Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

Abstract

We study the following second-order super-half-linear impulsive differential equations with delay [𝑟(𝑡)𝜑_𝛾(𝑥′(𝑡))]′+𝑝(𝑡)𝜑_𝛾(𝑥(𝑡−𝜎))+𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))=𝑒(𝑡), 𝑡≠𝜏_𝑘, 𝑥(𝑡⁺)=𝑎_𝑘𝑥(𝑡),𝑥′(𝑡⁺)=𝑏_𝑘𝑥′(𝑡), 𝑡=𝜏_𝑘, where 𝑡≥𝑡₀∈ℝ, 𝜑_∗(𝑢)=|𝑢|^∗−1𝑢, 𝜎 is a nonnegative constant, {𝜏_𝑘} denotes the impulsive moments sequence with 𝜏₁<𝜏₂<⋯<𝜏_𝑘<⋯, lim_𝑘→∞𝜏_𝑘=∞, and 𝜏_𝑘+1−𝜏_𝑘>𝜎. By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

1. Introduction

We consider the following second-order super-half-linear impulsive differential equations with delay 𝑟(𝑡)𝜑_𝛾𝑥^′(𝑡)^′+𝑝(𝑡)𝜑_𝛾(𝑥(𝑡−𝜎))+𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))=𝑒(𝑡),𝑡≠𝜏_𝑘,𝑥𝑡⁺=𝑎_𝑘𝑥(𝑡),𝑥^′𝑡⁺=𝑏_𝑘𝑥^′(𝑡),𝑡=𝜏_𝑘,𝑘=1,2,…, (1.1) where 𝑡≥𝑡₀∈ℝ, 𝜑_∗(𝑢)=|𝑢|^∗−1𝑢, 𝜎 is a nonnegative constant, {𝜏_𝑘} denotes the impulsive moments sequence with 𝜏₁<𝜏₂<⋯<𝜏_𝑘<⋯, lim_𝑘→∞𝜏_𝑘=∞, and 𝜏_𝑘+1−𝜏_𝑘>𝜎.

Let 𝐽⊂ℝ be an interval, we define 𝑃𝐿𝐶(𝐽,ℝ)∶=𝑦∶𝐽→ℝ𝑦iscontinuouseverywhereexcepteach𝜏_𝑘𝑦𝜏atwhich_+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 nonimpulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1, 2]).

In the last decades, there has been an increasing interest in obtaining sufficient conditions for oscillation and/or nonoscillation of different classes impulsive differential equations with delay (constant or variable), 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, see [10–15].

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 [16] gave the first research in this direction recently. They considered the second-order delay differential equations with impulses 𝑥^′′(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,…, (1.3) and established some interval oscillation criteria which developed some known results for the equation without delay or impulses [13, 17, 18].

Motivated mainly by [16], in this paper, we study the interval oscillation of the delay impulsive equation (1.1). By using classical inequalities, Riccati transformation, and two classes of functions (introduced first by Philos [19]), we establish some interval oscillation criteria which generalize and improve some known results of [13, 16–18]. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

2. Main Results

Throughout the paper, we always assume that the following conditions hold: (𝐴₁)𝑟(𝑡)∈𝐶([𝑡₀,∞),(0,∞)) and is nondecreasing, 𝑝(𝑡),𝑞(𝑡),𝑒(𝑡)∈𝑃𝐿𝐶([𝑡₀,∞),ℝ); (𝐴₂)𝛾 is a quotient of odd positive integers 𝑎_𝑘, 𝑏_𝑘 are real constants satisfying 𝑏_𝑘≥𝑎_𝑘>0, 𝑘=1,2,…; (𝐴₃)𝑓∈𝐶(ℝ,ℝ), 𝑥𝑓(𝑥)>0 and there exist some positive constants 𝜂 and 𝛼 such that 𝑓(𝑥)/𝜑_𝛼(𝑥)>𝜂 for all 𝑥≠0 with 𝛼≥𝛾.

We introduce the following notations at intervals [𝑐₁,𝑑₁] and [𝑐₂,𝑑₂]𝑘(𝑠)∶=max𝑖∶𝑡₀<𝜏_𝑖<𝑠,𝑀_𝑗∶=max𝑟(𝑡)∶𝑡∈𝑐_𝑗,𝑑_𝑗Ω,𝑗=1,2,_𝑗𝑐_𝑗,𝑑_𝑗∶=𝑤∈𝐶¹𝑐_𝑗,𝑑_𝑗∣𝑤(𝑡)≢0,𝑤𝑐_𝑗=𝑤𝑑_𝑗=0,𝑗=1,2. (2.1)

For two constants 𝑐,𝑑∉{𝜏_𝑘} with 𝑐<𝑑, 𝑘(𝑐)<𝑘(𝑑) and a function 𝜑∈𝐶([𝑐,𝑑],ℝ), we define an operator 𝑄∶𝐶([𝑐,𝑑],ℝ)→ℝ by 𝑄^𝑑_𝑐𝜑=𝜑𝜏_𝑘(𝑐)+1𝑏^𝛾_𝑘(𝑐)+1−𝑎^𝛾_𝑘(𝑐)+1𝑎^𝛾_𝑘(𝑐)+1𝜏_𝑘(𝑐)+1−𝑐^𝛾+^𝑘(𝑑)_{(𝑐)𝑖=𝑘+2}𝜑𝜏_𝑖𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝜏_𝑖−𝜏_𝑖−1^𝛾, (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 [20] and the second from [21].

Lemma 2.1. Let 𝜆 and 𝛿 be positive real numbers with 𝜆>𝛿. Then 𝐴𝑋^𝜆−𝛿+𝐵𝑋^−𝛿≥𝜆𝛿^−𝛿/𝜆(𝜆−𝛿)^{(𝛿/𝜆)−1}𝐴^𝛿/𝜆𝐵^{1−𝛿/𝜆}, (2.3) for all 𝐴,𝐵≥0 and 𝑋>0.

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 𝛾>0, 𝐵≥0, 𝐴>0 and 𝑦≥0. Put 1𝜆=1+𝛾,𝑋=𝐴^{𝛾/(𝛾+1)}𝛾𝑦,𝑌=𝛾+1^𝛾𝐵^𝛾𝐴^{−𝛾2/(𝛾+1)}. (2.5) It follows from Lemma 2.2 that 𝐵𝑦−𝐴𝑦^{(𝛾+1)/𝛾}≤𝛾^𝛾𝛾+1^𝛾+1𝐵^𝛾+1𝐴^𝛾. (2.6)

Lemma 2.3. Assume that for any 𝑇≥𝑡₀, there exists 𝑐_𝑗,𝑑_𝑗∉{𝜏_𝑘}, 𝑗=1,2, such that 𝑇<𝑐₁<𝑑₁≤𝑐₂<𝑑₂ and 𝑝(𝑡),𝑞(𝑡)≥0,𝑡∈𝑐₁−𝜎,𝑑₁∪𝑐₂−𝜎,𝑑₂⧵𝜏_𝑘,𝑒(𝑡)≤0,𝑡∈𝑐₁−𝜎,𝑑₁⧵𝜏_𝑘,𝑒(𝑡)≥0,𝑡∈𝑐₂−𝜎,𝑑₂⧵𝜏_𝑘. (2.7) If 𝑥(𝑡) is a nonoscillatory solution of (1.1), then there exist the following estimations of 𝑥(𝑡−𝜎)/𝑥(𝑡): (a)for𝑡∈𝜏_𝑖+𝜎,𝜏_𝑖+1,𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_𝑖−𝜎𝑡−𝜏_𝑖,(b)for𝑡∈𝜏_𝑖,𝜏_𝑖,𝑥()+𝜎𝑡−𝜎𝑥(𝑡)>𝑡−𝜏_𝑖𝑏_𝑖𝑡+𝜎−𝜏_𝑖,(c)for𝑡∈𝑐_𝑗,𝜏_{𝑘(𝑐𝑗)+1},𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_{𝑘(𝑐𝑗)}−𝜎𝑡−𝜏_{𝑘(𝑐𝑗)},(d)for𝑡∈𝜏_{𝑘(𝑑𝑗)},𝑑_𝑗,𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_{𝑘(𝑑𝑗)}𝑏_{𝑘(𝑑𝑗)}𝑡+𝜎−𝜏_{𝑘(𝑑𝑗)}, (2.8) where 𝑖=𝑘(𝑐_𝑗),…,𝑘(𝑑_𝑗)−1, 𝑗=1,2.

Proof. Without loss of generality, we assume that 𝑥(𝑡)>0 and 𝑥(𝑡−𝜎)>0 for 𝑡≥𝑡₀. In this case the selected interval of 𝑡 is [𝑐₁,𝑑₁]. From (1.1) and (2.7), we obtain 𝑟(𝑡)𝜑_𝛾𝑥^′(𝑡)^′=𝑒(𝑡)−𝑝(𝑡)𝜑_𝛾(𝑥(𝑡−𝜎))−𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))≤0. (2.9) Hence 𝑟(𝑡)𝜑_𝛾(𝑥′(𝑡)) is nonincreasing on the interval [𝑐₁,𝑑₁]⧵{𝜏_𝑘}.
Case (a) (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. (2.10) Since 𝑥(𝜏⁺_𝑖)>0, the function 𝜑_𝛾(⋅) is an increasing function and 𝑟(𝑠)𝜑_𝛾(𝑥′(𝑠)) is non-increasing on (𝜏_𝑖,𝜏_𝑖+1), we have 𝜑_𝛾(𝑥(𝑠))>𝜑_𝛾𝑥^′𝜉₁𝑠−𝜏_𝑖=𝑟𝜉₁𝜑_𝛾𝑥^′𝜉₁𝑟𝜉₁𝑠−𝜏_𝑖^𝛾≥𝑟(𝑠)𝜑_𝛾𝑥^′(𝑠)𝑟𝜉₁𝑠−𝜏_𝑖^𝛾,𝜉₁∈𝜏_𝑖,𝑠. (2.11) From (2.11) and the conditions 𝑟(𝑠)>0 and 𝑟(𝑠) is nondecreasing, we have 𝜑_𝛾(𝑥(𝑠))>𝜑_𝛾𝑥^′(𝑠)𝑠−𝜏_𝑖^𝛾=𝜑_𝛾𝑥^′(𝑠)𝑠−𝜏_𝑖,𝜉₁∈𝜏_𝑖,𝑠. (2.12) Thus 𝑥^′(𝑠)𝑥(𝑠)<1𝑠−𝜏_𝑖. (2.13) Integrating both sides of 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. (2.14)
Case (b) (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. (2.15) Using the impulsive condition of (1.1) and the monotone properties of 𝑟(𝑡), 𝜑_𝛾(⋅) and 𝑟(𝑡)𝜑_𝛾(𝑥′(𝑡)), we get 𝜑_𝛾𝑥(𝑡)−𝑎_𝑖𝑥𝜏_𝑖=𝑟𝜉₂𝜑_𝛾𝑥^′𝜉₂𝑟𝜉₂𝑡−𝜏_𝑖^𝛾≤𝑟𝜏⁺_𝑖𝜑_𝛾𝑥^′𝜏⁺_𝑖𝑟𝜉₂𝑡−𝜏_𝑖^𝛾=𝑟𝜏_𝑖𝜑_𝛾𝑏_𝑖𝑥^′𝜏_𝑖𝑡−𝜏_𝑖𝑟𝜉₂. (2.16) Since 𝑥(𝜏_𝑖)>0, we have 𝜑_𝛾𝑥(𝑡)𝑥𝜏_𝑖−𝑎_𝑖≤𝑟𝜏_𝑖𝑟𝜉₂𝜑_𝛾𝑏_𝑖𝑥^′𝜏_𝑖𝑥𝜏_𝑖𝑡−𝜏_𝑖. (2.17) In addition, 𝑥𝜏_𝑖>𝑥𝜏_𝑖−𝑥𝜏_𝑖−𝜎=𝑥^′𝜉₃𝜎,𝜉₃∈𝜏_𝑖−𝜎,𝜏_𝑖. (2.18) Similar to the analysis of (2.11)–(2.14), we have 𝑥^′𝜏_𝑖𝑥𝜏_𝑖<1𝜎. (2.19) From (2.17) and (2.19) and note that the monotone properties of 𝜑_𝛾(⋅) and 𝑟(𝑡), we get 𝑥(𝑡)𝑥𝜏_𝑖<𝑎_𝑖+𝑏_𝑖𝜎𝑡−𝜏_𝑖. (2.20) In view of (𝐴₃), we have 𝑥𝜏_𝑖𝑥(𝑡)>𝜎𝜎𝑎_𝑖+𝑏_𝑖𝑡−𝜏_𝑖≥𝜎𝑏_𝑖𝑡+𝜎−𝜏_𝑖>0. (2.21) On the other hand, using similar analysis of (2.11)–(2.19), we get 𝑥′(𝑠)𝑥(𝑠)<1𝑠−𝜏_𝑖𝜏+𝜎,𝑠∈_𝑖−𝜎,𝜏_𝑖. (2.22) Integrating (2.22) from 𝑡−𝜎 to 𝜏_𝑖, where 𝑡∈(𝜏_𝑖,𝜏_𝑖+𝜎), we have 𝑥(𝑡−𝜎)𝑥𝜏_𝑖>𝑡−𝜏_𝑖𝜎≥0. (2.23) From (2.21) and (2.23), we obtain 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_𝑖𝑏_𝑖𝑡+𝜎−𝜏_𝑖,𝑡∈𝜏_𝑖,𝜏_𝑖+𝜎. (2.24)
Case (c) (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 (2.14) of Case (a), we have 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝜏_𝑘(𝑐1)−𝜎𝑡−𝜏_𝑘(𝑐1),𝑡∈𝑐₁,𝜏_{𝑘(𝑐1)+1}. (2.25)
Case (d) (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 Case (b), 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. (2.26)
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.

Theorem 2.4. Assume that for any 𝑇≥𝑡₀, there exists 𝑐_𝑗,𝑑_𝑗∉{𝜏_𝑘}, 𝑗=1,2, such that 𝑇<𝑐₁<𝑑₁≤𝑐₂<𝑑₂ and (2.7) holds. If there exists 𝑤_𝑗(𝑡)∈Ω_𝑗(𝑐_𝑗,𝑑_𝑗) (𝑗=1,2) such that, for 𝑘(𝑐_𝑗)<𝑘(𝑑_𝑗), ^{𝜏𝑗𝑘(𝑐)+1}_𝑐𝑗𝑊_𝑗(𝑡)𝑡−𝜏_{𝑘(𝑐𝑗)}−𝜎^𝛾𝑡−𝜏_{𝑘(𝑐𝑗)}^𝛾+𝑑𝑡_{𝑘(𝑑𝑗)−1𝑐𝑖=𝑘𝑗+1𝜏𝑖𝜏+𝜎𝑖}𝑊_𝑗(𝑡)𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑊_𝑗(𝑡)𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾+𝑑𝑡_{𝑑𝑗𝜏𝑗)𝑘(𝑑}𝑊_𝑗(𝑡)𝑡−𝜏_{𝑘(𝑑𝑗)}^𝛾𝑏^𝛾_𝑘𝑑𝑗𝑡+𝜎−𝜏_{𝑘(𝑑𝑗)}^𝛾𝑑𝑡−^𝑑𝑗_𝑐𝑗𝑟(𝑡)𝑤^′_𝑗(𝑡)^𝛾+1𝑑𝑡≥𝑀_𝑗𝑄^𝑑𝑗_𝑐𝑗𝑤_𝑗^𝛾+1, (2.27) and for 𝑘(𝑐_𝑗)=𝑘(𝑑_𝑗), ^𝑑𝑗_𝑐𝑗𝑊_𝑗(𝑡)𝑡−𝑐_𝑖^𝛾𝑡−𝑐_𝑖+𝜎^𝛾−𝑟(𝑡)𝑤^′_𝑗(𝑡)^𝛾+1𝑑𝑡≥0, (2.28) where 𝑊_𝑗(𝑡)=(𝑝(𝑡)+𝛼𝛾^−𝛾/𝛼(𝛼−𝛾)^{𝛾/𝛼−1}𝜂^𝛾/𝛼𝑞^𝛾/𝛼(𝑡)|𝑒(𝑡)|^{1−𝛾/𝛼})𝑤_𝑗^𝛾+1(𝑡), 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].
We define 𝑢(𝑡)(𝑡)𝜑=𝑟^𝛾𝑥^′(𝑡)𝑥^𝛾(𝑡),for𝑡∈𝑐₁,𝑑₁. (2.29)
Differentiating 𝑢(𝑡) and in view of (1.1) we obtain, for 𝑡≠𝑡_𝑘, 𝑢^′(𝑡)(𝑡)𝑥=−𝑝^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)(𝑡)𝑓(𝑥(−𝑞𝑡−𝜎))𝑥^𝛾(𝑡)−|𝑒(𝑡)|𝑥^𝛾(𝑡)−𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}≤−𝑝(𝑡)+𝜂𝑞(𝑡)𝑥^𝛼−𝛾(𝑡−𝜎)+|𝑒(𝑡)|𝑥^−𝛾(𝑥𝑡−𝜎)^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)−𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}. (2.30) Putting 𝐴=𝜂𝑞(𝑡), 𝐵=|𝑒(𝑡)|, 𝑋=𝑥(𝑡−𝜎), 𝜆=𝛼, and 𝛿=𝛾, by Lemma 2.1, we see that 𝑢^′(𝑡)≤−𝑝(𝑡)+𝛼𝛾^−𝛾/𝛼𝛼−𝛾^{(𝛾/𝛼)−1}𝜂𝑞(𝑡)^𝛾/𝛼|𝑒(𝑡)|^{1−(𝛾/𝛼)}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)−𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑥=−^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑅(𝑡)−𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}, (2.31) where 𝑅(𝑡)=𝑝(𝑡)+𝛼𝛾^−𝛾/𝛼(𝛼−𝛾)^{𝛾/𝛼−1}(𝜂𝑞(𝑡))^𝛾/𝛼(𝑡)|𝑒(𝑡)|^{1−𝛾/𝛼}.
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 a 𝑤₁(𝑡)∈Ω₁(𝑐₁,𝑑₁), multiplying both sides of (2.31) by 𝑤₁^𝛾+1(𝑡) and then integrating it from 𝑐₁ to 𝑑₁, we obtain ^{𝜏1𝑘(𝑐)+1}_𝑐1𝑢^′(𝑡)𝑤₁^𝛾+1(𝑡)𝑑𝑡+^{𝜏1𝑘(𝑐)+2}_{𝜏1𝑘(𝑐)+1}𝑢^′(𝑡)𝑤₁^𝛾+1(𝑡)𝑑𝑡+⋯+^𝑑1_{𝜏1)𝑘(𝑑}𝑢^′(𝑡)𝑤₁^𝛾+1(𝑡)𝑑𝑡≤−^{𝜏1𝑘(𝑐)+1}_𝑐1𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑤₁^𝛾+1(𝑡)𝑑𝑡−^{𝜏1𝑘(𝑐)+2}_{𝜏1𝑘(𝑐)+1}𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑤₁^𝛾+1(𝑡)𝑑𝑡−⋯−^𝑑1_{𝜏1)𝑘(𝑑}𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑤₁^𝛾+1(𝑡)𝑑𝑡−^{𝜏1𝑘(𝑐)+1}_𝑐1𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)−𝑑𝑡_{𝜏1𝑘(𝑐)+1𝜏+𝜎1𝑘(𝑐)+1}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡−^{𝜏1𝑘(𝑐)+2}_{𝜏1𝑘(𝑐)+1+𝜎}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡−⋯−^{𝜏1)𝑘(𝑑}_{𝜏1𝑘(𝑑)−1+𝜎}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡−^𝑑1_{𝜏1)𝑘(𝑑}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡, (2.32) where 𝑊₁(𝑡)=𝑅(𝑡)𝑤₁^𝛾+1(𝑡). Using the integration by parts on the left side of above inequality and noting the condition 𝑤₁(𝑐₁)=𝑤₁(𝑑₁)=0, we obtain ^𝑘(𝑑1)_{𝑐𝑖=𝑘1+1}𝑤₁^𝛾+1𝜏_𝑖𝑢𝜏_𝑖−𝑢𝜏⁺_𝑖≤^{𝜏1𝑘(𝑐)+1}_𝑐1𝛾+1𝑤^𝛾₁(𝑡)𝑤^′₁(𝑡𝑢(𝑡−𝛾)|)|𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑤₁^𝛾+1(𝑡+)𝑑𝑡_{𝑘(𝑑1)−1𝑐𝑖=𝑘1+1𝜏𝑖+1𝜏𝑖}𝛾+1𝑤^𝛾₁(𝑡)𝑤^′₁(𝑡𝑢(𝑡−𝛾)|)|𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑤₁^𝛾+1(𝑡+)𝑑𝑡_{𝑑1𝜏1)𝑘(𝑑}𝛾+1𝑤^𝛾₁(𝑡)𝑤^′₁(𝑡𝑢(𝑡−𝛾)|)|𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑤₁^𝛾+1(𝑡−)𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)−𝑑𝑡_{𝑘(𝑑1)−1𝑐𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)−𝑑𝑡_{𝑑1𝜏1)𝑘(𝑑}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡. (2.33) Letting 𝑦=|𝑤^𝛾₁(𝑡)||𝑢(𝑡)|, 𝐵=(𝛾+1)|𝑤₁′(𝑡)|, 𝐴=𝛾/𝑟^1/𝛾(𝑡) and using (2.6), for the integrand function in above inequality we have that 𝛾+1𝑤^𝛾₁(𝑡)𝑤^′₁(𝑡𝑢(𝑡−𝛾)|)|𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}𝑤₁^𝛾+1(𝑡)≤𝑟(𝑡)𝑤^′₁(𝑡)^𝛾+1. (2.34) Meanwhile, for 𝑡=𝜏_𝑘, 𝑘=1,2,…, we have 𝑢𝜏⁺_𝑘=𝑏_𝑘𝑎_𝑘^𝛾𝑢𝜏_𝑘. (2.35) Hence ^𝑘(𝑑1)_{𝑐𝑖=𝑘1+1}𝑤₁^𝛾+1𝜏_𝑖𝑢𝜏_𝑖−𝑢𝜏⁺_𝑖=^𝑘(𝑑1)_{𝑐𝑖=𝑘1+1}𝑎^𝛾_𝑖−𝑏^𝛾_𝑖𝑎^𝛾_𝑖𝑤₁^𝛾+1𝜏_𝑖𝑢𝜏_𝑖. (2.36) Therefore, we get ^𝑘(𝑑1)_{𝑐𝑖=𝑘1+1}𝑎^𝛾_𝑖−𝑏^𝛾_𝑖𝑎^𝛾_𝑖𝑤₁^𝛾+1𝜏_𝑖𝑢𝜏_𝑖≤^𝑑1_𝑐1𝑟(𝑡)𝑤^′₁(𝑡)^𝛾+1𝑑𝑡−^{𝜏1𝑘(𝑐)+1}_𝑐1𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)−𝑑𝑡_{𝑘(𝑑1)−1𝑐𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)−𝑑𝑡_{𝑑1𝜏1)𝑘(𝑑}𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡. (2.37) 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. (2.38) In view of 𝑥(𝜏_𝑖−1)>0, and note that the monotone properties of 𝜑_𝛾(⋅), 𝑟(𝑡)𝜑_𝛾(𝑥′(𝑡)), and 𝑟(𝑡), we obtain 𝜑_𝛾(𝑥(𝑡))>𝜑_𝛾𝑥^′(𝜉)𝜑_𝛾𝑡−𝜏_𝑖−1≥𝑟(𝑡)𝑟(𝜉)𝜑_𝛾𝑥^′(𝑡)𝜑_𝛾𝑡−𝜏_𝑖−1. (2.39) This is 𝑟(𝑡)𝜑_𝛾𝑥^′(𝑡)𝜑_𝛾(𝑥(𝑡<𝑟(𝜉)))𝑡−𝜏_𝑖−1^𝛾. (2.40) Let 𝑡→𝜏⁻_𝑖, it follows 𝑢𝜏_𝑖=𝑟𝜏_𝑖𝜑_𝛾𝑥^′𝜏_𝑖𝜑_𝛾𝑥𝜏_𝑖<𝑀₁𝜏_𝑖−𝜏_𝑖−1^𝛾,𝑖=𝑘𝑐₁+2,…,𝑘𝑑₁. (2.41) Similar analysis on (𝑐₁,𝜏_{𝑘(𝑐1)+1}], we can get 𝑢𝜏_{𝑘(𝑐1)+1}<𝑀₁𝜏_{𝑘(𝑐1)+1}−𝑐₁^𝛾. (2.42) Then from (2.41), (2.42), and (𝐴₂), we have ^𝑘(𝑑1)_{𝑐𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝑤₁^𝛾+1𝜏_𝑖𝑢𝜏_𝑖<𝑀₁𝑤₁^𝛾+1𝜏_{𝑘(𝑐1)+1}𝜃𝑐₁+^𝑘(𝑑1)_{𝑐𝑖=𝑘1+2}𝑤₁^𝛾+1𝜏_𝑖𝜁𝜏_𝑖=𝑀₁𝑄^𝑑1_𝑐1𝑤₁^𝛾+1, (2.43) where 𝜃(𝑐₁)=(𝑏^𝛾_{𝑘(𝑐1)+1}−𝑎^𝛾_{𝑘(𝑐1)+1})/(𝑎^𝛾_{𝑘(𝑐1)+1}(𝜏_{𝑘(𝑐1)+1}−𝑐₁)^𝛾) and 𝜁(𝜏_𝑖)=(𝑏^𝛾_𝑖−𝑎^𝛾_𝑖)/(𝑎^𝛾_𝑖(𝜏_𝑖−𝜏_𝑖−1)^𝛾). From (2.37) and (2.43) and applying Lemma 2.3, we obtain ^{𝜏1𝑘(𝑐)+1}_𝑐1𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛾𝑡−𝜏_𝑘(𝑐1)^𝛾+𝑑𝑡_{𝑘(𝑑1)−1𝑐𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝑊₁(𝑡)𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑊₁(𝑡)(𝑡−𝜏_𝑖−𝜎)^𝛾𝑡−𝜏_𝑖^𝛾𝑑𝑡+^𝑑1_{𝜏1)𝑘(𝑑}𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑑1)^𝛾𝑏^𝛾_𝑘𝑑1𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛾−𝑑𝑡_𝑑1𝑐1𝑟(𝑡)𝑤^′₁(𝑡)^𝛾+1𝑑𝑡<𝑀₁𝑄^𝑑1_𝑐1𝑤₁^𝛾+1. (2.44) This contradicts (2.27).
Next we consider the case 𝑘(𝑐₁)=𝑘(𝑑₁). By the condition (S1) we know there is no impulsive moment in [𝑐₁,𝑑₁]. Multiplying both sides of (2.31) by 𝑤₁^𝛾+1(𝑡) and integrating it from 𝑐₁ to 𝑑₁, we obtain ^𝑑1_𝑐1𝑢^′(𝑡)𝑤₁^𝛾+1(𝑡)𝑑𝑡≤−^𝑑1_𝑐1𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(𝛾+1)/𝛾}𝑤₁^𝛾+1(𝑡)𝑑𝑡−^𝑑1_𝑐1𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡. (2.45) Using the integration by parts on the left-hand side and noting the condition 𝑤₁(𝑐₁)=𝑤₁(𝑑₁)=0, we obtain 𝛾+1𝑤^𝑑1_𝑐1^𝛾₁(𝑡)𝑤^′₁(𝑡)𝑢(𝑡)−𝛾𝑟^1/𝛾(𝑡)𝑤₁^𝛾+1(𝑡)|𝑢(𝑡)|^{(𝛾+1)/𝛾}𝑑𝑡−^𝑑1_𝑐1𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡≥0. (2.46) It follows that 𝛾+1𝑤^𝑑1_𝑐1^𝛾₁(𝑡)𝑤^′₁(𝑡𝑢(𝑡−𝛾)|)|𝑟^1/𝛾(𝑡)𝑤₁^𝛾+1(𝑡)|𝑢(𝑡)|^{(𝛾+1)/𝛾}𝑑𝑡−^𝑑1_𝑐1𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡≥0. (2.47) Letting 𝐴=(𝛾/𝑟^1/𝛾(𝑡))𝑤₁^𝛾+1(𝑡), 𝐵=(𝛾+1)|𝑤^𝛾₁(𝑡)𝑤^′₁(𝑡)|, and 𝑦=|𝑢(𝑡)| and applying the inequality (2.6), we get ^𝑑1_𝑐1𝑟(𝑡)𝑤^′₁(𝑡)^𝛾+1−𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑊₁(𝑡)𝑑𝑡≥0. (2.48) Using same way as Case (a) we get 𝑥(𝑡−𝜎)𝑥(𝑡)>𝑡−𝑐₁𝑡−𝑐₁𝑐+𝜎,𝑡∈₁,𝑑₁. (2.49) From (2.48) and (2.49) we obtain ^𝑑1_𝑐1𝑊₁(𝑡)𝑡−𝑐₁^𝛾𝑡−𝑐₁+𝜎^𝛾−𝑟(𝑡)𝑤^′₁(𝑡)^𝛾+1𝑑𝑡<0. (2.50) This contradicts our assumption (2.28).
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 2.5. When 𝛾=1, 𝑝(𝑡)=0, 𝑓(𝑥)=|𝑥|^𝛼−1𝑥 and the delay term 𝜎=0, (1.1) reduces to that studied by Liu and Xu [13]. Therefore our Theorem 2.4 generalizes Theorem 2.1 of [13].

Remark 2.6. When 𝛾=𝛼=1, 𝑟(𝑡)=1 and 𝑝(𝑡)=0, (1.1) reduces to the (1.3) studied by Huang and Feng [16]. Therefore our Theorem 2.4 extends Theorem 2.1 of [16].

Remark 2.7. When 𝑎_𝑘=𝑏_𝑘=1, for all 𝑘=1,2,..., the impulses in (1.1) disappear, Theorem 2.4 reduces to the main results of [17, 18].

In the following we will establish a Kemenev type interval oscillation criteria for (1.1) by the ideas of Philos [19] 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 exists ℎ₁,ℎ₂∈𝐿_loc(𝐷,ℝ) satisfying the following conditions:(𝐴₄)𝐻₁(𝑡,𝑡)=𝐻₂(𝑡,𝑡)=0, 𝐻₁(𝑡,𝑠)>0, 𝐻₂(𝑡,𝑠)>0 for 𝑡>𝑠; (𝐴₅) (𝜕/𝜕𝑡)𝐻₁(𝑡,𝑠)=ℎ₁(𝑡,𝑠)𝐻₁(𝑡,𝑠), (𝜕/𝜕𝑠)𝐻₂(𝑡,𝑠)=ℎ₂(𝑡,𝑠)𝐻₂(𝑡,𝑠).

We assume there exist 𝑐_𝑗,𝑑_𝑗,𝛿_𝑗∉{𝜏_𝑘,𝑘=1,2,…} (𝑗=1,2) which satisfy 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, (S1)𝑘(𝑐_𝑗)<𝑘(𝛿_𝑗)<𝑘(𝑑_𝑗); (S2) 𝑘(𝑐_𝑗)=𝑘(𝛿_𝑗)<𝑘(𝑑_𝑗); (S3)𝑘(𝑐_𝑗)<𝑘(𝛿_𝑗)=𝑘(𝑑_𝑗); (S4) 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: (S1)𝜏_{𝑘(𝛿𝑗)}+𝜎>𝛿_𝑗;    (S2) 𝜏_{𝑘(𝛿𝑗)}+𝜎≤𝛿_𝑗.

In the following theorem, we only consider the case of combination of (S1) with (S1). For the other cases, similar conclusions can be given and their proofs will be omitted here.

For convenience in the expressing blow, we define Π_1,𝑗1=∶𝐻₁𝛿_𝑗,𝑐_𝑗^{𝜏𝑗𝑘(𝑐)+1𝑐}_𝑗𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_{𝑘(𝑐𝑗)}−𝜎^𝛾𝑡−𝜏_{𝑘(𝑐𝑗)}^𝛾+𝑑𝑡_{𝑘(𝛿𝑗)−1𝑐𝑖=𝑘𝑗+1𝜏𝑖𝜏+𝜎𝑖}𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾+𝑑𝑡_{𝛿𝑗𝜏𝑗)𝑘(𝛿}𝐻₁𝑡,𝑐_𝑗𝑡−𝜏_{𝑘(𝛿𝑗)}^𝛾𝑏^𝛾_𝑘𝛿𝑗𝑡+𝜎−𝜏_{𝑘(𝛿𝑗)}^𝛾−1𝑑𝑡𝛾+1^{𝛾+1𝛿𝑗}_𝑐𝑗𝑟(𝑡)𝐻_𝑗𝑡,𝑐_𝑗ℎ_𝑗𝑡,𝑐_𝑗^𝛾+1,Π𝑑𝑡_2,𝑗1=∶𝐻₂𝑑_𝑗,𝛿_𝑗^{𝜏𝑗)𝑘(𝛿𝛿+𝜎}_𝑗𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_{𝑘(𝛿𝑗)}^𝛾𝑏^𝛾_𝑘𝛿𝑗𝑡+𝜎−𝜏_{𝑘(𝛿𝑗)}^𝛾𝑑𝑡+^{𝜏𝑗𝑘(𝛿)+1}_{𝜏𝑗)𝑘(𝛿+𝜎}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_{𝑘(𝛿𝑗)}−𝜎^𝛾𝑡−𝜏_{𝑘(𝛿𝑗)}^𝛾+𝑑𝑡_{𝑘(𝑑𝑗)−1𝛿𝑖=𝑘𝑗+1𝜏𝑖𝜏+𝜎𝑖}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾+𝑑𝑡_{𝑑𝑗𝜏𝑗)𝑘(𝑑}𝐻₂𝑑_𝑗,𝑡𝑡−𝜏_{𝑘(𝑑𝑗)}^𝛾𝑏^𝛾_{𝑘(𝑑𝑗)}𝑡+𝜎−𝜏_{𝑘(𝑑𝑗)}^𝛾−1𝑑𝑡𝛾+1^{𝛾+1𝑑𝑗}_𝛿𝑗𝑟(𝑡)𝐻₂𝑑_𝑗,𝑡ℎ₂𝑑_𝑗,𝑡^𝛾+1𝑑𝑡, (2.51) where 𝐻₁(𝑡,𝑐_𝑗)=𝐻₁(𝑡,𝑐_𝑗)𝑅(𝑡), 𝐻₂(𝑑_𝑗,𝑡)=𝐻₂(𝑑_𝑗,𝑡)𝑅(𝑡)  (𝑗=1,2) and 𝑅(𝑡)=𝑝(𝑡)+𝛼𝛾^−𝛾/𝛼(𝛼−𝛾)^{𝛾/𝛼−1}𝜂^𝛾/𝛼𝑞^𝛾/𝛼(𝑡)|𝑒(𝑡)|^{1−𝛾/𝛼}.

Theorem 2.8. Assume that for any 𝑇≥𝑡₀, there exists 𝑐_𝑗,𝑑_𝑗∉{𝜏_𝑘}, 𝑗=1,2, such that 𝑇<𝑐₁<𝑑₁≤𝑐₂<𝑑₂ and if there exists a pair of (𝐻₁,𝐻₂)∈ℋ such that Π_1,𝑗+Π_2,𝑗>𝑀_𝑗𝐻₁𝛿_𝑗,𝑐_𝑗𝑄^𝛿𝑗_𝑐𝑗𝐻₁⋅,𝑐_𝑗+𝑀_𝑗𝐻₂𝑑_𝑗,𝛿_𝑗𝑄^𝑑𝑗_𝛿𝑗𝐻₂𝑑_𝑗,⋅,𝑗=1,2, (2.52) 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]. Using the same proof as in Theorem 2.4, we can get (2.31). Multiplying both sides of (2.31) by 𝐻₁(𝑡,𝑐₁) and integrating it from 𝑐₁ to 𝛿₁, we have ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑢^′(𝑡)𝑑𝑡≤−^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝛾𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(1+𝛾)/𝛾}−𝑑𝑡_𝛿1𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑑𝑡, (2.53) where 𝐻₁(𝑡,𝑐₁)=𝐻₁(𝑡,𝑐₁)(𝑝(𝑡)+𝛼𝛾^−𝛾/𝛼(𝛼−𝛾)^{𝛾/𝛼−1}𝜂^𝛾/𝛼𝑞^𝛾/𝛼(𝑡)|𝑒(𝑡)|^{1−𝛾/𝛼}). Noticing impulsive moments 𝜏_{𝑘(𝑐1)+1},𝜏_{𝑘(𝑐1)+2},…,𝜏_𝑘(𝛿1) are in [𝑐₁,𝛿₁] and using the integration by parts 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)𝑐𝑖=𝑘1+1}𝑎^𝛾_𝑖−𝑏^𝛾_𝑖𝑎^𝛾_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. (2.54) Substituting (2.54) into (2.53), we have ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑑𝑡≤^𝑘(𝛿1)_{𝑐𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖−𝐻₁𝛿₁,𝑐₁𝑢𝛿₁+^{𝜏1𝑘(𝑐)+1}_𝑐1+^{𝜏1𝑘(𝑐)+2}_{𝜏1𝑘(𝑐)+1}+⋯+^𝛿1_{𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁ℎ₁𝑡,𝑐₁𝑢(𝑡−𝛾|)|𝑟^1/𝛾(𝑡)|𝑢(𝑡)|^{(𝛾+1)/𝛾}𝑑𝑡. (2.55) Letting 𝐴=𝛾/𝑟^1/𝛾(𝑡), 𝐵=|ℎ₁(𝑡,𝑐₁)|, 𝑦=|𝑢(𝑡)| and using (2.6) to the right side of above inequality, we have ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)𝑑𝑡≤^𝑘(𝛿1)_{𝑐𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖−𝐻₁𝛿₁,𝑐₁𝑢𝛿₁+1𝛾+1^𝛾+1𝛿1_𝑐1𝑟(𝑡)𝐻₁𝑡,𝑐₁ℎ₁𝑡,𝑐₁^𝛾+1𝑑𝑡. (2.56) Because there are different integration intervals in (2.56), we need to divide the integration interval [𝑐₁,𝛿₁] into several subintervals for estimating the function 𝑥(𝑡−𝜎)/𝑥(𝑡). Using Lemma 2.2, (2.6), we get estimation for the left-hand side of above inequality as follows, ^𝛿1_𝑐1𝐻₁𝑡,𝑐₁𝑥^𝛾(𝑡−𝜎)𝑥^𝛾(𝑡)>𝑑𝑡_{𝜏1𝑘(𝑐)+1𝑐1}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛾𝑡−𝜏_𝑘(𝑐1)^𝛾+𝑑𝑡_{𝑘(𝛿1)−1𝑐𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾+𝑑𝑡_{𝛿1𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝛿1)^𝛾𝑏^𝛾_𝑘𝛿1𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛾𝑑𝑡. (2.57) From (2.56) and (2.57), we have ^{𝜏1𝑘(𝑐)+1}_𝑐1𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛾𝑡−𝜏_𝑘(𝑐1)^𝛾+𝑑𝑡_{𝑘(𝛿1)−1𝑐𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾𝑑𝑡+^𝛿1_{𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝛿1)^𝛾𝑏^𝛾_𝑘𝛿1𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛾1𝑑𝑡−𝛾+1^𝛾+1𝛿1_𝑐1𝑟(𝑡)𝐻₁𝑡,𝑐₁ℎ₁𝑡,𝑐₁^𝛾+1<𝑑𝑡_{𝑘(𝛿1)𝑐𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖−𝐻₁𝛿₁,𝑐₁𝑢𝛿₁. (2.58) Multiplying both sides of (2.31) by 𝐻₂(𝑑₁,𝑡) and using similar analysis to the above, we can obtain ^{𝜏1)𝑘(𝛿}_𝛿^+𝜎₁𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝛿1)^𝛾𝑏^𝛾_𝑘𝛿1𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛾+^{𝜏1𝑘(𝛿)+1}_{𝜏1)𝑘(𝛿+𝜎}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝛿1)−𝜎^𝛾𝑡−𝜏_𝑘(𝛿1)^𝛾+𝑑𝑡_{𝑘(𝑑1)−1𝛿𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾𝑑𝑡+^𝑑1_{𝜏1)𝑘(𝑑}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝑑1)^𝛾𝑏^𝛾_𝑘𝑑1𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛾1𝑑𝑡−𝛾+1^𝛾+1𝑑1_𝛿1𝑟(𝑡)𝐻₂𝑑₁,𝑡ℎ₂𝑑₁,𝑡^𝛾+1<𝑑𝑡_{𝑘(𝑑1)𝛿𝑖=𝑘1+1}𝑏_𝑖^𝛾−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₂𝑑₁,𝜏_𝑖𝑢𝜏_𝑖+𝐻₂𝑑₁,𝛿₁𝑢𝛿₁. (2.59) Dividing (2.58) and (2.59) by 𝐻₁(𝛿₁,𝑐₁) and 𝐻₂(𝑑₁,𝛿₁), respectively, and adding them, we get Π_1,1+Π_2,1<1𝐻₁𝛿₁,𝑐₁^𝑘(𝛿1)_{𝑐𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖+1𝐻₂𝑑₁,𝛿₁^𝑘(𝑑1)_{𝛿𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₂𝑑₁,𝜏_𝑖𝑢𝜏_𝑖. (2.60)
On the other hand, similar to (2.43), we have ^𝑘(𝛿1)_{𝑐𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₁𝜏_𝑖,𝑐₁𝑢𝜏_𝑖≤𝑀₁𝑄^𝛿1_𝑐1𝐻₁⋅,𝑐₁,_{𝑘(𝑑1)𝛿𝑖=𝑘1+1}𝑏^𝛾_𝑖−𝑎^𝛾_𝑖𝑎^𝛾_𝑖𝐻₂𝑑₁,𝜏_𝑖𝑢𝜏_𝑖≤𝑀₁𝑄^𝑑1_𝛿1𝐻₂𝑑₁,⋅. (2.61) From (2.60) and (2.61), we can obtain a contradiction to the condition (2.52).
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 2.9. Let 𝐻₁(𝑡,𝑠)=𝐻₂(𝑡,𝑠)=𝐻(𝑡,𝑠), ℎ₁(𝑡,𝑠)=2ℎ^∗₁(𝑡,𝑠)/𝐻(𝑡,𝑠) and ℎ₂(𝑡,𝑠)=−2ℎ^∗₂(𝑡,𝑠)/𝐻(𝑡,𝑠), the conditions (𝐴₄), (𝐴₅) can be changed into (𝐴₆)𝐻(𝑡,𝑡)=0, 𝐻(𝑡,𝑠)>0, for 𝑡>𝑠; (𝐴₇)(𝜕/𝜕𝑡)𝐻(𝑡,𝑠)=2ℎ^∗₁(𝑡,𝑠)𝐻(𝑡,𝑠), (𝜕/𝜕𝑠)𝐻(𝑡,𝑠)=−2ℎ^∗₂(𝑡,𝑠)𝐻(𝑡,𝑠). We know that (𝐴₆) and (𝐴₇) are the main assumptions used in [13, 16] to obtain Kemenev type oscillation criteria. Therefore, Theorem 2.8 is a generalization of Theorem 2.3 in [13] and Theorem 2.5 in [16].

3. Examples

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

Example 3.1. Consider the following equation 𝜑_𝛾𝑥^′(𝑡)′+𝜈₁sin𝑡𝜑_𝛾𝑥𝜋𝑡−12+𝜈²₂cos𝑡𝜑_𝛼𝑥𝜋𝑡−12=−cos2𝑡,𝑡≠𝜏_𝑘,𝑖,𝑥𝑡⁺=𝑎_𝑘𝑥(𝑡),𝑥^′𝑡⁺=𝑏_𝑘𝑥^′(𝑡),𝑡=𝜏_𝑘,𝑖, (3.1) where 𝜏_𝑘,𝑖=2𝑘𝜋+(𝑖−1)(𝜋/2)+(−1)^(𝑖−1)(𝜋/8), 𝑖=1,2,  𝑘=1,2,..., 𝑡≥0, 𝛾=3, 𝛼=6, 𝜈₁ and 𝜈₂ are positive constants.
For any 𝑇>0, we can choose large 𝑛₀∈ℕ such that 𝑇<𝑐₁𝜋=2𝑛𝜋+12,𝑑₁𝜋=2𝑛𝜋+4,𝑐₂𝜋=2𝑛𝜋+3,𝑑₂𝜋=2𝑛𝜋+2,𝑛=𝑛₀,𝑛₀+1,…. (3.2) There are impulsive moments 𝜏_𝑛,1=2𝑛𝜋+𝜋/8 in [𝑐₁,𝑑₁] and 𝜏_𝑛,2=2𝑛𝜋+3𝜋/8 in [𝑐₂,𝑑₂]. From 𝜏_𝑛,2−𝜏_𝑛,1=𝜋/4>𝜋/12 and 𝜏_𝑛+1,1−𝜏_𝑛,2=13𝜋/8>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.
Let 𝑤(𝑡)=sin12𝑡. It is easy to get that 𝑊₁(𝑡)=(𝜈₁sin𝑡+2𝜈₂√cos𝑡cos2𝑡)sin⁴12𝑡. In view of ∑^{𝑘(𝑑1)−1}_{𝑖=𝑘(𝑐1)+1}=0 as 𝑘(𝑐₁)+1>𝑘(𝑑₁)−1, by a simple calculation, the left side of (2.27) is the following ^{𝜏1𝑘(𝑐)+1}_𝑐1𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛾𝑡−𝜏_𝑘(𝑐1)^𝛾+𝑑𝑡_{𝑘(𝑑1)−1𝑐𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝑊₁(𝑡)𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝑊₁(𝑡)𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾𝑑𝑡+^𝑑1_{𝜏1)𝑘(𝑑}𝑊₁(𝑡)𝑡−𝜏_𝑘(𝑑1)^𝛾𝑏^𝛾_𝑘𝑑1𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛾𝑑𝑡−^𝑑1_𝑐1𝑟(𝑡)𝑤_1′(𝑡)^𝛾+1=𝑑𝑡_{2𝑛𝜋+𝜋/82𝑛𝜋+𝜋/12}𝑊₁(𝑡)𝑡−2(𝑛−1)𝜋−3𝜋/8−𝜋/12𝑡−2(𝑛−1)𝜋−3𝜋/8³+𝑑𝑡_{2𝑛𝜋+𝜋/42𝑛𝜋+𝜋/8}𝑊₁(𝑡)𝑡−2𝑛𝜋−𝜋/8𝑏_𝑛,1(𝑡+𝜋/12−2𝑛𝜋−𝜋/8)³𝑑𝑡−^{2𝑛𝜋+𝜋/4}_{2𝑛𝜋+𝜋/12}12⁴cos⁴=12𝑡𝑑𝑡_{𝜋/8𝜋/12}𝑊₁(𝑡)𝑡+37𝜋/24𝑡+13𝜋/8³𝑑𝑡+^𝜋/4_𝜋/8𝑊₁(𝑡)𝑡−𝜋/8𝑏_𝑛,1(𝑡−𝜋/24)³𝑑𝑡−12^4𝜋/4_𝜋/12cos⁴12𝑡𝑑𝑡≈0.671𝜈₁+1.326𝜈₂+𝑏⁻³_𝑛,10.008𝜈₁+0.002𝜈₂−6⁴𝜋. (3.3) On the other hand, we have 𝑄^𝑑1_𝑐1𝑤⁴=24𝜋³𝑏³_𝑛,1−𝑎³_𝑛,1𝑎³_𝑛,1. (3.4) Thus condition (2.27) is satisfied for 𝑡∈[𝑐₁,𝑑₁] if 0.671𝜈₁+1.326𝜈₂+𝑏⁻³_𝑛,10.008𝜈₁+0.002𝜈₂−6⁴𝜋≥24𝜋³𝑏³_𝑛,1−𝑎³_𝑛,1𝑎³_𝑛,1. (3.5) Similarly, for 𝑡∈[𝑐₂,𝑑₂] we can get the following condition 0.012𝜈₁+0.003𝜈₂+𝑏⁻³_𝑛,20.012𝜈₁+0.001𝜈₂−6⁴𝜋≥24𝜋³𝑏³_𝑛,2−𝑎³_𝑛,2𝑎³_𝑛,2 (3.6) which ensures (2.27). Hence, by Theorem 2.4, if (3.5) and (3.6) hold, (3.1) is oscillatory. It is easy to see that (3.5) and (3.6) may be satisfied when 𝜈₁ or 𝜈₂ is large enough. Particularly, let 𝑎_𝑛,𝑖=𝑏_𝑛,𝑖, for 𝑖=1,2 and 𝑛=1,2,…, condition (3.5) and (3.6) become a simple form 0.671𝜈₁+1.326𝜈₂+𝑏⁻³_𝑛,10.008𝜈₁+0.002𝜈₂≥6⁴𝜋,0.012𝜈₁+0.003𝜈₂+𝑏⁻³_𝑛,20.012𝜈₁+0.001𝜈₂≥6⁴𝜋. (3.7)

Example 3.2. Consider the following equation 𝑥^′′(𝑡)+𝜇₁𝑝(𝑡)𝑥2𝑡−3+𝜇₂𝑞(𝑡)𝑥²2𝑡−3=𝑒(𝑡),𝑡≠𝜏_𝑘,𝑥𝑡⁺=𝑎_𝑘𝑥(𝑡),𝑥^′𝑡⁺=𝑏_𝑘𝑥^′(𝑡),𝑡=𝜏_𝑘, (3.8) where 𝜏_𝑘: 𝜏_𝑛,1=8𝑛+1/2, 𝜏_𝑛,2=8𝑛+3/2, 𝜏_𝑛,3=8𝑛+9/2, 𝜏_𝑛,4=8𝑛+(11/2) (𝑛=0,1,2,…), 𝛾=1, 𝛼=2, 𝜎=2/3, 𝑏_𝑘≥𝑎_𝑘>0, 𝜇₁,𝜇₂ (>0) is a constant. Clearly, 𝜏_𝑘+1−𝜏_𝑘>𝜎. In addition, let 𝑝(𝑡)=𝑞(𝑡)=(𝑡−8𝑛)³[],(),𝑡∈8𝑛,8𝑛+28𝑛+4−𝑡³[],(),𝑡∈8𝑛+2,8𝑛+4𝑡−8𝑛−4³[],(),𝑡∈8𝑛+4,8𝑛+68𝑛+8−𝑡³[],𝑒(𝑡)=(),𝑡∈8𝑛+6,8𝑛+8𝑡−8𝑛−2³[],(),𝑡∈8𝑛,8𝑛+48𝑛+6−𝑡³,𝑡∈[8𝑛+4,8𝑛+8]. (3.9) For any 𝑡₀>0, we choose 𝑛 large enough such that 𝑡₀<8𝑛 and let [𝑐₁,𝑑₁]=[8𝑛,8𝑛+2], [𝑐₂,𝑑₂]=[8𝑛+4,8𝑛+6], 𝛿₁=8𝑛+1 and 𝛿₂=8𝑛+5. Then 𝑝(𝑡), 𝑞(𝑡), and 𝑒(𝑡) on [𝑐₁,𝑑₁] and [𝑐₂,𝑑₂] satisfy (2.7). Let 𝐻₁(𝑡,𝑠)=𝐻₂(𝑡,𝑠)=(𝑡−𝑠)³, then ℎ₁(𝑡,𝑠)=−ℎ₂(𝑡,𝑠)=3/(𝑡−𝑠). By simple calculation, we get Π_1,11=∶𝐻₁𝛿₁,𝑐₁^{𝜏1𝑘(𝑐)+1}_𝑐1𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝑐1)−𝜎^𝛾𝑡−𝜏_𝑘(𝑐1)^𝛾+𝑑𝑡_{𝑘(𝛿1)−1𝑐𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾−𝛾𝑑𝑡_𝛾𝛾+1^𝛾+1𝛿1_𝑐1𝑟(𝑡)𝐻₁𝑡,𝑐₁ℎ₁𝑡,𝑐₁^𝛾+1+𝑑𝑡_{𝛿1𝜏1)𝑘(𝛿}𝐻₁𝑡,𝑐₁𝑡−𝜏_𝑘(𝛿1)^𝛾𝑏^𝛾_𝑘𝛿1𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛾=𝑑𝑡_{8𝑛+1/28𝑛}(𝑡−8𝑛)³𝜇₁(𝑡−8𝑛)³+2𝜇₂^1/2(𝑡−8𝑛)^3/2(8𝑛+2−𝑡)^1/2𝑡−8𝑛+11/6+𝑡−8𝑛+5/2𝑑𝑡_{8𝑛+18𝑛+1/2}(𝑡−8𝑛)³𝜇₁(𝑡−8𝑛)³+2𝜇₂^1/2(𝑡−8𝑛)^3/2(8𝑛+2−𝑡)^1/2(𝑡−8𝑛−1/2)𝑏_𝑛,1()9𝑡−8𝑛+1/6𝑑𝑡−8=𝜇₁₀^1/2𝑢⁶(𝑢+11/6)𝑢+5/2𝑑𝑢+2𝜇₂^1/2₀^1/2𝑢^9/2(2−𝑢)^1/2(𝑢+11/6)+𝜇𝑢+5/2𝑑𝑢₁𝑏_𝑛,1¹_1/2𝑢⁶(𝑢−1/2)𝑢+1/6𝑑𝑢+2𝜇₂^1/2𝑏_𝑛,1¹_1/2𝑢^9/2(2−𝑢)^1/2(𝑢−1/2)9𝑢+1/6𝑑𝑢−8≈0.0008+0.0005𝑏_𝑛,1𝜇₁+0.0081+0.1268𝑏_𝑛,1𝜇₂^1/2−98,Π_2,11=∶𝐻₂𝑑₁,𝛿₁^{𝜏1)𝑘(𝛿}_𝛿^+𝜎₁𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝛿1)^𝛾𝑏^𝛾_𝑘𝛿1𝑡+𝜎−𝜏_𝑘(𝛿1)^𝛾+^{𝜏1𝑘(𝛿)+1}_{𝜏1)𝑘(𝛿+𝜎}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝛿1)−𝜎^𝛾𝑡−𝜏_𝑘(𝛿1)^𝛾+𝑑𝑡_{𝑘(𝑑1)−1𝛿𝑖=𝑘1+1𝜏𝑖𝜏+𝜎𝑖}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑖^𝛾𝑏^𝛾_𝑖𝑡+𝜎−𝜏_𝑖^𝛾𝑑𝑡+^𝜏𝑖+1_{𝜏𝑖+𝜎}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑖−𝜎^𝛾𝑡−𝜏_𝑖^𝛾−𝛾𝑑𝑡_𝛾𝛾+1^𝛾+1𝑑1_𝛿1𝑟(𝑡)𝐻₂𝑑₁,𝑡ℎ₂𝑑₁,𝑡^𝛾+1𝑑𝑡+^𝑑1_{𝜏1)𝑘(𝑑}𝐻₂𝑑₁,𝑡𝑡−𝜏_𝑘(𝑑1)^𝛾𝑏^𝛾_𝑘𝑑1𝑡+𝜎−𝜏_𝑘(𝑑1)^𝛾=𝑑𝑡_{8𝑛+7/68𝑛+1}(8𝑛+2−𝑡)³𝜇₁(𝑡−8𝑛)³+2𝜇₂^1/2(𝑡−8𝑛)^3/2(8𝑛+2−𝑡)^1/2𝑡−8𝑛−1/2𝑏_𝑛,1()+𝑡−8𝑛+1/6𝑑𝑡_{8𝑛+3/28𝑛+7/6}(8𝑛+2−𝑡)³𝜇₁(𝑡−8𝑛)³+2𝜇₂^1/2(𝑡−8𝑛)^3/2(8𝑛+2−𝑡)^1/2𝑡−8𝑛−7/6+𝑡−8𝑛−1/2𝑑𝑡_{8𝑛+28𝑛+3/2}(8𝑛+2−𝑡)³𝜇₁(𝑡−8𝑛)³+2𝜇₂^1/2(𝑡−8𝑛)^3/2(8𝑛+2−𝑡)^1/2𝑡−8𝑛−3/2𝑏_𝑛,2()9𝑡−8𝑛−5/6𝑑𝑡−8=𝜇₁𝑏_𝑛,11^7/6𝑢³(2−𝑢)³(𝑢−1/2)𝑢+1/6𝑑𝑢+2𝜇₂^1/2𝑏_𝑛,11^7/6𝑢^3/2(2−𝑢)^7/2(𝑢−1/2)𝑢+1/6𝑑𝑢+𝜇₁^3/2_7/6𝑢³(2−𝑢)³(𝑢−7/6)𝑢−1/2𝑑𝑢+2𝜇₂^1/23/2_7/6𝑢^3/2(2−𝑢)^7/2(𝑢−7/6)+𝜇𝑢−1/2𝑑𝑢₁𝑏_𝑛,2²_3/2𝑢³(2−𝑢)³(𝑢−3/2)𝑢−5/6𝑑𝑢+2𝜇₂^1/2𝑏_𝑛,2²_3/2𝑢^3/2(2−𝑢)^7/2(𝑢−3/2)9𝑢−5/6𝑑𝑢−8≈0.0389+0.0754𝑏_𝑛,1+0.0087𝑏_𝑛,2𝜇₁+0.0394+0.1281𝑏_𝑛,1+0.0046𝑏_𝑛,2𝜇₂^1/2−98. (3.10) Then the left side of the inequality (2.52) is Π_1,1+Π_2,1≈0.0497+0.0759𝑏_𝑛,1+0.0087𝑏_𝑛,2𝜇₁+0.0475+0.2541𝑏_𝑛,1+0.0046𝑏_𝑛,2𝜇₂^1/2−94. (3.11)
Because 𝑀₁=𝑀₂=1, 𝜏_{𝑘(𝑐1)+1}=𝜏_𝑘(𝛿1)=𝜏_𝑛,1=8𝑛+1/2∈(𝑐₁,𝛿₁) and 𝜏_{𝑘(𝛿1)+1}=𝜏_𝑘(𝑑1)=𝜏_𝑛,2=8𝑛+3/2∈(𝛿₁,𝑑₁), it is easy to get that the right side of the inequality (2.52) for 𝑗=1 is 𝑀₁𝐻₁𝛿₁,𝑐₁𝑄^𝛿1_𝑐1𝐻₁⋅,𝑐₁+𝑀₁𝐻₂𝑑₁,𝛿₁𝑄^𝑑1_𝛿1𝐻₂𝑑₁=𝑏,⋅^𝑛,1−𝑎_𝑛,14𝑎²_𝑛,1+29𝑏_𝑛,2−𝑎_𝑛,28𝑎_𝑛,2. (3.12) Thus (2.52) is satisfied for 𝑗=1 if 0.0497+0.0759𝑏_𝑛,1+0.0087𝑏_𝑛,2𝜇₁+0.0475+0.2541𝑏_𝑛,1+0.0046𝑏_𝑛,2𝜇₂^1/2>94+𝑏_𝑛,1−𝑎_𝑛,14𝑎²_𝑛,1+29𝑏_𝑛,2−𝑎_𝑛,28𝑎_𝑛,2. (3.13) When 𝑗=2, with the similar argument above we get that the left side of inequality (2.52) is Π_1,2+Π_2,2=𝜇₁₀^1/2𝑢⁶(𝑢+11/6)𝑢+5/2𝑑𝑢+2𝜇₂^1/2₀^1/2𝑢^9/2(2−𝑢)^1/2(𝑢+11/6)+𝜇𝑢+5/2𝑑𝑢₁𝑏_𝑛,3¹_1/2𝑢⁶(𝑢−1/2)𝑢+1/6𝑑𝑢+2𝜇₂^1/2𝑏_𝑛,3¹_1/2𝑢^9/2(2−𝑢)^1/2(𝑢−1/2)+𝜇𝑢+1/6𝑑𝑢₁𝑏_𝑛,41^7/6𝑢³(2−𝑢)³(𝑢−1/2)𝑢+1/6𝑑𝑢+2𝜇₂^1/2𝑏_𝑛,41^7/6𝑢^3/2(2−𝑢)^7/2(𝑢−1/2)𝑢+1/6𝑑𝑢+𝜇₁^3/2_7/6𝑢³(2−𝑢)³(𝑢−7/6)𝑢−1/2𝑑𝑢+2𝜇₂^1/23/2_7/6𝑢^3/2(2−𝑢)^7/2(𝑢−7/6)+𝜇𝑢−1/2𝑑𝑢₁𝑏_𝑛,4²_3/2𝑢³(2−𝑢)³(𝑢−3/2)𝑢−5/6𝑑𝑢+2𝜇₂^1/2𝑏_𝑛,4²_3/2𝑢^3/2(2−𝑢)^7/2(𝑢−3/2)9𝑢−5/6𝑑𝑢−4≈0.0497+0.0759𝑏_𝑛,3+0.0087𝑏_𝑛,4𝜇₁+0.0475+0.2541𝑏_𝑛,3+0.0046𝑏_𝑛,4𝜇₂^1/2−94 (3.14) and the right side of the inequality (2.52) is 𝑀₂𝐻₂𝛿₂,𝑐₂𝑄^𝛿2_𝑐2𝐻₁⋅,𝑐₂+𝑀₂𝐻₂𝑑₂,𝛿₂𝑄^𝑑2_𝛿2𝐻₂𝑑₂=𝑏,⋅^𝑛,3−𝑎_𝑛,34𝑎²_𝑛,3+29𝑏_𝑛,4−𝑎_𝑛,48𝑎_𝑛,4. (3.15) Therefore (2.52) is satisfied for 𝑗=2 if 0.0497+0.0759𝑏_𝑛,3+0.0087𝑏_𝑛,4𝜇₁+0.0475+0.2541𝑏_𝑛,4+0.0046𝑏_𝑛,4𝜇₂^1/2>94+𝑏_𝑛,3−𝑎_𝑛,34𝑎²_𝑛,3+29𝑏_𝑛,4−𝑎_𝑛,48𝑎_𝑛,4. (3.16) Hence, by Theorem 2.8, (3.8) is oscillatory if (3.13) and (3.16) hold. Particularly, let 𝑎_𝑛,𝑖=𝑏_𝑛,𝑖, for 𝑖=1,2,3,4, condition (3.13) and (3.16) become a simple form 0.0497+0.0759𝑏_𝑛,1+0.0087𝑏_𝑛,2𝜇₁+0.0475+0.2541𝑏_𝑛,1+0.0046𝑏_𝑛,2𝜇₂^1/2>94,0.0497+0.0759𝑏_𝑛,3+0.0087𝑏_𝑛,4𝜇₁+0.0475+0.2541𝑏_𝑛,3+0.0046𝑏_𝑛,4𝜇₂^1/2>94. (3.17)

Acknowledgments

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