Oscillation Criteria for Even Order Neutral Equations with Distributed Deviating Argument

Abstract

We present new oscillation criteria for the even order neutral delay differential equations with distributed deviating argument [𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)]^′+∫^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[𝑥(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)=0,𝑡≥𝑡₀, where 𝑍(𝑡)=𝑥(𝑡)+𝑞(𝑡)𝑥(𝑡−𝜏). Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those by Wang et al. (2005).

1. Introduction

In this paper, we are concerned with the oscillation behavior of the even order neutral delay differential equations of the form

[𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)]^′+^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[𝑥(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)=0,𝑡≥𝑡₀, (1.1) where 𝑍(𝑡)=𝑥(𝑡)+𝑞(𝑡)𝑥(𝑡−𝜏), 𝜏⩾0 and 𝑛  is an even positive integer. We assume that

(𝐴₁)𝑟,𝑞∈𝐶(𝐼,𝑅) and 0≤𝑞(𝑡)≤1, 𝑟(𝑡)>0 for 𝑡∈𝐼, ∫^∞(1/𝑟(𝑠))𝑑𝑠=∞, 𝐼=[𝑡₀,∞);(𝐴₂)𝜓∈𝐶¹(𝑅,𝑅), 𝜓(𝑥)>0 for 𝑥≠0;(𝐴₃)𝑓∈𝐶(𝑅,𝑅), 𝑥𝑓(𝑥)>0 for 𝑥≠0;(𝐴₄)𝑝∈𝐶(𝐼×[𝑎,𝑏],[0,∞) and 𝑝(𝑡,𝜉) is not eventually zero on any half linear [𝑡_𝑢,∞)×[𝑎,𝑏],𝑡_𝑢≥𝑡₀;(𝐴₅)𝑔∈𝐶(𝐼×[𝑎,𝑏],[0,∞)), 𝑔(𝑡,𝜉)≤𝑡 for 𝜉∈[𝑎,𝑏], 𝑔(𝑡,𝜉) has a continuous and positive partial derivative on 𝐼×[𝑎,𝑏] with respect to 𝑡 and nondecreasing with respect to 𝜉, respectively, liminf_𝑡→∞𝑔(𝑡,𝜉)=∞ for 𝜉∈[𝑎,𝑏];(𝐴₆)𝜎∈𝐶([𝑎,𝑏],𝑅) is nondecreasing, and the integral of (1.1) is in the sense of Riemann-Stieltijes.

We restrict our attention to those solutions 𝑥(𝑡) of (1.1) which exist on some half linear [𝑡_𝑥,∞) and satisfy sup{|𝑥(𝑡)|∶𝑡⩾𝑡_𝑥}≠0 for any 𝑇⩾𝑡₀. As usual, such a solution of (1.1) is called oscillatory if the set of its zeros is unbounded from above; otherwise, it is said to be nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.

The oscillatory behavior of solutions of higher-order neutral differential equations is of both theoretical and practical interest. There have been some results on the oscillatory and asymptotic behavior of even order neutral equations. We mention here [1–12]. The oscillation problem for nonlinear delay equation such as

[𝑟(𝑡)𝑥^′(𝑡)]^′+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0,𝑡>𝑡₀ (1.2) as well as for the the linear ordinary differential equation

[𝑟(𝑡)𝑥^′(𝑡)]^′+𝑝(𝑡)𝑥^′(𝑡)+𝑞(𝑡)𝑥(𝑡)=0,𝑡>𝑡₀ (1.3) and the neutral delay differential equation

(𝑥(𝑡)+𝑞(𝑡)𝑥(𝑡−𝜎))^′′+𝑝(𝑡)𝑥(𝑡−𝜏)=0 (1.4) has been studied by many authors with different methods. In [13], Rogovchenko established some general oscillation criteria for second-order nonlinear differential equation: (𝑟(𝑡)𝑥^′(𝑡))^′+𝑝(𝑡)𝑥^′(𝑡)+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡≥𝑡₀. (1.5) In [14], the authors discussed the following neutral equations of the form [𝑥(𝑡)+𝑐(𝑡)𝑥(𝑡−𝜏)]^(𝑛)+^𝑏_𝑎𝑝(𝑡,𝜉)𝑥[𝑔(𝑡,𝜉)]𝑑𝜎(𝜉)=0,𝑡≥𝑡₀ (1.6) and obtained the following results.

Theorem 1 A (see [14,Theorem 2]). Assume that there exist functions 𝐻(𝑡,𝑠)∈𝐶^′(𝐷;𝑅),ℎ(𝑡,𝑠)∈𝐶(𝐷₀;𝑅), and 𝜌(𝑡)∈𝐶^′([𝑡₀,∞),(0,∞)), such that(𝐼)𝐻(𝑡,𝑡)=0,𝐻(𝑡,𝑠)>0;(𝐼𝐼)𝐻^′_𝑠(𝑡,𝑠)≤0, and −𝐻^′_𝑠(𝑡,𝑠)−𝐻(𝑡,𝑠)(𝜌^′(𝑠)/𝜌(𝑠))=ℎ(𝑡,𝑠)𝐻(𝑡,𝑠), and0<inf_𝑠≥𝑡0[lim_𝑡→∞inf𝐻(𝑡,𝑠)𝐻(𝑡,𝑡₀)]≤∞;lim_𝑡→∞1sup𝐻(𝑡,𝑡₀)^𝑡_𝑡0𝜌(𝑠)ℎ²(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑑𝑠<∞. (C1)(C2) If there exists a function 𝜑(𝑡)∈𝐶([𝑡₀,∞),𝑅) satisfying lim_𝑡→∞1sup𝐻(𝑡,𝑢)^𝑡_𝑢[𝐻(𝑡,𝑠)𝜌(𝑠)^𝑏_𝑎−𝑝(𝑠,𝜉)(1−𝑐[𝑔(𝑠,𝜉)])𝑑𝜎(𝜉)𝜌(𝑠)ℎ²(𝑡,𝑠)2𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≥𝜑(𝑢),𝑢≥𝑡₀,lim_𝑡→∞1sup𝐻(𝑡,𝑡₀)^𝑡_𝑡0𝑔^′(𝑢,𝑎)𝑔^𝑛−2(𝑢,𝑎)𝜑𝜌(𝑢)²⁺(𝑢)𝑑𝑢=∞,𝜑₊(𝑢)=max_𝑢≥𝑡0{𝜑(𝑢),0}, (C3)(C4) then every solution of (1.6) is oscillatory.

We will use the function class 𝑊 to study the oscillation criteria for (1.1). Let 𝐷={(𝑡,𝑠)∣𝑡≥𝑠≥𝑡₀}, and 𝐷₀={(𝑡,𝑠)𝑡>𝑠≥𝑡₀}. We say that a continuous function 𝐻(𝑡,𝑠)⊂𝐶^′(𝐷,ℜ) belongs to the class 𝑊 if

(𝐴₇)𝐻(𝑡,𝑡)=0𝑎𝑛𝑑𝐻(𝑡,𝑠)>0  for  −∞<𝑠<𝑡<+∞;(𝐴₈)𝐻has a continuous partial derivative𝜕𝐻/𝜕𝑠satisfying, for some ℎ∈𝐿_loc(𝐷,𝑅), the condition 𝜕𝐻/𝜕𝑠=−ℎ(𝑡,𝑠)𝐻(𝑡,𝑠).

The purpose of this paper is to further improve Theorem A by Wang et al. [14], using a generalized Riccati transformation and developing ideas exploited by the Rogovchenko and Tuncay [13], we establish some new oscillation criteria for (1.1), which remove condition (C₂) in Theorem A by Wang et al. [14]; this complements and extends the results in [14].

In addition, we will make use of the following conditions.

(𝑆₁) There exists a positive real number 𝑀 such that |𝑓(±𝑢𝑣)|≥𝑀𝑓(𝑢)𝑓(𝑣) for 𝑢𝑣>0.

Lemma 1.1. If 𝑎>0,𝑏≥0, then −𝑎𝑥²𝑎+𝑏𝑥≤−2𝑥²+𝑏².2𝑎 (1.7)

Lemma 1.2 (Kiguradze [15]). Let u(t ) be a positive and 𝑛 times differentiable function on 𝑅. If 𝑢^(𝑛)(𝑡) is of constant sign and identically zero on any ray [𝑡,+∞) for (𝑡₁>0), then there exists a 𝑡_𝑢≥𝑡₁ and an integer 𝑙(0≤𝑙≤𝑛), with 𝑛+𝑙 even for 𝑢(𝑡)𝑢^(𝑛)(𝑡)≥0 or 𝑛+𝑙 odd for 𝑢(𝑡)𝑢^(𝑛)(𝑡)≤0, and for 𝑡≥𝑡_𝑢, 𝑢(𝑡)𝑢^(𝑘)(𝑡)>0,0≤𝑘≤𝑙;(−1)^𝑘−1𝑢(𝑡)𝑢^(𝑘)(𝑡)>0,𝑙≤𝑘≤𝑛. (1.8)

Lemma 1.3 (Philos [16]). Suppose that the conditions of Lemma 1.2 are satisfied, and 𝑢^(𝑛−1)(𝑡)𝑢^(𝑛)(𝑡)≤0,𝑡≥𝑡_𝑢, (1.9) then there exists a constant 𝜃 in (0,1) such that for sufficiently large 𝑡, and there exists a constant 𝑀_𝜃>0 satisfying |𝑢^′(𝑡2)|≥𝑀_𝜃𝑡^𝑛−2|𝑢^(𝑛−1)(𝑡)|, (1.10) where 𝑀_𝜃=𝜃/(𝑛−2)!.

2. When 𝑓(𝑥) Is Monotone

In this section, we will deal with the oscillation for (1.1) under the assumptions (𝐴₁)-(𝐴₈),(𝑆₁) and the following assumption.

(𝐴₉)𝑓^′(𝑥) exists, 𝑓^′(𝑥)≥𝐾₁ and 𝜓(𝑥)≤𝐿⁻¹ for 𝑥≠0.

Theorem 2.1. Let (𝑆₁), (𝐴₁)-(𝐴₉) hold. Equation (1.1) is oscillatory provided that 𝜌(𝑡)∈𝐶¹([𝑡₀,∞),𝑅) such that lim_𝑡→∞1sup𝐻(𝑡,𝑡₀)^𝑡_𝑡0ℎ[𝐻(𝑡,𝑠)𝑄(𝑠)−²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠=∞, (2.1) where 𝑄(𝑡)=𝜌(𝑡)𝑀^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)−(𝜌^′(𝑡))²𝑟(𝑡)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑡,𝑎)𝑔^′.(𝑡,𝑎)𝜌(𝑡) (2.2)

Proof. Suppose to the contrary that there exists a solution 𝑥(𝑡) of (1.1) such that 𝑥(𝑡)>0,𝑥(𝑡−𝜏)>0,𝑥[𝑔(𝑡,𝜉)]>0,𝑡≥𝑡₁,𝜉∈[𝑎,𝑏]for𝑡⩾𝑡₁⩾𝑡₀. (2.3) From (1.1), we also have 𝑍(𝑡)>0 and [(𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)]^′≤0 for 𝑡≥𝑡₁.
It follows that the function 𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡) is decreasing and we claim that
𝑍^(𝑛−1)(𝑡)≥0for𝑡≥𝑡₁. (2.4) Otherwise, if there exist a 𝑡₁̃≥𝑡₁ such that 𝑍^(𝑛−1)(𝑡₁̃)<0, then for all 𝑡≥𝑡₁̃, 𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)≤𝑟(𝑡₁̃)𝜓(𝑥(𝑡₁̃))𝑍^(𝑛−1)(𝑡₁̃)=−𝐶(𝐶>0), (2.5) which implies that 𝑍^(𝑛−1)(𝑡)≤−𝐶/𝑟(𝑡)𝜓(𝑥(𝑡)),𝑡≥𝑡₁; integrating the above inequality from 𝑡₁̃ to t, we have 𝑍^(𝑛−2)(𝑡)≤𝑍^(𝑛−2)(𝑡₁̃)−CL^𝑡𝑡₁̃1𝑟(𝑡)𝑑𝑠. (2.6) Let 𝑡→∞; from (𝐴₁), we get lim_𝑡→∞𝑍^(𝑛−2)(𝑡)=−∞, which implies that 𝑍^(𝑛−1)(𝑡) and 𝑍^(𝑛−2)(𝑡) are negative for all large t; from Lemma 1.2, no two consecutive derivative can be eventually negative, for this would imply that lim_𝑡→∞𝑍(𝑡)=−∞, which is a contradiction. Hence 𝑍^(𝑛−1)(𝑡)≥0 for 𝑡≥𝑡₁. Using this fact together with 𝑥(𝑡)≤𝑍(𝑡), we have that 𝑥(𝑡)≥[1−𝑞(𝑡)]𝑍(𝑡),𝑡≥𝑡₁. (2.7) Now from (𝐴₁),(𝑆₁), and (2.7), we get 𝑓[𝑥(𝑔(𝑡,𝜉))]≥𝑀𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑓[𝑍(𝑔(𝑡,𝜉))],𝑡≥𝑡₁, (2.8) and thus, from(1.1), we get 0=[𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)]^′+^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[𝑥(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)≥[(𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)]^′+𝑀^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑓[𝑍(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉). (2.9) Further, observing that 𝑔(𝑡,𝜉) is nondecreasing with respect to 𝜉 and 𝑍^(𝑛−1)(𝑡)>0 for 𝑡≥𝑡₁, from Lemma 1.2, we have 𝑍^′(𝑡)≥0, 𝑡≥𝑡₁, and so 𝑍(𝑔(𝑡,𝜉))≥𝑍(𝑔(𝑡,𝑎)),𝑡≥𝑡₁,𝜉∈[𝑎,𝑏]. (2.10) So, 𝑓[𝑍(𝑔(𝑡,𝜉))]≥𝑓[𝑍(𝑔(𝑡,𝑎))] for 𝑡≥𝑡₁ and 𝜉∈[𝑎,𝑏]. Thus [𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)]^′+𝑀𝑓[𝑍(𝑔(𝑡,𝑎))]^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)≤0,𝑡⩾𝑡₁. (2.11) Define 𝑤(𝑡)=𝜌(𝑡)𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)𝑓[𝑍(𝑔(𝑡,𝑎)/2)],𝑡⩾𝑡₁. (2.12) From (1.1), (2.11), and Lemma 1.3 we get 𝑤^′𝜌(𝑡)=^′(𝑡)𝜌(𝑡)𝑤(𝑡)+𝜌(𝑡)(𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡))^′𝑓[𝑍(𝑔(𝑡,𝑎)/2)]−𝜌(𝑡)𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)𝑓²𝑓[𝑍(𝑔(𝑡,𝑎)/2)]^′[𝑍(𝑔(𝑡,𝑎)2)]𝑍^′(𝑔(𝑡,𝑎)2)12𝑔^′≤𝜌(𝑡,𝑎)_′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)𝑀^𝑏_𝑎1𝑝(𝑡,𝜉)𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)−2𝐾₁𝐿𝑀_𝜃𝑔^(𝑛−2)(𝑡,𝑎)𝑔^′(𝑡,𝑎)𝑤𝜌(𝑡)𝑟(𝑡)²(𝑡). (2.13) Then, by Lemma 1.1 we get 𝑤^′(𝑡)≤−𝜌(𝑡)𝑀^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)+(𝜌^′(𝑡))²𝑟(𝑡)𝐾₁𝐿𝑀_𝜃𝑔^(𝑛−2)(𝑡,𝑎)𝑔^′−1(𝑡,𝑎)𝜌(𝑡)4𝐾₁𝐿𝑀_𝜃𝑔^(𝑛−2)(𝑡,𝑎)𝑔^′(𝑡,𝑎)𝑤𝜌(𝑡)𝑟(𝑡)²1(𝑡)=−𝑄(𝑡)−4𝐾₁𝐿𝑀_𝜃𝑔^(𝑛−2)(𝑡,𝑎)𝑔^′(𝑡,𝑎)𝑤𝜌(𝑡)𝑟(𝑡)²(𝑡). (2.14) Let 𝑄(𝑡)=𝜌(𝑡)𝑀^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)−(𝜌^′(𝑡))²𝑟(𝑡)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑡,𝑎)𝑔^′.(𝑡,𝑎)𝜌(𝑡) (2.15) That is, 𝑄(𝑡)≤−𝑤^′𝐾(𝑡)−¹𝐿𝑀_𝜃𝑔^𝑛−2(𝑡,𝑎)𝑔^′(𝑡,𝑎)𝑤4𝜌(𝑡)𝑟(𝑡)²(𝑡). (2.16) Integrating by parts for any 𝑡>𝑇≥𝑡₁, and using properties (𝐴₇) and (𝐴₈), we obtain ^𝑡_𝑇𝐻(𝑡,𝑠)𝑄(𝑠)𝑑𝑠≤−^𝑡_𝑇𝐻(𝑡,𝑠)𝑤^′(𝑠)𝑑𝑠−^𝑡_𝑇𝐾𝐻(𝑡,𝑠)¹𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤4𝜌(𝑠)𝑟(𝑠)²(𝑠)=𝐻(𝑡,𝑇)𝑤(𝑇)+^𝑡_𝑇𝑤(𝑠)𝜕𝐻(𝑡,s)𝜕𝑠𝑑𝑠−^𝑡_𝑇𝐾𝐻(𝑡,𝑠)¹𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤4𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠=𝐻(𝑡,𝑇)𝑤(𝑇)−^𝑡_𝑇−𝑤(𝑠)𝜕𝐻(𝑡,s)𝜕𝑠𝑑𝑠−^𝑡_𝑇𝐾𝐻(𝑡,𝑠)¹𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤4𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠=𝐻(𝑡,𝑇)𝑤(𝑇)−^𝑡_𝑇[ℎ(𝑡,𝑠)𝐾𝐻(𝑡,𝑠)𝑤(𝑠)+𝐻(𝑡,𝑠)¹𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤4𝜌(𝑠)𝑟(𝑠)²−(𝑠)]𝑑𝑠=𝐻(𝑡,𝑇)𝑤(𝑇)_𝑡𝑇(𝐻(𝑡,𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)4𝛽𝜌(𝑠)𝑟(𝑠)𝑤(𝑠)+ℎ(𝑡,𝑠)𝛽𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′)(𝑠,𝑎)²+𝛽𝑑𝑠𝑀_𝜃𝐾₁𝐿^𝑡_𝑇ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′−(𝑠,𝑎)𝑑𝑠(𝛽−1)𝐾₁𝑀_𝜃𝐿4𝛽^𝑡_𝑇𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠. (2.17) We obtain ^𝑡_𝑇[𝐻(𝑡,𝑠)𝑄(𝑠)−𝛽ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≤𝐻(𝑡,𝑇)𝑤(𝑇)−(𝛽−1)𝐾₁𝑀_𝜃𝐿4𝛽^𝑡_𝑇𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²−(𝑠)𝑑𝑠_𝑡𝑇(𝐻(𝑡,𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)4𝛽𝜌(𝑠)𝑟(𝑠)𝑤(𝑠)+ℎ(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′)(𝑠,𝑎)²𝑑𝑠. (2.18) From (𝐴₈),𝐻^′(𝑡,𝑠)≤0, for 𝑡₁≥𝑡₀,𝐻(𝑡,𝑡₁)≤𝐻(𝑡,𝑡₀),^𝑡_𝑡1[𝐻(𝑡,𝑠)𝑄(𝑠)−𝛽ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≤𝐻(𝑡,𝑡₁)𝑤(𝑡₁)≤𝐻(𝑡,𝑡₀)𝑤(𝑡₁), (2.19) which implies that 1𝐻(𝑡,𝑡₀)^𝑡_𝑡0[𝐻(𝑡,𝑠)𝑄(𝑠)−𝛽ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≤𝑤(𝑡₁1)+𝐻(𝑡,𝑡₀)^𝑡1_𝑡0ℎ[𝐻(𝑡,𝑠)𝑄(𝑠)−²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≤𝑤(𝑡₁)+^𝑡1_𝑡0𝑄(𝑠)𝑑𝑠<∞. (2.20) Let 𝑡→∞, and taking upper limits, we have lim_𝑡→∞1sup𝐻(𝑡,𝑡₀)^𝑡_𝑡0ℎ[𝐻(𝑡,𝑠)𝑄(𝑠)−²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠<∞, (2.21) which contradicts the assumption (2.1). This complete the proof of Theorem 2.1.

From Theorem 2.1, we have the following oscillation result.

Corollary 2.2. If condition (2.1) of Theorem 2.1 is replaced by lim_𝑡→∞1sup𝐻(𝑡,𝑡₀)^𝑡_𝑡0[𝐻(𝑡,𝑠)𝑄(𝑠)]𝑑𝑠=∞,lim_𝑡→∞1sup𝐻(𝑡,𝑡₀)^𝑡_𝑡0ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑑𝑠<∞, (2.22) where 𝑄(𝑡) is defined by (2.2), then (1.1) is oscillatory.

Remark 2.3. By introducing various 𝐻(𝑡,𝑠) from Theorem 2.1 or Corollary 2.2, we can obtain some oscillatory criteria of (1.1). For example, let 𝐻(𝑡,𝑠)=(𝑡−𝑠)^𝑚−1,𝑡≥𝑠≥𝑡₀, in which 𝑚>2 is a integer. By choosing ℎ(𝑡,𝑠)=(𝑡−𝑠)^(𝑚−3)/2(𝑚−1), (2.23) it is clear that the conditions of (𝐴₇) and (𝐴₈) hold; then, from Theorem 2.1 and Corollary 2.2, we have the following.

Corollary 2.4. Assume that there exists a function 𝜌(𝑡)∈𝐶^′([𝑡₀,∞),(0,∞)) such that lim_𝑡→∞1sup𝑡^𝑚−1𝑡_𝑡0[(𝑡−𝑠)^𝑚−1𝑄(𝑠)]−𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)(𝑡−𝑠)^𝑚−3(𝑚−1)²𝑑𝑠=∞, (2.24) where 𝑄(𝑡) is defined by (2.2), then (1.1) is oscillatory.

Corollary 2.5. Assume that there exists a function 𝜌(𝑡)∈𝐶^′([𝑡₀,∞),(0,∞)) such that lim_𝑡→∞1sup𝑡^𝑚−1𝑡_𝑡0(𝑡−𝑠)^𝑚−1𝑄(𝑠)𝑑𝑠=∞,lim_𝑡→∞1sup𝑡^𝑚−1𝑡_𝑡0𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)(𝑡−𝑠)^𝑚−3(𝑚−1)²𝑑𝑠<∞, (2.25) where 𝑄(𝑡) is defined by (2.2), then (1.1) is oscillatory.

Theorem 2.6. Assume that the conditions of Theorem 2.1 hold, and 0<inf_𝑠≥𝑡0[lim_𝑡→∞inf𝐻(𝑡,s)𝐻(𝑡,𝑡₀)]≤∞. (2.26) If there exists a function 𝜑(𝑡)∈𝐶([𝑡₀,∞),ℜ) satisfying lim_𝑡→∞1sup𝐻(𝑡,𝑢)^𝑡_𝑢ℎ[𝐻(𝑡,𝑠)𝑄(𝑠)−²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≥𝜑(𝑢),𝑢≥𝑡₀,lim_𝑡→∞sup^𝑡_𝑡0𝑔^𝑛−2(𝑢,𝑎)𝑔^′(𝑢,𝑎)𝜑𝜌(𝑢)𝑟(𝑢)²⁺(𝑢)𝑑𝑢=∞,𝜑₊(𝑢)=max_𝑢≥𝑡0{𝜑(𝑢),(0)}, (2.27)(2.28) where 𝑄(𝑡) is defined by (2.2), then (1.1) is oscillatory.

Proof. Assume that there exists a nonoscillatory solution 𝑥(𝑡) of (1.1) on [𝑡₀,∞), such that 𝑥(𝑡)≠0 on [𝑡₀,∞). Without loss of generality, assume that 𝑥(𝑡)>0, 𝑡≥𝑡₀. Then, proceeding as in the proof of Theorem 2.1, for 𝑡>𝑢≥𝑡₁≥𝑡₀, we have 1𝐻(𝑡,𝑢)^𝑡_𝑢[𝐻(𝑡,𝑠)𝑄(𝑠)−𝛽ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′1(𝑠,𝑎)]𝑑𝑠≤𝑤(𝑢)−𝐻(𝑡,𝑢)(𝛽−1)𝐾₁𝑀_𝜃𝐿4𝛽^𝑡_𝑢𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠. (2.29) Let 𝑡→∞, and taking upper limits, we have lim_𝑡→∞1sup𝐻(𝑡,𝑢)^𝑡_𝑢[𝐻(𝑡,𝑠)𝑄(𝑠)−𝛽ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≤𝑤(𝑢)−lim_𝑡→∞1inf𝐻(𝑡,𝑢)(𝛽−1)𝐾₁𝑀_𝜃𝐿4𝛽^𝑡_𝑢𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠, (2.30) thus, from (2.27), we have 𝑤(𝑢)≥𝜑(𝑢)+lim_𝑡→∞1inf𝐻(𝑡,𝑢)(𝛽−1)𝐾₁𝑀_𝜃𝐿4𝛽^𝑡_𝑢𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠, (2.31) then 𝑤(𝑢)≥𝜑(𝑢), and lim_𝑡→∞1inf𝐻(𝑡,𝑢)^𝑡_𝑢𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠<4𝛽(𝛽−1)𝐾₁𝑀_𝜃𝐿(𝑤(𝑢)−𝜑(𝑢))<∞. (2.32) Now we can claim that ^∞_𝑡1𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠<∞,𝑡<𝑡₁. (2.33) In fact, assume the contrary, that ^∞_𝑡1𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠=∞,𝑡<𝑡₁. (2.34) From (2.26), there exists a constant 𝜌>0 such that inf_𝑠≥𝑡0[lim_𝑡→∞inf𝐻(𝑡,s)𝐻(𝑡,𝑡₀)]>𝜌>0, (2.35) this is lim_𝑡→∞inf𝐻(𝑡,s)𝐻(𝑡,𝑡₀)>𝜌>0, (2.36) and there exists a 𝑇₂≥𝑡₁ such that 𝐻(𝑡,𝑇)/𝐻(𝑡,𝑡₀)≥𝜌, for all 𝑡≥𝑇₂. On the other hand, by virtue of (2.34), for any positive number 𝛼, there exists a 𝑇₁≥𝑡₁, such that, for all 𝑡≥𝑇₁^𝑡_𝑡1𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²𝛼(𝑠)𝑑𝑠>𝜌. (2.37) Using integration by parts, we conclude that, for all 𝑡≥𝑇>𝑡₁, 1𝐻(𝑡,𝑡₁)^𝑡_𝑡1𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²=1(𝑠)𝑑𝑠𝐻(𝑡,𝑡₁)^𝑡_𝑡1𝐻(𝑡,𝑠)𝑑(^𝑠_𝑡1𝑔^𝑛−2(𝑢,𝑎)𝑔^′(𝑢,𝑎)𝑤𝜌(𝑢)𝑟(𝑢)²=1(𝑠)𝑑𝑢)𝐻(𝑡,𝑡₁)^𝑡_𝑡1(^𝑠_𝑡1𝑔^𝑛−2(𝑢,𝑎)𝑔^′(𝑢,𝑎)𝑤𝜌(𝑢)𝑟(𝑢)²(𝑠)𝑑𝑢)(−𝜕𝐻≥𝛼𝜕𝑠)𝑑𝑠𝜌𝐻(𝑡,𝑇)𝐻(𝑡,𝑡₁)≥𝛼. (2.38) Since 𝛼 is an arbitrary positive constant, lim_𝑡→∞1inf𝐻(𝑡,𝑡₁)^𝑡_𝑡1𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠=∞, (2.39) which contradicts (2.32), consequently, (2.33) holds, and, by virtue of 𝜔(𝑢)≥𝜑(𝑢) for 𝑢≥𝑡₁≥𝑡₀,lim_𝑡→∞sup^𝑡_𝑡0𝑔^′(𝑢,𝑎)𝑔^𝑛−2(𝑢,𝑎)𝜑𝜌(𝑢)𝑟(𝑢)²⁺(𝑢)𝑑𝑢≤lim_𝑡→∞sup^𝑡_𝑡0𝑔^′(𝑢,𝑎)𝑔^𝑛−2(𝑢,𝑎)𝜔𝜌(𝑢)𝑟(𝑢)²⁺(𝑢)𝑑𝑢<∞, (2.40) which contradicts (2.28), and therefore, (1.1) is oscillatory.

Remark 2.7. Choosing 𝐻 as in Remark 2.3, it is not difficult to see that condition (2.26) is satisfied because, for any 𝑠≥𝑡₀, lim_𝑡→∞𝐻(𝑡,𝑠)𝐻(𝑡,𝑡₀)=lim_𝑡→∞(𝑡−𝑠)^𝑛−1(𝑡−𝑡₀)^𝑛−1=1. (2.41)

Consequently, one immediately derives from Theorem 2.6 the following useful corollary for the oscillation of (1.1).

Corollary 2.8. Assume that there exist functions 𝜌(𝑡)∈𝐶([𝑡₀,∞),(0,∞)) and 𝜑(𝑡)∈𝐶([𝑡₀,∞),ℜ) satisfying lim_𝑡→∞1sup𝑡^𝑚−1𝑡_𝑡0[(𝑡−𝑠)^𝑚−1𝑄(𝑠)−𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)(𝑡−𝑠)^𝑚−3(𝑚−1)²]𝑑𝑠≥𝜑(𝑢),𝑢≥0,lim_𝑡→∞sup^𝑡_𝑡0𝑔^𝑛−2(𝑢,𝑎)𝑔^′(𝑢,𝑎)𝜑𝜌(𝑢)𝑟(𝑢)²⁺(𝑢)𝑑𝑢=0,𝜑₊(𝑢)=max_𝑢≥𝑡0{𝜑(𝑢),(0)}, (2.42) where 𝑄(𝑡) is defined by (2.2), then (1.1) is oscillatory.

Theorem 2.9. Assume that the conditions of Theorem 2.1 and (2.26) hold, and lim_𝑡→∞1inf𝐻(𝑡,𝑢)^𝑡_𝑢ℎ[𝐻(𝑡,𝑠)𝑄(𝑠)−²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝛽𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≥𝜑(𝑢),𝑢≥𝑡₀,lim_𝑡→∞sup^𝑡_𝑡0𝑔^𝑛−2(𝑢,𝑎)𝑔^′(𝑢,𝑎)𝜑𝜌(𝑢)𝑟(𝑢)²⁺(𝑢)𝑑𝑢=∞,𝜑₊(𝑢)=max_𝑢≥𝑡0{𝜑(𝑢),0}, (2.43) then (1.1) is oscillatory.

Proof. Assume that there exists a nonoscillatory solution 𝑥(𝑡) of (1.1) on [𝑡₀,∞), such that 𝑥(𝑡)≠0 on [𝑡₀,∞). Without loss of generality, assume that 𝑥(𝑡)>0, 𝑡≥𝑡₀. Then, proceeding as in the proof of Theorem 2.1, for 𝑡>𝑢≥𝑡₁≥𝑡₀, we have 1𝐻(𝑡,𝑇)^𝑡_𝑇[𝐻(𝑡,𝑠)𝑄(𝑠)−𝛽ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′1(𝑠,𝑎)]𝑑𝑠≤𝑤(𝑇)−𝐻(𝑡,𝑇)(𝛽−1)𝐾₁𝑀_𝜃𝐿4𝛽^𝑡_𝑇𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠. (2.44) Let 𝑡→∞, and taking lower limits, we have lim_𝑡→∞1inf𝐻(𝑡,𝑢)^𝑡_𝑢[𝐻(𝑡,𝑠)𝑄(𝑠)−𝛽ℎ²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝐾₁𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≤𝑤(𝑢)−lim_𝑡→∞1sup𝐻(𝑡,𝑢)(𝛽−1)𝐾₁𝑀_𝜃𝐿4𝛽^𝑡_𝑢𝐻(𝑡,𝑠)𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)𝑤𝜌(𝑠)𝑟(𝑠)²(𝑠)𝑑𝑠. (2.45)
The following proof is similar to Theorem 2.6, so we omit the details. This completes the proof of Theorem 2.9.

3. When 𝑓(𝑥) Is Not Monotone

In this section, we will deal with the oscillation for (1.1) under the assumptions (𝐴₁)-(𝐴₈) and the following assumption:

(𝐴₁₀)𝑓(𝑥)/𝑥≥𝐾₂ and 𝜓(𝑥)≤𝐿⁻¹ for 𝑥≠0.

Theorem 3.1. Let (𝐴₁)-(𝐴₈) and (𝐴₁₀) hold. Equation (1.1) is oscillatory provided that 𝜌(𝑡)∈𝐶¹([𝑡₀,∞),𝑅) such that lim_𝑡→∞1sup𝐻(𝑡,𝑡₀)^𝑡_𝑡0[𝐻(𝑡,𝑠)𝑄₂ℎ(𝑠)−²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝛽𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠=∞, (3.1) where 𝑄₂(𝑡)=𝜌(𝑡)𝐾₂^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[1−𝑞(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)−(𝜌^′(𝑡))²𝑟(𝑡)𝐿𝑀_𝜃𝑔^𝑛−2(𝑡,𝑎)𝑔^′,(𝑡,𝑎)𝜌(𝑡) (3.2) then (1.1) is oscillatory.

Proof. Let 𝑥(𝑡) be an eventually positive solution of (1.1). As in the proof of Theorem 2.1, there exists 𝑡₁⩾𝑡₀, such that (2.3), (2.4), and (2.7) hold. Thus, from (1.1) and (𝐴₁₀), we get 0=(𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡))^′+^𝑏_𝑎𝑝(𝑡,𝜉)𝑓[𝑥(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)≥(𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡))^′+𝐾₂^𝑏_𝑎𝑝(𝑡,𝜉)𝑥[𝑔(𝑡,𝜉)]𝑑𝜎(𝜉)=(𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡))^′+𝐾₂^𝑏_𝑎𝑝(𝑡,𝜉){𝑍[𝑔(𝑡,𝜉)]−𝑞[𝑔(𝑡,𝜉)]𝑥[𝑔(𝑡,𝜉)−𝜏]}𝑑𝜎(𝜉). (3.3) Noting that 𝑍[𝑔(𝑡,𝜉)]≥𝑍[𝑔(𝑡,𝜉)−𝜏]≥𝑥[𝑔(𝑡,𝜉)−𝜏]. (3.4) Thus, (3.3) implies that (𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡))^′+𝐾₂^𝑏_𝑎𝑝(𝑡,𝜉)[1−𝑞(𝑔(𝑡,𝜉))]𝑍[𝑔(𝑡,𝜉)]𝑑𝜎(𝜉)≤0,𝑡≥𝑡₁. (3.5) From (2.10) and (3.5) we get (𝑟(𝑡)𝜓(𝑥(𝑡))𝑧^(𝑛−1)(𝑡))^′+𝐾₂𝑍[𝑔(𝑡,𝑎)]^𝑏_𝑎𝑝(𝑡,𝜉)[1−𝑞(𝑔(𝑡,𝜉))]𝑑𝜎(𝜉)≤0,𝑡≥𝑡₁. (3.6) Define 𝑤(𝑡)=𝜌(𝑡)𝑟(𝑡)𝜓(𝑥(𝑡))𝑍^(𝑛−1)(𝑡)𝑍[(𝑔(𝑡,𝑎)/2)],𝑡⩾𝑡₁. (3.7) Differentiating (3.7) and using (3.6), Lemma 1.1, and 1.3 we get 𝑤^′𝜌(𝑡)≤^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)[𝐾₂^𝑏_𝑎𝑀𝑝(𝑡,𝜉){1−𝑞(𝑔(𝑡,𝜉))}𝑑𝜎(𝜉)]−^𝜃𝐿𝑔^𝑛−2(𝑡,𝑎)𝑔^′(𝑡,𝑎)𝑤2𝑟(𝑡)𝜌(𝑡)²(𝑡)≤−𝐾₂𝜌(𝑡)[^𝑏_𝑎𝑝(𝑡,𝜉){1−𝑞(𝑔(𝑡,𝜉))}𝑑𝜎(𝜉)]+(𝜌^′(𝑡))²𝑟(𝑡)𝑀_𝜃𝐿𝑔^𝑛−2(𝑡,𝑎)𝑔^′−𝑀(𝑡,𝑎)𝜌(𝑡)_𝜃𝐿𝑔^𝑛−2(𝑡,𝑎)𝑔^′(𝑡,𝑎)𝑤4𝜌(𝑡)𝑟(𝑡)²(𝑡)=−𝑄₂𝑀(𝑡)−^𝜃𝐿𝑔^𝑛−2(𝑡,𝑎)𝑔^′(𝑡,𝑎)𝑤4𝜌(𝑡)𝑟(𝑡)²(𝑡). (3.8) The rest proof is similar to that of Theorem 2.1 and hence is omitted. This completes the proof of Theorem 3.1.

Theorem 3.2. Assume that the conditions of Theorem 2.1 and (2.26) hold; if there exists a function 𝜑(𝑡)∈𝐶([𝑡₀,∞),ℜ) satisfying lim_𝑡→∞1sup𝐻(𝑡,𝑢)^𝑡_𝑢[𝐻(𝑡,𝑠)𝑄₂ℎ(𝑠)−²(𝑡,𝑠)𝜌(𝑠)𝑟(𝑠)𝛽𝐿𝑀_𝜃𝑔^𝑛−2(𝑠,𝑎)𝑔^′(𝑠,𝑎)]𝑑𝑠≥𝜑(𝑢),𝑢≥𝑡₀,lim_𝑡→∞sup^𝑡_𝑡0𝑔^𝑛−2(𝑢,𝑎)𝑔^′(𝑢,𝑎)𝜑𝜌(𝑢)𝑟(𝑢)²⁺(𝑢)𝑑𝑢=∞,𝜑₊(𝑢)=max_𝑢≥𝑡0{𝜑(𝑢),(0)}, (3.9) then (1.1) is oscillatory.

Theorem 3.3. Let all assumptions of Theorem 2.6 be satisfied except that limsup in condition Theorem 3.2 is replaced with liminf, then (1.1) is oscillatory.

Acknowledgment

This research was partial supported by the NNSF of China (10771118).