Interval Oscillation Criteria of Second Order Mixed Nonlinear Impulsive Differential Equations with Delay

Abstract

We study the following second order mixed nonlinear impulsive differential equations with delay (𝑟(𝑡)Φ_𝛼(𝑥^′(𝑡)))^′+𝑝₀(𝑡)Φ_𝛼(𝑥(𝑡))+∑^𝑛_𝑖=1𝑝_𝑖(𝑡)Φ_𝛽𝑖(𝑥(𝑡−𝜎))=𝑒(𝑡),𝑡≥𝑡₀,𝑡≠𝜏_𝑘,𝑥(𝜏⁺_𝑘)=𝑎_𝑘𝑥(𝜏_𝑘),𝑥′(𝜏⁺_𝑘𝑏)=_𝑘𝑥′(𝜏_𝑘),𝑘=1,2,…, where Φ_∗(𝑢)=|𝑢|^∗−1𝑢, 𝜎 is a nonnegative constant, {𝜏_𝑘} denotes the impulsive moments sequence, and 𝜏_𝑘+1−𝜏_𝑘>𝜎. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

1. Introduction

We consider the following second order impulsive differential equations with delay𝑟(𝑡)Φ_𝛼𝑥^′(𝑡)^′+𝑝₀(𝑡)Φ_𝛼(𝑥(𝑡))+^𝑛_𝑖=1𝑝_𝑖(𝑡)Φ_𝛽𝑖(𝑥(𝑡−𝜎))=𝑒(𝑡),𝑡≥𝑡₀,𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=𝑎_𝑘𝑥𝜏_𝑘,𝑥′𝜏⁺_𝑘=𝑏_𝑘𝑥′𝜏_𝑘,𝑘=1,2,…, (1.1) where Φ_∗(𝑢)=|𝑢|^∗−1𝑢, 𝜎 is a nonnegative constant, {𝜏_𝑘} denotes the impulsive moments sequence, and 𝜏_𝑘+1−𝜏_𝑘>𝜎, for all 𝑘∈ℕ.

Let 𝐽⊂ℝ be an interval, and we definePLC(𝐽,ℝ)∶=𝑦∶𝐽⟶ℝ∣𝑦iscontinuouseverywhereexcepteach𝜏_𝑘atwhich𝑦𝜏⁺_𝑘and𝑦𝜏⁻_𝑘existand𝑦𝜏⁻_𝑘=𝑦𝜏_𝑘,𝑘∈ℕ. (1.2) For given 𝑡₀ and 𝜙∈PLC([𝑡₀−𝜎,𝑡₀],ℝ), we say 𝑥∈PLC([𝑡₀−𝜎,∞),ℝ) is a solution of (1.1) with initial value 𝜙 if 𝑥(𝑡) satisfies (1.1) for 𝑡≥𝑡₀ and 𝑥(𝑡)=𝜙(𝑡) for 𝑡∈[𝑡₀−𝜎,𝑡₀].

A solution of (1.1) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, this solution is said to be oscillatory.

Impulsive differential equation is an adequate mathematical apparatus for the simulation of processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth. Because it has more richer theory than its corresponding without impulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1, 2]).

In the last decades, there is constant interest in obtaining new sufficient conditions for oscillation or nonoscillation of the solutions of various impulsive differential equations, see, for example, [1–9] and the references cited therein.

In recent years, interval oscillation of impulsive differential equations was also arousing the interest of many researchers. In 2007, Özbekler and Zafer [10] investigated the following equations:𝑚(𝑡)𝜑_𝛼𝑦′′+𝑞(𝑡)𝜑_𝛽𝑦=𝑓(𝑡),𝑡≠𝜃_𝑖,Δ𝑚(𝑡)𝜑_𝛼𝑦′+𝑞_𝑖𝜑_𝛽𝑦=𝑓_𝑖,𝑡=𝜃_𝑖,(),𝑚(𝑡)(𝑡)(𝑡)𝜑𝑖∈ℕ𝑦′′+𝑠𝑦′+𝑞_𝛽𝑦=𝑓(𝑡),𝑡≠𝜃_𝑖,Δ𝑚(𝑡)𝑦′+𝑞_𝑖𝜑_𝛽𝑦=𝑓_𝑖,𝑡=𝜃_𝑖,𝛽≥1, (1.3)(1.4) where 𝜑_∗(𝑢)=|𝑢|^∗−1𝑢, 𝛽≥𝛼, {𝑞_𝑖} and {𝑓_𝑖} are sequences of real numbers. In 2009, they further gave a research [11] for equations of the form𝑟(𝑡)𝜑_𝛼𝑥′′+𝑝(𝑡)𝜑_𝛼𝑥′+𝑞(𝑡)𝜑_𝛽(𝑥)=𝑒(𝑡),𝑡≠𝜃_𝑖,Δ𝑟(𝑡)𝜑_𝛼𝑥′+𝑞_𝑖𝜑_𝛽(𝑥)=𝑒_𝑖,𝑡=𝜃_𝑖, (1.5) and obtained some interval oscillation results which improved and extended the earlier ones for the equations without impulses.

For the mixed type Emden-Fowler equations𝑟(𝑡)𝑥′(𝑡)′+𝑝(𝑡)𝑥(𝑡)+^𝑛_𝑖=1𝑝_𝑖(𝑡)|𝑥(𝑡)|^𝛼𝑖−1𝑥(𝑡)=𝑒(𝑡),𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=𝑎_𝑘𝑥𝜏_𝑘,𝑥′𝜏⁺_𝑘=𝑏_𝑘𝑥′𝜏_𝑘,𝑘∈ℕ, (1.6) Liu and Xu [12] established some interval oscillation results. Recently, Özbekler and Zafer [13] investigated the more general cases𝑟(𝑡)Φ_𝛼𝑥′(𝑡)′+𝑞(𝑡)Φ_𝛼(𝑥(𝑡))+^𝑛_𝑘=1𝑞_𝑘(𝑡)Φ_𝛽𝑘(𝑥(𝑡))=𝑒(𝑡),𝑡≠𝜃_𝑖,𝑥𝜃⁺_𝑖=𝑎_𝑖𝑥𝜃_𝑖,𝑥′𝜃⁺_𝑖=𝑏_𝑖𝑥′𝜃_𝑖, (1.7) where 𝛽₁>⋯>𝛽_𝑚>𝛼>𝛽_𝑚+1>⋯>𝛽_𝑛>0.

However, for the impulsive equations, almost all of interval oscillation results in the existing literature were established only for the case of “without delay.” In other words, for the case of “with delay” the study on the interval oscillation is very scarce. To the best of our knowledge, Huang and Feng [14] gave the first research in this direction recently. They considered second order delay differential equations with impulses𝑥^′′(𝑡)+𝑝(𝑡)𝑓(𝑥(𝑡−𝜏))=𝑒(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘𝑥𝑡,𝑘=1,2,…,_+𝑘=𝑎_𝑘𝑥𝑡_𝑘,𝑥′𝑡⁺_𝑘=𝑏_𝑘𝑥′𝑡_𝑘,𝑘=1,2,… (1.8) and established some interval oscillation criteria which developed some known results for the equations without delay or impulses [15–17].

Motivated mainly by [13, 14], in this paper, we study the interval oscillation of the delay impulsive (1.1). By using some inequalities, Riccati transformation and ℋ functions (introduced first by Philos [18]), we establish some interval oscillation criteria which generalize and improve some known results. Moreover, examples are considered to illustrate the main results.

2. Main Results

Throughout the paper, we always assume that the following conditions hold:(𝐴₁)the exponents satisfy that 𝛽₁>⋯>𝛽_𝑚>𝛼>𝛽_𝑚+1>⋯>𝛽_𝑛>0;(𝐴₂)𝑟(𝑡)∈𝐶([𝑡₀,∞),(0,∞)) is nondecreasing, 𝑒(𝑡),𝑝_𝑖(𝑡)∈PLC([𝑡₀,∞),ℝ),𝑖=0,1,…,𝑛;(𝐴₃){𝑎_𝑘} and {𝑏_𝑘} are real constant sequences such that 𝑏_𝑘≥𝑎_𝑘>0,𝑘∈ℕ.

It is clear that all solutions of (1.1) are oscillatory if there exists a subsequence {𝑘_𝑖} of {𝑘} such that 𝑎_𝑘𝑖≤0 for all 𝑖∈ℕ. So, we assume 𝑎_𝑘>0 for all 𝑘∈ℕ in condition (𝐴₃).

In this section, intervals [𝑐₁,𝑑₁] and [𝑐₂,𝑑₂] are considered to establish oscillation criteria. For convenience, we introduce the following notations (see [12]). Let𝑘(𝑠)=max𝑖∶𝑡₀<𝜏_𝑖<𝑠,𝑟_𝑗=max𝑟(𝑡)∶𝑡∈𝑐_𝑗,𝑑_𝑗,Ω𝑐_𝑗,𝑑_𝑗=𝑤_𝑗∈𝐶¹𝑐_𝑗,𝑑_𝑗∶𝑤_𝑗(𝑡)≢0,𝑤_𝑗𝑐_𝑗=𝑤_𝑗𝑑_𝑗=0,𝑗=1,2. (2.1)

For two constants 𝑐,𝑑∉{𝜏_𝑘} with 𝑐<𝑑 and 𝑘(𝑐)<𝑘(𝑑) and a function 𝜑∈𝐶([𝑐,𝑑],ℝ), we define an operator 𝑄∶𝐶([𝑐,𝑑],ℝ)→ℝ by𝑄^𝑑_𝑐𝜑=𝜑𝜏_𝑘(𝑐)+1𝑏^𝛼_𝑘(𝑐)+1−𝑎^𝛼_𝑘(𝑐)+1𝑎^𝛼_𝑘(𝑐)+1𝜏_𝑘(𝑐)+1−𝑐^𝛼+^𝑘(𝑑)_{𝑖=𝑘(𝑐)+2}𝜑𝜏_𝑖𝑏^𝛼_𝑖−𝑎^𝛼_𝑖𝑎^𝛼_𝑖𝜏_𝑖−𝜏_𝑖−1^𝛼, (2.2) where ∑^𝑡_𝑠=0 if 𝑠>𝑡.

In the discussion of the impulse moments of 𝑥(𝑡) and 𝑥(𝑡−𝜎), we need to consider the following cases for 𝑘(𝑐_𝑗)<𝑘(𝑑_𝑗)(S1)𝜏_{𝑘(𝑐𝑗)}+𝜎<𝑐_𝑗 and 𝜏_{𝑘(𝑑𝑗)}+𝜎>𝑑_𝑗,(S2)𝜏_{𝑘(𝑐𝑗)}+𝜎<𝑐_𝑗 and 𝜏_{𝑘(𝑑𝑗)}+𝜎<𝑑_𝑗,(S3)𝜏_{𝑘(𝑐𝑗)}+𝜎>𝑐_𝑗 and 𝜏_{𝑘(𝑑𝑗)}+𝜎>𝑑_𝑗,(S4)𝜏_{𝑘(𝑐𝑗)}+𝜎>𝑐_𝑗 and 𝜏_{𝑘(𝑑𝑗)}+𝜎<𝑑_𝑗,

and the cases for 𝑘(𝑐_𝑗)=𝑘(𝑑_𝑗)(S1)𝜏_{𝑘(𝑐𝑗)}+𝜎<𝑐_𝑗,(S2)𝑐_𝑗<𝜏_{𝑘(𝑐𝑗)}+𝜎<𝑑_𝑗,(S3)𝜏_{𝑘(𝑐𝑗)}+𝜎>𝑑_𝑗.

Combining (S∗) with (S∗), we can get 12 cases. In order to save space, throughout the paper, we study (1.1) under the case of combination of (S1) with (S1) only. The discussions for other cases are similar and omitted.

The following preparatory lemmas will be useful to prove our theorems. The first is derived from [19] and second is from [20].

Lemma 2.1. For any given n-tuple {𝛽₁,𝛽₂,…,𝛽_𝑛} satisfying (𝐴₁), then there exists an n-tuple {𝜂₁,𝜂₂,…,𝜂_𝑛} such that ^𝑛_𝑖=1𝛽_𝑖𝜂_𝑖=𝛼,^𝑛_𝑖=1𝜂_𝑖=𝜆,0<𝜂_𝑖<1, (2.3) where 𝜆∈(0,1].

Lemma 2.2. Suppose 𝑋 and 𝑌 are nonnegative, then 𝜆𝑋𝑌^𝜆−1−𝑋^𝜆≤(𝜆−1)𝑌^𝜆,𝜆>1, (2.4) where equality holds if and if 𝑋=𝑌.

Let 𝛼>0, 𝐵≥0, 𝐴>0 and 𝑦≥0. Put1𝜆=1+𝛼,𝑋=𝐴^{𝛼/(𝛼+1)}𝛼𝑦,𝑌=𝛼+1^𝛼𝐵^𝛼𝐴^{−𝛼2/(𝛼+1)}. (2.5) It follows from Lemma 2.2 that𝐵𝑦−𝐴𝑦^{(𝛼+1)/𝛼}≤𝛼^𝛼(𝛼+1)^𝛼+1𝐵^𝛼+1𝐴^𝛼. (2.6)

Theorem 2.3. Assume that for any 𝑇≥𝑡₀, there exist 𝑐_𝑗,𝑑_𝑗∉{𝜏_𝑘},𝑗=1,2, such that 𝑇<𝑐₁<𝑑₁≤𝑐₂<𝑑₂ and for 𝑗=1,2𝑝_𝑖(𝑡)≥0,𝑡∈𝑐_𝑗−𝜎,𝑑_𝑗⧵𝜏_𝑘(),𝑖=0,1,2,…,𝑛;−1^𝑗𝑒(𝑡)≥0,𝑡∈𝑐_𝑗−𝜎,𝑑_𝑗⧵𝜏_𝑘. (2.7) If there exist 𝑤_𝑗(𝑡)∈Ω(𝑐_𝑗,𝑑_𝑗) and 𝜌(𝑡)∈𝐶¹([𝑐_𝑗,𝑑_𝑗],(0,∞)) such that, for 𝑘(𝑐_𝑗)<𝑘(𝑑_𝑗), 𝑗=1,2, ^{𝜏𝑗𝑘(𝑐)+1}_𝑐𝑗𝑊_𝑗(𝑡)𝑡−𝜏_{𝑘(𝑐𝑗)}−𝜎^𝛼𝑡−𝜏_{𝑘(𝑐𝑗)}^𝛼+𝑑𝑡_{𝑘(𝑑𝑗)−1𝑖=𝑘(𝑐𝑗)+1𝜏𝑖𝜏+𝜎𝑖}𝑊_𝑗(𝑡)𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑊_𝑗(𝑡)𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝑑𝑗𝜏𝑗)𝑘(𝑑}𝑊_𝑗(𝑡)𝑡−𝜏_{𝑘(𝑑𝑗)}^𝛼𝑏^𝛼_{𝑘(𝑑𝑗)}𝑡+𝜎−𝜏_{𝑘(𝑑𝑗)}^𝛼+𝑑𝑡_{𝑑𝑗𝑐𝑗}𝜌(𝑡)𝑝₀(𝑡)𝑤_𝑗(𝑡)^𝛼+1−𝑟(𝑡)𝑤^′_𝑗(𝑡+𝜌)^′(𝑡)𝑤_𝑗(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1𝑑𝑡≥𝜌_𝑗𝑟_𝑗𝑄^𝑑𝑗_𝑐𝑗𝑤_𝑗^𝛼+1, (2.8) where 𝜌_𝑗 is maximum value of 𝜌(𝑡) on [𝑐_𝑗,𝑑_𝑗] and, for 𝑘(𝑐_𝑗)=𝑘(𝑑_𝑗), 𝑗=1,2, ^𝑑𝑖_𝑐𝑖𝑊_𝑗(𝑡)𝑡−𝑐_𝑖^𝛼𝑡−𝑐_𝑖+𝜎^𝛼+𝜌(𝑡)𝑝₀(𝑡)𝑤_𝑗(𝑡)^𝛼+1−𝑟(𝑡)𝑤^′_𝑗(𝑡+𝜌)^′(𝑡)𝑤_𝑗(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1𝑑𝑡≥0, (2.9) where 𝑊_𝑗(𝑡)=𝜂^−𝜂0₀|𝑒(𝑡)|^𝜂0∏^𝑛_𝑖=1𝜂^−𝜂𝑖_𝑖(𝑝_𝑖(𝑡))^𝜂𝑖|𝑤_𝑗(𝑡)|^𝛼+1 with 𝜂₀=1−∑^𝑛_𝑖=1𝜂_𝑖 and 𝜂₁,𝜂₂,…,𝜂_𝑛 are positive constants satisfying conditions of Lemma 2.1, then (1.1) is oscillatory.

Proof. Assume, to the contrary, that 𝑥(𝑡) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that 𝑥(𝑡)>0 and 𝑥(𝑡−𝜎)>0 for 𝑡≥𝑡₀. In this case the interval of 𝑡 selected for the following discussion is [𝑐₁,𝑑₁]. Define 𝑢(𝑡)(𝑡)𝑟(𝑡)Φ=𝜌^𝛼𝑥′(𝑡)Φ_𝛼(𝑥(𝑡𝑐)),𝑡∈₁,𝑑₁. (2.10) It follows, for 𝑡≠𝜏_𝑘, that 𝑢′(𝑡)=−𝜌(𝑡)𝑝₀(𝑡)−𝜌(𝑡)^𝑛_𝑖=1𝑝_𝑖(𝑡)Φ_{𝛽𝑖−𝛼}(𝑥(+|𝑒(𝑡𝑡−𝜎)))|Φ_𝛼(𝑥(Φ𝑡−𝜎))^𝛼(𝑥(𝑡−𝜎))Φ_𝛼(𝑥(𝑡+𝜌))_′(𝑡)𝜌(𝑡)𝑢(𝑡)−𝛼𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(𝛼+1)/𝛼}. (2.11) Now, let 𝑣₀=𝜂₀⁻¹|𝑒(𝑡)|Φ_𝛼(𝑥(𝑡−𝜎)),𝑣_𝑖=𝜂_𝑖⁻¹𝑝_𝑖(𝑡)Φ_{𝛽𝑖−𝛼}(𝑥(𝑡−𝜎)),𝑖=1,2,…,𝑛, (2.12) where 𝜂₁,𝜂₂,…,𝜂_𝑛 are positive constants satisfying conditions of Lemma 2.1 and 𝜂₀=1−∑^𝑛_𝑖=1𝜂_𝑖. Employing in (2.11) the arithmetic-geometric mean inequality (see [20]) ^𝑛_𝑖=0𝜂_𝑖𝑣_𝑖≥^𝑛_𝑖=0𝑣^𝜂𝑖_𝑖 (2.13) and in view of (2.3), we have that 𝑢′(𝑡)≤−𝜌(𝑡)𝑝₀(𝑡)(𝑡)𝜓(𝑡)Φ−𝜌^𝛼(𝑥(𝑡−𝜎))Φ_𝛼(𝑥(𝑡−𝛼))𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(𝛼+1)/𝛼}+𝜌′(𝑡)𝜌(𝑡)𝑢(𝑡), (2.14) where 𝜓(𝑡)=𝜂^−𝜂0₀|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂^−𝜂𝑖_𝑖𝑝_𝑖(𝑡)^𝜂𝑖. (2.15)
First, we consider the case 𝑘(𝑐₁)<𝑘(𝑑₁).
In this case, we assume impulsive moments in [𝑐₁,𝑑₁] are 𝜏_{𝑘(𝑐1)+1},𝜏_{𝑘(𝑐1)+2},…,𝜏_𝑘(𝑑1). Choosing 𝑤₁(𝑡)∈Ω(𝑐₁,𝑑₁), multiplying both sides of (2.14) by |𝑤₁(𝑡)|^𝛼+1 and then integrating it from 𝑐₁ to 𝑑₁, we obtain ^{𝜏1𝑘(𝑐)+1}_𝑐1+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖+1_𝜏𝑖+^𝑑1_{𝜏1)𝑘(𝑑}𝑢^′(𝑡)𝑤₁(𝑡)^𝛼+1≤𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖+1_𝜏𝑖+^𝑑1_{𝜏1)𝑘(𝑑}𝜌^′(𝑡)𝜌(𝑡)𝑢(𝑡)−𝛼𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(1+𝛼)/𝛼}𝑤₁(𝑡)^𝛼+1−𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖_𝜏^+𝜎_𝑖+^𝜏𝑖+1_{𝜏𝑖+𝜎}+^𝑑1_{𝜏1)𝑘(𝑑}𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑊₁(𝑡)−𝑑𝑡_𝑑1𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1𝑑𝑡, (2.16) where 𝑊₁(𝑡)=𝜌(𝑡)𝜓(𝑡)|𝑤₁(𝑡)|^𝛼+1. Using the integration by parts formula in the left-hand side of above inequality and noting the condition 𝑤₁(𝑐₁)=𝑤₁(𝑑₁)=0, we obtain ^𝑘(𝑑1)_{𝑖=𝑘(𝑐1)+1}𝑤₁𝜏_𝑖^𝛼+1𝑢𝜏_𝑖−𝑢𝜏⁺_𝑖≤^{𝜏1𝑘(𝑐)+1}_𝑐1+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖+1_𝜏𝑖+^𝑑1_{𝜏1)𝑘(𝑑}×(𝛼+1)Φ_𝛼𝑤₁(𝑡)𝑤^′₁(𝑡)𝑢(𝑡)+𝜌^′(𝑡)𝜌(𝑡)𝑢(𝑡)𝑤₁(𝑡)^𝛼+1−𝛼𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(1+𝛼)/𝛼}𝑤₁(𝑡)^𝛼+1−𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖_𝜏^+𝜎_𝑖+^𝜏𝑖+1_{𝜏𝑖+𝜎}+^𝑑1_{𝜏1)𝑘(𝑑}𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑊₁(𝑡)−𝑑𝑡_𝑑1𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1≤𝑑𝑡,_{𝜏1𝑘(𝑐)+1𝑐1}+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖+1_𝜏𝑖+^𝑑1_{𝜏1)𝑘(𝑑}×(𝛼+1)𝑤₁(𝑡)^𝛼|𝑢(𝑡)|𝑤^′₁(𝑡+𝜌)^′(𝑡)𝑤₁(𝑡)()𝜌(𝑡)−𝛼𝛼+1𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(1+𝛼)/𝛼}𝑤₁(𝑡)^𝛼+1−𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖_𝜏^+𝜎_𝑖+^𝜏𝑖+1_{𝜏𝑖+𝜎}+^𝑑1_{𝜏1)𝑘(𝑑}𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑊₁(𝑡)−𝑑𝑡_𝑑1𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1𝑑𝑡. (2.17) Letting 𝑦=|𝑤₁(𝑡)|^𝛼|𝑢(𝑡)|, 𝐵=(𝛼+1)(|𝑤^′₁(𝑡)|+|𝜌′(𝑡)||𝑤₁(𝑡)|/(𝛼+1)|𝜌(𝑡)|), 𝐴=𝛼/(𝜌(𝑡)𝑟(𝑡))^1/𝛼 and using (2.6), we have for the integrand function in above inequality that (𝛼+1)𝑤₁(𝑡)^𝛼𝑤^′₁(𝑡𝑢(𝑡−𝛼)|)|𝑟^1/𝛼(𝑡)|𝑢(𝑡)|^{(1+𝛼)/𝛼}𝑤₁(𝑡)^𝛼+1≤𝜌(𝑡)𝑟(𝑡)𝑤^′₁(𝑡+𝜌)^′(𝑡)𝑤₁(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1. (2.18)
In view of the impulse condition in (1.1) and the definition of 𝑢 we have, for 𝑡=𝜏_𝑘, 𝑘=1,2,…, that 𝑢𝜏⁺_𝑘=𝑏^𝛼_𝑘𝑎^𝛼_𝑘𝑢𝜏_𝑘. (2.19) From (2.19), we have ^𝑘(𝑑1)_{𝑖=𝑘(𝑐1)+1}𝑤₁𝜏_𝑖^𝛼+1𝑢𝜏_𝑖−𝑢𝜏⁺_𝑖=^𝑘(𝑑1)_{𝑖=𝑘(𝑐1)+1}𝑏1−^𝛼𝑖𝑎^𝛼_𝑖𝑤₁𝜏_𝑖^𝛼+1𝑢𝜏_𝑖. (2.20) Therefore, we get ^𝑘(𝑑1)_{𝑖=𝑘(𝑐1)+1}𝑏1−^𝛼𝑖𝑎^𝛼_𝑖𝑤₁𝜏_𝑖^𝛼+1𝑢𝜏_𝑖≤^𝑑1_𝑐1𝜌(𝑡)𝑟(𝑡)𝑤^′₁(𝑡+𝜌)^′(𝑡)𝑤₁(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1−𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}+^{𝑘(𝑑1)−1}_{𝑐𝑖=𝑘1+1}^𝜏𝑖_𝜏^+𝜎_𝑖+^𝜏𝑖+1_{𝜏𝑖+𝜎}+^𝑑1_{𝜏1)𝑘(𝑑}𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑊₁(𝑡)𝑑𝑡−^𝑑1_𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1𝑑𝑡. (2.21) On the other hand, for 𝑡∈[𝑐₁,𝑑₁]⧵{𝜏_𝑖}, 𝑟(𝑡)Φ_𝛼(𝑥′(𝑡))^′=𝑒(𝑡)−𝑝₀(𝑡)Φ_𝛼(𝑥(𝑡))−^𝑛_𝑖=1𝑝_𝑖(𝑡)Φ_𝛽𝑖(𝑥(𝑡−𝜎))≤0. (2.22) Hence 𝑟(𝑡)Φ_𝛼(𝑥′(𝑡)) is nonincreasing on [𝑐₁,𝑑₁]⧵{𝜏_𝑘}.
Because there are different integration intervals in (2.21), we will estimate 𝑥(𝑡−𝜎)/𝑥(𝑡) in each interval of 𝑡 as follows.
Case 1. 𝑡∈(𝜏_𝑖,𝜏_𝑖+1]⊂[𝑐₁,𝑑₁], for 𝑖=𝑘(𝑐₁)+1,…,𝑘(𝑑₁)−1.
Subcase 1. If 𝜏_𝑖+𝜎<𝑡≤𝜏_𝑖+1, then (𝑡−𝜎,𝑡)⊂(𝜏_𝑖,𝜏_𝑖+1]. Thus there is no impulsive moment in (𝑡−𝜎,𝑡). For any 𝑠∈(𝑡−𝜎,𝑡), we have 𝑥(𝑠)−𝑥𝜏⁺_𝑖=𝑥′𝜉₁𝑠−𝜏_𝑖,𝜉₁∈𝜏_𝑖,𝑠. (2.23) Since 𝑥(𝜏⁺_𝑖)>0, 𝑟(𝑠) is nondecreasing, function Φ_𝛼(⋅) is an increasing function and 𝑟(𝑡)Φ_𝛼(𝑥′(𝑡)) is nonincreasing on (𝜏_𝑖,𝜏_𝑖+1), we have Φ_𝛼(𝑥(𝑠≥𝑟𝜉))¹𝑟(𝑠)Φ_𝛼(𝑥(𝑠>𝑟𝜉))¹𝑟(𝑠)Φ_𝛼𝑥′𝜉₁𝑠−𝜏_𝑖=𝑟𝜉₁Φ_𝛼𝑥^′𝜉₁𝑟(𝑠)𝑠−𝜏_𝑖^𝛼≥𝑟(𝑠)Φ_𝛼𝑥′(𝑠)𝑟(𝑠)𝑠−𝜏_𝑖^𝛼=Φ_𝛼𝑥^′(𝑠)𝑠−𝜏_𝑖,𝜉₁∈𝜏_𝑖,𝑠. (2.24) Therefore, 𝑥′(𝑠)𝑥(𝑠)<1𝑠−𝜏_𝑖. (2.25) Integrating both sides of the above inequality from 𝑡−𝜎 to 𝑡, we obtain 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_𝑖−𝜎𝑡−𝜏_𝑖,𝑡∈𝜏_𝑖+𝜎,𝜏_𝑖+1. (2.26)
Subcase 2. If 𝜏_𝑖<𝑡<𝜏_𝑖+𝜎, then 𝜏_𝑖−𝜎<𝑡−𝜎<𝜏_𝑖<𝑡<𝜏_𝑖+𝜎. There is an impulsive moment 𝜏_𝑖 in (𝑡−𝜎,𝑡). For any 𝑡∈(𝜏_𝑖,𝜏_𝑖+𝜎), we have 𝑥(𝑡)−𝑥𝜏⁺_𝑖=𝑥′𝜉₂𝑡−𝜏_𝑖,𝜉₂∈𝜏_𝑖,𝑡. (2.27) Using the impulsive condition of (1.1) and the monotone properties of 𝑟(𝑡), Φ_𝛼(⋅) and 𝑟(𝑡)Φ_𝛼(𝑥′(𝑡)), we get Φ_𝛼𝑥(𝑡)−𝑎_𝑖𝑥𝜏_𝑖=𝑟𝜉₂Φ_𝛼𝑥′𝜉₂𝑟𝜉₂𝑡−𝜏_𝑖^𝛼≤𝑟𝜏_𝑖Φ_𝛼𝑥′𝜏⁺_𝑖𝑟𝜉₂𝑡−𝜏_𝑖^𝛼=𝑟𝜏_𝑖Φ_𝛼𝑏_𝑖𝑥′𝜏_𝑖𝑡−𝜏_𝑖𝑟𝜉₂. (2.28) Since 𝑥(𝜏_𝑖)>0, we have Φ_𝛼𝑥(𝑡)𝑥𝜏_𝑖−𝑎_𝑖≤𝑟𝜏_𝑖𝑟𝜉₂Φ_𝛼𝑏_𝑖𝑥′𝜏_𝑖𝑥𝜏_𝑖𝑡−𝜏_𝑖. (2.29) In addition, 𝑥𝜏_𝑖>𝑥𝜏_𝑖−𝑥𝜏_𝑖−𝜎=𝑥′𝜉₃𝜎,𝜉₃∈𝜏_𝑖−𝜎,𝜏_𝑖. (2.30) Using the same analysis as (2.24) and (2.25), we have 𝑥′𝜏_𝑖𝑥𝜏_𝑖<1𝜎. (2.31) From (2.29) and (2.31) and note that the monotone properties of Φ_𝛼(⋅) and 𝑟(𝑡), we get Φ_𝛼𝑥(𝑡)𝑥𝜏_𝑖−𝑎_𝑖<𝑟𝜏_𝑖𝑟𝜉₂Φ_𝛼𝑏_𝑖𝜎𝑡−𝜏_𝑖≤Φ_𝛼𝑏_𝑖𝜎𝑡−𝜏_𝑖. (2.32) Then, 𝑥(𝑡)𝑥𝜏_𝑖<𝑎_𝑖+𝑏_𝑖𝜎𝑡−𝜏_𝑖. (2.33) In view of (𝐴₃), we have 𝑥𝜏_𝑖𝑥(𝑡)>𝜎𝜎𝑎_𝑖+𝑏_𝑖𝑡−𝜏_𝑖≥𝜎𝑏_𝑖𝑡+𝜎−𝜏_𝑖>0. (2.34) On the other hand, similar to the above analysis, we get 𝑥′(𝑠)𝑥(𝑠)<1𝑠−𝜏_𝑖𝜏+𝜎,𝑠∈_𝑖−𝜎,𝜏_𝑖. (2.35) Integrating (2.35) from 𝑡−𝜎 to 𝜏_𝑖, where 𝑡∈(𝜏_𝑖,𝜏_𝑖+𝜎), we have 𝑥(𝑡−𝜎)𝑥𝜏_𝑖>𝑡−𝜏_𝑖𝜎≥0. (2.36) From (2.34) and (2.36), we obtain 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_𝑖𝑏_𝑖𝑡+𝜎−𝜏_𝑖,𝑡∈𝜏_𝑖,𝜏_𝑖+𝜎. (2.37)
Case 2 (𝑡∈[𝑐₁,𝜏_{𝑘(𝑐1)+1}]). Since 𝜏_𝑘(𝑐1)+𝜎<𝑐₁, then 𝑡−𝜎∈[𝑐₁−𝜎,𝜏_{𝑘(𝑐1)+1}−𝜎]⊂(𝜏_𝑘(𝑐1),𝜏_{𝑘(𝑐1)+1}−𝜎]. So, there is no impulsive moment in (𝑡−𝜎,𝑡). Similar to (2.26) of Subcase 1, we have 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_𝑘(𝑐1)−𝜎𝑡−𝜏_𝑘(𝑐1),𝑡∈𝑐₁,𝜏_{𝑘(𝑐1)+1}. (2.38)
Case 3 (𝑡∈(𝜏_𝑘(𝑑1),𝑑₁]). Since 𝜏_𝑘(𝑑1)+𝜎>𝑑₁, then 𝑡−𝜎∈(𝜏_𝑘(𝑑1)−𝜎,𝑑₁−𝜎]⊂(𝜏_𝑘(𝑑1)−𝜎,𝜏_𝑘(𝑑1)). Hence, there is an impulsive moment 𝜏_𝑘(𝑑1) in (𝑡−𝜎,𝑡). Making a similar analysis of Subcase 2, we obtain 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_𝑘(𝑑1)𝑏_𝑘(𝑑1)𝑡+𝜎−𝜏_𝑘(𝑑1)≥0,𝑡∈𝜏_𝑘(𝑑1),𝑑₁. (2.39) From (2.21), (2.26), (2.37), (2.38), and (2.39) we get ^𝑘(𝑑1)_{𝑖=𝑘(𝑐1)+1}𝑏1−^𝛼𝑖𝑎^𝛼_𝑖𝑤₁𝜏_𝑖^𝛼+1𝑢𝜏_𝑖<^𝑑1_𝑐1𝜌(𝑡)𝑟(𝑡)𝑤^′₁(𝑡+𝜌)^′(𝑡)𝑤₁(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1𝑑𝑡−^{𝜏1𝑘(𝑐)+1}_𝑐1𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛼𝑡−𝜏_𝑘(𝑐1)^𝛼−𝑑𝑡_{𝑘(𝑑1)−1𝑖=𝑘(𝑐1)+1𝜏𝑖𝜏+𝜎𝑖}𝑊₁(𝑡)𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡−^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑊₁(𝑡)𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼−𝑑𝑡_{𝑑1𝜏1)𝑘(𝑑}𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑑1)^𝛼𝑏^𝛼_𝑘(𝑑1)𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛼𝑑𝑡−^𝑑1_𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1𝑑𝑡. (2.40)
On the other hand, for 𝑡∈(𝜏_𝑖−1,𝜏_𝑖]⊂[𝑐₁,𝑑₁], 𝑖=𝑘(𝑐₁)+2,…,𝑘(𝑑₁), we have 𝑥(𝑡)−𝑥𝜏_𝑖−1=𝑥′(𝜉)𝑡−𝜏_𝑖−1,𝜉∈𝜏_𝑖−1,𝑡. (2.41) In view of 𝑥(𝜏_𝑖−1)>0 and the monotone properties of Φ_𝛼(⋅), 𝑟(𝑡)Φ_𝛼(𝑥′(𝑡)) and 𝑟(𝑡), we obtain Φ_𝛼(𝑥(𝑡))>Φ_𝛼𝑥′(𝜉)Φ_𝛼𝑡−𝜏_𝑖−1≥𝑟(𝑡)𝑟(𝜉)Φ_𝛼𝑥′(𝑡)Φ_𝛼𝑡−𝜏_𝑖−1. (2.42) This is 𝜌(𝑡)𝑟(𝑡)Φ_𝛼𝑥′(𝑡)Φ_𝛼(𝑥(𝑡<𝜌))¹𝑟(𝜉)𝑡−𝜏_𝑖−1^𝛼. (2.43) Letting 𝑡→𝜏⁻_𝑖, we have 𝑢𝜏_𝑖(𝑡)𝑟𝜏=𝜌^𝑖Φ_𝛼𝑥′𝜏_𝑖Φ_𝛼𝑥𝜏_𝑖<𝜌₁𝑟₁𝜏_𝑖−𝜏_𝑖−1^𝛼,𝑖=𝑘𝑐₁+2,…,𝑘𝑑₁. (2.44) Using similar analysis on (𝑐₁,𝜏_{𝑘(𝑐1)+1}], we get 𝑢𝜏_{𝑘(𝑐1)+1}<𝜌₁𝑟₁𝜏_{𝑘(𝑐1)+1}−𝑐₁^𝛼. (2.45) Then from (2.44), (2.45), and (𝐴₃), we have ^𝑘(𝑑1)_{𝑖=𝑘(𝑐1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝑤₁𝜏_𝑖^𝛼+1𝑢𝜏_𝑖<𝜌₁𝑟₁𝑤₁𝜏_{𝑘(𝑐1)+1}^𝛼+1𝜃𝑐₁+^𝑘(𝑑1)_{𝑐𝑖=𝑘1+2}𝑤₁𝜏_𝑖^𝛼+1𝜁𝜏_𝑖=𝜌₁𝑟₁𝑄^𝑑1_𝑐1𝑤₁^𝛼+1, (2.46) where 𝜃(𝑐₁)=(𝑏^𝛼_{𝑘(𝑐1)+1}−𝑎^𝛼_{𝑘(𝑐1)+1})/𝑎^𝛼_{𝑘(𝑐1)+1}(𝜏_{𝑘(𝑐1)+1}−𝑐₁)^𝛼 and 𝜁(𝜏_𝑖)=(𝑏^𝛼_𝑖−𝑎^𝛼_𝑖)/𝑎^𝛼_𝑖(𝜏_𝑖−𝜏_𝑖−1)^𝛼.
From (2.40) and (2.46), we obtain ^{𝜏1𝑘(𝑐)+1}_𝑐1𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛼𝑡−𝜏_𝑘(𝑐1)^𝛼+𝑑𝑡_{𝑘(𝑑1)−1𝑖=𝑘(𝑐1)+1𝜏𝑖𝜏+𝜎𝑖}𝑊₁(𝑡)𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑊₁(𝑡)𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝑑1𝜏1)𝑘(𝑑}𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑑1)^𝛼𝑏^𝛼_𝑘(𝑑1)𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛼+𝑑𝑡_𝑑1𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1𝑑𝑡−^𝑑1_𝑐1𝜌(𝑡)𝑟(𝑡)𝑤^′₁(𝑡+𝜌)^′(𝑡)𝑤₁(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1𝑑𝑡<𝜌₁𝑟₁𝑄^𝑑1_𝑐1𝑤₁^𝛼+1. (2.47) This contradicts (2.8).
Next we consider the case 𝑘(𝑐₁)=𝑘(𝑑₁). From the condition (S1) we know that there is no impulsive moment in [𝑐₁,𝑑₁]. Multiplying both sides of (2.47) by |𝑤₁(𝑡)|^𝛼+1 and integrating it from 𝑐₁ to 𝑑₁, we obtain ^𝑑1_𝑐1𝑢^′(𝑡)𝑤₁(𝑡)^𝛼+1𝑑𝑡≤−^𝑑1_𝑐1𝛼𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(𝛼+1)/𝛼}𝑤₁(𝑡)^𝛼+1−𝑑𝑡_𝑑1𝑐1𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑊₁(𝑡)𝑑𝑡−^𝑑1_𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1𝑑𝑡. (2.48) Similar to the proof of (2.21), we have ^𝑑1_𝑐1𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑊₁(𝑡)+𝜌(𝑡)𝑝₀(𝑡)𝑤_𝑗(𝑡)^𝛼+1−𝜌(𝑡)𝑟(𝑡)𝑤^′_𝑗(𝑡+𝜌)^′(𝑡)𝑤_𝑗(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1𝑑𝑡≤0. (2.49) Using same way as Subcase 1, we get 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝑐₁𝑡−𝑐₁𝑐+𝜎,𝑡∈₁,𝑑₁. (2.50) From (2.49) and (2.50) we obtain ^𝑑1_𝑐1𝑊₁(𝑡)𝑡−𝑐₁^𝛼𝑡−𝑐₁+𝜎^𝛼+𝜌(𝑡)𝑝₀(𝑡)𝑤_𝑗(𝑡)^𝛼+1−𝜌(𝑡)𝑟(𝑡)𝑤^′_𝑗(𝑡+𝜌)^′(𝑡)𝑤_𝑗(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1𝑑𝑡<0. (2.51) This contradicts condition (2.9).
When 𝑥(𝑡)<0, we can choose interval [𝑐₂,𝑑₂] to study (1.1). The proof is similar and will be omitted. Therefore, we complete the proof.

Remark 2.4. In article [14], the authors obtained the following inequalities: 𝑥(𝑡)𝑥𝑡_𝑗>𝑎_𝑗𝑡_𝑗+𝜏−𝑡𝜏≥0,𝑡∈𝑡_𝑗,𝑡_𝑗+𝜏, (2.52) See [14, equation (2.9)], 𝑥(𝑡)−𝜏𝑥𝑡_𝑗>𝑡−𝑡_𝑗𝜏≥0,𝑡∈𝑡_𝑗,𝑡_𝑗+𝜏. (2.53) See [14, equation (2.10)].
Dividing [14, equation (2.10)] by [14, equation (2.9)], they obtained 𝑥(𝑡)−𝜏𝑥(𝑡)>𝑡−𝑡_𝑗𝑎_𝑗𝑡_𝑗𝑡+𝜏−𝑡≥0,𝑡∈_𝑗,𝑡_𝑗+𝜏. (2.54) See [14, equation (2.11)]
This is an error. Moreover, similar errors appeared many times in the later arguments, for example, in inequalities (2.15), (2.19), and (2.20) in [14]. Moreover, the above substitution can lead to some divergent integrals, for example, the integrals in (2.22), (2.24) in [14]. Therefore, the conditions of their Theorems 2.1–2.5 must be defective. In the proof of our Theorem 2.3, this error is remedied.

Remark 2.5. When 𝜎=0, that is, the delay disappears, (1.1) reduces to (1.7) studied by Özbekler and Zafer [13]. In this case, our result with 𝜌(𝑡)=1 is Theorem 2.1 of [13].

Remark 2.6. When 𝜎=0, that is, the delay disappears in (1.1) and 𝛼=1, our result reduces to Theorem 2.1 of [12].

Remark 2.7. When 𝑎_𝑘=𝑏_𝑘=1 for all 𝑘=1,2,… and 𝜎=0, that is, both impulses and delay disappear in (1.1), our result with 𝛼=1 and 𝜌(𝑡)=1 reduces to Theorem 1 of [21].

In the following we will establish a Kong-type interval oscillation criteria for (1.1) by the ideas of Philos [18] and Kong [22].

Let 𝐷={(𝑡,𝑠)∶𝑡₀≤𝑠≤𝑡}, 𝐻₁,𝐻₂∈𝐶¹(𝐷,ℝ), then a pair function 𝐻₁,𝐻₂ is said to belong to a function set ℋ, defined by (𝐻₁,𝐻₂)∈ℋ, if there exist ℎ₁,ℎ₂∈𝐿_loc(𝐷,ℝ) satisfying the following conditions:(𝐶₁)𝐻₁(𝑡,𝑡)=𝐻₂(𝑡,𝑡)=0,𝐻₁(𝑡,𝑠)>0,𝐻₂(𝑡,𝑠)>0for𝑡>𝑠;(𝐶₂)(𝜕/𝜕𝑡)𝐻₁(𝑡,𝑠)=ℎ₁(𝑡,𝑠)𝐻₁(𝑡,𝑠),(𝜕/𝜕𝑠)𝐻₂(𝑡,𝑠)=ℎ₂(𝑡,𝑠)𝐻₂(𝑡,𝑠).

We assume that there exist 𝑐_𝑗,𝑑_𝑗,𝛿_𝑗∉{𝜏_𝑘,𝑘=1,2,….}(𝑗=1,2) such that 𝑇<𝑐₁<𝛿₁<𝑑₁≤𝑐₂<𝛿₂<𝑑₂ for any 𝑇≥𝑡₀. Noticing whether or not there are impulsive moments of 𝑥(𝑡) in [𝑐_𝑗,𝛿_𝑗] and [𝛿_𝑗,𝑑_𝑗], we should consider the following four cases, namely, (S5) 𝑘(𝑐_𝑗)<𝑘(𝛿_𝑗)<𝑘(𝑑_𝑗); (S6) 𝑘(𝑐_𝑗)=𝑘(𝛿_𝑗)<𝑘(𝑑_𝑗); (S7) 𝑘(𝑐_𝑗)<𝑘(𝛿_𝑗)=𝑘(𝑑_𝑗) and (S8) 𝑘(𝑐_𝑗)=𝑘(𝛿_𝑗)=𝑘(𝑑_𝑗). Moreover, in the discussion of the impulse moments of 𝑥(𝑡−𝜎), it is necessary to consider the following two cases: (S5) 𝜏_{𝑘(𝛿𝑗)}+𝜎>𝛿_𝑗 and (S6) 𝜏_{𝑘(𝛿𝑗)}+𝜎≤𝛿_𝑗. In the following theorem, we only consider the case of combination of (S5) with (S5). For the other cases, similar conclusions can be given and the proofs will be omitted here.

For convenience in the expression below, we define, for 𝑗=1,2,Π_1,𝑗1=∶𝐻₁𝛿_𝑗,𝑐_𝑗×^{𝜏𝑗𝑘(𝑐)+1}_𝑐𝑗𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_{𝑘(𝑐𝑗)}−𝜎^𝛼𝑡−𝜏_{𝑘(𝑐𝑗)}^𝛼+𝑑𝑡_{𝑘(𝛿𝑗)−1𝑖=𝑘(𝑐𝑗)+1𝜏𝑖𝜏+𝜎𝑖}𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝛿𝑗𝜏𝑗)𝑘(𝛿}𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_{𝑘(𝛿𝑗)}^𝛼𝑏^𝛼_{𝑘(𝛿𝑗)}𝑡+𝜎−𝜏_{𝑘(𝛿𝑗)}^𝛼𝑑𝑡+^𝛿𝑗_𝑐𝑗𝜌(𝑡)𝑝₀(𝑡)𝐻₁𝑡,𝑐_𝑗−1𝑑𝑡(𝛼+1)^{𝛼+1𝛿𝑗}_𝑐𝑗𝜌(𝑡)𝑟(𝑡)𝐻₁𝑡,𝑐_𝑗ℎ₁𝑡,𝑐_𝑗+𝜌^′(𝑡)𝜌(𝑡)^𝛼+1,Π𝑑𝑡_2,𝑗1=∶𝐻₂𝑑_𝑗,𝛿_𝑗^{𝜏𝑗)𝑘(𝛿𝛿+𝜎}_𝑗𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_{𝑘(𝛿𝑗)}^𝛼𝑏^𝛼_{𝑘(𝛿𝑗)}𝑡+𝜎−𝜏_{𝑘(𝛿𝑗)}^𝛼+𝑑𝑡_{𝜏𝑗𝑘(𝛿)+1𝜏𝑗)𝑘(𝛿+𝜎}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_{𝑘(𝛿𝑗)}−𝜎^𝛼𝑡−𝜏_{𝑘(𝛿𝑗)}^𝛼+𝑑𝑡_{𝑘(𝑑𝑗)−1𝑖=𝑘(𝛿𝑗)+1𝜏𝑖𝜏+𝜎𝑖}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝑑𝑗𝜏𝑗)𝑘(𝑑}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_{𝑘(𝑑𝑗)}^𝛼𝑏^𝛼_{𝑘(𝑑𝑗)}𝑡+𝜎−𝜏_{𝑘(𝑑𝑗)}^𝛼−1𝑑𝑡(𝛼+1)^{𝛼+1𝑑𝑗}_𝛿𝑗𝜌(𝑡)𝑟(𝑡)𝐻₂𝑑_𝑗,𝑡ℎ₂𝑑_𝑗+𝜌,𝑡^′(𝑡)𝜌(𝑡)^𝛼+1+𝑑𝑡_{𝑑𝑗𝛿𝑗}𝜌(𝑡)𝑝₀(𝑡)𝐻₂𝑑_𝑗,𝑡𝑑𝑡, (2.55) where 𝐻₁(𝑡,𝑐_𝑗)=𝐻₁(𝑡,𝑐_𝑗)𝜓(𝑡), 𝐻₂(𝑑_𝑗,𝑡)=𝐻₂(𝑑_𝑗,𝑡)𝜓(𝑡) and 𝜓(𝑡)=𝜂^−𝜂0₀|𝑒(𝑡)|^𝜂0∏^𝑛_𝑖=1𝜂^−𝜂𝑖_𝑖(𝑝_𝑖(𝑡))^𝜂𝑖 with 𝜂₀=1−∑^𝑛_𝑖=1𝜂_𝑖 and 𝜂₁,𝜂₂,…,𝜂_𝑛 are positive constants satisfying conditions of Lemma 2.1.

Theorem 2.8. Assume (2.7) holds. If there exists a pair of (𝐻₁,𝐻₂)∈ℋ such that Π_1,𝑗+Π_2,𝑗>𝜌_𝑗𝑟_𝑗𝐻₁𝛿_𝑗,𝑐_𝑗𝑄^𝛿𝑗_𝑐𝑗𝐻₁⋅,𝑐_𝑗+𝜌_𝑗𝑟_𝑗𝐻₂𝑑_𝑗,𝛿_𝑗𝑄^𝑑𝑗_𝛿𝑗𝐻₂𝑑_𝑗,⋅,𝑗=1,2, (2.56) then (1.1) is oscillatory.

Proof. Assume, to the contrary, that 𝑥(𝑡) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that 𝑥(𝑡)>0 and 𝑥(𝑡−𝜎)>0 for 𝑡≥𝑡₀. In this case the interval of 𝑡 selected for the following discussion is [𝑐₁,𝑑₁]. Similar to the proof of Theorem 2.3, we can get (2.14) and (2.19). Multiplying both sides of (2.14) by 𝐻₁(𝑡,𝑐₁) and integrating it from 𝑐₁ to 𝛿₁, we have ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑢′(𝑡)𝑑𝑡≤^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝜌^′(𝑡)𝜌(𝑡)𝑢(𝑡)−𝛼𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(1+𝛼)/𝛼}−𝑑𝑡_𝛿1𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑑𝑡−^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝜌(𝑡)𝑝₀(𝑡)𝑑𝑡, (2.57) where 𝐻₁(𝑡,𝑐₁)=𝐻₁(𝑡,𝑐₁)𝜌(𝑡)𝜓(𝑡), 𝜓(𝑡)=𝜂^−𝜂0₀|𝑒(𝑡)|^𝜂0∏^𝑛_𝑖=1𝜂^−𝜂𝑖_𝑖(𝑝_𝑖(𝑡))^𝜂𝑖 with 𝜂₀=1−∑^𝑛_𝑖=1𝜂_𝑖 and 𝜂₁,𝜂₂,…,𝜂_𝑛 are positive constants satisfying conditions of Lemma 2.1.
Noticing impulsive moments 𝜏_{𝑘(𝑐1)+1},𝜏_{𝑘(𝑐1)+2},…,𝜏_𝑘(𝛿1) are in [𝑐₁,𝛿₁] and using the integration by parts formula on the left-hand side of above inequality, we obtain ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑢′(𝑡)𝑑𝑡=^{𝜏1𝑘(𝑐)+1}_𝑐1+^{𝜏1𝑘(𝑐)+2}_{𝜏1𝑘(𝑐)+1}+⋯+^𝛿1_{𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁(𝑡)=𝑑𝑢_{𝑘(𝛿1)𝑖=𝑘(𝑐1)+1}𝑏1−^𝛼𝑖𝑎^𝛼_𝑖𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖+𝐻₁𝛿₁,𝑐₁𝑢𝛿₁−^{𝜏1𝑘(𝑐)+1}_𝑐1+^{𝜏1𝑘(𝑐)+2}_{𝜏1𝑘(𝑐)+1}+⋯+^𝛿1_{𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁ℎ₁𝑡,𝑐₁𝑢(𝑡)𝑑𝑡. (2.58) Substituting (2.58) into (2.57), we obtain ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑑𝑡≤^𝑘(𝛿1)_{𝑖=𝑘(𝑐1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖−𝐻₁𝛿₁,𝑐₁𝑢𝛿₁+^{𝜏1𝑘(𝑐)+1}_𝑐1+^{𝜏1𝑘(𝑐)+2}_{𝜏1𝑘(𝑐)+1}+⋯+^𝛿1_{𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁×ℎ₁𝑡,𝑐₁+𝜌^′(𝑡)𝜌(𝑡)𝑢(𝑡−𝛼|)|𝜌(𝑡)𝑟(𝑡)^1/𝛼|𝑢(𝑡)|^{(𝛼+1)/𝛼}−𝑑𝑡_𝛿1𝑐1𝜌(𝑡)𝑝₀(𝑡)𝐻₁𝑡,𝑐₁𝑑𝑡. (2.59) Letting 𝐴=𝛼/(𝜌(𝑡)𝑟(𝑡))^1/𝛼, 𝐵=|ℎ₁(𝑡,𝑐₁)+𝜌′(𝑡)/𝜌(𝑡)|, 𝑦=|𝑢(𝑡)| and using (2.6) to the right-hand side of above inequality, we have ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)𝑑𝑡≤^𝑘(𝛿1)_{𝑖=𝑘(𝑐1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖−𝐻₁𝛿₁,𝑐₁𝑢𝛿₁+1(𝛼+1)^𝛼+1𝛿1_𝑐1𝜌(𝑡)𝑟(𝑡)𝐻₁𝑡,𝑐₁ℎ₁𝑡,𝑐₁+𝜌^′(𝑡)𝜌(𝑡)^𝛼+1−𝑑𝑡_𝛿1𝑐1𝜌(𝑡)𝑝₀(𝑡)𝐻₁𝑡,𝑐₁𝑑𝑡. (2.60) Similar to the proof of Theorem 2.3, we need to divide the integration interval [𝑐₁,𝛿₁] into several subintervals for estimating the function 𝑥(𝑡−𝜎)/𝑥(𝑡). Using the methods of (2.26), (2.37), (2.38), and (2.39) we estimate the left-hand side of above inequality as follows: ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛼(𝑡−𝜎)𝑥^𝛼(𝑡)>𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛼𝑡−𝜏_𝑘(𝑐1)^𝛼+𝑑𝑡_{𝑘(𝛿1)−1𝑖=𝑘(𝑐1)+1𝜏𝑖𝜏+𝜎𝑖}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝛿1𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝛿1)^𝛼𝑏^𝛼_𝑘(𝛿1)𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛼𝑑𝑡. (2.61) From (2.60) and (2.61), we have ^{𝜏1𝑘(𝑐)+1}_𝑐1𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛼𝑡−𝜏_𝑘(𝑐1)^𝛼+𝑑𝑡_{𝑘(𝛿1)−1𝑖=𝑘(𝑐1)+1𝜏𝑖𝜏+𝜎𝑖}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝛿1𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝛿1)^𝛼𝑏^𝛼_𝑘(𝛿1)𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛼−1𝑑𝑡(𝛼+1)^𝛼+1𝛿1_𝑐1𝜌(𝑡)𝑟(𝑡)𝐻₁𝑡,𝑐₁ℎ₁𝑡,𝑐₁+𝜌^′(𝑡)𝜌(𝑡)^𝛼+1𝑑𝑡+^𝛿1_𝑐1𝜌(𝑡)𝑝₀(𝑡)𝐻₁𝑡,𝑐₁<𝑑𝑡_{𝑘(𝛿1)𝑖=𝑘(𝑐1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖−𝐻₁𝛿₁,𝑐₁𝑢𝛿₁. (2.62) On the other hand, multiplying both sides of (2.14) by 𝐻₂(𝑑₁,𝑡) and using similar analysis to the above, we can obtain ^{𝜏1)𝑘(𝛿}_𝛿^+𝜎₁𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝛿1)^𝛼𝑏^𝛼_𝑘(𝛿1)𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛼+^{𝜏1𝑘(𝛿)+1}_{𝜏1)𝑘(𝛿+𝜎}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝛿1)−𝜎^𝛼𝑡−𝜏_𝑘(𝛿1)^𝛼+𝑑𝑡_{𝑘(𝑑1)−1𝑖=𝑘(𝛿1)+1𝜏𝑖𝜏+𝜎𝑖}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝑑1𝜏1)𝑘(𝑑}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝑑1)^𝛼𝑏^𝛼_𝑘(𝑑1)𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛼𝑑𝑡+^𝑑1_𝛿1𝜌(𝑡)𝑝₀(𝑡)𝐻₂𝑑₁−1,𝑡𝑑𝑡(𝛼+1)^𝛼+1𝑑1_𝛿1𝜌(𝑡)𝑟(𝑡)𝐻₂𝑑₁,𝑡ℎ₂𝑑₁+𝜌,𝑡^′(𝑡)𝜌(𝑡)^𝛼+1<𝑑𝑡_{𝑘(𝑑1)𝑖=𝑘(𝛿1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₂𝑑₁,𝜏_𝑖𝑢𝜏_𝑖+𝐻₂𝑑₁,𝛿₁𝑢𝛿₁. (2.63) Dividing (2.62) and (2.63) by 𝐻₁(𝛿₁,𝑐₁), and 𝐻₂(𝑑₁,𝛿₁) respectively, and adding them, we get Π_1,1+Π_2,1<1𝐻₁𝛿₁,𝑐₁^𝑘(𝛿1)_{𝑖=𝑘(𝑐1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖+1𝐻₂𝑑₁,𝛿₁^𝑘(𝑑1)_{𝑖=𝑘(𝛿1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₂𝑑₁,𝜏_𝑖𝑢𝜏_𝑖. (2.64) Using the same methods as (2.46), we have ^𝑘(𝛿1)_{𝑖=𝑘(𝑐1)+1}𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖≤𝜌₁𝑟₁𝑄^𝛿1_𝑐1𝐻₁⋅,𝑐₁,_{𝑘(𝑑1)𝑖=𝑘(𝛿1)+1}𝑟_𝑖𝑏^𝛼_𝑖𝑎^𝛼_𝑖−1𝐻₂𝑑₁,𝜏_𝑖𝑢𝜏_𝑖≤𝜌₁𝑟₁𝑄^𝑑1_𝛿1𝐻₂𝑑₁,⋅. (2.65) From (2.64), (2.65), we can obtain a contradiction to the condition (2.56).
When 𝑥(𝑡)<0, we choose interval [𝑐₂,𝑑₂] to study (1.1). The proof is similar and will be omitted. Therefore, we complete the proof.

Remark 2.9. When 𝜎=0, that is, the delay disappears and 𝛼=1 in (1.1), our result Theorem 2.8 reduces to Theorem 2.2 of [12].

3. Examples

In this section, we give two examples to illustrate the effectiveness and nonemptiness of our results.

Example 3.1. Consider the following delay differential equation with impulse: 𝑥^′′(𝑡)+𝜇₁𝑥𝜋𝑡−12^3/2𝑥𝜋𝑡−12+𝜇₂𝑥𝜋𝑡−12^−1/2𝑥𝜋𝑡−()12=−sin2𝑡,𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=𝑎_𝑘𝑥𝜏_𝑘,𝑥′𝜏⁺_𝑘=𝑏_𝑘𝑥′𝜏_𝑘,𝑡=𝜏_𝑘, (3.1) where 𝜏_𝑘∶𝜏_𝑛,1=2𝑛𝜋+5𝜋/18,𝜏_𝑛,2=2𝑛𝜋+11𝜋/18,𝑛∈ℕ and 𝜇₁,𝜇₂ are positive constants.

For any 𝑇>0, we can choose large 𝑛₀ such that 𝑇<𝑐₁=2𝑛𝜋+𝜋/6,𝑑₁=2𝑛𝜋+𝜋/3,𝑐₂=2𝑛𝜋+𝜋/2,𝑑₂=2𝑛+2𝜋/3,𝑛=𝑛₀,𝑛₀+1,…. There are impulsive moments 𝜏_𝑛,1 in [𝑐₁,𝑑₁] and 𝜏_𝑛,2 in [𝑐₂,𝑑₂]. From 𝜏_𝑛,2−𝜏_𝑛,1=𝜋/3>𝜋/12 and 𝜏_𝑛+1,1−𝜏_𝑛,2=5𝜋/3>𝜋/12 for all 𝑛>𝑛₀, we know that condition 𝜏_𝑘+1−𝜏_𝑘>𝜎 is satisfied. Moreover, we also see the conditions (S1) and (2.7) are satisfied.

We can choose 𝜂₀=𝜂₁=𝜂₂=1/3 such that Lemma 2.1 holds. Let 𝑤₁(𝑡)=𝑤₂(𝑡)=sin(6𝑡) and 𝜌(𝑡)=1. It is easy to verify that 𝑊₁(𝑡)=3(𝜇₁𝜇₂)^1/3|sin(2𝑡)|^1/3sin²(6𝑡). By a simple calculation, the left side of (2.8) is the following:^{𝜏1𝑘(𝑐)+1}_𝑐1𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛼𝑡−𝜏_𝑘(𝑐1)^𝛼+𝑑𝑡_{𝑘(𝑑1)−1𝑖=𝑘(𝑐1)+1𝜏𝑖𝜏+𝜎𝑖}𝑊₁(𝑡)𝑡−𝜏_𝑖^𝛼𝑏^𝛼_𝑖𝑡+𝜎−𝜏_𝑖^𝛼𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑊₁(𝑡)𝑡−𝜏_𝑖−𝜎^𝛼𝑡−𝜏_𝑖^𝛼+𝑑𝑡_{𝑑1𝜏1)𝑘(𝑑}𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑑1)^𝛼𝑏^𝛼_𝑘𝑑1𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛼𝑑𝑡+^𝑑1_𝑐1𝜌(𝑡)𝑝₀(𝑡)𝑤₁(𝑡)^𝛼+1−𝑑𝑡_𝑑1𝑐1𝜌(𝑡)𝑟(𝑡)𝑤^′₁(𝑡+𝜌)^′(𝑡)𝑤₁(𝑡)(𝛼+1)𝜌(𝑡)^𝛼+1=𝑑𝑡_{2𝑛𝜋+5𝜋/182𝑛𝜋+𝜋/6}𝑊₁(𝑡)𝑡−2(𝑛−1)𝜋−11𝜋/18−𝜋/12()+𝑡−2𝑛−1𝜋−11𝜋/18𝑑𝑡_{2𝑛𝜋+𝜋/32𝑛𝜋+5𝜋/18}𝑊₁(𝑡)𝑡−2𝑛𝜋−5𝜋/18𝑏_𝑛,1(𝑡−2𝑛𝜋−5𝜋/18+𝜋/12)𝑑𝑡−^{2𝑛𝜋+𝜋/3}_{2𝑛𝜋+𝜋/6}36cos²=6𝑡𝑑𝑡_{5𝜋/18𝜋/6}𝑊₁(𝑡)𝑡+47𝜋/36𝑡+25𝜋/18𝑑𝑡+^𝜋/3_5𝜋/18𝑊₁(𝑡)𝑡−5𝜋/18𝑏_𝑛,1(𝑡−7𝜋/36)𝑑𝑡−3𝜋≈3√³𝜇₁𝜇₂0.199+0.715𝑏_𝑛,1−3𝜋. (3.2) On the other hand, we have𝑄^𝑑1_𝑐1𝑤²₁=27𝑏4𝜋^𝑛,1−𝑎_𝑛,1𝑎_𝑛,1. (3.3) Thus if3√³𝜇₁𝜇₂0.199+0.715𝑏_𝑛,1≥3𝜋+27𝑏4𝜋^𝑛,1−𝑎_𝑛,1𝑎_𝑛,1, (3.4) the condition (2.8) is satisfied in [𝑐₁,𝑑₁]. Similarly, we can show that for 𝑡∈[𝑐₂,𝑑₂] the condition (2.8) is satisfied if3√³𝜇₁𝜇₂0.057+0.003𝑏_𝑛,2≥3𝜋+27𝑏4𝜋^𝑛,2−𝑎_𝑛,2𝑎_𝑛,2. (3.5) Hence, by Theorem 2.3, (3.1) is oscillatory, if (3.4) and (3.5) hold. Particularly, let 𝑎_𝑘=𝑏_𝑘, for all 𝑘∈ℕ, condition (3.4) and (3.5) become√³𝜇₁𝜇₂0.199+0.715𝑏_𝑛,1≥𝜋,₃√𝜇₁𝜇₂0.057+0.003𝑏_𝑛,2≥𝜋. (3.6)

Example 3.2. Consider the following equation: 𝑥^′′(𝑡)+𝜇₁𝑝₁(𝑡𝑥2)𝑡−3^3/2𝑥2𝑡−3+𝜇₂𝑝₂(𝑡𝑥2)𝑡−3^−1/2𝑥2𝑡−3=𝑒(𝑡),𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=𝑎_𝑘𝑥𝜏_𝑘,𝑥^′𝜏⁺_𝑘=𝑏_𝑘𝑥^′𝜏_𝑘,𝑘=1,2,…, (3.7) where 𝜇₁,𝜇₂ are positive constants; 𝜏_𝑘∶𝜏_𝑛,1=9𝑛+3/2, 𝜏_𝑛,2=9𝑛+5/2, 𝜏_𝑛,3=9𝑛+15/2, 𝜏_𝑛,4=9𝑛+17/2(𝑛=0,1,2,…) and 𝜏_𝑘+1−𝜏_𝑘>𝜎=2/3. In addition, let 𝑝₁(𝑡)=𝑝₂(𝑡)=(𝑡−9𝑛)³[],3,𝑡∈9𝑛,9𝑛+3₃[],(),𝑡∈9𝑛+3,9𝑛+69𝑛+9−𝑡³[],𝑒(𝑡)=(),𝑡∈9𝑛+6,9𝑛+9𝑡−9𝑛−3³,𝑡∈[9𝑛,9𝑛+9]. (3.8)

For any 𝑡₀>0, we choose 𝑛 large enough such that 𝑡₀<9𝑛, and let [𝑐₁,𝑑₁]=[9𝑛+1,9𝑛+3], [𝑐₂,𝑑₂]=[9𝑛+7,9𝑛+9], 𝛿₁=9𝑛+2 and 𝛿₂=9𝑛+8. It is easy to see that condition (2.7) in Theorem 2.8 is satisfied. Letting 𝐻₁(𝑡,𝑠)=𝐻₂(𝑡,𝑠)=(𝑡−𝑠)³, we get ℎ₁(𝑡,𝑠)=−ℎ₂(𝑡,𝑠)=3/(𝑡−𝑠). By simple calculation, we haveΠ_1,1=3√³𝜇₁𝜇₂^9𝑛+3/2_9𝑛+1(𝑡−9𝑛−1)³(9𝑛+3−𝑡)(𝑡−9𝑛)²𝑡−9𝑛−1/6+𝑡−9𝑛+1/2𝑑𝑡_{9𝑛+29𝑛+3/2}(𝑡−9𝑛−1)³(9𝑛+3−𝑡)(𝑡−9𝑛)²𝑡−9−3/2𝑏_𝑛,1()−9𝑡−9𝑛−5/6𝑑𝑡8=3√³𝜇₁𝜇₂₁^3/2𝑢²(𝑢−1)³(3−𝑢)(𝑢−1/6)𝑢+1/2𝑑𝑢+²_3/2𝑢²(𝑢−1)³(3−𝑢)(𝑢−3/2)𝑏_𝑛,1()−9𝑢−5/6𝑑𝑢8≈3√³𝜇₁𝜇₂0.32+0.290𝑏_𝑛,1−98,Π_2,1=3√³𝜇₁𝜇₂^9𝑛+13/6_9𝑛+2(9𝑛+3−𝑡)⁴(𝑡−9𝑛)²𝑡−9𝑛−3/2𝑏_𝑛,1()+𝑡−9𝑛+5/6𝑑𝑡_{9𝑛+5/29𝑛+13/6}(9𝑛+3−𝑡)⁴(𝑡−9𝑛)²𝑡−9𝑛−13/6+𝑡−9𝑛−3/2𝑑𝑡_{9𝑛+39𝑛+5/2}(9𝑛+3−𝑡)⁴(𝑡−9𝑛)²𝑡−9𝑛−5/2𝑏_𝑛,2()−9𝑡−9𝑛−11/6𝑑𝑡8=3√³𝜇₁𝜇₂₂^13/6𝑢²(3−𝑢)⁴(𝑢−3/2)𝑏_𝑛,1(𝑢+5/6)𝑑𝑡+^5/2_13/6𝑢²(3−𝑢)⁴(𝑢−13/6)+𝑢−3/2𝑑𝑡_35/2𝑢²(3−𝑢)⁴(𝑢−5/2)𝑏_𝑛,2()−9𝑢−11/6𝑑𝑡8≈3√³𝜇₁𝜇₂0.102𝑏_𝑛,1+0.056+0.005𝑏_𝑛,2−98. (3.9) Then the left-hand side of the inequality (2.56) isΠ_1,1+Π_2,1≈3√³𝜇₁𝜇₂0.376+0.392𝑏_𝑛,1+0.005𝑏_𝑛,2−94. (3.10)

Because 𝑟₁=𝑟₂=1, 𝜏_{𝑘(𝑐1)+1}=𝜏_𝑘(𝛿1)=𝜏_𝑛,1=9𝑛+3/2∈(𝑐₁,𝛿₁) and 𝜏_{𝑘(𝛿1)+1}=𝜏_𝑘(𝑑1)=𝜏_𝑛,2=9𝑛+5/2∈(𝛿₁,𝑑₁), it is easy to get that the right-hand side of the inequality (2.56) for 𝑗=1 is𝑟₁𝐻₁𝛿₁,𝑐₁𝑄^𝛿1_𝑐1𝐻₁⋅,𝑐₁+𝑟₁𝐻₂𝑑₁,𝛿₁𝑄^𝑑1_𝛿1𝐻₂𝑑₁=𝑏,⋅^𝑛,1−𝑎_𝑛,14𝑎_𝑛,1+𝑏_𝑛,2−𝑎_𝑛,24𝑎_𝑛,2. (3.11) Thus (2.56) is satisfied with 𝑗=1 if3√³𝜇₁𝜇₂0.376+0.392𝑏_𝑛,1+0.005𝑏_𝑛,2>94+𝑏_𝑛,1−𝑎_𝑛,14𝑎_𝑛,1+𝑏_𝑛,2−𝑎_𝑛,24𝑎_𝑛,2. (3.12)

When 𝑗=2, with the same argument as above we get that the left-hand side of inequality (2.56) isΠ_1,2+Π_2,2=3√³𝜇₁𝜇₂₇^15/2(𝑢−9)²(𝑢−3)(𝑢−7)³(𝑢−19/6)+𝑢−5/2𝑑𝑢_815/2(𝑢−9)²(𝑢−3)(𝑢−7)³(𝑢−15/2)𝑏_𝑛,3()+𝑢−41/6𝑑𝑢_849/6(𝑢−3)(9−𝑢)⁵(𝑢−15/2)𝑏_𝑛,3(𝑢−41/6)𝑑𝑢+^17/2_49/6(𝑢−3)(9−𝑢)⁵(𝑢−19/6)+𝑢−5/2𝑑𝑢_917/2(𝑢−3)(9−𝑢)⁵(𝑢−17/2)𝑏_𝑛,4()−9𝑢−47/6𝑑𝑢4≈3√³𝜇₁𝜇₂0.400+0.724𝑏_𝑛,3+0.001𝑏_𝑛,4−94, (3.13) and the right-hand side of the inequality (2.56) is𝑟₂𝐻₂𝛿₂,𝑐₂𝑄^𝛿2_𝑐2𝐻₁⋅,𝑐₂+𝑟₂𝐻₂𝑑₂,𝛿₂𝑄^𝑑2_𝛿2𝐻₂𝑑₂=𝑏,⋅^𝑛,3−𝑎_𝑛,34𝑎_𝑛,3+𝑏_𝑛,4−𝑎_𝑛,44𝑎_𝑛,4. (3.14) Therefore, (2.51) is satisfied with 𝑗=2 if3√³𝜇₁𝜇₂0.400+0.724𝑏_𝑛,3+0.001𝑏_𝑛,4>94+𝑏_𝑛,3−𝑎_𝑛,34𝑎_𝑛,3+𝑏_𝑛,4−𝑎_𝑛,44𝑎_𝑛,4. (3.15) Hence, by Theorem 2.8, (3.7) is oscillatory if3√³𝜇₁𝜇₂0.376+0.392𝑏_𝑛,1+0.005𝑏_𝑛,2>94+𝑏_𝑛,1−𝑎_𝑛,14𝑎_𝑛,1+𝑏_𝑛,2−𝑎_𝑛,24𝑎_𝑛,2,3√³𝜇₁𝜇₂0.400+0.724𝑏_𝑛,3+0.001𝑏_𝑛,4>94+𝑏_𝑛,3−𝑎_𝑛,34𝑎_𝑛,3+𝑏_𝑛,4−𝑎_𝑛,44𝑎_𝑛,4. (3.16) Particularly, when 𝑎_𝑘=𝑏_𝑘, for all 𝑘∈ℕ, condition (3.16) becomes3√³𝜇₁𝜇₂0.376+0.392𝑏_𝑛,1+0.005𝑏_𝑛,2>94,3√³𝜇₁𝜇₂0.400+0.724𝑏_𝑛,3+0.001𝑏_𝑛,4>94. (3.17)

Acknowledgments

The authors thank the anonymous reviewers for their detailed and insightful comments and suggestions for improvement of the paper. They were supported by the NNSF of China (11161018), the NSF of Guangdong Province (10452408801004217).