Oscillation Theorems for Second-Order Forced Neutral Nonlinear Differential Equations with Delayed Argument

Abstract

We are concerned with the oscillation of the forced second-order neutral nonlinear differential equations with delayed argument in the form (𝑟(𝑡)(𝑥(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡)))^′)^′+𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))+∑^𝑛_𝑖=1𝑞_𝑖(𝑡)|𝑥(𝑡)|^𝜆𝑖sng𝑥(𝑡)=𝑒(𝑡). No restriction is imposed on the potentials 𝑝(𝑡), 𝑞_𝑖(𝑡), and𝑒(𝑡) to be nonnegative. Our methodology is somewhat different from those of previous authors.

1. Introduction

In this paper, we study the oscillatory behavior of the forced neutral nonlinear functional differential equation of the form

(𝑟(𝑡)(𝑥(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡)))^′)^′+𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))+^𝑛_𝑖=1𝑞_𝑖(𝑡)|𝑥(𝑡)|^𝜆𝑖sgn𝑥(𝑡)=𝑒(𝑡), (1.1) where 𝑡≥𝑡₀. In this paper, we assume that

(𝐼₁)𝑟(𝑡)∈𝐶([𝑡₀,∞),(0,∞)),𝑟^′(𝑡)≥0,∫^∞(1/𝑟(𝑡))𝑑𝑡=∞,(𝐼₂)𝑎(𝑡)∈𝐶([𝑡₀,∞),[0,1)),(𝐼₃)𝜎(𝑡)∈𝐶([𝑡₀,∞),ℝ) is nondecreasing, 𝜎(𝑡)≤𝑡 for 𝑡≥𝑡₀, and lim_𝑡→∞𝜎(𝑡)=∞,(𝐼₄)𝜏(𝑡)∈𝐶([𝑡₀,∞),ℝ),𝜏(𝑡)≤𝑡 for 𝑡≥𝑡₀ and lim_𝑡→∞𝜏(𝑡)=∞,(𝐼₅)𝑝(𝑡),𝑞_𝑖(𝑡), and 𝑒(𝑡) are continuous functions defined on [0,∞), 𝑝(𝑡)>0, 𝜆₁>⋯>𝜆_𝑚>1>𝜆_𝑚+1>⋯>𝜆_𝑛>0(𝑛>𝑚≥1),(𝐼₆)𝑓(𝑥) is nondecreasing, 𝑓(𝑥)/𝑥≥𝑀>0, and 𝑥≠0.

No restriction is imposed on the potentials 𝑝(𝑡),𝑞_𝑖(𝑡), and 𝑒(𝑡) to be nonnegative. As usual, a solution of (1.1) is called oscillatory if it is defined on some ray [𝑇,∞) with 𝑇≥0 and has unbounded set of zeros. (1.1) is called oscillatory if all of its solutions on some ray are oscillatory.

In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of second-order linear and nonlinear delay differential equations (see, for example, [1–22] and the references therein). Let us consider the familiar forced Emden-Fowler equation

𝑥^′′(𝑡)+𝑝(𝑡)|𝑥(𝑡)|^𝜆sgn𝑥(𝑡)=𝑒(𝑡),𝑡≥𝑡₀. (1.2) When 𝜆₁>1, (1.2) is known as the superlinear equation, and when 0<𝜆₁<1, it is known as the sublinear equation. The oscillation of (1.2) has been the subject of much attention during the last 50 years; see the seminal book by Agarwal, et al. [23]. Here, we refer to the papers [1–3] and the references cited therein. In this case, one can usually establish oscillation criteria for more general nonlinear equations by using a technique introduced by Kartsatos [9] where it is additionally assumed that 𝑓 is the second derivative of an oscillatory function. This approach has been expressed in [5, 6]. Sun [4] has extended these results to delay differential equations of the form of (1.2), where 𝜆≥1 and the potentials 𝑝(𝑡) and 𝑒(𝑡) are allowed to change sign. However, Sun [4] does not say anything else for the oscillation of equation (1.2) with 0<𝜆<1. Later, employing the arguments in [4], Çakmak and Tiryaki [7] have established similar oscillation criteria for the equation of the form

𝑥^′′(𝑡)+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=𝑒(𝑡), (1.3) where 𝑓(𝑥) is assumed to satisfy certain growth conditions.

Very recently, Sun et al. [13, 14] obtained some new oscillation criteria for the equations in the form

(𝑟(𝑡)𝑥^′(𝑡))^′+𝑝(𝑡)𝑥(𝑡)+^𝑛_𝑖=1𝑞_𝑖(𝑡)|𝑥(𝑡)|^𝜆𝑖sgn𝑥(𝑡)=0,(𝑟(𝑡)𝑥^′(𝑡))^′+𝑝(𝑡)𝑥(𝑡)+^𝑛_𝑖=1𝑞_𝑖(𝑡)|𝑥(𝑡)|^𝜆𝑖sgn𝑥(𝑡)=𝑒(𝑡), (1.4) where 𝜆₁>⋯>𝜆_𝑚>1>𝜆_𝑚+1>⋯>𝜆_𝑛>0(𝑛>𝑚≥1). He also established oscillation theorems when 𝑛>1. When 𝑛=1, this approach was initiated by Agarwal and Grace [1, pages 244–249] for higher-order equations and subsequently developed in papers of Ou and Wong [15], Q.Yang [18], X.Yang [19], as well as Sun and Agarwal [16, 17].

In [24], Xu and Meng studied the oscillation of the equation

(𝑟(𝑡)(𝑥(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡)))^′)^′+^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖(𝑦(𝜏_𝑖(𝑡)))=0,𝑡≥𝑡₀, (1.5) by using the generalized Riccati technique and the function class 𝒴.

The purpose of this paper is to give some new oscillation criteria for (1.1), which can be regarded as further investigation for the (1.1) including the papers of Sun and Wong [13], Xu and Meng [24]. These criteria do not assume that 𝑟(𝑡),𝑝(𝑡),𝑞_𝑖(𝑡), and 𝑒(𝑡) are of definite sign. Our methodology is somewhat different from those of previous authors, and the results we obtained are more general than those of Sun and Wong [13].

2. Main Results

We will need the following lemmas that have been proved in [13].

Lemma 2.1 (see [13]). Let 𝜆_𝑖,𝑖=1,2,…,𝑛, be n-tuple satisfying 𝜆₁>⋯>𝜆_𝑚>1>𝜆_𝑚+1>⋯>𝜆_𝑛>0. Then there exists an n-tuple (𝑘₁,𝑘₂,…,𝑘_𝑛) satisfying ^𝑛_𝑖=1𝜆_𝑖𝑘_𝑖=1,( (𝑎)) which also satisfies either ^𝑛_𝑖=1𝑘_𝑖<1,0<𝑘_𝑖<1,( (𝑏)) or ^𝑛_𝑖=1𝑘_𝑖=1,0<𝑘_𝑖<1.( (𝑐))

Lemma 2.2 (see [13]). Let 𝑢,𝐴,𝐵,𝐶,𝑎𝑛𝑑𝐷 be positive real numbers. Then(i)𝐴𝑢^𝛼+𝐵≥𝛼(𝛼−1)^1/𝛼−1𝐴^1/𝛼𝐵^1−1/𝛼𝑢,𝛼>1,(ii)𝐶𝑢−𝐷𝑢^𝛼≥(𝛼−1)𝛼^{𝛼/(1−𝛼)}𝐶^{𝛼/(𝛼−1)}𝐷^1/(1−𝛼),0<𝛼<1.

Remark 2.3. For a given set of exponents 𝜆_𝑖 satisfying 𝜆₁>⋯>𝜆_𝑚>1>𝜆_𝑚+1>⋯>𝜆_𝑛>0, Lemma 2.1 ensures the existence of an n-tuple (𝑘₁,𝑘₂,…,𝑘_𝑛) such that either (𝑎) and (𝑏) hold or (𝑎) and (𝑐) hold. When n = 2 and 𝜆₁>1>𝜆₂>0, in the first case, we have that𝑘₁=1−𝜆₂(1−𝑘₀)𝜆₁−𝜆₂,𝑘₂=𝜆₁(1−𝑘₀)−1𝜆₁−𝜆₂,( (𝑑)) where 𝑘₀ can be any positive number satisfying 0<𝑘₀<(𝜆₁−1)/𝜆₁. This will ensure that 0<𝑘₁,𝑘₂<1, and conditions (𝑎) and (𝑏) are satisfied. In the second case, we simply solve (𝑎) and (𝑐) and obtain 𝑘₁=1−𝜆₂𝜆₁−𝜆₂,𝑘₂=𝜆₁−1𝜆₁−𝜆₂.( (𝑒))

Theorem 2.4. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂, and 𝑞_𝑖(𝑡)≥0,𝑡∈[𝜏(𝑎₁),𝑏₁]∪[𝜏(𝑎₂),𝑏₂],𝑖=1,…,𝑛,𝑒(𝑡)≤0,𝑡∈[𝜏(𝑎₁),𝑏₁],𝑒(𝑡)≥0,𝑡∈[𝜏(𝑎₂),𝑏₂]. (2.1) Let 𝐷(𝑎_𝑗,𝑏_𝑗)={𝑢∈𝐶¹[𝑎_𝑗,𝑏_𝑗]∶𝑢^𝜈+1>0,𝜈>0𝑖𝑠𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,𝑡∈(𝑎_𝑗,𝑏_𝑗),𝑎𝑛𝑑𝑢(𝑎_𝑗)=𝑢(𝑏_𝑗)=0}, for j = 1, 2. Assume that there exists a positive, nondecreasing function 𝜌∈𝐶¹([𝑡₀,∞),ℝ) such that, for some 𝐻∈𝐷(𝑎_𝑗,𝑏_𝑗) and for some 𝜃≥1, ^𝑏𝑗_𝑎𝑗[𝐻^𝜈+1(𝑡)𝜌(𝑡)𝑅(𝑡)−𝜃𝜌(𝑡)𝑟(𝑡)𝐻^𝜈−1(𝑡)𝐴²(𝑡)4]𝑑𝑡>0, (2.2) for j = 1, 2, then (1.1) is oscillatory, where 𝑅(𝑡)=𝑀𝑝(𝑡)[1−𝑎(𝜏(𝑡))]𝜏(𝑡)−𝜏(𝑎_𝑗)𝑡−𝜏(𝑎_𝑗)+𝑎₀(1−𝑎(𝑡))|𝑒(𝑡)|^𝑘0𝑛_𝑖=1𝑞^𝑘𝑖_𝑖𝜌(𝑡),𝐴(𝑡)=𝐻(𝑡)^′(𝑡)𝜌(𝑡)+(𝜈+1)𝐻^′(𝑡), (2.3)𝑎₀=∏^𝑛_𝑖=0𝑘^−𝑘𝑖_𝑖, and 𝑘₀,𝑘₁,…,𝑘_𝑛 are positive constants satisfying (𝑎) and (𝑏) of Lemma 2.1.

Proof. Assume to the contrary that there exists a solution 𝑥(𝑡) of (1.1) such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0,𝑥(𝜎(𝑡))>0, when 𝑡≥𝑡₀>0, for some 𝑡₀ depending on the solution 𝑥(𝑡). Set 𝑧(𝑡)=𝑥(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡)). (2.4)
By assumption, we have that 𝑧(𝑡)>0 for 𝑡≥𝑡₀≥0, and from (2.4) it follows that (𝑟(𝑡)𝑧^′(𝑡))^′=𝑒(𝑡)−𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))−^𝑛_𝑖=1𝑞_𝑖(𝑡)|𝑥(𝑡)|^𝜆𝑖sgn𝑥(𝑡)≤0,𝑡≥𝑡₀≥0. (2.5) It is not difficult to show that 𝑧^′(𝑡) is eventually positive. In fact, first, we know that 𝑧^′(𝑡)≢0 for sufficiently large 𝑡, since 𝑧(𝑡) is nontrivial. Second, if there exists an 𝑡₁≥𝑡₀ such that 𝑟(𝑡₁)𝑧^′(𝑡₁)=𝐶<0, then 𝑟(𝑡)𝑧^′(𝑡)≤𝐶 for 𝑡≥𝑡₁≥𝑡₀, that is, 𝑧^′(𝑡)≤𝐶/𝑟(𝑡), and hence, 𝑧(𝑡)≤𝑧(𝑡₁)+∫^𝑡_𝑡1(𝐶/𝑟(𝑡))𝑑𝑡→−∞ as 𝑡→∞, which contradicts the fact that 𝑧(𝑡)>0. Without loss of generality; say 𝑧^′(𝑡)>0, 𝑡≥𝑡₀≥0. Thus we have that 𝑥(𝑡)≥𝑧(𝑡)−𝑎(𝑡)𝑧(𝑡),𝑡≥𝑡₀≥0. (2.6) Define 𝑤(𝑡)=𝜌(𝑡)𝑟(𝑡)𝑧^′(𝑡)𝑧(𝑡),𝑡≥𝑡₀. (2.7) It follows from (2.7) that 𝑤(𝑡) satisfies the following differential equality: 𝑤^′𝜌(𝑡)=^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))−𝑤𝑧(𝑡)²(𝑡)𝜌(𝑡)𝑟(𝑡)+𝜌(𝑡)𝑒(𝑡)−𝑧(𝑡)𝜌(𝑡)∑^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥^𝜆𝑖(𝑡).𝑧(𝑡) (2.8) Using (2.6), we have that 𝑤^′𝜌(𝑡)≤^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)𝑝(𝑡)𝑓((1−𝑎(𝜏(𝑡)))𝑧(𝜏(𝑡)))−𝑤𝑧(𝑡)²(𝑡)𝜌(𝑡)𝑟(𝑡)+𝜌(𝑡)𝑒(𝑡)𝑧(𝑡)−𝜌(𝑡)^𝑛_𝑖=1𝑞_𝑖(𝑡)(1−𝑎(𝑡))^𝜆𝑖𝑧^𝜆𝑖−1(𝑡), (2.9) and by the condition 𝑓(𝑥)/𝑥≥𝑀>0, we have that 𝑤^′𝜌(𝑡)≤^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝑀𝜌(𝑡)𝑝(𝑡)(1−𝑎(𝜏(𝑡)))𝑧(𝜏(𝑡))−𝑤𝑧(𝑡)²(𝑡)𝜌(𝑡)𝑟(𝑡)+𝜌(𝑡)𝑒(𝑡)𝑧(𝑡)−𝜌(𝑡)^𝑛_𝑖=1𝑞_𝑖(𝑡)(1−𝑎(𝑡))^𝜆𝑖𝑧^𝜆𝑖−1(𝑡). (2.10) By assumption, we can choose 𝑎₁,𝑏₁≥𝑡₀ such that 𝑏₁≥𝜏(𝑎₁),𝜏²(𝑎₁)=𝜏(𝜏(𝑎₁))≥𝑡₀,𝑞_𝑖(𝑡)≥0, 𝑖=1,2,…,𝑛, for 𝑡∈[𝜏(𝑎₁),𝑏₁], and 𝑒(𝑡)≤0 for 𝑡∈[𝜏(𝑎₁),𝑏₁]. Recall the arithmetic-geometric mean inequality (see [25]) ^𝑛_𝑖=0𝑘_𝑖𝑢_𝑖≥^𝑛_𝑖=0𝑢^𝑘𝑖_𝑖,𝑢_𝑖≥0, (2.11) where 𝑘₀=1−∑^𝑛_𝑖=1𝑘_𝑖 and 𝑘_𝑖>0,𝑖=1,2,…,𝑛 are chosen to satisfy (𝑎) and (𝑏) of Lemma 2.1 for the given 𝜆₁,𝜆₂,…,𝜆_𝑛>0. Now return to (2.10) and identify 𝑢₀=𝑘₀⁻¹|𝑒(𝑡)|𝑧⁻¹(𝑡) and 𝑢_𝑖=𝑘_𝑖⁻¹𝑞_𝑖(𝑡)𝑧^𝜆𝑖−1(𝑡)(1−𝑎(𝑡))^𝜆𝑖 in (2.11) to obtain 𝑤^′𝜌(𝑡)≤^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝑀𝜌(𝑡)𝑝(𝑡)(1−𝑎(𝜏(𝑡)))𝑧(𝜏(𝑡))−𝑤𝑧(𝑡)₂(𝑡)𝜌(𝑡)𝑟(𝑡)−𝜌(𝑡)(1−𝑎(𝑡))𝑘^−𝑘0₀|𝑒(𝑡)|^𝑘0𝑛_𝑖=1𝑘^−𝑘𝑖_𝑖𝑞^𝑘𝑖_𝑖(𝑡). (2.12) From (1.1), we can easily obtain 𝑧^′′(𝑡)≤0, for 𝑡∈[𝜏(𝑎₁),𝑏₁]. Therefore, we have that, for 𝑡∈[𝜏(𝑎₁),𝑏₁], 𝑧(𝑡)−𝑧(𝜏(𝑎₁))=𝑧^′(𝑠)(𝑡−𝜏(𝑎₁))≥𝑧^′(𝑡)(𝑡−𝜏(𝑎₁)). (2.13) Noting that 𝑧(𝑡)>0 for 𝑡≥𝜏(𝑎₁), we get by (2.13) that 𝑧(𝑡)≥𝑧^′(𝑡)(𝑡−𝜏(𝑎₁)),𝑡∈[𝜏(𝑎₁),𝑏₁], (2.14) that is, 𝑧^′(𝑡)≤1𝑧(𝑡)𝑡−𝜏(𝑎₁),𝑡∈[𝜏(𝑎₁),𝑏₁]. (2.15) Integrating (2.15) from 𝜏(𝑡) to 𝑡>𝑎₁, we obtain 𝑧(𝜏(𝑡))≥𝑧(𝑡)𝜏(𝑡)−𝜏(𝑎₁)𝑡−𝜏(𝑎₁),𝑡∈(𝑎₁,𝑏₁]. (2.16) By using (2.16) in (2.12), we have that, for 𝑡∈(𝑎₁,𝑏₁], 𝑤^′𝜌(𝑡)≤^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝑀𝜌(𝑡)𝑝(𝑡)[1−𝑎(𝜏(𝑡))]𝜏(𝑡)−𝜏(𝑎₁)𝑡−𝜏(𝑎₁)−𝑤²(𝑡)𝜌(𝑡)𝑟(𝑡)−𝜌(𝑡)(1−𝑎(𝑡))𝑘^−𝑘0₀|𝑒(𝑡)|^𝑘0𝑛_𝑖=1𝑘^−𝑘𝑖_𝑖𝑞^𝑘𝑖_𝑖=𝜌(𝑡)_′(𝑡)𝑤𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)𝑅(𝑡)−²(𝑡).𝜌(𝑡)𝑟(𝑡) (2.17) Multiplying both sides of (2.17) by 𝐻^𝜈+1(𝑡) as given in the hypothesis of Theorem 2.4 and integrating (2.17) from 𝑎₁ to 𝑏₁, we obtain ^𝑏1_𝑎1𝐻^𝜈+1(𝑡)𝜌(𝑡)𝑅(𝑡)𝑑𝑡≤^𝑏1_𝑎1𝐻^𝜈+1𝜌(𝑡)^′(𝑡)𝜌(𝑡)𝑤(𝑡)𝑑𝑡−^𝑏1_𝑎1𝐻^𝜈+1(𝑡)𝑤^′(𝑡)𝑑𝑡−^𝑏1_𝑎1𝐻^𝜈+1𝑤(𝑡)²(𝑡)𝜌(𝑡)𝑟(𝑡)𝑑𝑡. (2.18) Using the integration by parts formula, we have that ^𝑏1_𝑎1𝐻^𝜈+1(𝑡)𝑤^′(𝑡)𝑑𝑡=𝐻^𝜈+1(𝑡)𝑤(𝑡)∣^𝑏1_𝑎1−^𝑏1_𝑎1(𝜈+1)𝐻^𝜈(𝑡)𝐻^′(𝑡)𝑤(𝑡)𝑑𝑡=−^𝑏1_𝑎1(𝜈+1)𝐻^𝜈(𝑡)𝐻^′(𝑡)𝑤(𝑡)𝑑𝑡, (2.19) where 𝐻(𝑎₁)=𝐻(𝑏₁)=0. Substituting (2.19) into (2.18), we obtain ^𝑏1_𝑎1𝐻^𝜈+1(𝑡)𝜌(𝑡)𝑅(𝑡)𝑑𝑡≤^𝑏1_𝑎1𝐻^𝜈+1𝜌(𝑡)^′(𝑡)𝜌(𝑡)𝑤(𝑡)𝑑𝑡+^𝑏1_𝑎1(𝜈+1)𝐻^𝜈(𝑡)𝐻^′−(𝑡)𝑤(𝑡)𝑑𝑡_𝑏1𝑎1𝐻^𝜈+1𝑤(𝑡)²(𝑡)=𝜌(𝑡)𝑟(𝑡)𝑑𝑡_𝑏1𝑎1𝐴(𝑡)𝐻^𝜈(𝑡)𝑤(𝑡)𝑑𝑡−^𝑏1_𝑎1𝐻^𝜈+1𝑤(𝑡)²(𝑡)𝜌(𝑡)𝑟(𝑡)𝑑𝑡. (2.20) Then ^𝑏1_𝑎1𝐻^𝜈+1(𝑡)𝜌(𝑡)𝑅(𝑡)𝑑𝑡≤−^𝑏1_𝑎1[−𝐴(𝑡)𝐻^𝜈(𝑡)𝑤(𝑡)+𝐻^𝜈+1𝑤(𝑡)²(𝑡)𝜌(𝑡)𝑟(𝑡)]𝑑𝑡=−^𝑏1_𝑎1[𝐻^𝜈+1(𝑡)𝜃𝜌(𝑡)𝑟(𝑡)𝑤(𝑡)−𝜃𝜌(𝑡)𝑟(𝑡)4𝐻^𝜈+1𝐻(𝑡)^𝜈(𝑡)𝐴(𝑡)]²+𝑑𝑡_𝑏1𝑎1[𝜃𝜌(𝑡)𝑟(𝑡)4𝐻^𝜈+1𝐻(𝑡)^𝜈(𝑡)𝐴(𝑡)]²𝑑𝑡−^𝑏1_𝑎1(𝜃−1)𝐻^𝜈+1(𝑡)𝑤𝜃𝜌(𝑡)𝑟(𝑡)²(𝑡)𝑑𝑡. (2.21) From the hypothesis of Theorem 2.4 and (2.21), we have that ^𝑏𝑖_𝑎𝑖[𝐻^𝜈+1(𝑡)𝜌(𝑡)𝑅(𝑡)−(𝜃𝜌(𝑡)𝑟(𝑡)4𝐻^𝜈+1𝐻(𝑡)^𝜈(𝑡)𝐴(𝑡))²]𝑑𝑡=^𝑏𝑖_𝑎𝑖[𝐻^𝜈+1(𝑡)𝜌(𝑡)𝑅(𝑡)−𝜃𝜌(𝑡)𝑟(𝑡)𝐻^𝜈−1(𝑡)𝐴²(𝑡)4]𝑑𝑡≤0, (2.22) which contradicts (2.2). When 𝑥(𝑡) is eventually negative, we can obtain similar contradiction using the interval [𝜏(𝑎₂),𝑏₂] instead of [𝜏(𝑎₁),𝑏₁]. This completes the proof.

Remark 2.5. Let 𝑟(𝑡)=1, 𝑞_𝑖(𝑡)=0, and 𝑎(𝑡)=0 for 𝑖=1,2,…,𝑛. It is easy to see that Theorem 2.4 reduces to Theorem 1 of [7].

In Theorem 2.6, we do not impose any restriction on signs of those coefficients corresponding to sublinear terms of (1.1), that is, 𝑞_𝑙(𝑡) for 𝑙=𝑚+1,…,𝑛. If it is nonpositive, we can easily see that Theorem 2.4 is invalid. However, the following theorem is valid for this case.

Theorem 2.6. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and 𝑞_𝑖(𝑡)≥0,𝑡∈[𝜏(𝑎₁),𝑏₁]∪[𝜏(𝑎₂),𝑏₂],𝑖=1,…,𝑚,𝑒(𝑡)<0,𝑡∈[𝜏(𝑎₁),𝑏₁],𝑒(𝑡)>0,𝑡∈[𝜏(𝑎₂),𝑏₂]. (2.23) Assume that there exists a positive, nondecreasing function 𝜌∈𝐶¹([𝑡₀,∞),ℝ) such that, for some 𝐻∈𝐷(𝑎_𝑗,𝑏_𝑗) and for some 𝜃≥1, ^𝑏𝑗_𝑎𝑗[𝐻^𝜈+1(𝑡)𝜌(𝑡)𝑅(𝑡)−𝜃𝜌(𝑡)𝑟(𝑡)𝐻^𝜈−1(𝑡)𝐴²(𝑡)4]𝑑𝑡>0, (2.24) for j = 1, 2, then (1.1) is oscillatory, where 𝜌𝐴(𝑡)=𝐻(𝑡)^′(𝑡)𝜌(𝑡)+(𝜈+1)𝐻^′(𝑡),𝑅(𝑡)=𝑝(𝑡)𝑀[1−𝑎(𝜏(𝑡))]𝜏(𝑡)−𝜏(𝑎_𝑗)𝑡−𝜏(𝑎_𝑗)+^𝑚_𝑖=1𝜇_𝑖(𝛽_𝑖|𝑒(𝑡)|)^{1−1/𝜆𝑖}𝑞^1/𝜆𝑖_𝑖−(𝑡)_{𝑛𝑙=𝑚+1}𝛾_𝑙(𝛿_𝑙|𝑒(𝑡)|)^{1−1/𝜆𝑙}𝑞^1/𝜆𝑙_𝑙(𝑡), (2.25) with ∑^𝑚_𝑖=1𝛽_𝑖+∑^𝑛_𝑙=𝑚+1𝛿_𝑙=1 for 𝛽_𝑖>0,𝛿_𝑙>0, 𝜇_𝑖=𝜆_𝑖(𝜆_𝑖−1)^{1/𝜆𝑖−1},𝑖=1,…,𝑚,𝛾_𝑙=𝜆_𝑙(1−𝜆_𝑙)^{1/𝜆𝑙−1}, and 𝑞_𝑙(𝑡)=max{−𝑞_𝑙(𝑡),0},𝑙=𝑚+1,…,𝑛.

Proof. Assume to the contrary that there exists a solution 𝑥(𝑡) of (1.1) such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, when 𝑡≥𝑡₀>0, for some 𝑡₀ depending on the solution 𝑥(𝑡). When 𝑥(𝑡) is eventually negative, the proof follows the same argument using the interval [𝜏(𝑎₂),𝑏₂] instead of [𝜏(𝑎₁),𝑏₁]. Apply the assumption of 𝛽_𝑖 and 𝛿_𝑙, then (1.1) is rearranged as (𝑟(𝑡)(𝑥(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡)))^′)^′++𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))_𝑚𝑖=1[𝑞_𝑖(𝑡)|𝑥(𝑡)|^𝜆𝑖−𝛽_𝑖𝑒(𝑡)]+^𝑛_𝑙=𝑚+1[𝑞_𝑙(𝑡)|𝑥(𝑡)|^𝜆𝑙−𝛿_𝑙𝑒(𝑡)]=0. (2.26) Noting the assumption (2.23) and applying Lemma 2.2(i) to the first summation term in (2.26), we get that (𝑟(𝑡)(𝑥(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡)))^′)^′+𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))+𝑥(𝑡)^𝑚_𝑖=1𝜇_𝑖(𝛽_𝑖|𝑒(𝑡)|)^{1−1/𝜆𝑖}𝑞^1/𝜆𝑖_𝑖+(𝑡)_{𝑛𝑙=𝑚+1}(𝑞_𝑙(𝑡)|𝑥(𝑡)|^𝜆𝑙−𝛿_𝑙𝑒(𝑡))≤0. (2.27) Introduce the Riccati substitution as (2.4) and apply Lemma 2.2(ii) to each of the nonlinear terms in the last sum in (2.27). Here 𝑢=𝑥(𝑡),𝛼=𝜆_𝑙,𝐷=𝑞_𝑙(𝑡), and 𝐶=𝛼(1−𝛼)^1/𝛼−1(𝛿_𝑙|𝑒(𝑡)|)^1−1/𝛼𝑞_𝑙^1/𝛼(𝑡). (2.28) We can obtain from (2.27) the following Riccati inequality: 𝑤^′𝜌(𝑡)≤^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)𝑝(𝑡)𝑀[1−𝑎(𝜏(𝑡))]𝜏(𝑡)−𝜏(𝑎₁)𝑡−𝜏(𝑎₁)−𝑤²(𝑡)𝜌(𝑡)𝑟(𝑡)−𝜌(𝑡)^𝑚_𝑖=1𝜇_𝑖(𝛽_𝑖|𝑒(𝑡)|)^{1−1/𝜆𝑖}𝑞^1/𝜆𝑖_𝑖(𝑡)+𝜌(𝑡)^𝑛_𝑙=𝑚+1𝛾_𝑙(𝛿_𝑙|𝑒(𝑡)|)^{1−1/𝜆𝑙}𝑞^1/𝜆𝑙_𝑙=𝜌(𝑡)_′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)𝑤𝑅(𝑡)−²(𝑡),𝜌(𝑡)𝑟(𝑡) (2.29) where 𝑞_𝑙(𝑡)=max{−𝑞_𝑙(𝑡),0}. The remaining argument is the same as that in Theorem 2.4 and this completes the proof.

Remark 2.7. Let 𝑓(𝑥)=𝑥,𝑎(𝑡)=0, and 𝜏(𝑡)=𝑡,𝑛=1. Theorems 2.4 and 2.6 reduce to Theorem 1 of [8, 11].

We will use the function class 𝒴 to study the oscillatory of (1.1). We say that a function Φ=Φ(𝑡,𝑠,𝑙) belongs to the function class 𝒴, denoted by Φ∈𝒴 if Φ∈𝐶(𝐸,ℝ), where 𝐸={(𝑡,𝑠,𝑙)∶𝑡₀≤𝑙≤𝑠≤𝑡<∞}, which satisfies Φ(𝑡,𝑡,𝑙)=0, andΦ(𝑡,𝑙,𝑙)=0,Φ(𝑡,𝑠,𝑙)≠0 for 𝑙<𝑠<𝑡, and has the partial derivative 𝜕Φ/𝜕𝑠 on 𝐸 such that 𝜕Φ/𝜕𝑠 is locally integrable with respect to 𝑠 in 𝐸.

We defined the operator 𝐵[⋅;𝑙,𝑡] by

𝐵[𝑔;𝑙,𝑡]=^𝑡_𝑙Φ²(𝑡,𝑠,𝑙)𝑔(𝑠)𝑑𝑠,for𝑡≥𝑠≥𝑙≥𝑡₀,𝑔(𝑠)∈𝐶[𝑡₀,∞), (2.30) and the function 𝜑=𝜑(𝑡,𝑠,𝑙) is defined by

𝜕Φ(𝑡,𝑠,𝑙)𝜕𝑠=𝜑(𝑡,𝑠,𝑙)Φ(𝑡,𝑠,𝑙). (2.31) It is easy to verify that 𝐵[⋅;𝑙,𝑡] is a linear operator and satisfies

𝐵[𝑔^′;𝑙,𝑡]=−2𝐵[𝑔𝜑;𝑙,𝑡],for𝑔∈𝐶¹[𝑡₀,∞). (2.32)

Theorem 2.8. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and satisfies (2.1). Assume that there exists a function Φ∈𝒴, such that, for each 𝑡≥𝑡₀ and for some 𝜃≥1, limsup_𝑡→∞𝐵[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4𝜌(2𝜑−^′(𝑠))𝜌(𝑠)²;𝑡₀,𝑡]>0, (2.33) for j=1, 2, then (1.1) is oscillatory, where the operator B is defined by (2.30), 𝜑=𝜑(𝑡,𝑠,𝑡₀) is defined by (2.31), 𝑅(𝑠)=𝑀𝑝(𝑠)[1−𝑎(𝜏(𝑠))]𝜏(𝑠)−𝜏(𝑎_𝑗)𝑠−𝜏(𝑎_𝑗)+𝑎₀(1−𝑎(𝑠))|𝑒(𝑠)|^𝑘0𝑛_𝑖=1𝑞^𝑘𝑖_𝑖(𝑠), (2.34)𝑎₀=∏^𝑛_𝑖=0𝑘^−𝑘𝑖_𝑖, and 𝑘₀,𝑘₁,…,𝑘_𝑛 are positive constants satisfying (𝑎) and (𝑏) of Lemma 2.1.

Proof. We proceed as in Theorem 2.4. Assume to the contrary that there exists a solution 𝑥(𝑡) of (1.1) such that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0, when 𝑡⩾𝑡₀>0, for some 𝑡₀ depending on the solution 𝑥(𝑡). From the proof of Theorem 2.4, we obtain (2.16) for all 𝑡∈[𝜏(𝑎₁),𝑏₁]. Applying 𝐵[⋅;𝑡₀,𝑡],(𝑡≥𝑡₀) to (2.17), we have that 𝐵[𝑤^′(𝑠);𝑡₀𝜌,𝑡]≤−𝐵[^′(𝑠)𝜌(𝑠)𝑤(𝑠);𝑡₀,𝑡]−𝐵[𝜌(𝑠)𝑅(𝑠);𝑡₀𝑤,𝑡]−𝐵[²(𝑠)𝜌(𝑠)𝑟(𝑠);𝑡₀,𝑡]. (2.35) By (2.32) and above inequality, we have, for 𝑡≥𝑡₀, that 𝐵[𝜌(𝑠)𝑅(𝑠);𝑡₀,𝑡]≤2𝐵[𝑤(𝑠)𝜑;𝑡₀𝜌,𝑡]−𝐵[^′(𝑠)𝜌(𝑠)𝑤(𝑠);𝑡₀𝑤,𝑡]−𝐵[²(𝑠)𝜌(𝑠)𝑟(𝑠);𝑡₀,𝑡]=𝐵[−(𝑤(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)𝜌𝜃𝜌(𝑠)𝑟(𝑠)(𝜑−^′(𝑠)2𝜌(𝑠)))²+𝜃𝜌(𝑠)𝑟(𝑠)4𝜌(2𝜑−^′(𝑠))𝜌(𝑠)²−𝜃−1𝜃𝑤²(𝑠)𝜌(𝑠)𝑟(𝑠);𝑡₀,𝑡]≤𝐵[𝜃𝜌(𝑠)𝑟(𝑠)4𝜌(2𝜑−^′(𝑠))𝜌(𝑠)²;𝑡₀,𝑡]. (2.36)
That is, 𝐵[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4𝜌(2𝜑−^′(𝑠))𝜌(𝑠)²;𝑡₀,𝑡]≤0. (2.37) Taking the super limit in the above inequality, we have that limsup_𝑡→∞𝐵[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4𝜌(2𝜑−^′(𝑠))𝜌(𝑠)²;𝑡₀,𝑡]≤0, (2.38) which contradicts assumption (2.33). When 𝑥(𝑡) is eventually negative, the proof follows the same argument using the interval [𝜏(𝑎₂),𝑏₂] instead of [𝜏(𝑎₁),𝑏₁]. This completes the proof.

Theorem 2.9. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and satisfies (2.23). Assume that there exists a function Φ∈𝒴 such that, for each 𝑡≥𝑡₀ and for some 𝜃≥1, limsup_𝑡→∞𝐵[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4𝜌(2𝜑−^′(𝑠))𝜌(𝑠)²;𝑡₀,𝑡]>0, (2.39) for j = 1, 2, then (1.1) is oscillatory, where the operator B is defined by (2.30), 𝜑=𝜑(𝑡,𝑠,𝑡₀) is defined by (2.31), and 𝑅(𝑡) is defined by (2.25).

The proof of Theorem 2.9 can be completed by following the proofs of Theorems 2.6 and 2.8 with suitable changes, we omit it here.

3. Corollaries

As Theorems 2.4–2.9 are rather general, it is convenient for applications to derive a number of oscillation criteria with the appropriate choice of the functions 𝐻,𝜌, and Φ(𝑡,𝑠,𝑙).

With an appropriate choice of the functions 𝐻(𝑡),𝜌(𝑡), one can derive from Theorems 2.4 and 2.6 a number of oscillation criteria for (1.1). For example, we consider the simple case 𝑛=2; hence ,𝜆₁>1>𝜆₂>0. Let 𝑏_𝑗=𝑎_𝑗+𝜋/𝑐_𝑗,𝑏_𝑗=𝑎_𝑗+𝜋/4𝑑_𝑗,𝑗=1,2, where 𝑎_𝑗,𝑏_𝑗 are given in Theorem 2.4. This determines 𝑐_𝑗,𝑑_𝑗,𝑗=1,2. Choose 𝐻(𝑡)=sin[𝑐_𝑗(𝑡−𝑎_𝑗)], 𝜌(𝑡)=sin^𝑘[𝑑_𝑗(𝑡−𝑎_𝑗)], and k being natural number. We obtain from Theorems 2.4 and 2.6 the following corollaries.

Corollary 3.1. Let 𝑘₀,𝑘₁,𝑘₂ be chosen to satisfy (𝑎), (𝑏) of Lemma 2.1 for 𝜆₁>1>𝜆₂>0. If for any 𝑇≥𝑡₀, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂, then it holds that 𝑞₁𝑞(𝑡)≥0,₂(𝑡)≥0,𝑡∈[𝜏(𝑎₁),𝑏₁]∪[𝜏(𝑎₂),𝑏₂],𝑒(𝑡)≤0,𝑡∈[𝜏(𝑎₁),𝑏₁],𝑒(𝑡)≥0,𝑡∈[𝜏(𝑎₂),𝑏₂], (3.1) and for 𝑐_𝑗=𝜋²/(𝑏_𝑗−𝑎_𝑗)²,16𝑑_𝑗=𝜋²/(𝑏_𝑗−𝑎_𝑗)², and 𝜃≥1, one has that ^𝑏𝑗_𝑎𝑗[(𝑝(𝑡)𝑀(1−𝑎(𝜏(𝑡)))𝜏(𝑡)−𝜏(𝑎_𝑗)𝑡−𝜏(𝑎_𝑗)+𝑘^−𝑘0₀𝑘^−𝑘1₁𝑘^−𝑘2₂|𝑒(𝑡)|^𝑘0𝑞^𝑘1₁(𝑡)𝑞^𝑘2₂(𝑡)(1−𝑎(𝑡)))×sin^𝜈+1[𝑐_𝑗(𝑡−𝑎_𝑗)]sin^𝑘[𝑑_𝑗(𝑡−𝑎_𝑗)]−𝜃𝑟(𝑡)𝑐_𝑗sin^𝜈−1[𝑐_𝑗(𝑡−𝑎_𝑗)]4sin^𝑘[𝑑_𝑗(𝑡−𝑎_𝑗)]×(𝑘𝑑_𝑗sin[𝑐_𝑗(𝑡−𝑎_𝑗)]sin^𝑘−1[𝑑_𝑗(𝑡−𝑎_𝑗)]cos[𝑑_𝑗(𝑡−𝑎_𝑗)]+(𝜈+1)𝑐_𝑗sin^𝑘[𝑑_𝑗(𝑡−𝑎_𝑗)]cos[𝑐_𝑗(𝑡−𝑎_𝑗)])²]𝑑𝑡>0, (3.2) then all solutions of the equation (𝑟(𝑡)(𝑥(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡)))^′)^′+𝑝(𝑡)𝑓(𝑥(𝜏(𝑡)))+𝑞₁(𝑡)|𝑥(𝑡)|^𝜆1sgn𝑥(𝑡)+𝑞₂(𝑡)|𝑥(𝑡)|^𝜆2sgn𝑥(𝑡)=𝑒(𝑡) (3.3) are oscillatory.

Corollary 3.2. Assume that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and 𝑞₁(𝑡)≥0,𝑡∈[𝜏(𝑎₁),𝑏₁]∪[𝜏(𝑎₂),𝑏₂],𝑒(𝑡)<0,𝑡∈[𝜏(𝑎₁),𝑏₁],𝑒(𝑡)>0,𝑡∈[𝜏(𝑎₂),𝑏₂]. (3.4) Also assume that, for 𝜆₁>1>𝜆₂>0, and that there exists some 𝜆∈(0,1) such that ^𝑏𝑗_𝑎𝑗[(𝑝(𝑡)𝑀(1−𝑎(𝜏(𝑡)))𝜏(𝑡)−𝜏(𝑎_𝑗)𝑡−𝜏(𝑎_𝑗)+𝜆₁(𝜆₁−1)^1/𝜆1−1(𝜆|𝑒(𝑡)|)^1−1/𝜆1𝑞^1/𝜆1₁(𝑡)(1−𝑎(𝑡))−𝜆₂(1−𝜆₂)^1/𝜆2−1((1−𝜆)|𝑒(𝑡)|)^1−1/𝜆2𝑞^1/𝜆2₂(𝑡)(1−𝑎(𝑡)))×sin^𝜈+1[𝑐_𝑗(𝑡−𝑎_𝑗)]sin^𝑘[𝑑_𝑗(𝑡−𝑎_𝑗)]−𝜃𝑟(𝑡)𝑐_𝑗sin^𝜈−1[𝑐_𝑗(𝑡−𝑎_𝑗)]4sin^𝑘[𝑑_𝑗(𝑡−𝑎_𝑗)]×(𝑘𝑑_𝑗sin[𝑐_𝑗(𝑡−𝑎_𝑗)]sin^𝑘−1[𝑑_𝑗(𝑡−𝑎_𝑗)]cos[𝑑_𝑗(𝑡−𝑎_𝑗)]+(𝜈+1)𝑐_𝑗sin^𝑘[𝑑_𝑗(𝑡−𝑎_𝑗)]cos[𝑐_𝑗(𝑡−𝑎_𝑗)])²]𝑑𝑡>0, (3.5) where 𝑞₂(𝑡)=max{−𝑞₂(𝑡),0}, then all solutions of (3.3) are oscillatory.

If we choose Φ(𝑡,𝑠,𝑙)=𝜌(𝑠)(𝑡−𝑠)^𝛼(𝑠−𝑙)^𝛽 for 𝛼,𝛽>1/2 and 𝜌(𝑠)∈𝐶¹([𝑡₀,(0,∞))), then we have that

𝜌𝜑(𝑡,𝑠,𝑙)=^′(𝑠)+𝜌(𝑠)𝛽𝑡−(𝛼+𝛽)𝑠+𝛼𝑙.(𝑡−𝑠)(𝑠−𝑙) (3.6) Thus by Theorems 2.8 and 2.9, we have two new oscillation results.

Corollary 3.3. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and satisfies (2.1). For each 𝑡≥𝑡₀ and for some 𝜃≥1, limsup_𝑡→∞^𝑡_𝑡0𝜌²(𝑡)(𝑡−𝑠)^2𝛼(𝑠−𝑡₀)^2𝛽[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4(2𝛽𝑡−(𝛼+𝛽)𝑠+𝛼𝑡₀(𝑡−𝑠)(𝑠−𝑡₀)+𝜌^′(𝑠))𝜌(𝑠)²]𝑑𝑠>0, (3.7) for j = 1, 2, then (1.1) is oscillatory, where 𝑅(𝑠)=𝑀𝑝(𝑠)[1−𝑎(𝜏(𝑠))]𝜏(𝑠)−𝜏(𝑎_𝑗)𝑠−𝜏(𝑎_𝑗)+𝑎₀(1−𝑎(𝑠))|𝑒(𝑠)|^𝑘0𝑛_𝑖=1𝑞^𝑘𝑖_𝑖(𝑠), (3.8)𝑎₀=∏^𝑛_𝑖=0𝑘^−𝑘𝑖_𝑖, and 𝑘₀,𝑘₁,…,𝑘_𝑛 are positive constants satisfying (𝑎) and (𝑏) of Lemma 2.1.

Corollary 3.4. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and satisfies (2.23). Assume that there exists a function Φ∈𝑌 such that, for each 𝑡≥𝑡₀ and for some 𝜃≥1, limsup_𝑡→∞^𝑡_𝑡0𝜌²(𝑡)(𝑡−𝑠)^2𝛼(𝑠−𝑡₀)^2𝛽[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4(2𝛽𝑡−(𝛼+𝛽)𝑠+𝛼𝑡₀(𝑡−𝑠)(𝑠−𝑡₀)+𝜌^′(𝑠))𝜌(𝑠)²]𝑑𝑠>0, (3.9) for j = 1, 2, then (1.1) is oscillatory, where 𝑅(𝑡) is defined by (2.25).

We say that a function 𝐻=𝐻(𝑡,𝑠) belongs to the function class 𝒳, if 𝐻∈𝐶(𝐷,ℝ⁺), where 𝐷={(𝑡,𝑠)∶𝑡₀≤𝑠≤𝑡<∞}, which satisfies 𝐻(𝑡,𝑡)=0,𝐻(𝑡,𝑠)>0 for 𝑡>𝑠 and has partial derivatives 𝜕𝐻/𝜕𝑠 and 𝜕𝐻/𝜕𝑡 on 𝐷 such that

𝜕𝐻𝜕𝑡=ℎ₁(𝑡,𝑠)𝐻(𝑡,𝑠),𝜕𝐻𝜕𝑠=−ℎ₂(𝑡,𝑠)𝐻(𝑡,𝑠), (3.10) where ℎ₁(𝑡,𝑠),ℎ₂(𝑡,𝑠) are locally integrable with respect to 𝑡 and 𝑠, respectively, in 𝐷.

If we choose Φ(𝑡,𝑠,𝑙)=𝐻₁(𝑠,𝑙)𝐻₂(𝑡,𝑠), for 𝐻₁,𝐻₂∈𝒳, then we have that

1𝜑(𝑡,𝑠,𝑙)=2(ℎ₁⁽¹⁾(𝑠,𝑙)𝐻₁−ℎ(𝑠,𝑙)²⁽²⁾(𝑡,𝑠)𝐻₂(𝑡,𝑠)), (3.11) where ℎ₁⁽¹⁾(𝑠,𝑙),ℎ₂⁽²⁾(𝑡,𝑠) are defined as the following:

𝜕𝐻₁(𝑠,𝑙)𝜕𝑠=ℎ₁⁽¹⁾(𝑠,𝑙)𝐻₁(𝑠,𝑙),𝜕𝐻₂(𝑡,𝑠)𝜕𝑠=−ℎ₂⁽²⁾(𝑡,𝑠)𝐻₂(𝑡,𝑠). (3.12) Thus by Theorems 2.8 and 2.9, we also have the two following oscillation results.

Corollary 3.5. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and satisfies (2.1). For each 𝑡≥𝑡₀ and for some 𝜃≥1, limsup_𝑡→∞^𝑡_𝑡0𝐻₁(𝑠,𝑡₀)𝐻₂(𝑡,𝑠)[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4(ℎ₁⁽¹⁾(𝑠,𝑡₀)𝐻₁(𝑠,𝑡₀)−ℎ₂⁽²⁾(𝑡,𝑠)𝐻₂−𝜌(𝑡,𝑠)^′(𝑠))𝜌(𝑠)²]𝑑𝑠>0, (3.13) for j = 1, 2, then (1.1) is oscillatory, where ℎ₁⁽¹⁾(𝑠,𝑡₀),ℎ₂⁽²⁾(𝑡,𝑠) are defined by (3.12), 𝑅(𝑠)=𝑀𝑝(𝑠)[1−𝑎(𝜏(𝑠))]𝜏(𝑠)−𝜏(𝑎_𝑗)𝑠−𝜏(𝑎_𝑗)+𝑎₀(1−𝑎(𝑠))|𝑒(𝑠)|^𝑘0𝑛_𝑖=1𝑞^𝑘𝑖_𝑖(𝑠), (3.14)𝑎₀=∏^𝑛_𝑖=0𝑘^−𝑘𝑖_𝑖, and 𝑘₀,𝑘₁,…,𝑘_𝑛 are positive constants satisfying (𝑎) and (𝑏) of Lemma 2.1.

Corollary 3.6. Suppose that, for any 𝑇≥0, there exist constants 𝑎₁,𝑏₁,𝑎₂,𝑏₂ such that 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ and satisfies (2.23). Assume that there exists a function Φ∈𝒴 such that, for each 𝑡≥𝑡₀ and for some 𝜃≥1, limsup_𝑡→∞^𝑡_𝑡0𝐻₁(𝑠,𝑡₀)𝐻₂(𝑡,𝑠)[𝜌(𝑠)𝑅(𝑠)−𝜃𝜌(𝑠)𝑟(𝑠)4(ℎ₁⁽¹⁾(𝑠,𝑡₀)𝐻₁(𝑠,𝑡₀)−ℎ₂⁽²⁾(𝑡,𝑠)𝐻₂−𝜌(𝑡,𝑠)^′(𝑠))𝜌(𝑠)²]𝑑𝑠>0, (3.15) for j = 1, 2, then (1.1) is oscillatory, where ℎ₁⁽¹⁾(𝑠,𝑡₀),ℎ₂⁽²⁾(𝑡,𝑠) is defined by (3.12) and 𝑅(𝑡) are defined by (2.25).

Acknowledgments

The aouthors thank the referees for their helpful comments to improve this paper. This research was supported by the NNSF of China (Grant no. 10771118) and NSF of Shandong (Grants no. ZR2009AM011 and no. ZR2009AQ010).