New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

Abstract

We establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms [𝑥(𝑡)+∑_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′′+𝑟(𝑡)[𝑥(𝑡)+∑_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′+∑_𝑖∈𝑃𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))−∑_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))=0 and [𝑥(𝑡)+∑_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′′+𝑟(𝑡)[𝑥(𝑡)+∑_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′+∑_𝑖∈𝑃𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))−∑_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))=𝑓(𝑡). The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al. (2009).

1. Introduction

In this paper, we consider the oscillation of all solutions of the second-order neutral delay differential equations with positive and negative coefficients having the forms[𝑥(𝑡)+_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′′+𝑟(𝑡)[𝑥(𝑡)+_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′+_𝑖∈𝑃𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))−_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))=0,[𝑥(𝑡)+_𝑖∈𝑅𝑐_i(𝑡)𝑥(𝛼_𝑖(𝑡))]^′′+𝑟(𝑡)[𝑥(𝑡)+_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′+_𝑖∈𝑃𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))−_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))=𝑓(𝑡). (1.1)(1.2)

We introduce the following class of functions 𝐷([𝑡₀,∞)) equipped with the functions satisfying the following properties:(𝑃₁)𝑓∈𝐶¹([𝑡₀,∞),ℝ) is strictly increasing and lim_𝑡→∞𝑓(𝑡)=∞ holds,(𝑃₁)𝑓(𝑡)≤𝑡 holds for all 𝑡≥𝑡₀.

In this paper, we make the following assumptions:(𝐻₁)𝑅,𝑃,𝑄 are bounded starting segments of positive integers; that is, 𝑅={1,2,…,𝑅₀},𝑃={1,2,…,𝑃₀},𝑄={1,2,…,𝑄₀},𝑅₀,𝑃₀,𝑄₀∈ℕ; (1.3)(𝐻₂)𝑐_𝑖∈𝐶([𝑡₀,∞),ℝ⁺) for all 𝑖∈𝑅,𝑝_𝑖∈𝐶([𝑡₀,∞),ℝ⁺) for all 𝑖∈𝑃, and 𝑞_𝑖∈𝐶([𝑡₀,∞),ℝ⁺) for all 𝑖∈𝑄;(𝐻₃)𝛼_𝑖∈𝐷([𝑡₀,∞)) with liminf_𝑡→∞𝛼^′_𝑖(𝑡)>0 for all 𝑖∈R,𝛽_𝑖∈𝐷([𝑡₀,∞)) for all 𝑖∈𝑃, and 𝛾_𝑖∈𝐷([𝑡₀,∞)) for all 𝑖∈𝑄;(𝐻₄)𝑟∈𝐶¹([𝑡₀,∞),ℝ⁺), and 𝑟^′(𝑡)≤0;(𝐻₅)𝑓∈𝐶([𝑡₀,∞),ℝ) and that there exists a function 𝐹∈𝐶²([𝑡₀,∞),ℝ) which satisfies 𝐹^′′=𝑓 and lim_𝑡→∞𝐹(𝑡)=0.

In order to establish our main results, we will assume that there exists a mapping 𝜓∶𝑄→𝑃 satisfying the following conditions:(𝐴₁)𝛾_𝑖(𝑡)≥𝛽_𝜓(𝑖)(𝑡) for all 𝑡≥𝑡₀ and 𝑖∈𝑄;(𝐴₂)ℎ_𝑖∈𝐶([𝑡₀,∞),ℝ⁺) for all 𝑖∈𝑃, where ℎ_𝑖𝑝(𝑡)∶=^𝑖(𝑡)−_{𝑗∈𝑄,𝜓(𝑗)=𝑖}𝜌^′_𝑗(𝑡)𝑞_𝑗(𝜌_𝑗𝑝(𝑡)),𝑖∈𝜓(𝑄),_𝑖(𝑡),𝑖∉𝜓(𝑄), (1.4) and 𝜌_𝑖(𝑡)∶=𝛾_𝑖⁻¹(𝛽_𝜓(𝑖)(𝑡)) for all 𝑖∈𝑄 and 𝑡≥𝑡₀;(𝐴₃) there exists 𝑖₀∈𝑃 such that liminf_𝑡→∞ℎ_𝑖0(𝑡)>0 and limsup_𝑡→∞𝛽^′_𝑖0(𝑡)<∞.

A function 𝑥 is called a solution of (1.1) (or (1.2)) provided that 𝑥 satisfies (1.1)(or (1.2)) identically on [𝑡₀,∞),𝑥+∑_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥∘𝛼_𝑖∈𝐶²([𝑡₀,∞),ℝ) and 𝑥∈𝐶([𝑡₋₁,∞),ℝ), where 𝑡₋₁:=min{𝛼,𝛽,𝛾},𝛼:=min{𝛼_𝑖(𝑡₀)∶𝑖∈𝑅},𝛽:=min{𝛽_𝑖(𝑡₀)∶𝑖∈𝑃}, and 𝛾:=min{𝛾_𝑖(𝑡₀)∶𝑖∈𝑄}. We restrict our attention only to the nontrivial solution 𝑥, that is, to the solution 𝑥 such that sup{|𝑥(𝑡)|∶𝑡≥𝑡₁}>0 for all 𝑡₁≥𝑡₀. A nontrivial solution of (1.1) (or (1.2)) is called oscillatory if it has arbitrary large zeros, otherwise, it is called nonoscillatory.

The oscillation and nonoscillation of solutions of second-order neutral delay differential equations have been studied by many authors; see [1–10]. However, to the best of our knowledge,there seem to be fewoscillation results for (1.1) and (1.2).

Recently,Manojlovićet al. [4] andWeng and Sun[10] have studied oscillation and asymptotic behavior of all solutions of the following equations:[𝑥(𝑡)+^𝑙_𝑖=1𝑐_𝑖(𝑡)𝑥(𝑡−𝜏_𝑖)]^′′+^𝑚_𝑖=1𝑝_𝑖(𝑡)𝑥(𝑡−𝛿_𝑖)−^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥(𝑡−𝜎_𝑖)=0,𝑡≥𝑡₀,[𝑥(𝑡)+^𝑙_𝑖=1𝑐_𝑖(𝑡)𝑥(𝑡−𝜏_𝑖)]^′′+^𝑚_𝑖=1𝑝_𝑖(𝑡)𝑥(𝑡−𝛿_𝑖)−^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥(𝑡−𝜎_𝑖)=𝑓(𝑡),𝑡≥𝑡₀, (1.5) and several well-known results have been obtained.

By using weaker conditions than in [4, 10], Karpuz et al. [1] have established oscillation criteria for differential equation [𝑥(𝑡)+_𝑖∈𝑅𝑟_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡))]^′′+_𝑖∈𝑃𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))−_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))=𝑓(𝑡). (1.6)

In this paper, we shall continue in the direction to study the oscillatory properties of (1.1) and (1.2). We establish new oscillation criteria for (1.1) and (1.2), which extend and improve the corresponding results in [1, 4, 10]. We also give two examples to illustrate our main results.

2. Main Results

The following properties of the set 𝐿¹([𝑡₀,∞)) in [1] are needed for our subsequent discussion.

Property 1. If 𝑓∈𝐿¹([𝑡₀,∞)) and 𝑓∈𝐶([𝑡₀,∞),ℝ⁺), then liminf_t→∞𝑓(𝑡)=0.

Corollary 2.1. Suppose that 𝑓∈𝐿¹([𝑡₀,∞)) and lim_𝑡→∞𝑓(𝑡) exists; then lim_𝑡→∞𝑓(𝑡)=0.

Property 2. If 𝑓∈𝐶([𝑡₀,∞),ℝ) and 𝑓∈𝐿¹([𝑡₁,∞)), where 𝑡₁≥𝑡₀, then we have 𝑓∈𝐿¹([𝑡₀,∞)).

Property 3. Let 𝑡₁ be such that 𝑔(𝑡₁)≥𝑡₀. Suppose 𝑔∈𝐷([𝑡₁,∞)) with limsup_𝑡→∞𝑔^′(𝑡)<∞ and 𝑓∈𝐶([𝑡₀,∞),ℝ⁺). If 𝑓∘𝑔∈𝐿¹([𝑡₁,∞)) holds, then 𝑓∈𝐿¹([𝑡₀,∞)).

Property 4. Let 𝑡₁ be such that 𝑔(𝑡₁)≥𝑡₀. Suppose 𝑔∈𝐷([𝑡₁,∞)) with liminf_𝑡→∞𝑔^′(𝑡)>0,𝑓∘𝑔∈𝐶([𝑡₁,∞),ℝ⁺), and 𝑓∈𝐶([𝑡₀,∞),ℝ⁺). If 𝑓∈𝐿¹([𝑡₀,∞)) holds, then 𝑓∘𝑔∈𝐿¹([𝑡₁,∞)) holds.

For simplicity, we denote the set of bounded functions by 𝐵([𝑡₀,∞)):={𝑓∈𝐶([𝑡₀,∞),ℝ)∶‖𝑓‖<∞}, (2.1) where ‖𝑓‖:=sup{|𝑓(𝑡)|,𝑡≥𝑡₀}. (2.2) For an arbitrary function 𝜓∶𝑄→𝑃, which satisfies (𝐴₁)-(𝐴₃), we denote the function 𝜑∶[𝑡₀,∞)→ℝ⁺ by 𝜑(𝑡):=_𝑖∈𝑄^𝑡_{𝜌𝑖(𝑡)}𝑞_𝑖(𝑣)𝑑𝑣,𝑡≥𝑡₀. (2.3)

In this section, for convenience, we suppose that 𝑞_𝑖≡0 holds for all 𝑖∈𝑄 on [𝑡₋₁,𝑡₀).

We start with the following Theorem.

Theorem 2.2. Assume that (𝐻₁)-(𝐻₄) hold and there exists a mapping 𝜓∶𝑄→𝑃 which satisfies (𝐴₁)-(𝐴₃) and that ∑_𝑖∈𝑅𝑐_𝑖∈𝐵([𝑡₀,∞)). If 𝜑∈𝐿¹([𝑡₀,∞)), then every solution 𝑥 of (1.1) is oscillatory.

Proof. Suppose that 𝑥 is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that 𝑥(𝑡)>0 for 𝑡≥𝑡₀. Therefore, we may assume existence of 𝑡₁≥𝑡₀ such that ^∞_𝑡1𝜑(𝑢)𝑑𝑢<1,∀𝑡≥𝑡₁,𝑥(𝛼_𝑖(𝑡))>0,∀𝑖∈𝑅,𝑥(𝛽_𝑖(𝑡))>0,∀𝑖∈𝑃. (2.4)
Now, we set 𝑤(𝑡):=𝑥(𝑡)+_𝑖∈𝑅𝑐_𝑖(𝑡)𝑥(𝛼_𝑖(𝑡)),𝑡≥𝑡₁,𝑧(𝑡,𝑠):=𝑤(𝑡)+^𝑡_𝑠𝑟(𝑢)𝑤(𝑢)𝑑𝑢−_𝑖∈𝑄^𝑡_𝑠^𝑢_{𝜌𝑖(𝑢)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣𝑑𝑢,𝑡≥𝑠≥𝑡₁. (2.5)(2.6) By 𝑧^′(𝑡,𝑠), we denote differential of functions with respect to the first component. Considering (2.5), we rewrite (1.1) in the form 𝑤^′′(𝑡)+𝑟(𝑡)𝑤^′(𝑡)+_𝑖∈𝑃𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))−_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))=0 (2.7) on [𝑡₁,∞). By Leibnitz’s rule, (2.7) and (𝐻₄), we have 𝑧^′′(𝑡,𝑡₁)=𝑤^′′(𝑡)+𝑟^′(𝑡)𝑤(𝑡)+𝑟(𝑡)𝑤^′(𝑡)−_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))+_𝑖∈𝑄𝜌^′_𝑖(𝑡)𝑞_𝑖(𝜌_𝑖(𝑡))𝑥(𝛽_𝜓(𝑖)(𝑡))≤𝑤^′′(𝑡)+𝑟(𝑡)𝑤^′(𝑡)−_𝑖∈𝑄𝑞_𝑖(𝑡)𝑥(𝛾_𝑖(𝑡))+_𝑖∈𝑄𝜌^′_𝑖(𝑡)𝑞_𝑖(𝜌_𝑖(𝑡))𝑥(𝛽_𝜓(𝑖)(𝑡))=−_𝑖∈𝑃𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))+_𝑖∈𝑄𝜌^′_𝑖(𝑡)𝑞_𝑖(𝜌_𝑖(𝑡))𝑥(𝛽_𝜓(𝑖)(𝑡))=−_{𝑖∈𝜓(𝑄)}𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))−_{𝑖∉𝜓(𝑄)}𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))+_{𝑖∈𝜓(𝑄)𝑗∈𝑄,𝜓(𝑗)=𝑖}𝜌^′_𝑗(𝑡)𝑞_𝑗(𝜌_𝑗(𝑡))𝑥(𝛽_𝑖(𝑡))=−_{𝑖∈𝜓(𝑄)}[𝑝_𝑖(𝑡)−_{𝑗∈𝑄,𝜓(𝑗)=𝑖}𝜌^′_𝑗(𝑡)𝑞_𝑗(𝜌_𝑗(𝑡))]𝑥(𝛽_𝑖(𝑡))−_{𝑖∉𝜓(𝑄)}𝑝_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡)).=−_𝑖∈𝑃ℎ_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))≤0,∀𝑡≥𝑡₁, (2.8) which implies 𝑧^′(⋅,𝑡₁) and 𝑧(⋅,𝑡₁) is eventually strictly monotonic on [𝑡₁,∞). Hence there exists 𝑡₂≥𝑡₁ such that either 𝑧^′(𝑡,𝑡₁)<0 or 𝑧^′(𝑡,𝑡₁)>0 holds for all 𝑡≥𝑡₂.
We consider the following two possible cases:Case 1 (𝑧^′(𝑡,𝑡₁)>0forall𝑡≥𝑡₂). Integrating (2.8) from 𝑡₂ to ∞, we have ∞>𝑧^′(𝑡₂,𝑡₁)≥𝑧^′(𝑡₂,𝑡₁)−lim_𝑡→∞𝑧^′(𝑡,𝑡₁)=−^∞_𝑡2𝑧^′′(𝑢,𝑡₁)𝑑𝑢=_𝑖∈𝑃^∞_𝑡2ℎ_𝑖(𝑢)𝑥(𝛽_𝑖(𝑢))𝑑𝑢, (2.9) which implies that ∑_𝑖∈𝑃ℎ_𝑖(𝑡)⋅(𝑥∘𝛽_𝑖)(𝑡)∈𝐿¹[𝑡₂,∞). Therefore, for 𝑖₀∈𝑃 for which (𝐴₃) holds, we have 𝑥∘𝛽_𝑖0∈𝐿¹[𝑡₂,∞). Then we conclude that 𝑥∈𝐿¹[𝑡₀,∞) by Property 3. Hence ∑_𝑖∈𝑅𝑐_𝑖∈𝐵([𝑡₀,∞)).(𝐻₃) and Property 4 imply that 𝑤∈𝐿¹[𝑡₁,∞). Since 𝑟(𝑡) is bounded, 𝑏(𝑡)=𝑟(𝑡)𝑤(𝑡) is also integrable in [𝑡₁,∞). So we obtain that there exists a constant 𝑀>0 such that ^𝑡_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢≤𝑀,∀𝑡≥𝑡₁. (2.10) Let 𝑢(𝑡)=𝑤(𝑡)+^𝑡_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢,∀𝑡≥𝑡₂. (2.11) From (2.6), we have 𝑢^′(𝑡)=𝑧^′(𝑡,𝑡₁)+_𝑖∈𝑄^𝑡_{𝜌𝑖(𝑡)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣>0. (2.12) Then 𝑢(𝑡) is bounded and monotonous and lim_𝑡→∞𝑢(𝑡) exists. We can suppose that lim_𝑡→∞𝑢(𝑡)=𝜇>0 since 𝑢(𝑡)>𝑥(𝑡)>0and𝑢^′(𝑡)>0.
So there exists 𝑡₃≥𝑡₂ such that 𝑢(𝑡)>𝜇−𝜀/2,𝑡≥𝑡₃, for arbitrary 𝜀∈(0,𝜇−𝑀); by 𝑢(𝑡)=𝑤(𝑡)+^𝑡_𝑡3𝜀𝑟(𝑢)𝑤(𝑢)𝑑𝑢>𝜇−2, (2.13) we have 𝑤(𝑡)>−^𝑡_𝑡3𝜀𝑟(𝑢)𝑤(𝑢)𝑑𝑢+𝜇−21>𝜇−2=1(𝜇−𝑀)−𝑀2(𝜇−𝑀). (2.14) This implies that 𝑤(𝑡)∉𝐿¹([𝑡₁,∞), which is a contradiction.Case 2 (𝑧^′(𝑡,𝑡₁)<0forall𝑡≥𝑡₂). Since 𝑧^′(⋅,𝑡₁) is nonincreasing by (2.8), the inequality 𝑧^′(𝑡,𝑡₁)≤𝑧^′(𝑡₂,𝑡₁) implies that lim_𝑡→∞𝑧(𝑡,𝑡₁)=−∞. Hence 𝑧(⋅,𝑡₁)∉𝐵([𝑡₁,∞)). We claim that 𝑥∈𝐵([𝑡₀,∞)). On contrary, there exists 𝑡₃≥𝑡₂ such that 𝑧(𝑡₃,𝑡₁)<0 and 𝑥(𝑡₃)=sup{𝑥(𝑡)∶𝑡∈[𝑡₀,𝑡₃)}. We get the following contradiction: 0>𝑧(𝑡₃,𝑡₁)=𝑤(𝑡₃)+^𝑡3_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢−_𝑖∈𝑄^𝑡3_𝑡1^𝑢_{𝜌𝑖(𝑢)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣𝑑𝑢≥𝑥(𝑡₃)+𝑥(𝑡₃)^𝑡3_𝑡1𝑟(𝑢)𝑑𝑢−_𝑖∈𝑄^𝑡3_𝑡1^𝑢_{𝜌𝑖(𝑢)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣𝑑𝑢≥𝑥(𝑡₃)[1+^𝑡3_𝑡1𝑟(𝑢)𝑑𝑢−^𝑡3_𝑡1𝜑(𝑢)𝑑𝑢]>0, (2.15) since ^𝑡3_𝑡1𝜑(𝑢)𝑑𝑢<^∞_𝑡1𝜑(𝑢)𝑑𝑢<1. (2.16) Thus ‖𝑥‖<∞. Accordingly, by (2.4) and (2.6), it follows that 𝑧(𝑡,𝑡₁)≥𝑤(𝑡)+^𝑡_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢−‖𝑥‖^𝑡_𝑡1𝜑(𝑢)𝑑𝑢≥−‖𝑥‖^𝑡_𝑡1𝜑(𝑢)𝑑𝑢>−‖𝑥‖,∀𝑡≥𝑡₂. (2.17) Therefore, ‖𝑥‖≥‖𝑧(⋅,𝑡₁)‖ holds and we see that 𝑧(⋅,𝑡₁)∈𝐵([𝑡₁,∞)), which is a contradiction.
Therefore, we completed the proof by considering both possible cases.

Remark 2.3. When 𝑟(𝑡)=0, and Theorem 2.2 reduces to Theorem 3.1 in [1]. So Theorem 2.2 extends and improves the corresponding results in [1, 4, 10].

Theorem 2.4. Assume that (𝐻₁)-(𝐻₅) hold and there exists a mapping 𝜓∶𝑄→𝑃 which satisfies (𝐴₁)-(𝐴₃). Furthermore, assume that ∑_𝑖∈𝑅𝑐_𝑖∈𝐵([𝑡₀,∞)). If 𝜑∈𝐿¹([𝑡₀,∞)), then every solution 𝑥 of (1.2) is oscillatory or tends to zero asymptotically.

Proof. Suppose that 𝑥 is a nonoscillatory solution of (1.2). Without loss of generality, we assume that 𝑥(𝑡)>0 for 𝑡≥𝑡₀. Therefore, we may assume existence of a constant 𝜀>0 and 𝑡₁≥𝑡₀ such that (2.4) and 𝐹(𝑡)≤𝜀 hold for all 𝑡≥𝑡₁. Now, for 𝑡≥𝑠≥𝑡₁, set 𝑊(𝑡):=𝑤(𝑡)−𝐹(𝑡),𝑍(𝑡,𝑠):=𝑊(𝑡)+^𝑡_𝑠𝑟(𝑢)𝑤(𝑢)𝑑𝑢−_𝑖∈𝑄^𝑡_𝑠^𝑢_{𝜌𝑖(𝑢)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣𝑑𝑢+𝜀, (2.18)(2.19) where 𝑤(𝑡) is defined on the interval [𝑡₁,∞) as in (2.5). Then as in (2.8), we have 𝑍^′′(𝑡,𝑡₁)≤−_𝑖∈𝑃ℎ_𝑖(𝑡)𝑥(𝛽_𝑖(𝑡))≤0,∀𝑡≥𝑡₁. (2.20) Thus there exists 𝑡₂≥𝑡₁ satisfying either 𝑍^′(𝑡,𝑡₁)>0 or 𝑍^′(𝑡,𝑡₁)<0 for all 𝑡≥𝑡₂.
We consider the following two possible cases.Case 1 (𝑍^′(𝑡,𝑡₁)>0forall𝑡≥𝑡₂). In this case, one can show that 𝑤∈𝐿¹([𝑡₁,∞)) as shown in above proofs. Since 𝑟(𝑡) is bounded, 𝑟(𝑡)𝑤(𝑡) is also integrable in [𝑡₁,∞). Let 𝑣(𝑡)=𝑊(𝑡)+^𝑡_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢,𝑡≥𝑡₂. (2.21) By (2.19), we have 𝑣^′(𝑡)=𝑍^′(𝑡,𝑠)+_𝑖∈𝑄^𝑡_{𝜌𝑖(𝑡)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣>0,𝑡≥𝑡₂; (2.22) then 𝑣(𝑡) is bounded and monotonous and lim_𝑡→∞𝑣(𝑡) exists. By (2.21), we can obtain that lim_𝑡→∞𝑤(𝑡) exists. Letting 𝑎(𝑡)=𝑤(𝑡)+^𝑡_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢, (2.23) we can obtain that lim_𝑡→∞𝑎(𝑡) exists. Suppose that lim_𝑡→∞𝑎(𝑡)=𝜃, where 𝜃∈[0,∞). We claim 𝜃=0. Suppose that 𝜃∈(0,∞). By 𝑟(𝑡)𝑤(𝑡)∈𝐿¹([𝑡₁,∞)), we see that there exists 𝑁>0, such that ^𝑡_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢≤𝑁. (2.24) Because lim_𝑡→∞𝑎(𝑡)=𝜃>0, there exists 𝑡₃≥𝑡₂, such that 𝑎(𝑡)>𝜃−(1/2)𝜖,𝑡≥𝑡₃, for arbitrary 𝜖∈(0,𝜃−𝑁). But by 𝑤(𝑡)+^𝑡_𝑡3𝜖𝑟(𝑢)𝑤(𝑢)𝑑𝑢>𝜃−2, (2.25) we have 𝑤(𝑡)>−^𝑡_𝑡3𝜖𝑟(𝑢)𝑤(𝑢)𝑑𝑢+𝜃−21>𝜃−2=1(𝜃−𝑁)−𝑁2(𝜃−𝑁); (2.26) this implies that 𝑤(𝑡)∉𝐿¹([𝑡₂,∞), which is a contradiction. Therefore, lim_𝑡→∞𝑎(𝑡)=0. Since 0<𝑥(𝑡)<𝑎(𝑡) for 𝑡≥𝑡₂, we have that lim_𝑡→∞𝑥(𝑡)=0.Case 2 (𝑍^′(𝑡,𝑡₁)<0forall𝑡≥𝑡₂). Then we have that 𝑍(⋅,𝑡₁)∉𝐵([𝑡₁,∞)) by (2.20). We claim that 𝑥∈𝐵([𝑡₀,∞)). On contrary, there exists 𝑡₃≥𝑡₂ such that 𝑧(𝑡₃,𝑡₁)<0 and 𝑥(𝑡₃)=sup{𝑥(𝑡)∶𝑡∈[𝑡₀,𝑡₃)} hold and lim_𝑡→∞𝑍(𝑡,𝑡₁)<0.
Taking (2.4), (2.5), (2.18), and (2.19) into account, we get the following contradiction: 0>𝑍(𝑡₃,𝑡₁)=𝑊(𝑡₃)+^𝑡3_𝑡1𝑟(𝑢)𝑤(𝑢)𝑑𝑢−_𝑖∈𝑄^𝑡3_𝑡1^𝑢_{𝜌𝑖(𝑢)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣𝑑𝑢+𝜖≥𝑥(𝑡₃)[1+^𝑡3_𝑡1𝑟(𝑢)𝑑𝑢−^𝑡3_𝑡1𝜑(𝑢)𝑑𝑢]>0. (2.27) Hence ‖𝑥‖<∞. Accordingly, using (2.18), (2.19), and the fact that 𝑤(𝑡)>0 on [𝑡₂,∞), we have that 𝑍(𝑡,𝑡₂):=𝑊(𝑡)+^𝑡_𝑡2𝑟(𝑢)𝑤(𝑢)𝑑𝑢−_𝑖∈𝑄^𝑡_𝑠^𝑢_{𝜌𝑖(𝑢)}𝑞_𝑖(𝑣)𝑥(𝛾_𝑖(𝑣))𝑑𝑣𝑑𝑢+𝜖≥𝑤(𝑡)+^𝑡_𝑡2𝑟(𝑢)𝑤(𝑢)𝑑𝑢−‖𝑥‖^𝑡_𝑡2𝜑(𝑢)𝑑𝑢≥−‖𝑥‖,∀𝑡≥𝑡₂. (2.28) Thus ‖𝑥‖≥‖𝑍(⋅,𝑡₁)‖ and 𝑍(⋅,𝑡₁)∈𝐵([𝑡₁,∞)), which is a contradiction.Therefore, we completed the proof by considering both possible cases.

Remark 2.5. If there exists 𝐹∈𝐶²([𝑡₀,∞),ℝ) such that 𝐹^′′=𝑓 on [𝑡₀,∞) and 𝑙₀∶=lim_𝑡→∞𝐹(𝑡) exists and is finite, then 𝐺:=𝐹−𝑙₀ on [𝑡₀,∞) satisfies (𝐻₅).

Corollary 2.6. If all conditions of Theorem 2.4 hold, then every nonoscillatory solution of (1.2) converges to zero at infinity.

Remark 2.7. When 𝑟(𝑡)=0, Theorem 2.4 reduces to Theorem 3.3 in [1]. So Theorem 2.4 extends and improves the corresponding results in [1, 4, 10].

3. Examples

In this section, we provide two examples to illustrate our main results.

Example 3.1. Consider the following equation: [𝑥(𝑡)+3𝑥(𝑡−𝜋)]^′′+11+𝑡²[𝑥(𝑡)+3𝑥(𝑡−𝜋)]^′+(𝑡+1)𝑥(𝑡−2𝜋)−𝑒^{−𝑡−2𝜋}𝑥(𝑡−𝜋)=0,𝑡≥1. (3.1) Here, we have 𝑐₁(𝑡)=3,𝛼₁=𝑡−𝜋,𝛽₁=𝑡−2𝜋,𝛾₁𝑝=𝑡−𝜋,₁(𝑡)=𝑡+1,𝑞₁(𝑡)=𝑒^{−𝑡−2𝜋}1,𝑟(𝑡)=1+𝑡². (3.2) Set the function 𝜓 with 𝜓(1)=1, then 𝜌₁(𝑡)=𝑡−𝜋.
By simple calculation, we have ℎ₁(𝑡)=𝑡+1−𝑒^{−𝑡−𝜋},𝜑(𝑡)=𝑒^{−𝑡−𝜋}−𝑒^{−𝑡−2𝜋},liminf_𝑡→∞ℎ₁(𝑡)=∞,_∞1𝜑(𝑢)𝑑𝑢=𝑒^−1−𝜋−𝑒^−1−2𝜋. (3.3) Therefore, according to Theorem 2.2, every solution 𝑥 of (3.1) is oscillatory. Clearly, the known results in [1–10] cannot be applied to (3.1).

Example 3.2. Consider the following equation: [𝑥(𝑡)+𝑒^−𝑡𝑡𝑥(2)]^′′+11+𝑡²[𝑥(𝑡)+𝑒^−𝑡𝑡𝑥(2)]^′𝑡+𝑥(2𝑡−2𝜋)+𝑥(2−𝜋)−𝑒^−𝑡𝑡𝑥(2−3𝜋23)−4𝑒^−𝑡𝑡𝑥(2−𝜋)=𝑒^−𝑡−1𝑡³1sin𝑡,𝑡≥0. (3.4)
Here, we have 𝑐𝑅={1},𝑃={1,2},𝑄={1,2},₁(𝑡)=𝑒^−𝑡,𝑝₁(𝑡)=𝑝₂(𝑡)=1,𝑞₁(𝑡)=𝑒^−𝑡,𝑞₂3(𝑡)=−4𝑒^−𝑡,𝛼₁𝑡(𝑡)=2,𝛽₁𝑡(𝑡)=2−2𝜋,𝛽₂𝑡(𝑡)=2𝛾−𝜋,₁𝑡(𝑡)=2−3𝜋2,𝛾₂𝑡(𝑡)=2−𝜋,𝑓(𝑡)=𝑒^−𝑡+1𝑡³1sin𝑡1,𝑟(𝑡)=1+𝑡². (3.5) Set the function 𝜓∶𝑄→𝑃 with 𝜓(𝑖)=1 for 𝑖=1,2; then 𝜌₁(𝑡)=𝑡−𝜋and𝜌₂(𝑡)=𝑡−2𝜋.
By simple calculation, we have 𝐹(𝑡)=𝑒^−𝑡1+𝑡sin𝑡,ℎ₁(𝑡)=1−𝑒^−𝑡+𝜋−34𝑒^{−𝑡+2𝜋},ℎ₂(𝑡)=1,𝜑(𝑡)=^𝑡_𝑡−𝜋𝑒^−𝑣3𝑑𝑢+4^𝑡_𝑡−2𝜋𝑒^−𝑣𝑑𝑣,liminf_𝑡→∞ℎ₂(𝑡)=1,^∞₀3𝜑(𝑢)𝑑𝑢=4(𝑒^2𝜋−1)+𝑒^𝜋−1,lim_𝑡→∞𝐹(𝑡)=1. (3.6) Therefore, according to Theorem 2.4 and Remark 2.5, every solution 𝑥 of (3.4) is oscillatory or tends to zero asymptotically. Clearly, the known results in [1–10] cannot be applied to (3.4).

Acknowledgment

The authors gratefully acknowledge the referee who pointed out the important reference [1] and gave some valuable suggestions which improved the results in this paper. Project supported by the National Natural Science Foundation of China (10771118), STPF of University in Shandong Province of China (J09LA04) and the Research Foundation of Qufu Normal University (XJ0620).