Oscillation Criteria for Second-Order Nonlinear Dynamic Equations on Time Scales

Abstract

This paper is concerned with oscillation of second-order nonlinear dynamic equations of the form (𝑟(𝑡)((𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡)))^Δ)^𝛾)^Δ+𝑓₁(𝑡,𝑦(𝛿₁(𝑡)))+𝑓₂(𝑡,𝑦(𝛿₂(𝑡)))=0 on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria.

1. Introduction

The theory of time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis. Not only can this theory of the so-called “dynamic equations" unify theories of differential equations and difference equations but also extend these classical cases to cases “in between", for example, to the so-called 𝑞-difference equations. A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ with the topology and ordering inherited form ℝ, and the cases when this time scale is equal to ℝ or to the integers ℤ represent the classical theories of differential and difference equations. Of course many other interesting time scales exist, and they give rise to plenty of applications. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales, we refer the reader to [1–14].

In 2006, Wu et al. [1] considered the second-order nonlinear neutral dynamic equation with variable delays 𝑟(𝑡)𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡))^Δ𝛾Δ+𝑓𝑡,𝑦(𝛿(𝑡))=0,𝑡∈𝕋, (1.1) where 𝛾≥1 is a quotient of odd positive integers. In 2007, Saker et al. [2] also discussed (1.1) for an odd positive integer 𝛾≥1. In 2010, Zhang and Wang [3] extended and complemented some results in [1, 2] for 𝛾≥1 and gave some new results for 0<𝛾<1. In 2011, Saker [4] considered (1.1) in different conditions. In 2010, Sun et al. [5] considered the second-order quasiliner neutral delay dynamic equation 𝑟(𝑡)𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡))^Δ𝛾Δ+𝑞₁𝑥^𝛼𝜏₁(𝑡)+𝑞₂𝑥^𝛽𝜏₂(𝑡)=0,𝑡∈𝕋, (1.2) where 𝛾, 𝛼, and 𝛽 are quotients of odd positive integers with 0<𝛼<𝛾<𝛽.

In this paper, we study the second-order nonlinear dynamic equation 𝑟(𝑡)𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡))^Δ𝛾Δ+𝑓₁𝑡,𝑦𝛿₁(𝑡)+𝑓₂𝑡,𝑦𝛿₂(𝑡)=0, (1.3) on a time scale 𝕋, where 𝑝∈𝐶_rd(𝕋,[0,1)), 𝑓_𝑖∈𝐶(𝕋×ℝ,ℝ),𝑖=1,2,𝛾>0 is a quotient of odd positive integers.

The paper is organized as follows. In the next section, we give some preliminaries and lemmas. In Section 3, we will use the Riccati transformation technique to prove our main results. In Section 4, we present two examples to illustrate our results.

2. Preliminaries and Lemmas

For convenience, we recall some concepts related to time scales. More details can be found in [6].

Definition 2.1. Let 𝕋 be a time scale, for 𝑡∈𝕋 the forward jump operator is defined by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡}, the backward jump operator by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}, and the graininess function by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡, where inf∅∶=sup𝕋 and sup∅∶=inf𝕋. If 𝜎(𝑡)>𝑡, 𝑡 is said to be right-scattered, otherwise, it is right-dense. If 𝜌(𝑡)<𝑡, 𝑡 is said to be left-scattered, otherwise, it is left-dense. The set 𝕋^𝜅 is defined as follows. If 𝕋 has a left-scattered maximum 𝑚, then 𝕋^𝜅=𝕋−{𝑚}, otherwise, 𝕋^𝜅=𝕋.

Definition 2.2. For a function 𝑓∶𝕋→ℝ and 𝑡∈𝕋^𝜅, one defines the delta-derivative 𝑓^Δ(𝑡) of 𝑓(𝑡) to be the number (provided it exists) with the property that given any 𝜀>0, there is a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)∩𝕋 for some 𝛿) such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]≤𝜀|𝜎(𝑡)−𝑠|,∀𝑠∈𝑈. (2.1) We say that 𝑓 is delta-differentiable (or in short, differentiable) on 𝕋^𝜅 provided 𝑓^Δ(𝑡) exists, for all 𝑡∈𝕋^𝜅.

It is easily seen that if 𝑓 is continuous at 𝑡∈𝕋 and 𝑡 is right-scattered, then 𝑓 is differentiable at 𝑡 with 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜇(𝑡). (2.2) Moreover, if 𝑡 is right-dense then 𝑓 is differential at 𝑡 if the limit lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠)𝑡−𝑠 (2.3) exists as a finite number. In this case 𝑓^Δ(𝑡)=lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠).𝑡−𝑠 (2.4) In addition, if 𝑓^Δ≥0, then 𝑓 is nondecreasing. A useful formula is 𝑓^𝜎(𝑡)=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡),where𝑓^𝜎(𝑡)∶=𝑓(𝜎(𝑡)). (2.5) We will make use of the following product and quotient rules for the derivative of the product 𝑓𝑔 and the quotient 𝑓/𝑔 (where 𝑔𝑔^𝜎≠0) of two differentiable functions 𝑓 and 𝑔: 𝑓𝑔^Δ=𝑓^Δ𝑔+𝑓^𝜎𝑔^Δ=𝑓𝑔^Δ+𝑓^Δ𝑔^𝜎,𝑓𝑔^Δ=𝑓^Δ𝑔−𝑓𝑔^Δ𝑔𝑔^𝜎. (2.6)

Definition 2.3. Let 𝑓∶𝕋→ℝ be a function, 𝑓 is called right-dense continuous (rd-continuous) if it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋. A function 𝐹∶𝕋→ℝ is called an antiderivative of 𝑓 provided 𝐹^Δ(𝑡)=𝑓(𝑡) holds for all 𝑡∈𝕋^𝑘. By the antiderivative, the Cauchy integral of 𝑓 is defined as ∫^𝑏_𝑎𝑓(𝑠)Δ𝑠=𝐹(𝑏)−𝐹(𝑎), and ∫^∞_𝑎𝑓(𝑠)Δ𝑠=lim_𝑡→∞∫^𝑡_𝑎𝑓(𝑠)Δ𝑠.
Let 𝐶_rd(𝕋,ℝ) denote the set of all rd-continuous functions mapping 𝕋 to ℝ. It is shown in [6] that every rd-continuous function has an antiderivative. An integration by parts formula is ^𝑏_𝑎𝑓(𝑡)𝑔^Δ(𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑓^Δ(𝑡)𝑔^𝜎(𝑡)Δ𝑡. (2.7) In (1.3), we assume that 𝕋 is a time scale and(ℎ₁)𝜏(𝑡),𝛿_𝑖(𝑡)∈𝐶_rd(𝕋,𝕋), lim_𝑡→∞𝜏(𝑡)=∞, 𝜏(𝑡)≤𝑡, lim_𝑡→∞𝛿_𝑖(𝑡)=∞, and 𝑡≤𝛿_𝑖(𝑡), 𝑖=1,2,(ℎ₂)𝑟(𝑡)∈𝐶_rd(𝕋,ℝ⁺), ∫^∞(1/𝑟(𝑡))^1/𝛾Δ𝑡=∞, 𝑝(𝑡)∈𝐶_rd(𝕋,[0,1)), where ℝ⁺=(0,∞),(ℎ₃)𝑓_𝑖(𝑡,𝑢)∶𝕋×ℝ→ℝ is continuous function such that 𝑢𝑓_𝑖(𝑡,𝑢)>0 for all 𝑢≠0, there exist 𝑞_𝑖(𝑡)∈𝐶_rd(𝕋,ℝ⁺) (𝑖=1,2), quotients of odd positive integers 𝛼 and 𝛽 such that |𝑢𝑓₁(𝑡,𝑢)|≥𝑞₁(𝑡)|𝑢|^𝛼+1, |𝑢𝑓₂(𝑡,𝑢)|≥𝑞₂(𝑡)|𝑢|^𝛽+1, and 0<𝛼<𝛾<𝛽.

Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume throughout that the time scale 𝕋 under consideration satisfies inf𝕋=𝑡₀ and sup𝕋=∞. For 𝑇∈𝕋, let [𝑇,∞)_𝕋∶={𝑡∈𝕋∶𝑡≥𝑇}. Throughout this paper, these assumptions will be supposed to hold. Let 𝜏^∗(𝑡)=min{𝜏(𝑡),𝛿₁(𝑡),𝛿₂(𝑡)}, 𝑇₀=min{𝜏^∗(𝑡)∶𝑡≥𝑡₀} and 𝜏^∗₋₁(𝑡)=sup{𝑠≥𝑡₀∶𝜏^∗(𝑠)≤𝑡}. Clearly 𝜏^∗₋₁(𝑡)≥𝑡 for 𝑡≥𝑇₀, 𝜏^∗₋₁(𝑡) is nondecreasing and coincides with the inverse of 𝜏^∗(𝑡) when the latter exists.

By a solution of (1.3), we mean a nontrivial real-valued function 𝑦(𝑡) which has the properties [𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡))]∈𝐶¹_rd[𝜏^∗₋₁(𝑡₀),∞) and 𝑟(𝑡)([𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡))])^Δ)^𝛾∈𝐶¹_rd[𝜏^∗₋₁(𝑡₀),∞). Our attention is restricted to those solutions of (1.3) that exist on some half line [𝑡_𝑦,∞) and satisfy sup{|𝑦(𝑡)|∶𝑡≥𝑡₁}>0 for any 𝑡₁≥𝑡_𝑦. A solution 𝑦(𝑡) of (1.3) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

For convenience, we use the notation 𝑥(𝜎(𝑡))=𝑥^𝜎(𝑡), 𝑥(𝛿_𝑖(𝑡))=𝑥^𝛿𝑖(𝑡)(𝑖=1,2) and 𝑥^Δ(𝜎(𝑡))=(𝑥^Δ(𝑡))^𝜎, and set 𝑥(𝑡)∶=𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡)). (2.8) Then (1.3) becomes 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑓₁𝑡,𝑦𝛿₁(𝑡)+𝑓₂𝑡,𝑦𝛿₂(𝑡)=0. (2.9) Now, we give the first lemma. Set 𝑅_𝑇(𝑡)=^𝑡_𝑇Δ𝑠(𝑟(𝑠))^1/𝛾. (2.10)

Lemma 2.4. Let conditions (ℎ₁)−(ℎ₃) hold. If 𝑦(𝑡) is an eventually positive solution of (1.3), then there exists 𝑇∈𝕋 sufficiently large such that 𝑥(𝑡)>0,𝑥^Δ(𝑡)>0, (𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾)^Δ<0, 𝑥(𝑡)>𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)𝑥^Δ(𝑡), and (𝑥^𝛿𝑖(𝑡)/𝑥^𝜎(𝑡))>(𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡))/(𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)+𝜇(𝑡)) (𝑖=1,2) for 𝑡∈[𝑇,∞)_𝕋.

Proof. If 𝑦(𝑡) is an eventually positive solution of (1.3), then by (ℎ₁) there exists a 𝑇∈[𝑡₀,∞)_𝕋 such that 𝑦(𝑡)>0,𝑦(𝜏(𝑡))>0,𝑦𝛿_𝑖(𝑡)>0,for𝑡≥𝑇,𝑖=1,2. (2.11) From (2.8), (1.3), and (ℎ₂), we see that 𝑥(𝑡)≥𝑦(𝑡). Also by (1.3) and (ℎ₃), we have 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ≤−𝑞₁(𝑡)𝑦^𝛼𝛿₁(𝑡)−𝑞₂(𝑡)𝑦^𝛽𝛿₂(𝑡)<0,for𝑡≥𝑇, (2.12) which implies that 𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾 is decreasing on [𝑇,∞)_𝕋.
We claim that 𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾>0 on [𝑇,∞)_𝕋. Assume not, there is a 𝑡₁∈[𝑇,∞)_𝕋 such that 𝑟(𝑡₁)(𝑥^Δ(𝑡₁))^𝛾<0. Since 𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾≤𝑟(𝑡₁)(𝑥^Δ(𝑡₁))^𝛾 for 𝑡≥𝑡₁, we have 𝑥^Δ(𝑡)≤𝑟𝑡₁^1/𝛾𝑥^Δ𝑡₁1𝑟(𝑡)^1/𝛾. (2.13) Integrating the inequality above form 𝑡₁ to 𝑡(≥𝑡₁), by (ℎ₂) we get 𝑥(𝑡)≤𝑥𝑡₁+𝑟𝑡₁^1/𝛾𝑥^Δ𝑡₁^𝑡_𝑡11𝑟(𝑠)^1/𝛾Δ𝑠⟶−∞(𝑡⟶∞), (2.14) and this contradicts the fact that 𝑥(𝑡)>0, for all 𝑡≥𝑇. Thus we have 𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾>0 on [𝑇,∞)_𝕋 and so 𝑥^Δ(𝑡)>0 on [𝑇,∞)_𝕋.
Note that 𝑥(𝑡)>𝑥(𝑡)−𝑥(𝑇)=^𝑡_𝑇𝑥^Δ(𝑠)=^𝑡_𝑇𝑟(𝑠)𝑥^Δ(𝑠)^𝛾1/𝛾𝑟^1/𝛾(𝑠)>𝑥Δ𝑠𝑟(𝑡)^Δ(𝑡)^{𝛾1/𝛾𝑡}_𝑇1𝑟^1/𝛾(𝑠)Δ𝑠=𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)𝑥^Δ(𝑡), (2.15) we have 𝑥^𝜎(𝑡)𝑥(𝑡)=𝑥(𝑡)+𝜇(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)(𝑡)𝑥=1+𝜇^Δ(𝑡)𝑥(𝑡)(𝑡)1<1+𝜇𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)=𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)+𝜇(𝑡)𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡). (2.16) Since 𝛿_𝑖(𝑡)≥𝑡 and 𝑥^Δ(𝑡)>0, we get 𝑥^𝛿𝑖(𝑡)𝑥^𝜎(𝑡)=𝑥^𝛿𝑖(𝑡)𝑥(𝑡)⋅𝑥(𝑡)𝑥^𝜎(𝑡)≥𝑥(𝑡)𝑥^𝜎(𝑡)>𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)+𝜇(𝑡),𝑖=1,2. (2.17) The proof is complete.

Remark 2.5. By 𝑥(𝑡)≥𝑦(𝑡) on [𝑇,∞)_𝕋, 𝑥^Δ>0, (1.3), (2.8) and (ℎ₁)−(ℎ₃), we get 0≥𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑞₁(𝑡)𝑦𝛿₁(𝑡)^𝛼+𝑞₂(𝑡)𝑦𝛿₂(𝑡)^𝛽=𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑞₁(𝑡)𝑥𝛿₁(𝑡)−𝑝𝛿₁(𝑡)𝑦𝜏𝛿₁(𝑡)^𝛼+𝑞₂(𝑡)𝑥𝛿₂(𝑡)−𝑝𝛿₂(𝑡)𝑦𝜏𝛿₂(𝑡)^𝛽≥𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑞₁(𝑡)𝑥𝛿₁(𝑡)−𝑝𝛿₁(𝑡)𝑥𝜏𝛿₁(𝑡)^𝛼+𝑞₂(𝑡)𝑥𝛿₂(𝑡)−𝑝𝛿₂(𝑡)𝑥𝜏𝛿₂(𝑡)^𝛽≥𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑞₁(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥𝛿₁(𝑡)^𝛼+𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥𝛿₂(𝑡)^𝛽. (2.18)

Lemma 2.6 (see [3]). Let 𝑔(𝑢)=𝐵𝑢−𝐴𝑢^{(𝛾+1)/𝛾}, where 𝐴>0 and B are constants, 𝛾 is a quotient of odd positive integers. Then 𝑔 attains its maximum value on ℝ at 𝑢^∗=(𝐵𝛾/(𝐴(𝛾+1)))^𝛾, and max_𝑢∈ℝ𝑔=𝑔𝑢^∗=𝛾^𝛾𝛾+1^𝛾+1𝐵^𝛾+1𝐴^𝛾. (2.19)

Lemma 2.7 (see [11]). 𝑥 and 𝑧 are delta-differentiable on 𝕋. For 𝑥≠0 and any 𝑡∈𝕋, one has 𝑥^Δ(𝑡𝑧)²(𝑡)𝑥(𝑡)^Δ=𝑧^Δ(𝑡)²−𝑥(𝑡)𝑥^𝜎(𝑡𝑧(𝑡))𝑥(𝑡)^Δ2. (2.20)

3. Main Results

In this section, by employing the Riccati transformation technique we will establish oscillation criteria for (1.3) in two cases: 𝛾≥1 and 0<𝛾<1. Set 𝑄(𝑠)=𝑞₁(𝑠)1−𝑝𝛿₁(𝑠)^{𝛼(𝛽−𝛾)/(𝛽−𝛼)}⋅𝑞₂(𝑠)1−𝑝𝛿₂(𝑠)^{𝛽(𝛾−𝛼)/(𝛽−𝛼)},𝑄₁(𝑠)=𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)+𝜇(𝑠)^𝛾𝑧(𝑠)𝑄(𝑠),𝑄₂(𝑠)=𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)+𝜇(𝑠)^𝛾𝑧²(𝑠)^𝜎𝑄(𝑠),𝐶(𝑡,𝑠)=𝐻^Δ_𝑠()()𝑧𝑡,𝑠+𝐻𝑡,𝑠^Δ(𝑠)𝑧(𝜎(𝑠(𝑠)=𝑧)),𝐴^Δ(𝑠)𝑧(𝜎(𝑠+𝐻))^Δ𝑠(𝑡,𝑠)𝐻(),𝑡,𝑠 (3.1)(I)𝛾≥1.

Theorem 3.1. Assume that (ℎ₁)−(ℎ₃) hold and 𝛾≥1. Furthermore, assume that there exists a positive rd-continuous Δ-differentiable function 𝑧(𝑡) such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑄₁(𝑠)−1𝛾+1^𝛾+1𝑟(𝑠)𝑧^Δ(𝑠)^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=∞, (3.2) then (1.3) is oscillatory.

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive (note that in the case when 𝑦(𝑡) is eventually negative, the proof is similar, since the substitution 𝑌(𝑡)=−𝑦(𝑡) transforms (1.3) into the same form). Then, by (ℎ₁)−(ℎ₃) there exists 𝑇≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0, and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8). Define the function 𝑤(𝑡) by the Riccati substitution 𝑤(𝑡)𝑧(𝑡)𝑟(𝑡)𝑥∶=^Δ(𝑡)^𝛾𝑥^𝛾(𝑡),for𝑡≥𝑇, (3.3) then 𝑤(𝑡)>0 and 𝑤^Δ(𝑡)=𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ𝑧(𝑡)𝑥^𝛾(𝑡)+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧(𝑡)𝑥^𝛾(𝑡)^Δ=𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ𝑧(𝑡)𝑥^𝛾(𝑡)+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧^Δ(𝑡)𝑥^𝛾(𝑡)−𝑧(𝑡)𝑥^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾. (3.4) By (1.3), 𝑥^Δ(𝑡)>0, and (2.18), we obtain 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ𝑥^𝛾(𝑡)𝑞≤−¹(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥^𝛿1(𝑡)^𝛼(𝑥^𝜎(𝑡))^𝛾−𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥^𝛿2(𝑡)^𝛽(𝑥^𝜎(𝑡))^𝛾. (3.5) Noting that 0<𝛼<𝛾<𝛽, we have 𝛽−𝛾𝛽−𝛼<1,𝛾−𝛼𝛽−𝛼<1. (3.6) By Young’s inequality 𝑎^𝜒𝑏^1−𝜒≤𝜒𝑎+1−𝜒𝑏,0<𝜒<1, (3.7) with 𝜒=𝛽−𝛾𝑞𝛽−𝛼,𝑎=¹(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥^𝛿1(𝑡)^𝛼(𝑥^𝜎(𝑡))^𝛾,𝑞𝑏=²(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥^𝛿2(𝑡)^𝛽(𝑥^𝜎(𝑡))^𝛾, (3.8) we have 𝑞₁(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥^𝛿1(𝑡)^𝛼(𝑥^𝜎(𝑡))^𝛾+𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥^𝛿2(𝑡)^𝛽(𝑥^𝜎(𝑡))^𝛾≥𝛽−𝛾𝑞𝛽−𝛼¹(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥^𝛿1(𝑡)^𝛼(𝑥^𝜎(𝑡))^𝛾+𝛾−𝛼𝑞𝛽−𝛼²(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥^𝛿2(𝑡)^𝛽(𝑥^𝜎(𝑡))^𝛾≥𝑞₁(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥^𝛿1(𝑡)^𝛼(𝑥^𝜎(𝑡))^{𝛾(𝛽−𝛾)/(𝛽−𝛼)}⋅𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥^𝛿2(𝑡)^𝛽(𝑥^𝜎(𝑡))^{𝛾(𝛾−𝛼)/(𝛽−𝛼)}=𝑞₁(𝑡)1−𝑝𝛿₁(𝑡)^{𝛼(𝛽−𝛾)/(𝛽−𝛼)}⋅𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^{𝛽(𝛾−𝛼)/(𝛽−𝛼)}⋅𝑥^𝛿1(𝑡)^{(𝛼𝛽−𝛼𝛾)/(𝛽−𝛼)}⋅𝑥^𝛿2(𝑡)^{(𝛽𝛾−𝛽𝛼)/(𝛽−𝛼)}(𝑥^𝜎(𝑡))^𝛾. (3.9) By 𝛾=((𝛼𝛽−𝛼𝛾)/(𝛽−𝛼))+((𝛽𝛾−𝛽𝛼)/(𝛽−𝛼)) and Lemma 2.4, we get 𝑞₁(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥^𝛿1(𝑡)^𝛼(𝑥^𝜎(𝑡))^𝛾+𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥^𝛿2(𝑡)^𝛽(𝑥^𝜎(𝑡))^𝛾≥𝑞₁(𝑡)1−𝑝𝛿₁(𝑡)^{𝛼(𝛽−𝛾)/(𝛽−𝛼)}⋅𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^{𝛽(𝛾−𝛼)/(𝛽−𝛼)}⋅𝑥^𝛿1(𝑡)𝑥^𝜎(𝑡)^{(𝛼𝛽−𝛼𝛾)/(𝛽−𝛼)}⋅𝑥^𝛿2(𝑡)𝑥^𝜎(𝑡)^{(𝛽𝛾−𝛽𝛼)/(𝛽−𝛼)}>𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)+𝜇(𝑡)^𝛾𝑄(𝑡). (3.10) In view of 𝑥^Δ(𝑡)>0 and (3.5)–(3.10), for all 𝑡≥𝑇, we obtain 𝑤^Δ(𝑡)(𝑡)𝑅<−𝑧^𝑇(𝑡)𝑟^1/𝛾(𝑡)𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)+𝜇(𝑡)^𝛾𝑄(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧(𝑡)𝑥^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾=−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧(𝑡)𝑥^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾. (3.11) Using 𝛾≥1, Lemma 2.4 and the Keller’s chain rule, we get 𝑥^𝛾(𝑡)^Δ=𝛾¹₀𝑥(𝑡)+ℎ𝜇(𝑡)𝑥^Δ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)=𝛾𝑥^Δ(𝑡)¹₀(1−ℎ)𝑥(𝑡)+ℎ𝑥^𝜎(𝑡)^𝛾−1𝑑ℎ≥𝛾𝑥^Δ(𝑡)¹₀((1−ℎ)𝑥(𝑡)+ℎ𝑥(𝑡))^𝛾−1𝑑ℎ=𝛾𝑥^𝛾−1(𝑡)𝑥^Δ(𝑡). (3.12) Also from Lemma 2.4 and 𝜎(𝑡)≥𝑡, we have 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾≥𝑟(𝜎(𝑡))𝑥^Δ(𝜎(𝑡))^𝛾. (3.13) By (3.11)–(3.13), we get 𝑤^Δ(𝑡)<−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧(𝑡)𝛾𝑥^𝛾−1(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾≤−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝑧(𝑡)𝛾𝑟^1/𝛾(𝑡)𝑟^𝜎(𝑡)^{(𝛾+1)/𝛾}𝑥^Δ(𝜎(𝑡))^𝛾+1(𝑥^𝜎(𝑡))^𝛾+1=−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝑧(𝑡)𝛾𝑟^1/𝛾(𝑡)(𝑧^𝜎(𝑡))^{(𝛾+1)/𝛾}𝑤^𝜎(𝑡)^{(𝛾+1)/𝛾}. (3.14) Setting 𝑧𝐵=^Δ(𝑡)𝑧^𝜎(𝑡)𝑧(𝑡)𝛾,𝐴=𝑟^1/𝛾(𝑡)(𝑧^𝜎(𝑡))^{(𝛾+1)/𝛾},𝑢=𝑤^𝜎(𝑡), (3.15) then by Lemma 2.6, from (3.14) we obtain that for all 𝑡≥𝑇, 𝑤^Δ(𝑡)<−𝑄₁(𝑡)+1𝛾+1^𝛾+1𝑟(𝑡)𝑧^Δ(𝑡)^𝛾+1𝑧^𝛾(𝑡). (3.16) Integrating the above inequality from 𝑇 to 𝑡(≥𝑇), we get ^𝑡_𝑇𝑄₁(𝑠)−1𝛾+1^𝛾+1𝑟(𝑠)𝑧^Δ(𝑠)^𝛾+1𝑧^𝛾(𝑠)Δ𝑠<𝑤(𝑇)−𝑤(𝑡)<𝑤(𝑇). (3.17) Taking lim sup on both sides of the above inequality as 𝑡→∞, we obtain a contradiction to condition (3.2). The proof is complete.

The following theorem gives new oscillation criteria for (1.3) which can be considered as the extension of Philos-type oscillation criterion. Define 𝐷={(𝑡,𝑠)∈𝕋²∶𝑡≥𝑠≥0} and ℋ_∗=𝐻(𝑡,𝑠)∈𝐶¹𝐷,ℝ₊∶𝐻(𝑡,𝑡)=0,𝐻(𝑡,𝑠)>0,𝐻^Δ_𝑠(𝑡,𝑠)≥0,for𝑡>𝑠≥0. (3.18)

Theorem 3.2. Assume that (ℎ₁)−(ℎ₃) hold and 𝛾≥1. Furthermore, assume that there exist a positive rd-continuous Δ-differentiable function 𝑧(𝑡) and a function 𝐻∈ℋ_∗ such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐻(𝑡,𝑠)𝑄₁(𝑠)−𝐶^𝛾+1(𝑡,𝑠)𝑟(𝑠)(𝑧(𝜎(𝑠)))^𝛾+1𝐻^𝛾(𝑡,𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=∞, (3.19) then (1.3) is oscillatory.

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive. Then, by (ℎ₁)−(ℎ₃) there exists 𝑇≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0, and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8). Define 𝑤(𝑡) as in (3.3). Proceeding as in the proof of Theorem 3.1, we can get (3.14). From (3.14), for function 𝐻∈ℋ_∗ and all 𝑡≥𝑇 we have ^𝑡_𝑇𝐻(𝑡,𝑠)𝑄₁(𝑠)Δ𝑠<−^𝑡_𝑇𝐻(𝑡,𝑠)𝑤^Δ(𝑠)Δ𝑠+^𝑡_𝑇𝐻(𝑡,𝑠)𝑤^𝜎(𝑠)𝑧^Δ(𝑠)𝑧^𝜎(𝑠)−Δ𝑠_𝑡𝑇𝐻()𝑧(𝑠)𝛾𝑡,𝑠𝑟^1/𝛾(𝑠)(𝑧^𝜎(𝑠))^{(𝛾+1)/𝛾}𝑤^𝜎(𝑠)^{(𝛾+1)/𝛾}Δ𝑠. (3.20) Using the integration by parts formula (2.7), we obtain −^𝑡_𝑇𝐻(𝑡,𝑠)𝑤^Δ(𝑠)Δ𝑠=−𝐻(𝑡,𝑠)𝑤(𝑠)|^𝑡_𝑇+^𝑡_𝑇𝐻^Δ_𝑠(𝑡,𝑠)𝑤^𝜎(𝑠)()𝑤(𝑇)+Δ𝑠=𝐻𝑡,𝑇^𝑡_𝑇𝐻^Δ_𝑠(𝑡,𝑠)𝑤^𝜎(𝑠)Δ𝑠. (3.21) It follows that ^𝑡_𝑇𝐻(𝑡,𝑠)𝑄₁(𝑠)Δ𝑠<𝐻(𝑡,𝑇)𝑤(𝑇)+^𝑡_𝑇𝐻^Δ_𝑠()()𝑧𝑡,𝑠+𝐻𝑡,𝑠^Δ(𝑠)𝑧^𝜎(𝑠)𝑤^𝜎(𝑠)−Δ𝑠_𝑡𝑇𝐻()𝑧(𝑠)𝛾𝑡,𝑠𝑟^1/𝛾(𝑠)(𝑧^𝜎(𝑠))^{(𝛾+1)/𝛾}𝑤^𝜎(𝑠)^{(𝛾+1)/𝛾}()𝑤(𝑇)+Δ𝑠=𝐻𝑡,𝑇^𝑡_𝑇𝐶(𝑡,𝑠)𝑤^𝜎(𝑠)−Δ𝑠_𝑡𝑇𝐻(𝑡,𝑠)𝑧(𝑠)𝛾𝑟^1/𝛾(𝑠)(𝑧^𝜎(𝑠))^{(𝛾+1)/𝛾}𝑤^𝜎(𝑠)^{(𝛾+1)/𝛾}Δ𝑠. (3.22) Setting ()𝐻()𝑧(𝑠)𝛾𝐵=𝐶𝑡,𝑠,𝐴=𝑡,𝑠𝑟^1/𝛾(𝑠)(𝑧^𝜎(𝑠))^{(𝛾+1)/𝛾},𝑢=𝑤^𝜎(𝑠), (3.23) by Lemma 2.6 we obtain that for all 𝑡≥𝑇, ^𝑡_𝑇𝐻(𝑡,𝑠)𝑄₁(𝑠)Δ𝑠<𝐻(𝑡,𝑇)𝑤(𝑇)+^𝑡_𝑇[𝐶(𝑡,𝑠)]^𝛾+1𝑧^𝜎(𝑠)^𝛾+1𝑟(𝑠)𝐻^𝛾(𝑡,𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠. (3.24) That is, 1𝐻(𝑡,𝑇)^𝑡_𝑇𝐻(𝑡,𝑠)𝑄₁(𝑠)−[𝐶(𝑡,𝑠)]^𝛾+1𝑟(𝑠)𝑧^𝜎(𝑠)^𝛾+1𝐻^𝛾(𝑡,𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠<𝑤(𝑇). (3.25) Taking lim sup on both sides of the above inequality as 𝑡→∞, we obtain a contradiction to condition (3.19). The proof is complete.

Theorem 3.3. Assume that (ℎ₁)−(ℎ₃) hold and 𝛾≥1. Then (1.3) is oscillatory if for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑄(𝑠)𝑅^𝛾_𝑇(𝑠)Δ𝑠=∞. (3.26)

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive. Then, by (ℎ₁)−(ℎ₃) there exists 𝑇≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0, and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8). Set 𝜙(𝑡)=𝑟^1/𝛾(𝑡)𝑥^Δ(𝑡). By Lemma 2.4, we get 𝜙>0,(𝜙^𝛾)^Δ<0. Using 𝛾≥1 and the keller’s chain rule, we get 𝜙^𝛾(𝑡)^Δ=𝛾¹₀𝜙(𝑡)+ℎ𝜇(𝑡)𝜙^Δ(𝑡)^𝛾−1𝑑ℎ𝜙^Δ(𝑡)=𝛾¹₀(1−ℎ)𝜙(𝑡)+ℎ𝜙^𝜎(𝑡)^𝛾−1𝑑ℎ𝜙^Δ(𝑡)<0. (3.27) So we have 𝜙^Δ(𝑡)<0 and there is a constant 𝐿>0 such that 𝜙(𝑡)≤𝐿 for 𝑡≥𝑇. Then (1.3) becomes (𝜙^𝛾)^Δ(𝑡)+𝑓₁(𝑡,𝑦(𝛿₁(𝑡)))+𝑓₂(𝑡,𝑦(𝛿₂(𝑡)))=0. By (2.18), we have 𝜙0≥^𝛾(𝑡)^Δ𝜙^𝜎(𝑡)^𝛾+𝑞₁(𝑡)1−𝑝𝛿₁(𝑡)^𝛼𝑥^𝛿1(𝑡)^𝛼𝜙^𝜎(𝑡)^𝛾+𝑞₂(𝑡)1−𝑝𝛿₂(𝑡)^𝛽𝑥^𝛿2(𝑡)^𝛽𝜙^𝜎(𝑡)^𝛾. (3.28) Similar to the proof of (3.10), we get 𝜙0≥^𝛾(𝑡)^Δ𝜙^𝜎(𝑡)^𝛾(𝑡)𝑥+𝑄^𝛿1(𝑡)𝜙^𝜎(𝑡)^{(𝛼𝛽−𝛼𝛾)/(𝛽−𝛼)}⋅𝑥^𝛿2(𝑡)𝜙^𝜎(𝑡)^{(𝛽𝛾−𝛽𝛼)/(𝛽−𝛼)}. (3.29) Using the Keller’s chain rule and 𝜙^Δ(𝑡)<0, we get 𝜙^𝜎≤𝜙 and (𝜙^𝛾(𝑡))^Δ≥𝛾𝜙^𝜎(𝑡)^𝛾−1𝜙^Δ(𝑡). (3.30) From 𝛿_1,2(𝑡)≥𝑡 and 𝑥^Δ(𝑡)>0, it follows that 0≥𝛾𝜙^Δ(𝑡)𝜙^𝜎(𝑡)(𝑡)𝑥+𝑄^𝛿1(𝑡)𝜙^𝜎(𝑡)^{(𝛼𝛽−𝛼𝛾)/(𝛽−𝛼)}⋅𝑥^𝛿2(𝑡)𝜙^𝜎(𝑡)^{(𝛽𝛾−𝛽𝛼)/(𝛽−𝛼)}≥𝛾𝜙^Δ(𝑡)𝐿(𝑡)𝑥(𝑡)+𝑄𝜙^𝜎(𝑡)^𝛾≥𝛾𝜙^Δ(𝑡)𝐿(𝑡)𝑥(𝑡)+𝑄𝜙(𝑡)^𝛾. (3.31) By Lemma 2.4, we get 𝑥(𝑡)𝜙(𝑡)=𝑥(𝑡)𝑟^1/𝛾(𝑡)𝑥^Δ(𝑡)>𝑅_𝑇(𝑡). (3.32) It follows that 0>𝛾𝜙^Δ(𝑡)𝐿+𝑄(𝑡)𝑅^𝛾_𝑇(𝑡). (3.33) Integrating the above inequality from 𝑇 to 𝑡(≥𝑇), we obtain ^𝑡_𝑇𝑄(𝑠)𝑅^𝛾_𝑇(𝑠)𝛾Δ𝑠<−𝐿^𝑡_𝑇𝜙^Δ(𝑠)𝛾Δ𝑠=𝐿𝜙(𝑇)(𝑡)<𝛾−𝜙𝐿𝜙(𝑇). (3.34) Taking lim sup on both sides of the above inequality as 𝑡→∞, we obtain a contradiction to condition (3.26). The proof is complete.

Theorem 3.4. Assume that (ℎ₁)−(ℎ₃) hold and 𝛾≥1. Furthermore, assume that there exists a positive rd-continuous Δ-differentiable function 𝑧(𝑡) such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑄₁(𝑠)−𝑟^1/𝛾(𝑠)𝑧^Δ(𝑠)²4𝛾𝑅_𝑇(𝑠)^𝛾−1𝑧(𝑠)Δ𝑠=∞, (3.35) then (1.3) is oscillatory.

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive. Then, by (ℎ₁)−(ℎ₃) there exists 𝑇≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0, and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8). Define 𝑤(𝑡) as in (3.3). By (2.6), we obtain 𝑤^Δ(𝑡)=𝑧^Δ(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝑥^𝛾(𝑡)^𝜎(𝑡)𝑟(𝑡)𝑥+𝑧^Δ(𝑡)^𝛾𝑥^𝛾(𝑡)^Δ=𝑧^Δ(𝑡)𝑧^𝜎(𝑡)𝑤^𝜎(𝑡)(𝑡)𝑟(𝑡)𝑥+𝑧^Δ(𝑡)^𝛾Δ𝑥^𝛾(𝑡)−𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝑥^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾=𝑧^Δ(𝑡)𝑧^𝜎(𝑡)𝑤^𝜎(𝑡)(𝑡)𝑟(𝑡𝑥+𝑧)^Δ(𝑡)^𝛾Δ𝑥^𝜎(𝑡)^𝛾−𝑟^𝜎(𝑡)𝑥^Δ(𝑡)^𝜎𝛾𝑥^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾. (3.36) By Lemma 2.4, 𝜎(𝑡)≥𝑡, and (3.5)–(3.13), for all 𝑡≥𝑇 we obtain 𝑤^Δ(𝑡)<−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)(𝑡)𝑟−𝑧^𝜎(𝑡)𝑥^Δ(𝜎(𝑡))^𝛾𝛾𝑥^𝛾−1(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾=−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝛾𝑧(𝑡)𝑟^𝜎(𝑡)𝑥^Δ(𝜎(𝑡))^𝛾(𝑥^𝜎(𝑡))^𝛾2𝑥^𝜎(𝑡)^𝛾𝑥^Δ(𝑡)𝑥(𝑡)𝑥^Δ(𝜎(𝑡))^𝛾<−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)(𝑡)𝑤−𝛾𝑧^𝜎(𝑡)𝑧^𝜎(𝑡)²𝑥^𝛾−1(𝑡)𝑥^Δ(𝑡)^𝛾−1𝑟(𝑡)<−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)(𝑡)1−𝛾𝑧𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)²𝑅_𝑇(𝑡)𝑟^1/𝛾(𝑡)^𝛾−1. (3.37) It follows that 𝑤^Δ(𝑡)<−𝑄₁(𝑡)+𝑧^Δ(𝑡)𝑧^𝜎(𝑡)𝑤^𝜎(𝑡)−𝛾𝑧(𝑡)𝑅_𝑇(𝑡)^𝛾−1𝑟^1/𝛾(𝑡)(𝑧^𝜎(𝑡))²𝑤^𝜎(𝑡)². (3.38) By completing the square, we have 𝑤^Δ(𝑡)<−𝑄₁(𝑡)+𝑟^1/𝛾(𝑡)𝑧^Δ(𝑡)²4𝛾𝑅_𝑇(𝑡)^𝛾−1𝑧(𝑡). (3.39) Integrating the above inequality from 𝑇 to 𝑡(≥𝑇), we get ^𝑡_𝑇𝑄₁(𝑠)−𝑟^1/𝛾(𝑠)𝑧^Δ(𝑠)²4𝛾𝑅_𝑇(𝑠)^𝛾−1𝑧(𝑠)Δ𝑠<𝑤(𝑇)−𝑤(𝑡)<𝑤(𝑇). (3.40) Taking lim sup on both sides of the above inequality as 𝑡→∞, we obtain a contradiction to condition (3.35). The proof is complete.

Theorem 3.5. Assume that (ℎ₁)−(ℎ₃) and 𝛾≥1 hold. Furthermore, assume that there exist a positive rd-continuous Δ-differentiable function 𝑧(𝑡) and a function 𝐻∈ℋ_∗ such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐻(𝑡,𝑠)𝑄₁(𝑠)−𝐻(𝑡,𝑠)𝐴²(𝑠)𝑟^1/𝛾(𝑠)𝑧^𝜎2(𝑠)4𝛾𝑅_𝑇(𝑠)^𝛾−1𝑧(𝑠)Δ𝑠=∞, (3.41) then (1.3) is oscillatory.

Proof. By (3.39), the proof is similar to Theorems 3.2 and 3.4, so we omit it.

Theorem 3.6. Assume that (ℎ₁)−(ℎ₃) hold and 𝛾≥1. Furthermore, assume that there exists a rd-continuous Δ-differentiable function 𝑧(𝑡) such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑄₂(𝑠)−1𝛾𝑧^Δ(𝑠)²𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)^1−𝛾Δ𝑠=∞, (3.42) then (1.3) is oscillatory.

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive. Then, by (ℎ₁)−(ℎ₃) there exists 𝑇≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0, and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8). Define the function 𝑣(𝑡) by the Riccati substitution 𝑣(𝑡)𝑧∶=²(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝑥^𝛾(𝑡),for𝑡≥𝑇, (3.43) then 𝑣(𝑡)>0. From (3.5) and (3.10), it follows that for all 𝑡≥𝑇𝑣^Δ(𝑡)=𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ𝑧²(𝑡)𝑥^𝛾(𝑡)^𝜎+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝑧²(𝑡)𝑥^𝛾(𝑡)^Δ≤−𝑄₂(𝑡)+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾(𝑥^𝛾(𝑡))^Δ𝑥^𝛾(𝑡)^Δ𝑧²(𝑡)𝑥^𝛾(𝑡)^Δ. (3.44) By (3.12) and Lemma 2.7, we obtain 𝑣^Δ(𝑡)≤−𝑄₂(𝑡)+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾(𝑥^𝛾(𝑡))^Δ𝑧^Δ(𝑡)²−𝑥^𝛾(𝑡)𝑥^𝜎(𝑡)^𝛾𝑧(𝑡)𝑥^𝛾(𝑡)^Δ2<−𝑄₂(𝑡)+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝑧^Δ(𝑡)²(𝑥^𝛾(𝑡))^Δ<−𝑄₂(𝑡)+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝑧^Δ(𝑡)²𝛾𝑥^𝛾−1(𝑡)𝑥^Δ(𝑡). (3.45) It follows from Lemma 2.4 that 𝑣^Δ(𝑡)<−𝑄₂(𝑡)+1𝛾𝑧^Δ(𝑡)²𝑟^1/𝛾(𝑡)𝑅_𝑇(𝑡)^1−𝛾. (3.46) Integrating the above inequality from 𝑇 to 𝑡(≥𝑇), we get ^𝑡_𝑇𝑄₂(𝑠)−1𝛾𝑧^Δ(𝑠)²𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)^1−𝛾Δ𝑠<𝑣(𝑇)−𝑣(𝑡)<𝑣(𝑇). (3.47) Taking lim sup on both sides of the above inequality as 𝑡→∞, we obtain a contradiction to condition (3.42). The proof is complete.

From Theorem 3.6, we can establish different sufficient conditions for the oscillation of (1.3) by using different choices of 𝑧(𝑡). For instance, if 𝑧(𝑡)=1 or 𝑧(𝑡)=√𝑡, we have the following results.

Corollary 3.7. Assume that (ℎ₁)−(ℎ₃) hold and 𝛾≥1. Then (1.3) is oscillatory if for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)+𝜇(𝑠)^𝛾𝑄(𝑠)Δ𝑠=∞. (3.48)

Corollary 3.8. Assume that (ℎ₁)−(ℎ₃) hold and 𝛾≥1. Then (1.3) is oscillatory if for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑄₃(𝑠)−1√𝑠+𝜎(𝑠)²𝑟^1/𝛾(𝑠)1𝛾𝑅_𝑇(𝑠)^1−𝛾Δ𝑠=∞, (3.49) where 𝑄₃(𝑠)=(𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)/(𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)+𝜇(𝑠)))^𝛾𝜎(𝑠)𝑄(𝑠).
(II)0<𝛾<1.
Theorem  3.1^′. Assume that (ℎ₁)−(ℎ₃) hold and 0<𝛾<1. Furthermore, assume that there exists a positive rd-continuous Δ-differentiable function 𝑧(𝑡) such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑄₁(𝑠)−1𝛾+1^𝛾+1𝑟(𝑠)𝑧^Δ(𝑠)^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=∞, (3.50) then (1.3) is oscillatory.

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive. Then, by (ℎ₁)−(ℎ₃) there exists T≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0 and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8). Define 𝑤(𝑡) as in (3.3). Proceeding as in the proof of Theorem 3.1, we get 𝑤^Δ(𝑡)=𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ𝑧(𝑡)𝑥^𝛾(𝑡)+𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧^Δ(𝑡)𝑥^𝛾(𝑡)−𝑧(𝑡)𝑥^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾. (3.51) Using 0<𝛾<1, Lemma 2.4 and the Keller’s chain rule, we get 𝑥^𝛾(𝑡)^Δ=𝛾¹₀𝑥(𝑡)+ℎ𝜇(𝑡)𝑥^Δ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)≥𝛾𝑥^Δ(𝑡)¹₀(1−ℎ)𝑥^𝜎(𝑡)+ℎ𝑥^𝜎(𝑡)^𝛾−1𝑥𝑑ℎ=𝛾^𝜎(𝑡)^𝛾−1𝑥^Δ(𝑡). (3.52) By (3.10), (3.51), and (3.52), we get 𝑤^Δ(𝑡)<−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧(𝑡)𝛾𝑥^𝜎(𝑡)^𝛾−1𝑥^Δ(𝑡)𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾. (3.53) Since −𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑧(𝑡)𝛾𝑥^𝜎(𝑡)^𝛾−1𝑥^Δ(𝑡)𝑥^𝛾(𝑡)(𝑥^𝜎(𝑡))^𝛾𝑟=−^𝜎(𝑡)^{(𝛾+1)/𝛾}𝑥^Δ(𝑡)^𝜎𝛾+1𝑧(𝑡)𝛾𝑥^Δ(𝑡)𝑥^𝛾(𝑡)𝑥^𝜎(𝑡)(𝑟^𝜎(𝑡))^1/𝛾𝑥^Δ(𝑡)^𝜎((𝑟For3.13))≤−^𝜎(𝑡)^{(𝛾+1)/𝛾}𝑥^Δ(𝑡)^𝜎𝛾+1𝑧(𝑡)𝛾𝑥^Δ(𝑡)𝑥^𝛾(𝑡)𝑥^𝜎(𝑡)𝑟^1/𝛾(𝑡)𝑥^Δ(𝑡)𝑧(𝑡)𝛾<−𝑟^1/𝛾(𝑡)(𝑧^𝜎(𝑡))^{(𝛾+1)/𝛾}𝑤^𝜎(𝑡)^{(𝛾+1)/𝛾}, (3.54) it follows that 𝑤^Δ(𝑡)<−𝑄₁(𝑡)+𝑤^𝜎(𝑡)𝑧^Δ(𝑡)𝑧^𝜎(𝑡)−𝑧(𝑡)𝛾𝑟^{(1/𝛾)(𝑡)}(𝑧^𝜎(𝑡))^{(𝛾+1)/𝛾}𝑤^𝜎(𝑡)^{(𝛾+1)/𝛾}. (3.55) It is easy to see (3.55) is of the same form as (3.14). The following is similar to the proof of Theorem 3.1 and hence omitted.

For 𝛾∈(0,1), Theorem 3.2 also holds. Its proof is similar to those of Theorems  3.1^′ and 3.2.

Theorem  3.2^′. Assume that (ℎ₁)−(ℎ₃) hold and 0<𝛾<1. Furthermore, assume that there exist a positive rd-continuous Δ-differentiable function 𝑧(𝑡) and a function 𝐻∈ℋ_∗ such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐻(𝑡,𝑠)𝑄₁(𝑠)−𝐶^𝛾+1(𝑡,𝑠)𝑟(𝑠)(𝑧(𝜎(𝑠)))^𝛾+1𝐻^𝛾(𝑡,𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=∞, (3.56)then (1.3) is oscillatory.

Theorem  3.3^′. Assume that (ℎ₁)−(ℎ₃)hold and 0<𝛾<1. Then (1.3) is oscillatory if for all sufficiently large 𝑇∈𝕋,limsup_𝑡→∞^𝑡_𝑇𝑄(𝑠)𝑅^𝛾_𝑇(𝑠)Δ𝑠=∞. (3.57)

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive. Then, by (ℎ₁)−(ℎ₃) there exists 𝑇≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0, and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8), 𝜙 is defined as in Theorem 3.3. Similar to the proof of Theorem 3.3, we get 𝜙0≥^𝛾(𝑡)^Δ𝜙^𝛾(𝑡)(𝑡)𝑥+𝑄^𝛿1(𝑡)𝜙(𝑡)^{(𝛼𝛽−𝛼𝛾)/(𝛽−𝛼)}⋅𝑥^𝛿2(𝑡)𝜙(𝑡)^{(𝛽𝛾−𝛽𝛼)/(𝛽−𝛼)}. (3.58) Using the Keller’s chain rule, 0<𝛾<1, and 𝜙^Δ(𝑡)<0, we get 𝜙^𝛾(𝑡)^Δ=𝛾¹₀𝜙(𝑡)+ℎ𝜇(𝑡)𝜙^Δ(𝑡)^𝛾−1𝑑ℎ𝜙^Δ(𝑡)≥𝛾¹₀(1−ℎ)𝜙(𝑡)+ℎ𝜙(𝑡)^𝛾−1𝑑ℎ𝜙^Δ(𝑡)=𝛾𝜙^𝛾−1(𝑡)𝜙^Δ(𝑡). (3.59) From 𝛿_1,2(𝑡)≥𝑡 and 𝑥^Δ(𝑡)>0, it follows that 0≥𝛾𝜙^Δ(𝑡)𝐿(𝑡)𝑥(𝑡)+𝑄𝜙(𝑡)^𝛾. (3.60) It is easy to see (3.31) is of the same form as (3.60). The following is similar to the proof of Theorem 3.3 and hence omitted.

Last, we give a theorem which holds for all 𝛾>0, a quotient of add positive integers.

Theorem 3.9. Assume that (ℎ₁)−(ℎ₃) hold. Furthermore, assume that there exists a positive rd-continuous Δ-differentiable function 𝑧(𝑡) such that for all sufficiently large 𝑇∈𝕋, limsup_𝑡→∞^𝑡_𝑇𝑄₁(𝑠)−𝑧^Δ𝑅^𝛾_𝑇(𝑠)Δ𝑠=∞, (3.61) then (1.3) is oscillatory.

Proof. Suppose to the contrary that 𝑦(𝑡) is a nonoscillatory solution of (1.3). Without loss of generality, we may assume that 𝑦(𝑡) is eventually positive. Then, by (ℎ₁)−(ℎ₃) there exists 𝑇≥𝑡₀ sufficiently large such that 𝑦(𝑡)>0, 𝑦(𝜏(𝑡))>0, 𝑦(𝛿_1,2(𝑡))>0, and Lemma 2.4 holds for 𝑡≥𝑇, where 𝑥(𝑡) is defined by (2.8). Define 𝑤(𝑡) as in (3.3). Then, 𝑤(𝑡)>0 and because (1/𝑥^𝛾)^Δ=−((𝑥^𝛾)^Δ/𝑥^𝛾(𝑥^𝛾)^𝜎)<0 we get 𝑤^Δ(𝑡)=𝑧(𝑡)⋅𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ1𝑥^𝛾(𝑡)+𝑧(𝑡)⋅𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎1𝑥^𝛾(𝑡)^Δ<𝑧^Δ(𝑡)⋅𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎+𝑧(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ1𝑥^𝛾(𝑡). (3.62) From (3.5), (3.10), and (3.13), we obtain 𝑤^Δ(𝑡)<𝑧^Δ(𝑡)⋅𝑟(𝑡)𝑥^Δ(𝑡)^𝛾+𝑧(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ1𝑥^𝛾(𝑡)<𝑧^Δ(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)^𝛾−𝑄₁(𝑡)<𝑧^Δ(𝑡)𝑅^𝛾_𝑇(𝑡)−𝑄₁(𝑡). (3.63) Integrating the above inequality from 𝑇 to 𝑡(≥𝑇), we get ^𝑡_𝑇𝑄₁(𝑠)−𝑧^Δ(𝑠)𝑅^𝛾_𝑇(𝑠)Δ𝑠<𝑤(𝑇)−𝑤(𝑡)<𝑤(𝑇). (3.64) Taking lim sup on both sides of the above inequality as 𝑡→∞, we obtain a contradiction to condition (3.61). The proof is complete.

4. Examples

In this section, we give two examples to illustrate our main results. To obtain the conditions for oscillation we will use the following facts: ^∞₁Δ𝑠𝑠^𝛾=∞if0≤𝛾≤1,^∞₁Δ𝑠𝑠^𝛾<∞,if𝛾>1. (4.1) We first give an example to show Theorems 3.1 and  3.1^′.

Example 4.1. Consider the equation 1(𝑡+𝜎(𝑡))^𝛾𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡))^Δ𝛾Δ+(𝜎(𝑡))^2𝛾1−𝑝𝛿₁(𝑡)^𝛼𝑡^2𝛾+1𝑦^𝛼𝛿₁(𝑡)+(𝜎(𝑡))^2𝛾1−𝑝𝛿₂(𝑡)^𝛽𝑡^2𝛾+1𝑦^𝛽𝛿₂(𝑡)=0,𝑡∈𝕋, (4.2) where 𝕋=[1,∞) is a time scale, 𝑝(𝑡) satisfies (ℎ₂), 𝜏(𝑡) and 𝛿_1,2(𝑡) satisfy (ℎ₁), 𝑟(𝑡)=1/(𝑡+𝜎(𝑡))^𝛾, and 𝛾 is a quotient of odd positive integers.
We choose 𝑞₁(𝑡)=(𝜎(𝑡))^2𝛾/((1−𝑝(𝛿₁(𝑡)))^𝛼𝑡^2𝛾+1), 𝑞₂(𝑡)=(𝜎(𝑡))^2𝛾/((1−𝑝(𝛿₂(𝑡)))^𝛽𝑡^2𝛾+1), and 𝑧=1, then 𝑧^Δ=0, ∫(𝑠+𝜎(𝑠))Δ𝑠=𝑡²+𝑐, and ∫^∞₁(Δ𝑡/𝑟^1/𝛾(𝑡))=∞. For any sufficiently large 𝑇∈𝕋 and 𝑠>𝑇, there exists a constant 𝑘>0 sufficiently large such that 𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)+𝜇(𝑠)^𝛾=𝑠²−𝑇²(1/(𝑠+𝜎(𝑠)))𝑠²−𝑇²(1/(𝑠+𝜎(𝑠)))+𝜎(𝑠)−𝑠^𝛾>𝑠²𝑘𝜎²(𝑠)^𝛾,limsup_𝑡→∞^𝑡_𝑇𝑄₁(𝑠)−1𝛾+1^𝛾+1𝑟(𝑠)𝑧^Δ(𝑠)^𝛾+1𝑧^𝛾(𝑠)Δ𝑠≥limsup_𝑡→∞𝑘^−𝛾𝑡_𝑇1𝑠Δ𝑠=∞. (4.3) Hence, by Theorems 3.1 and  3.1^′, (4.2) is oscillatory.

The second example illustrates Corollary 3.7.

Example 4.2. Consider the equation 𝑦(𝜏(𝑡𝑦(𝑡)+))𝛿⁻¹(𝑡)^ΔΔ+𝜎(𝑡)𝑡²𝑦^1/3(𝛿(𝑡+𝜎(𝑡)))𝑡²𝑦^5/3(𝛿(𝑡))=0,𝑡∈𝕋, (4.4) where 𝕋=[1,∞) is a time scale, 𝛾=1, 𝛼=1/3, 𝛽=5/3, 𝛿(𝑡), and 𝜏(𝑡) satisfy (ℎ₁), 𝛿(𝑡) has an inverse function 𝛿⁻¹(𝑡), and 𝑝(𝑡)=1/𝛿⁻¹(𝑡) satisfies (ℎ₂).
We choose 𝑞₁(𝑡)=𝑞₂(𝑡)=𝜎(𝑡)/𝑡² and 𝑧=1. For any sufficiently large 𝑇∈𝕋 and 𝑠>𝑇, there exists a constant 𝑘>0 sufficiently large such that 𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)(𝑠)=+𝜇𝑠−𝑇(𝑠)>𝑠𝑠−𝑇+𝜎−𝑠(𝑠),𝑘𝜎limsup_𝑡→∞^𝑡_𝑇𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)𝑅_𝑇(𝑠)𝑟^1/𝛾(𝑠)+𝜇(𝑠)^𝛾𝑄(𝑠)Δ𝑠≥limsup_𝑡→∞^𝑡_𝑇11𝑘𝑠1−𝑠²Δ𝑠=limsup_𝑡→∞1𝑘^𝑡_𝑇1𝑠−2𝑠²+1𝑠³Δ𝑠=∞. (4.5) Hence, by Corollary 3.7, (4.4) is oscillatory.

Acknowledgment

This project was supported by the NNSF of China (nos. 10971231, 11071238, and 11271379).