Oscillation of Third-Order Neutral Delay Differential Equations

Abstract

The purpose of this paper is to examine oscillatory properties of the third-order neutral delay differential equation [𝑎(𝑡)(𝑏(𝑡)(𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡)))′)′]′+𝑞(𝑡)𝑥(𝜏(𝑡))=0. Some oscillatory and asymptotic criteria are presented. These criteria improve and complement those results in the literature. Moreover, some examples are given to illustrate the main results.

1. Introduction

This paper is concerned with the oscillation and asymptotic behavior of the third-order neutral differential equation𝑎(𝑡)𝑏(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))^′′′+𝑞(𝑡)𝑥(𝜏(𝑡))=0.( (𝐸)) We always assume that (H1)𝑎(𝑡),𝑏(𝑡),𝑝(𝑡),𝑞(𝑡)∈𝐶([𝑡₀,∞)),𝑎(𝑡)>0,𝑏(𝑡)>0,𝑞(𝑡)>0,(H2)𝜏(𝑡),𝜎(𝑡)∈𝐶([𝑡₀,∞)),𝜏(𝑡)≤𝑡,𝜎(𝑡)≤𝑡,lim_𝑡→∞𝜏(𝑡)=lim_𝑡→∞𝜎(𝑡)=∞.

We set 𝑧(𝑡)∶=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡)). By a solution of (𝐸), we mean a nontrivial function 𝑥(𝑡)∈𝐶([𝑇_𝑥,∞)),𝑇_𝑥≥𝑡₀, which has the properties 𝑧(𝑡)∈𝐶¹([𝑇_𝑥,∞)), 𝑏(𝑡)𝑧^′(𝑡)∈𝐶¹([𝑇_𝑥,∞)), 𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′∈𝐶¹([𝑇_𝑥,∞)) and satisfies (𝐸) on [𝑇_𝑥,∞). We consider only those solutions 𝑥(𝑡) of (𝐸) which satisfy sup{|𝑥(𝑡)|∶𝑡≥𝑇}>0 for all 𝑇≥𝑇_𝑥. We assume that (𝐸) possesses such a solution. A solution of (𝐸) is called oscillatory if it has arbitrarily large zeros on [𝑇_𝑥,∞); otherwise, it is called nonoscillatory. Equation (𝐸) is said to be almost oscillatory if all its solutions are oscillatory or convergent to zero asymptotically.

Recently, great attention has been devoted to the oscillation of differential equations; see, for example, the papers [1–30]. Hartman and Wintner [9], Hanan [10], and Erbe [8] studied a particular case of (𝐸), namely, the third-order differential equation 𝑥^′′′(𝑡)+𝑞(𝑡)𝑥(𝑡)=0. (1.1) Equation (𝐸) with 𝑝(𝑡)=0 plays an important role in the study of the oscillation of third-order trinomial delay differential equation 𝑥^′′′(𝑡)+𝑝(𝑡)𝑥^′(𝑡)+𝑔(𝑡)𝑥(𝜏(𝑡))=0, (1.2) see [6, 12, 27]. Baculíková and Džurina [21, 22], Candan and Dahiya [25], Grace et al. [28], and Saker and Džurina [30] examined the oscillation behavior of (𝐸) with 𝑝(𝑡)=0. It seems that there are few results on the oscillation of (𝐸) with a neutral term. Baculíková and Džurina [23, 24] and Thandapani and Li [17] investigated the oscillation of (𝐸) under the assumption 𝑏(𝑡)=1,^∞_𝑡01𝑎(𝑡)d𝑡=∞,𝑎^′(𝑡)≥0. (1.3) Graef et al. [13] and Candan and Dahiya [26] considered the oscillation of 𝑎(𝑡)𝑏(𝑡)𝑥(𝑡)+𝑝₁𝑥(𝑡−𝜎)^′′′+𝑞(𝑡)𝑥(𝑡−𝜏)=0,0≤𝑝₁<1. (𝐸₁)

In this paper, we shall further the investigation of the oscillations of (𝐸) and 𝐸₁. Three cases:^∞_𝑡01𝑎(𝑡)d𝑡=∞,^∞_𝑡01𝑏(𝑡)d𝑡=∞,_∞𝑡01𝑎(𝑡)d𝑡<∞,^∞_𝑡01𝑏(𝑡)d𝑡=∞,_∞𝑡01𝑎(𝑡)d𝑡<∞,^∞_𝑡01𝑏(𝑡)d𝑡<∞, (1.4)(1.5)(1.6) are studied.

In the following, all functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all 𝑡 large enough. Without loss of generality, we can deal only with the positive solutions of (𝐸).

2. Main Results

In this section, we will give the main results.

Theorem 2.1. Assume that (1.4) holds, 0≤𝑝(𝑡)≤𝑝₁<1. If for some function 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), for all sufficiently large 𝑡₁≥𝑡₀ and for 𝑡₃>𝑡₂>𝑡₁, one has limsup_𝑡→∞^𝑡_𝑡3𝜌(𝑠)𝑞(𝑠)(𝜏(𝑠∫1−𝑝))^{𝑡𝜏(𝑠)2}∫^𝑣_𝑡1(1/𝑎(𝑢))d𝑢/𝑏(𝑣)d𝑣∫^𝑠_𝑡1((𝑢−𝑎(𝑠)𝜌1/𝑎))d𝑢^′(𝑠)²4𝜌(𝑠)d𝑠=∞,_∞𝑡01𝑏(𝑣)^∞_𝑣1𝑎(𝑢)^∞_𝑢𝑞(𝑠)d𝑠d𝑢d𝑣=∞, (2.1)(2.2) then (𝐸) is almost oscillatory.

Proof. Assume that 𝑥 is a positive solution of (𝐸). Based on the condition (1.4), there exist two possible cases: (1)𝑧(𝑡)>0,𝑧^′(𝑡)>0,(𝑏(𝑡)𝑧^′(𝑡))^′>0,[𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′<0, (2)𝑧(𝑡)>0,𝑧^′(𝑡)<0,(𝑏(𝑡)𝑧^′(𝑡))^′>0,[𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′<0 for 𝑡≥𝑡₁, 𝑡₁ is large enough.Assume that case (1) holds. We define the function 𝜔 by 𝜔(𝑡)(𝑡)𝑎(𝑡)𝑏(𝑡)𝑧=𝜌^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡),𝑡≥𝑡₁. (2.3) Then, 𝜔(𝑡)>0 for 𝑡≥𝑡₁. Using 𝑧^′(𝑡)>0, we have 𝑥(𝑡)≥1−𝑝(𝑡)𝑧(𝑡). (2.4) Since 𝑏(𝑡)𝑧^′(𝑡)≥^𝑡_𝑡1𝑎(𝑠)𝑏(𝑠)𝑧^′(𝑠)^′𝑎(𝑠)≥𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′𝑡_𝑡11𝑎(𝑠)d𝑠, (2.5) we have that 𝑏(𝑡)𝑧^′(𝑡)∫^𝑡_𝑡1(1/𝑎(𝑠))d𝑠^′≤0. (2.6) Thus, we get 𝑧(𝑡)=𝑧𝑡₂+^𝑡_𝑡2𝑏(𝑠)𝑧^′(𝑠)∫^𝑠_𝑡1((𝑢∫1/𝑎))d𝑢^𝑠𝑡1(1/𝑎(𝑢))d𝑢𝑏(𝑠)≥𝑏(𝑡)𝑧d𝑠_′(𝑡)∫^𝑡_𝑡1(1/𝑎(𝑢))d𝑢^𝑡𝑡₂∫^𝑠_𝑡1(1/𝑎(𝑢))d𝑢𝑏(𝑠)d𝑠, (2.7) for 𝑡≥𝑡₂>𝑡₁. Differentiating (2.3), we obtain 𝜔(𝑡)=𝜌^′(𝑡)𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)(𝑡)𝑎(𝑡𝑏(𝑡)𝑧+𝜌)^′(𝑡)^′′𝑏(𝑡)𝑧^′(𝑡)(𝑡)𝑎(𝑡)𝑏(𝑡)𝑧−𝜌^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)². (2.8) It follows from (𝐸), (2.3), and (2.4) that 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)𝑞(𝑡)(𝜏(𝑡𝑧(𝜏(𝑡−𝜌1−𝑝))))𝑏(𝑡)𝑧^′(𝑡)−𝜔²(𝑡)𝜌(𝑡)𝑎(𝑡), (2.9) that is, 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)𝑞(𝑡)(𝜏(𝑡𝑧(𝜏(𝑡−𝜌1−𝑝))))𝑏(𝜏(𝑡))𝑧^′(𝜏(𝑡𝑏(𝜏(𝑡𝑧))))^′(𝜏(𝑡))𝑏(𝑡)𝑧^′(𝑡)−𝜔²(𝑡)𝜌(𝑡)𝑎(𝑡), (2.10) which follows from (2.6) and (2.7) that 𝜔^′(𝑡)≤𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)𝑞(𝑡)(𝜏(𝑡∫−𝜌1−𝑝))^{𝑡𝜏(𝑡)2}∫^𝑠_𝑡1(1/𝑎(𝑢))d𝑢/𝑏(𝑠)d𝑠∫_𝑡^𝜏(𝑡)₁((𝑢∫1/𝑎))d𝑢^{𝑡𝜏(𝑡)1}(1/𝑎(𝑢))d𝑢∫^𝑡_𝑡1((𝑢−𝜔1/𝑎))d𝑢²(𝑡)𝜌(𝑡)𝑎(𝑡)=𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)(𝑡)𝑞(𝑡(𝜏(𝑡∫−𝜌)1−𝑝))^{𝑡𝜏(𝑡)2}∫^𝑠_𝑡1(1/𝑎(𝑢))d𝑢/𝑏(𝑠)d𝑠∫^𝑡_𝑡1((𝑢−𝜔1/𝑎))d𝑢²(𝑡)𝜌(𝑡)𝑎(𝑡). (2.11) Hence, we have 𝜔^′(𝑡)(𝑡)𝑞(𝑡)(𝜏(𝑡∫≤−𝜌1−𝑝))^{𝑡𝜏(𝑡)2}∫^𝑠_𝑡1(1/𝑎(𝑢))d𝑢/𝑏(𝑠)d𝑠∫^𝑡_𝑡1((𝑢+𝑎(𝑡)𝜌1/𝑎))d𝑢^′(𝑡)²(𝑡).4𝜌 (2.12) Integrating the last inequality from 𝑡₃(>𝑡₂) to 𝑡, we get ^𝑡_𝑡3𝜌(𝑠)𝑞(𝑠)(𝜏(𝑠∫1−𝑝))^{𝑡𝜏(𝑠)2}∫^𝑣_𝑡1(1/𝑎(𝑢))d𝑢/𝑏(𝑣)d𝑣∫^𝑠_𝑡1((𝑢−𝑎(𝑠)𝜌1/𝑎))d𝑢^′(𝑠)²(𝑠)𝑡4𝜌d𝑠≤𝜔₃, (2.13) which contradicts (2.1).
Assume that case (2) holds. Using the similar proof of [23, Lemma 2], we can get lim_𝑡→∞𝑥(𝑡)=0 due to condition (2.2). This completes the proof.

Theorem 2.2. Assume that (1.5) holds, 0≤𝑝(𝑡)≤𝑝₁<1. Further, assume that for some function 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), for all sufficiently large 𝑡₁≥𝑡₀ and for 𝑡₃>𝑡₂>𝑡₁, one has (2.1) and (2.2). If limsup_𝑡→∞^𝑡_𝑡2𝛿(𝑠)𝑞(𝑠)1−𝑝(𝜏(𝑠))_𝑡^𝜏(𝑠)₁d𝑣𝑏(𝑣)−14𝛿(𝑠)𝑎(𝑠)d𝑠=∞, (2.14) where 𝛿(𝑡)∶=^∞_𝑡1𝑎(𝑠)d𝑠, (2.15) then (𝐸) is almost oscillatory.

Proof. Assume that 𝑥 is a positive solution of (𝐸). Based on the condition (1.5), there exist three possible cases (1), (2) (as those of Theorem 2.1), and (3)𝑧(𝑡)>0,𝑧^′(𝑡)>0,(𝑏(𝑡)𝑧^′(t))^′<0,[𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′<0, for 𝑡≥𝑡₁, 𝑡₁ is large enough.
Assume that case (1) and case (2) hold, respectively. We can obtain the conclusion of Theorem 2.2 by applying the proof of Theorem 2.1.
Assume that case (3) holds. From [𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′<0, 𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))′ is decreasing. Thus, we get 𝑎(𝑠)𝑏(𝑠)𝑧^′(𝑠)^′≤𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′,𝑠≥𝑡≥𝑡₁. (2.16) Dividing the above inequality by 𝑎(𝑠) and integrating it from 𝑡 to 𝑙, we obtain 𝑏(𝑙)𝑧^′(𝑙)≤𝑏(𝑡)𝑧^′(𝑡)+𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′𝑙_𝑡d𝑠𝑎(𝑠). (2.17) Letting 𝑙→∞, we have 0≤𝑏(𝑡)𝑧^′(𝑡)+𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′∞_𝑡d𝑠𝑎(𝑠), (2.18) that is, −^∞_𝑡d𝑠𝑎(𝑠)𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)≤1. (2.19) Define function 𝜙 by 𝜙(𝑡)𝑎(𝑡)𝑏(𝑡)𝑧∶=^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡),𝑡≥𝑡₁. (2.20) Then, 𝜙(𝑡)<0 for 𝑡≥𝑡₁. Hence, by (2.19) and (2.20), we get −𝛿(𝑡)𝜙(𝑡)≤1. (2.21) Differentiating (2.20), we obtain 𝜙^′(𝑡)=𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′′𝑏(𝑡)𝑧^′(𝑡)−𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)². (2.22) Using 𝑧^′(𝑡)>0, we have (2.4). From (𝐸) and (2.4), we have 𝜙^′(𝑡)(𝑡)(𝜏(𝑡𝑧(𝜏(𝑡≤−𝑞1−𝑝))))𝑏(𝑡)𝑧^′(𝑡)−𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)^′𝑏(𝑡)𝑧^′(𝑡)². (2.23) In view of (3), we see that 𝑧(𝑡)≥𝑏(𝑡)^𝑡_𝑡1d𝑠𝑏(𝑠)𝑧^′(𝑡). (2.24) Hence, 𝑧(𝑡)∫^𝑡_𝑡1(d𝑠/𝑏(𝑠))^′≤0, (2.25) which implies that 𝑧(𝜏(𝑡))𝑧(𝑡)≥∫_𝑡^𝜏(𝑡)₁(d𝑠/𝑏(𝑠))∫^𝑡_𝑡1((𝑠.d𝑠/𝑏)) (2.26) By (2.20) and (2.23), (2.24), and (2.26), we obtain 𝜙^′(𝑡)≤−𝑞(𝑡)1−𝑝(𝜏(𝑡))_𝑡^𝜏(𝑡)₁d𝑠𝑏(𝑠)−𝜙²(𝑡)𝑎(𝑡). (2.27) Multiplying the last inequality by 𝛿(𝑡) and integrating it from 𝑡₂(>𝑡₁) to 𝑡, we have 𝜙(𝑡)𝛿(𝑡)−𝜙𝑡₂𝛿𝑡₂+^𝑡_𝑡2𝛿(𝑠)𝑞(𝑠)1−𝑝(𝜏(𝑠))_𝑡^𝜏(𝑠)₁d𝑣𝑏(𝑣)d𝑠+^𝑡_𝑡2𝜙²(𝑠)𝛿(𝑠)𝑎(𝑠)d𝑠+^𝑡_𝑡2𝜙(𝑠)𝑎(𝑠)d𝑠≤0, (2.28) which follows that ^𝑡_𝑡2𝛿(𝑠)𝑞(𝑠)1−𝑝(𝜏(𝑠))_𝑡^𝜏(𝑠)₁d𝑣𝑏(𝑣)−1(𝑠)𝑎(𝑠)𝑡4𝛿d𝑠≤1+𝜙₂𝛿𝑡₂ (2.29) due to (2.21), which contradicts (2.14). This completes the proof.

Theorem 2.3. Assume that (1.6) holds, 0≤𝑝(𝑡)≤𝑝₁<1. Further, assume that for some function 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), for all sufficiently large 𝑡₁≥𝑡₀ and for 𝑡₃>𝑡₂>𝑡₁, one has (2.1), (2.2), and (2.14). If ^∞_𝑡11𝑏(𝑣)^𝑣_𝑡11𝑎(𝑢)^𝑢_𝑡1𝜂(𝑠)𝑞(𝑠)𝜉(𝜏(𝑠))d𝑠d𝑢d𝑣=∞, (2.30) where 𝜂(𝑡)(𝜏(𝑡𝜉(𝜎(𝜏(𝑡∶=1−𝑝)))))𝜉(𝜏(𝑡(𝑡)))>0,𝜉∶=^∞_𝑡1𝑏(𝑠)d𝑠, (2.31) then (𝐸) is almost oscillatory.

Proof. Assume that 𝑥 is a positive solution of (𝐸). Based on the condition (1.6), there exist four possible cases (1), (2), (3) (as those of Theorem 2.2), and (4)𝑧(𝑡)>0,𝑧^′(𝑡)<0,(𝑏(𝑡)𝑧^′(𝑡))^′<0,[𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′<0, for 𝑡≥𝑡₁, 𝑡₁ is large enough.
Assume that case (1), case (2), and case (3) hold, respectively. We can obtain the conclusion of Theorem 2.3 by using the proof of Theorem 2.2.
Assume that case (4) holds. Since (𝑏(𝑡)𝑧^′(𝑡))^′<0, we get 𝑧^′(𝑠)≤𝑏(𝑡)𝑧^′(𝑡)𝑏(𝑠),𝑠≥𝑡, (2.32) which implies that 𝑧(𝑡)≥−𝜉(𝑡)𝑏(𝑡)𝑧^′(𝑡)≥𝐿𝜉(𝑡) (2.33) for some constant 𝐿>0. By (2.33), we obtain 𝑧(𝑡)𝜉(𝑡)^′≥0. (2.34) Using (2.34), we see that 𝑥(𝑡)(𝑡)(𝑡)𝑥(𝜎(𝑡(𝑡)(𝑡)𝑧(𝜎(𝑡≥(𝑡)𝜉(𝜎(𝑡=𝑧−𝑝))≥𝑧−𝑝))1−𝑝))𝜉(𝑡)𝑧(𝑡). (2.35) From (𝐸), (2.33), and (2.35), we have 𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′′(𝑡(𝜏(𝑡𝜉(𝜎(𝜏(𝑡+𝐿𝑞)1−𝑝)))))𝜉(𝜏(𝑡𝜉(𝜏(𝑡))))≤0. (2.36) Integrating the last inequality from 𝑡₁ to 𝑡, we get 𝑎(𝑡)𝑏(𝑡)𝑧^′(𝑡)^′+𝐿^𝑡_𝑡1𝑞(𝑠)(𝜏(𝑠𝜉(𝜎(𝜏(𝑠1−𝑝)))))𝜉(𝜏(𝑠𝜉(𝜏(𝑠))))d𝑠≤0. (2.37) Integrating again, we have 𝑏(𝑡)𝑧^′(𝑡)+𝐿^𝑡_𝑡11𝑎(𝑢)^𝑢_𝑡1𝑞(𝑠)(𝜏(𝑠𝜉(𝜎(𝜏(𝑠1−𝑝)))))𝜉(𝜏(𝑠𝜉(𝜏(𝑠))))d𝑠d𝑢≤0. (2.38) Integrating again, we obtain 𝑧𝑡₁≥𝐿^𝑡_𝑡11𝑏(𝑣)^𝑣_𝑡11𝑎(𝑢)^𝑢_𝑡1𝑞(𝑠)(𝜏(𝑠𝜉(𝜎(𝜏(𝑠1−𝑝)))))𝜉(𝜏(𝑠𝜉(𝜏(𝑠(𝑡),))))d𝑠d𝑢d𝑣+𝑧 (2.39) which contradicts (2.30). This completes the proof.

Theorem 2.4. Assume that (1.6) holds, 0≤𝑝(𝑡)≤𝑝₁<1. Further, assume that for some function 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), for all sufficiently large 𝑡₁≥𝑡₀ and for 𝑡₃>𝑡₂>𝑡₁, one has (2.1), (2.2) and (2.14). If ^∞_𝑡11𝑏(𝑣)^𝑣_𝑡11𝑎(𝑢)^𝑢_𝑡1𝑞(𝑠)d𝑠d𝑢d𝑣=∞, (2.40) then (𝐸) is almost oscillatory.

Proof. Assume that 𝑥 is a positive solution of (𝐸). Based on the condition (1.6), there exist four possible cases (1), (2), (3), and (4) (as those of Theorem 2.3).
Assume that case (1), case (2), and case (3) hold, respectively. We can obtain the conclusion of Theorem 2.4 by using the proof of Theorem 2.2.
Assume that case (4) holds. Then, lim_𝑡→∞𝑧(𝑡)=𝑙≥0(𝑙isfinite). Assume that 𝑙>0. Then, from the proof of [23, Lemma 2], we see that there exists a constant 𝑘>0 such that 𝑥(𝑡)≥𝑘𝑙. (2.41) The rest of the proof is similar to that of Theorem 2.3 and hence is omitted.

3. Examples

In this section, we will present some examples to illustrate the main results.

Example 3.1. Consider the third-order neutral delay differential equation 𝑡𝑥(𝑡)+𝑝₁𝑥𝑡2^′′′+𝜆𝑡²𝑥(𝑡)=0,𝜆>0,𝑡≥1, (3.1) where 𝑝₁∈[0,1).

Let 𝜌(𝑡)=𝑡. It follows from Theorem 2.1 that every solution 𝑥 of (3.1) is either oscillatory or lim_𝑡→∞𝑥(𝑡)=0, if 𝜆>1/(4𝑘(1−𝑝₁)) for some 𝑘∈(1/4,1).

Note that (3.1) is almost oscillatory, if 𝜆>2/(1−𝑝₁) due to [23, Corollary 3].

Example 3.2. Consider the third-order neutral delay differential equation 1𝑡𝑡^1/2𝑥(𝑡)+12𝑥(𝑡−𝜋)^′′′+𝑡^−1/22+38𝑡^−5/2𝑥𝑡−7𝜋2=0, (3.2)𝑡≥1.

Let 𝜌(𝑡)=1. It follows from Theorem 2.1 that every solution 𝑥 of (3.2) is almost oscillatory. One such solution is 𝑥(𝑡)=sin𝑡.

Example 3.3. Consider the third-order neutral delay differential equation 𝑡^4/3𝑥(𝑡)+𝑝₁𝑥𝑡2^′′′+𝜆𝑡^5/3𝑥(𝑡)=0,𝜆>0,𝑡≥1, (3.3) where 𝑝₁∈[0,1).

Let 𝜌(𝑡)=1. It follows from Theorem 2.2 that every solution 𝑥 of (3.3) is either oscillatory or lim_𝑡→∞𝑥(𝑡)=0, if 𝜆>1/(36𝑘(1−𝑝₁)) for some 𝑘∈(1/4,1).

Note that [22, Theorem 1] cannot be applied to (3.3) when 𝑝₁=0.

Example 3.4. Consider the third-order neutral delay differential equation 𝑡²𝑡²𝑥(𝑡)+13𝑥𝑡2^′′′+𝜆𝑡²𝑥(𝑡)=0,𝜆>0,𝑡≥1. (3.4)

Let 𝜌(𝑡)=1. It follows from Theorem 2.3 that every solution 𝑥 of (3.4) is either oscillatory or lim_𝑡→∞𝑥(𝑡)=0, if 𝜆>0.

4. Remarks

Remark 4.1. In [3], Agarwal et al. established a well-known result; see [4, Lemma 6.1]. Using [4, Lemma 6.1] and defining the function 𝜔 as in Theorem 2.1 with 𝜌(𝑡)=1, we can replace condition (2.1) with 𝑎(𝑡)𝑦^′(𝑡)^′(𝑡)(𝜏(𝑡∫+𝑞1−𝑝))^{𝑡𝜏(𝑡)2}∫^𝑠_𝑡1(1/𝑎(𝑢))d𝑢/𝑏(𝑠)d𝑠∫^𝑡_𝑡1((𝑢𝑦(𝑡)1/𝑎))d𝑢=0 (4.1) that is oscillatory. Similarly, we can replace condition (2.14) by 𝑎(𝑡)𝑦^′(𝑡)^′+𝑞(𝑡)1−𝑝(𝜏(𝑡))_𝑡^𝜏(𝑡)₁d𝑠𝑏(𝑠)𝑦(𝑡)=0 (4.2) that is oscillatory.

Remark 4.2. The results for (𝐸) can be extended to the nonlinear differential equations.

Remark 4.3. It is interesting to find a method to study (𝐸) for the case when ^∞_𝑡01𝑎(𝑡)d𝑡=∞,^∞_𝑡01𝑏(𝑡)d𝑡<∞. (4.3)

Remark 4.4. It is interesting to find other methods to present some sufficient conditions which guarantee that every solution of (𝐸) is oscillatory.

Acknowledgments

The authors would like to thank the referees and Professor P. J. Y. Wong for the comments which helped to improve the paper. This research is supported by NNSF of PR China (Grant nos. 61034007, 60874016, and 50977054). The second author would like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for their selfless guidance.