ₁𝑢−𝐻^′_𝜎𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢𝐴=𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝−𝐻^′𝐴𝑢²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝑢𝐴𝐻𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝=−^′𝑢𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝐴𝑢−𝐻𝜂𝐴𝑝−𝜇𝐵_′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝐴𝜂𝐴𝑝−𝜇𝐵𝜇𝑢²𝐻𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝=−^′𝜎𝐴^𝜎𝜂𝐴𝑝+𝜇𝐵𝐴𝑢𝜂𝐴𝑝−𝜇𝐵𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝐴𝑢𝐻𝜂𝐴𝑝−𝜇𝐵≤−^′𝜎𝐴^𝜎𝜂𝐴𝑝+𝜇𝐵𝐴𝜂𝐴𝑝−𝜇𝐵²𝑢²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝐴𝑢𝐻𝜂𝐴𝑝−𝜇𝐵=−^′𝜎𝐴^𝜎𝜂𝐴𝑝+𝜇𝐵𝐴𝜂𝐴𝑝−𝜇𝐵²×𝐻𝑢−𝜂𝐴𝑝−𝜇𝐵^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ2𝐻^′_𝜎𝐴^𝜎𝜂𝐴𝑝+𝜇𝐵²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ24𝐻^′_𝜎𝐴^𝜎𝐴≤𝐻𝜂𝐴𝑝+𝜇𝐵_′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ24𝐻^′𝐴min𝐴𝜂𝐴𝑝−𝜇𝐵,𝐴^𝜎𝜂𝐴𝑝+𝜇𝐵=𝑀₂. (45) For 𝑡≥𝑡₀, 𝑠∈[𝜎(𝑡₀),𝑡)_𝕋, and 𝑢(𝑠)>0, 𝐻^Δ₁𝑢−𝐻^′_𝜎𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢𝐴𝐻𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝=−^′𝐴𝑢²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ𝑢𝐴𝐻𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝=−^′𝐻𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝𝑢−^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ2𝐻^′𝐴²+𝐻^′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ24𝐻^′𝐴²≤𝐻𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝_′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ24𝐻^′𝐴²≤𝐻𝜂𝐴𝑝−𝜇𝐵_′𝐴𝐵+𝐻^′_𝜎𝐴^𝜎𝐵+𝜂𝐴𝑝𝐻^′𝐴^Δ24𝐻^′𝐴min𝐴𝜂𝐴𝑝−𝜇𝐵,𝐴^𝜎𝜂𝐴𝑝+𝜇𝐵=𝑀₂. (46) Hence, for all 𝑡≥𝑡₀, 𝑠∈[𝜎(𝑡₀),𝑡)_𝕋, we have 𝐻^Δ₁𝑢−𝐻^′_𝜎𝐴𝑢²−𝐴^𝜎+𝐴𝐵+𝜂𝐴^Δ𝐴𝑝𝑢𝐴𝜇𝑢−𝜇𝐵+𝜂𝐴𝑝≤𝑀₂. (47)
From (42), (44), and (47), we have ^𝑡_𝑡0𝐻^′_𝜎Φ₁Δ𝑠≤−𝐻𝑡,𝑡₀𝑢(𝑡)+^𝑡_𝜎𝑡0𝑀₂𝑠,𝑡₀+𝑡Δ𝑠𝜂𝑝₀𝐻^Δ₁𝑡₀,𝑡₀𝐴^𝜎𝑡₀+𝐻𝜎𝑡₀,𝑡₀𝐴^𝜎𝑡₀𝐵𝑡₀𝐴𝑡₀. (48)
For 𝑡∈𝕋₂, (13)^′ implies that −𝐻𝑡,𝑡₀𝑢(𝑡)≤𝐻𝑡,𝑡₀𝜂𝐴(𝑡)𝑝(𝑡)−𝜇(𝑡)𝐵(𝑡)𝜇(𝑡)=𝑁𝑡,𝑡₀. (49) Hence, ^𝑡_𝑡0𝐻𝜎(𝑠),𝑡₀Φ₁(𝑠)Δ𝑠≤𝑁𝑡,𝑡₀+^𝑡_𝜎𝑡0𝑀₂𝑠,𝑡₀+𝑡Δ𝑠𝜂𝑝₀𝐻^Δ₁𝑡₀,𝑡₀𝐴^𝜎𝑡₀+𝐻𝜎𝑡₀,𝑡₀𝐴^𝜎𝑡₀𝐵𝑡₀𝐴𝑡₀. (50)
Assume that condition (i) holds. Let 𝑡=𝑡_𝑛 in (50). Then, we obtain 1𝑁𝑡_𝑛,𝑡₀^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀Φ₁(𝑠)Δ𝑠−^𝑡𝑛_𝜎𝑡0𝑀₂𝑠,𝑡₀1Δ𝑠≤1+𝑁𝑡_𝑛,𝑡₀𝜂𝑝𝑡₀𝐻^Δ₁𝑡₀,𝑡₀𝐴^𝜎𝑡₀+𝐻𝜎𝑡₀,𝑡₀𝐴^𝜎𝑡₀𝐵𝑡₀𝐴𝑡₀. (51) Taking the limsup as 𝑛→∞ on both sides, we have limsup_𝑛→∞1𝑁𝑡_𝑛,𝑡₀^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀Φ₁(𝑠)−Δ𝑠_{𝑡𝑛𝜎𝑡0}𝑀₂𝑠,𝑡₀Δ𝑠<∞, (52) which contradicts (38).
The conclusions with conditions (ii) and (iii) can be proved similarly. We omit the details. The proof is complete.

When (𝐴,𝐵)=(1,0), Theorems 4 and 6 can be simplified as the following corollaries, respectively.

Corollary 7. Assume that (C1)–(C4) hold and that there exists a function 𝑞∈𝐶_𝑟𝑑(𝕋,ℝ) such that 𝑢𝑓(𝑡,𝑢)≥𝑞(𝑡)𝑢². Also, suppose that there exists 𝐻∈ℋ^∗ such that for any 𝑡₀∈𝕋, limsup_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻((𝑠𝑞(𝑠)−𝛾𝑡,𝜎))Δ𝑠₂₁𝜂4𝛾_2𝑡^𝜌(𝑡)₀𝐻^Δ₂(𝑡,𝑠)²𝐻((𝑠𝑝(𝑠)𝑡,𝜎))Δ𝑠+𝛾₁𝜂𝐻^Δ₂𝑡,𝜌(𝑡)𝑝𝜌(𝑡)=∞. (53) Then, (1) is oscillatory.

Corollary 8. Assume that (C1)–(C4) with 𝛾₁=𝛾₂=1 hold and that there exists a function 𝑞∈𝐶_𝑟𝑑(𝕋,ℝ) such that 𝑢𝑓(𝑡,𝑢)≥𝑞(𝑡)𝑢². Let 𝐻∈ℋ_∗, 𝕋₁ and 𝕋₂ be defined by (35) and (36). Then, (37) is oscillatory provided that there exists {𝑡_𝑛}^∞_𝑛=1⊂𝕋₂, 𝑡_𝑛→∞, such that for any 𝑡₀∈𝕋, one of the following holds(i)lim_𝑛→∞(𝐻(𝑡_𝑛,𝑡₀)𝑝(𝑡_𝑛))/𝜇(𝑡_𝑛)=∞ and limsup_𝑛→∞𝜇𝑡_𝑛𝐻𝑡_𝑛,𝑡₀𝑝𝑡_𝑛×^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀𝑞(𝑠)−𝜂Δ𝑠4^𝑡𝑛_𝜎𝑡0𝐻^Δ₁𝑠,𝑡₀²𝐻𝑠,𝑡₀𝑝(𝑠)Δ𝑠=∞, (54)(ii)limsup_𝑛→∞(𝐻(𝑡_𝑛,𝑡₀)𝑝(𝑡_𝑛))/𝜇(𝑡_𝑛)=∞ and lim_𝑛→∞𝜇𝑡_𝑛𝐻𝑡_𝑛,𝑡₀𝑝𝑡_𝑛^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀𝑞(𝑠)−𝜂Δ𝑠4^𝑡𝑛_𝜎𝑡0𝐻^Δ₁𝑠,𝑡₀²𝐻𝑠,𝑡₀𝑝(𝑠)Δ𝑠=∞, (55)(iii)limsup_𝑛→∞(𝐻(𝑡_𝑛,𝑡₀)𝑝(𝑡_𝑛))/𝜇(𝑡_𝑛)<∞ and limsup_𝑛→∞^𝑡𝑛_𝑡0𝐻𝜎(𝑠),𝑡₀𝑞(𝑠)−𝜂Δ𝑠4^𝑡𝑛_𝜎𝑡0𝐻^Δ₁𝑠,𝑡₀²𝐻𝑠,𝑡₀𝑝(𝑠)Δ𝑠=∞. (56)

Remark 9. When 𝜓(𝑥)≡1 and 𝑘(𝑦)=𝑦, Theorems 4 and 6 reduce to [16, Theorems 2.1 and 2.2], respectively. When 𝜓(𝑥)≡1, 𝑘(𝑦)=𝑦, 𝑓(𝑡,𝑢)=𝑞(𝑡)𝑢, and (𝐴,𝐵)=(1,0), Theorems 4 and 6 reduce to [8, Theorems 2.1 and 2.2], respectively.

3. Examples

In this section, we will show the application of our oscillation criteria in two examples. We first give an example to demonstrate Theorem 4 (or Corollary 7).

Example 10. Consider the equation 𝑡((𝑡𝑥2+sin𝑥))^Δ(𝑡)1+𝑥^Δ(𝑡)²2+𝑥^Δ(𝑡)^2Δ+𝑡²𝑡²+1𝑥(𝜎(𝑡))=0, (57) where 𝑝(𝑡)=𝑡, 𝜓(𝑥(𝑡))=2+sin𝑥(𝑡), 𝑘∘𝑥^Δ(𝑡)=(𝑥^Δ(𝑡)(1+(𝑥^Δ(𝑡))²))/(2+(𝑥^Δ(𝑡))²), and 𝑞(𝑡)=𝑡², so we have 𝛾₁=1, 𝛾₂=1/2, and 𝜂=3. Let (𝐴,𝐵)=(1,0) and 𝐻(𝑡,𝑠)=(𝑡−𝑠)², we have
(1) 𝕋=ℝ₊, limsup_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻((𝑠𝑞(𝑠)−𝛾𝑡,𝜎))Δ𝑠₂₁𝜂4𝛾_2𝑡^𝜌(𝑡)₀𝐻^Δ₂(𝑡,𝑠)²𝐻((𝑠𝑝(𝑠)𝑡,𝜎))Δ𝑠+𝛾₁𝜂𝐻^Δ₂𝑡,𝜌(𝑡)𝑝𝜌(𝑡)=limsup_𝑡→∞1𝑡−𝑡₀^2𝑡𝑡₀(𝑡−𝑠)²𝑠²3𝑑𝑠−2^𝑡_𝑡0𝑠𝑑𝑠=∞, (58) That is, (53) holds. By Corollary 7, we see that (57) is oscillatory;
(2) 𝕋=ℕ₀, limsup_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻((𝑠𝑞(𝑠)−𝛾𝑡,𝜎))Δ𝑠₂₁𝜂4𝛾_2𝑡^𝜌(𝑡)₀𝐻^Δ₂(𝑡,𝑠)²𝐻((𝑠𝑝(𝑠)𝑡,𝜎))Δ𝑠+𝛾₁𝜂𝐻^Δ₂𝑡,𝜌(𝑡)𝑝𝜌(𝑡)=limsup_𝑛→∞1(𝑛−𝑙)²×^𝑛−1_𝑘=𝑙(𝑛−𝑘−1)²𝑘²−32^𝑛−2_𝑘=𝑙(2𝑛−2𝑘−1)²𝑘(𝑛−𝑘−1)²−3(𝑛−1)≥limsup_𝑛→∞^𝑛−2_𝑘=𝑙𝑘²(𝑛−𝑙)²−^𝑛−2_𝑘=127𝑘2(𝑛−𝑙)²−3(𝑛−1)(𝑛−𝑙)²=∞, (59) that is, (53) holds. By Corollary 7, we see that (57) is oscillatory;
(3) 𝕋=ℎℕ₀, ℎ∈ℝ₊⧵{0}, limsup_𝑡→∞1𝐻𝑡,𝑡₀