Oscillation Criteria of Certain Third-Order Differential Equation with Piecewise Constant Argument

Abstract

We study the oscillation and asymptotic behavior of third-order nonlinear delay differential equation with piecewise constant argument of the form (𝑟₂(𝑡)(𝑟₁(𝑡)𝑥^′(𝑡))^′)^′+𝑝(𝑡)𝑥^′(𝑡)+𝑓(𝑡,𝑥([𝑡]))=0. We establish several sufficient conditions which insure that any solution of this equation oscillates or converges to zero. Some examples are given to illustrate the importance of our results.

1. Introduction

Let [⋅] denote the greatest-integer function. Consider the following third-order nonlinear delay differential equation with piecewise constant argument: 𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)′^′+𝑝(𝑡)𝑥^′(𝑡)+𝑓(𝑡,𝑥([𝑡]))=0,𝑡≥0, (1.1) where 𝑟₁(𝑡),𝑟₂(𝑡) are continuous on [0,∞) with 𝑟₁(𝑡),𝑟₂(𝑡)>0, 𝑝(𝑡) is continuously differentiable on [0,∞) with 𝑝(𝑡)≥0. We will show that every solution 𝑥(𝑡) of (1.1) oscillates or converges to zero, provided appropriate conditions are imposed.

Throughout this paper, we assume that 𝑥𝑓(𝑡,𝑥)≥0 and that there exist functions 𝑞(𝑡) and 𝜙(𝑥) such that (i)𝑞(𝑡) is continuous on [0,∞) with 𝑞(𝑡)>0,(ii)𝜙(𝑥) is continuously differentiable and nondecreasing on (−∞,+∞), 𝜙(𝑥)/𝑥≥𝐾>0 for 𝑥≠0, (iii)|𝑓(𝑡,𝑥)|≥𝑞(𝑡)|𝜙(𝑥)|,𝑥≠0,𝑡≥0, (iv)𝐾𝑞(𝑡)−𝑝^′(𝑡)≥0 and is not identically zero in any subinterval of [0,∞).

The delay functional differential equations provide a mathematical model for a physical or biological system in which the rate of change of the system depends upon its past history. In recent years, the oscillation theory and asymptotic behavior of delay functional differential equations and their applications have been and still are receiving intensive attention. In fact, in the last few years several monographs and hundreds of research papers have been written, see, for example [1–4]. In particular case, determining oscillation criteria for second-order delay differential equations has received a great deal of attention, while the study of oscillation and asymptotic behavior of the third-order delay differential equations has received considerably less attention in the literature.

The delay differential equations with piecewise continuous arguments can be looked as a special kind of delay functional differential equations. Since the delays of such equations are discontinuous, it need to be investigated individually. As is shown in [5], the solutions of differential equations with piecewise continuous arguments are determined by a finite set of initial data, rather than by an initial function as in the case of general functional differential equations. Moreover, the strong interest in such equations is motivated by the fact that they represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equation.

In the last few decades, there has been increasing interest in obtaining sufficient conditions for the oscillatory of solutions of different classes of the first-order differential equations with piecewise constant arguments, see [6–9] and the references therein. It is found that the presence of piecewise constant arguments plays an important role in the oscillation of the solution. For instant, all solution of 𝑥^′+2𝑥([𝑡])=0 are oscillatory (see [6]). But the corresponding ordinary differential equation 𝑥^′+2𝑥=0 has nonoscillatory solution 𝑥=𝑒^−2𝑡.

However, there are few results about the oscillation of higher order equations. As mentioned in [5, 10], there are reasons for investigating the higher order equations with piecewise constant arguments. For example, suppose a moving particle with time variable mass 𝑟(𝑡) is subjected to a restoring controller −𝜙(𝑥([𝑡])) which acts at sampled time [𝑡], then the second law of motion asserts that 𝑟(𝑡)𝑥^′(𝑡)^′+𝜙(𝑥([𝑡]))=0. (1.2)

In [10], the authors study a slightly more general second-order delay differential equations of the form: 𝑟(𝑡)𝑥^′(𝑡)^′+𝑓(𝑡,𝑥([𝑡]))=0,𝑡≥0. (1.3) Two sufficient conditions which insure that any solution of (1.3) oscillates are obtained.

However, as far as we know, there are not works studying the oscillation and asymptotic behavior of third-order delay differential equations with piecewise constant argument. Motivated by this fact, in the present paper, we will investigate the oscillatory and asymptotic behavior of a certain class of third-order equation (1.1) with damping. The main ideas we used here are based on the paper [3, 4, 10].

The rest of this paper is organized as follows. In Section 2 we give the definition of the solution of (1.1) and establish some lemmas which are useful in the proof of our main results. Section 3 is devoted to the presentation of several sufficient conditions for the oscillation of (1.1). In Section 4, several examples are given to illustrate the importance of our results.

2. Definitions and Preliminary Lemmas

Similar to [11], we give the following definition.

Definition 2.1. A solution of (1.1) on [0,∞) is a function 𝑥 that satisfies the conditions:(i)(𝑟₁(𝑡)𝑥^′(𝑡))^′ is continuous on [0,∞),(ii)𝑟₂(𝑡)(𝑟₁(𝑡)𝑥^′(𝑡))^′ is differentiable at each point 𝑡∈[0,∞), with the possible exception of the points [𝑡]∈[0,∞), where one-sided derivatives exist,(iii) Equation (1.1) is satisfied on each interval [𝑘,𝑘+1)⊂[0,∞) with 𝑘∈𝐍.

Our attention is restricted to those solutions of (1.1) which exist on the half line [0,∞) and satisfy sup{|𝑥(𝑡)|∶𝑇≤𝑡<∞}>0 for any 𝑇>0. We make a standing hypothesis that (1.1) does possess such solutions. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise. Equation (1.1) itself is called oscillatory if all its solutions are oscillatory.

Remark 2.2. If 𝑥 is a solution of (1.1), then 𝑦=−𝑥 is a solution of the equation 𝑟₂(𝑡)𝑟₁(𝑡)𝑦^′(𝑡)^′′+𝑝(𝑡)𝑦^′(𝑡)+𝑓𝑡,𝑦([𝑡])=0,𝑡≥0, (2.1) where 𝑓(𝑡,𝑦)=−𝑓(𝑡,−𝑦). It is easy to check that 𝑦𝑓(𝑡,𝑦)>0 and |𝑓(𝑡,𝑦)|≥𝑞(𝑡)|𝜙(𝑦)|, where 𝜙(𝑦)=−𝜙(−𝑦) with 𝜙(𝑦)/𝑦≥𝐾. Thus, concerning nonoscillatory solutions of (1.1), we can restrict our attention only to the positive ones.
For the sake of brevity, we denote the following two operators 𝐿₁𝑥(𝑡)=𝑟₁(𝑡)𝑥^′(𝑡),𝐿₂𝑥(𝑡)=𝑟₂(𝑡)𝐿₁𝑥(𝑡)^′. (2.2) Thus (1.1) can be written as 𝐿₂𝑥(𝑡)^′+𝑝(𝑡)𝑥^′(𝑡)+𝑓(𝑡,𝑥([𝑡]))=0,𝑡≥0. (2.3)
Similar to [12], we give the following definition.

Definition 2.3. Let 𝑥 be a solution of (1.1). We say that 𝑥 has property 𝑉₂ on [𝑇,∞),𝑇≥0, if 𝑥(𝑡)𝐿_𝑘𝑥(𝑡)>0,𝑖=1,2 for every 𝑡∈[𝑇,∞).

It is worth pointing out here that if 𝑥 has property 𝑉₂ on [𝑇,∞), then by (1.1), 𝑥(𝑡)(𝐿₂𝑥(𝑡))^′ is eventually nonpositive.

Define the functions 𝑅_𝑖(𝑡,𝑇)=^𝑡_𝑇𝑑𝑠𝑟_𝑖(𝑠),𝑖=1,2. (2.4)

We assume that 𝑅₁()𝑅𝑡,𝑇→∞,as𝑡→∞,₂(𝑡,𝑇)→∞,as𝑡→∞. (2.5)(2.6)

To obtain our main results we need the following lemmas.

Lemma 2.4. Suppose that 𝑟₁(𝑡)𝑟₂(𝑡)𝑦^′(𝑡)^′+𝑝(𝑡)𝑦(𝑡)=0 (2.7) is nonoscillatory. If 𝑥 is a nonoscillatory solution of (1.1), then 𝑥′(𝑡) does not change sign eventually.

Proof. Suppose that 𝑥(𝑡) is a nonoscillatory solution of (1.1) on [0,∞). Without loss of generality, we may assume that 𝑥(𝑡)>0 for 𝑡≥0. Let 𝑦(𝑡) be a nonoscillatory solution of (2.7). We will firstly consider the case that 𝑦(𝑡) is eventually negative, that is, there exists a constant 𝑇₀ such that 𝑦(𝑡)<0 for 𝑡≥𝑇₀. By (1.1) and (2.7), it is easy to see that 𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)−𝑟₂(𝑡)𝑦^′(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′=𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′′𝑦(𝑡)−𝑟₂(𝑡)𝑦^′(𝑡)^′𝑟₁(𝑡)𝑥^′(𝑡)=−𝑦(𝑡)𝑓(𝑡,𝑥(𝑘))≥−𝑦(𝑡)𝑞(𝑡)𝜙(𝑥(𝑘)),𝑡∈[𝑘,𝑘+1),𝑘∈𝐍,𝑘≥𝑇₀. (2.8)
Suppose to the contrary that 𝑥^′(𝑡) has arbitrarily large zeros, then there exist consecutive zeros of 𝑥^′(𝑡), 𝑡₁, and 𝑡₂, such that 𝑡₂>𝑡₁>𝑇₀ and 𝑥^′′(𝑡₁)≤0, 𝑥^′′(𝑡₂)≥0. If there exists an integer 𝑘₁ such that 𝑘₁−1≤𝑡₁<𝑘₁<𝑘₁+1<⋯<𝑘₁+𝑚<𝑡₂≤𝑘₁+𝑚+1, (2.9) see Figure 1(a), then integrating (2.8) we find 𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)𝑡=𝑡₂𝑡=𝑡₁=𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)−𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)𝑦^′(𝑡)𝑡=𝑡₂𝑡=𝑡₁=^𝑡2_𝑡1𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)−𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)𝑦^′(𝑡)^′=𝑑𝑡_𝑘1𝑡1𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)−𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)𝑦^′(𝑡)^′+𝑑𝑡_𝑘1𝑘+11𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)−𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)𝑦^′(𝑡)^′𝑑𝑡+⋯+^𝑡2_𝑘1+𝑚𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)−𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)𝑦^′(𝑡)^′𝑥𝑘𝑑𝑡≥𝜙₁−1^𝑘1_𝑡1𝑦(𝑡)𝑞(𝑡)𝑑𝑡+𝜙𝑥𝑘₁^𝑘1_𝑘⁺¹₁𝑦(𝑡𝑞(𝑡)𝑥𝑘)𝑑𝑡+⋯+𝜙₁+𝑚^𝑡2_𝑘1+𝑚𝑦(𝑡)𝑞(𝑡)𝑑𝑡>0. (2.10)
Note that (𝑟₁(𝑡)𝑥^′(𝑡))^′|^𝑡=𝑡_𝑖=𝑟₁(𝑡_𝑖)𝑥^′′(𝑡_𝑖)(𝑖=1,2), it follows from 𝑥^′′(𝑡₁)≤0,𝑥^′′(𝑡₂)≥0 that 𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)𝑡=𝑡₂𝑡=𝑡₁≤0, (2.11) which is contrary to (2.10).
If there exists an integer 𝑘₀ such that 𝑘₀≤𝑡₁<𝑡₂≤𝑘₀+1, see Figure 1(b), then integrating (2.8) directly from 𝑡₁ to 𝑡₂, we also get a contradiction.
Next we consider the case that 𝑦(𝑡) is eventually positive. Let 𝑇₁ be the constant such that 𝑦(𝑡)>0 for 𝑡≥𝑇₁. It follows from (1.1) and (2.7) that, 𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′𝑦(𝑡)−𝑟₂(𝑡)𝑦^′(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′≤−𝑦(𝑡)𝑞(𝑡)𝜙(𝑥(𝑘)), (2.12) where 𝑡∈[𝑘,𝑘+1) with 𝑘∈𝐍and𝑘≥𝑇₁. Suppose to the contrary that 𝑥^′(𝑡) has arbitrarily large zeros, then there exist consecutive zeros of 𝑥^′(𝑡), 𝑡₃, and 𝑡₄, such that 𝑡₄>𝑡₃>𝑇₁ and 𝑥^′′(𝑡₃)≥0,𝑥^′′(𝑡₄)≤0. Using the argument as above, we also arrive at a contradiction. Thus the proof is complete.

(a)

(b)

Figure 1 

The relative position of 𝑡1 and 𝑡2.

Remark 2.5. Lemma 2.4 shows that, the oscillatory or asymptotic behavior of (1.1) is linked to nonoscillation of the second-order homogeneous equation (2.7). The source of this interesting phenomenon can be explained as follows.
In (1.1), if we let 𝑦=𝑟₁𝑥^′, then 𝑦 verifies the following equation: 𝑟₁(𝑡)𝑟₂(𝑡)𝑦^′(𝑡)^′+𝑝(𝑡)𝑦(𝑡)+𝑟₁(𝑡)𝑓(𝑡,𝑥([𝑡]))=0. (2.13) Let 𝑥(𝑡) be any solution of (1.1). If 𝑥(𝑡) is nonoscillatory, then 𝑟₁(𝑡)𝑓(𝑡,𝑥([𝑡])) and thus 𝑟₁(𝑡)(𝑟₂(𝑡)𝑦^′(𝑡))^′+𝑝(𝑡)𝑦(𝑡) is eventually of one sign (see (2.13)). Hence, by the comparison method (as was shown by the proof of Lemma 2.4), the assumption that (2.7) is nonoscillatory guarantees that 𝑟₁(𝑡)𝑟₂(𝑡)𝑦^′(𝑡)^′+𝑝(𝑡)𝑦(𝑡)<0or(>0) (2.14) is nonoscillatory. This fact means that, under the hypotheses of Lemma 2.4, any solution 𝑥(𝑡) of (1.1) is either oscillatory or is monotone.

Lemma 2.6. Suppose that assumption (2.6) is satisfied and that 𝑥 is a nonoscillatory solution of (1.1) such that 𝑥(𝑡)𝐿₁𝑥(𝑡)>0 for every 𝑡≥𝑇≥0. Then 𝑥 has property 𝑉₂ on [𝑇₁,∞) for some 𝑇₁.

Proof. Assume without loss of generality that 𝑥(𝑡)>0,𝐿₁𝑥(𝑡)>0 for 𝑡≥𝑇. We assert firstly that for any integer 𝑘≥𝑇,𝐿₂𝑥(𝑘)≥0. If this is not true, then there exists an integer 𝑗≥𝑇 such that 𝐿₂𝑥(𝑗)∶=𝜇<0. By (1.1), we obtain for 𝑡∈[𝑗+𝑛,𝑗+𝑛+1),𝑛=0,1,2,…, that 𝐿₂𝑥(𝑡)^′=−𝑝(𝑡)𝑥^′(𝑡)−𝑓(𝑡,𝑥([𝑡]))<0. (2.15)
This implies that 𝐿₂𝑥(𝑡)≤𝐿₂𝑥(𝑗)=𝜇<0,𝑡∈[𝑗,𝑗+1), and hence 𝐿₂𝑥𝑗+1=𝐿₂𝑥𝑗+1⁻≤𝜇<0. (2.16) Using induction, we obtain that 𝐿₂𝑥(𝑗+𝑛)≤𝐿₂𝑥(𝑗)=𝜇<0,𝑛=0,1,2,…. Thus 𝐿₂𝑥(𝑡)≤𝐿₂𝑥𝑗+𝑛≤𝜇<0,𝑡∈𝑗+𝑛,𝑗+𝑛+1. (2.17) Integrating this inequality from 𝑗+𝑛 to 𝑡, we find 𝐿₁𝑥(𝑡)≤𝐿₁𝑥𝑗+𝑛+𝜇𝑅₂𝑡,𝑗+𝑛,𝑡∈𝑗+𝑛,𝑗+𝑛+1. (2.18) Letting 𝑡→(𝑗+𝑛+1)⁻, it follows from (2.18) and the continuity of 𝐿₁𝑥(𝑡) that 𝐿₁𝑥𝑗+𝑛+1≤𝐿₁𝑥𝑗+𝑛+𝜇𝑅₂𝑗+𝑛+1,𝑗+𝑛. (2.19) Consequently, 𝐿₁𝑥𝑗+𝑚+1=^𝑚_𝑛=0𝐿₁𝑥𝑗+𝑛+1−𝐿₁𝑥𝑗+𝑛+𝐿₁𝑥𝑗≤𝜇^𝑚_𝑛=0𝑅₂𝑗+𝑛+1,𝑗+𝑛+𝐿₁𝑥𝑗=𝜇𝑅₂𝑗+𝑚+1,𝑗+𝐿₁𝑥𝑗→−∞,as𝑚→∞. (2.20) This contradicts that 𝐿₁𝑥(𝑡)>0,𝑡≥𝑇. Therefore, 𝐿₂𝑥(𝑘)≥0 for any integer 𝑘≥𝑇. By the continuity and monotonicity of 𝐿₂𝑥(𝑡), we get that 𝐿₂𝑥(𝑡)≥0, 𝑡≥𝑇. Finally, since 𝐿^′₂𝑥(𝑡)>0, we conclude that 𝐿₂𝑥(𝑡) is eventually positive.
This completes the proof.

Lemma 2.7. Let 𝑥 be a solution of (1.1) such that 𝑥(𝑡)𝑥^′(𝑡)<0,𝑡≥𝑇, where 𝑇 is a constant. If lim_𝑡→∞𝑥(𝑡)=𝜆≠0, (2.21) then for any arbitrary constant 𝛿>1, there exists an integer 𝑘₀≥𝑇 such that for any 𝑡>𝑠≥𝑘₀, we have (𝑠)()(𝑠)(),()𝜆<𝑥<𝛿𝜆𝜆>0or𝛿𝜆<𝑥<𝜆𝜆<0−1^𝑥(𝑡)𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)−𝐿₂𝑥(𝑠)−𝑝(𝑠)𝑥(𝑠)+𝜆^𝑡_𝑠𝐾𝑞(𝑢)−𝑝^′(𝑢)𝑑𝑢≥0. (2.22)

Proof. Let us assume that 𝑥(𝑡) is eventually positive. The case when 𝑥(𝑡) is eventually negative can be similarly dealt with. For any constant 𝛿>1, it follows from (2.21) and 𝑥′(𝑡)<0 that, there exists an integer 𝑘₀≥𝑇 such that 𝜆<𝑥(𝑡)<𝛿𝜆 for 𝑡≥𝑘₀. By (1.1), we have 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)^′=𝑝^′(𝑡)𝑥(𝑡)−𝑓(𝑡,𝑥([𝑡]))≤𝑝^′(𝑡)𝑥(𝑡)−𝑞(𝑡)𝜙(𝑥([𝑡])),𝑡∈[𝑘,𝑘+1), (2.23) where 𝑘=𝑘₀,𝑘₀+1,𝑘₀+2,…. Since 𝑥([𝑡])>𝑥(𝑡) and 𝜙(𝑥) is nondecreasing, it follows from (2.23) that 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)^′≤𝑥(𝑡)𝑝^′(𝑡)(𝑡)𝜙(𝑥𝑡−𝑞([]))𝑥𝑡𝑝([])≤𝜆^′(𝑡)−𝐾𝑞(𝑡),𝑡∈[𝑘,𝑘+1). (2.24)
Integrating (2.24) from 𝜉 to 𝜁, where 𝑘≤𝜉<𝜁<𝑘+1, we have 𝐿₂𝑥(𝜁)+𝑝(𝜁)𝑥(𝜁)≤𝐿₂𝑥(𝜉)+𝑝(𝜉)𝑥(𝜉)+𝜆^𝜁_𝜉𝑝^′(𝑢)−𝐾𝑞(𝑢)𝑑𝑢. (2.25)
If there exists an integer 𝑘₁ such that 𝑘₁≤𝑠<𝑡<𝑘₁+1, then (2.22) follows from (2.25) directly. Otherwise, there exist an integer 𝑘₂ such that 𝑘₂≤𝑠<𝑘₂+1<𝑘₂+2<⋯<𝑘₂+𝑚<𝑡≤𝑘₂+𝑚+1. (2.26) Using (2.25) and the continuity of 𝐿₂𝑥(𝑡), we have that 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)−𝐿₂𝑥(𝑠)−𝑝(𝑠)𝑥(𝑠)=𝐿₂𝑥𝑘₂+1+𝑝𝑘₂+1𝑥𝑘₂+1−𝐿₂𝑥(𝑠)(𝑠)𝑥(𝑠)+−𝑝_{𝑚−1𝑛=1}𝐿₂𝑥𝑘₂+𝑛+1+𝑝𝑘₂+𝑛+1𝑥𝑘₂+𝑛+1−𝐿₂𝑥𝑘₂+𝑛−𝑝𝑘₂+𝑛𝑥𝑘₂+𝑛+𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)−𝐿₂𝑥𝑘₂+𝑚−𝑝𝑘₂+𝑚𝑥𝑘₂+𝑚≤𝜆^𝑘2_𝑠⁺¹𝑝^′(𝑢)−𝐾𝑞(𝑢)𝑑𝑢+𝜆^𝑚−1_𝑛=1^𝑘2_𝑘^+𝑛+1_2+𝑛𝑝^′(𝑢)−𝐾𝑞(𝑢)𝑑𝑢+𝜆^𝑡_𝑘2+𝑚𝑝^′(𝑢)−𝐾𝑞(𝑢)𝑑𝑢=𝜆^𝑡_𝑠𝑝^′(𝑢)−𝐾𝑞(𝑢)𝑑𝑢. (2.27) Thus we conclude that (2.22) holds for any 𝑘₀≤𝑠<𝑡.

3. Main Results

In this section we present some sufficient conditions which guarantee that every solution of (1.1) oscillates or converges to zero. For convenience, we let Φ(𝑡)=^∞_𝑡𝐾𝑞(𝜏)−𝑝^′(𝜏)𝑑𝜏,𝑡≥0. (3.1)

Theorem 3.1. Suppose that(i)(2.7) is nonoscillatory,(ii)(2.5) and (2.6) are satisfied, and for every 𝑇≥0, lim_𝑡→∞^𝑡_𝑇𝑞(𝑠)exp^𝑠_𝑇𝑝(𝑢)𝑅₂(𝑢,𝑇)𝑟₁(𝑢)𝑑𝑢𝑑𝑠=∞, (3.2)(iii)one of the following two conditions holds: Φ(𝑇)=∞𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑇≥0, (3.3) or 𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡≥0,Φ(𝑡)<∞,𝑎𝑛𝑑𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝛿>1𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡lim_𝑡→∞^𝑡_𝑇1𝑟₁(𝑠)^𝑡_𝑠Φ(𝑢)−𝛿𝑝(𝑢)𝑟₂(𝑢)𝑑𝑢𝑑𝑠=∞. (3.4)Then any solution 𝑥 of (1.1) is oscillatory or satisfies 𝑥(𝑡)→0 as 𝑡→∞.

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that 𝑥(𝑡)>0 for 𝑡≥0. By Lemma 2.4, there exists a constant 𝑇 such that one of the following cases holds: Case1∶𝑥(𝑡)>0,𝑥^′(𝑡)(𝑡)>0,𝑡≥𝑇,Case2∶𝑥>0,𝑥^′(𝑡)<0,𝑡≥𝑇. (3.5)(3.6)
We will firstly show that Case 1 is impossible. Indeed, if this case holds, then it follows from Lemma 2.6 that 𝑥(𝑡) has property 𝑉₂ on [𝑇₁,∞) for some 𝑇₁≥𝑇. We define 𝑤(𝑡)=𝐿₂𝑥(𝑡)𝜙(𝑥𝑡.([])) (3.7) Clearly, 𝑤(𝑡)>0, 𝑡≥𝑇₁. It follows from (1.1) that 𝑤^′(𝑡)𝑝(𝑡)𝑥=−^′(𝑡)+𝑓(𝑡,𝑥(𝑘))𝜙(𝑥(𝑘(𝑡)(𝑡)𝐿))≤−𝑞−𝑝²𝑥(𝑡)𝜙(𝑥(𝑘⋅𝑥))^′(𝑡)𝐿₂𝑥(𝑡),𝑡∈[𝑘,𝑘+1), (3.8) where 𝑘=[𝑇₁]+1,[𝑇₁]+2,…. Noting that 𝐿₁𝑥(𝑡)=𝐿₁𝑥𝑇₁+^𝑡_𝑇1𝐿₂𝑥(𝑢)𝑟₂(𝑢)𝑑𝑢≥𝐿₂𝑥(𝑡)𝑅₂𝑡,𝑇₁,𝑡≥𝑇₁, (3.9) we get from (3.8) that 𝑤^′(𝑡)(𝑡)(𝑡)𝑅≤−𝑞−𝑝²𝑡,𝑇₁𝑟₁(𝑡)𝑤(𝑡),𝑡∈[𝑘,𝑘+1). (3.10) Denoted by Ψ(𝑡)=𝑤(𝑡)exp^𝑡_𝑇1𝑝(𝑠)𝑅₂𝑠,𝑇₁𝑟₁(𝑠)𝑑𝑠, (3.11) then we get from (3.10) that Ψ^′(𝑡)≤−𝑞(𝑡)exp^𝑡_𝑇1𝑝(𝑠)𝑅₂𝑠,𝑇₁𝑟₁(𝑠)𝑑𝑠,𝑡∈[𝑘,𝑘+1). (3.12) Integrating (3.12), we have Ψ((𝑘+1)⁻)−Ψ(𝑘)≤−_𝑘^𝑘+1𝑞(𝑡)exp^𝑡_𝑇1𝑝(𝑠)𝑅₂𝑠,𝑇₁𝑟₁(𝑠)𝑑𝑠𝑑𝑡. (3.13) Using the increasing property of 𝑥(𝑡), it follows that Ψ()=𝐿𝑘+1²𝑥(𝑘+1)𝜙(𝑥(𝑘+1))exp_𝑇^𝑘+1₁𝑝(𝑠)𝑅₂𝑠,𝑇₁𝑟₁(𝑠)≤𝐿𝑑𝑠₂𝑥(𝑘+1)𝜙(𝑥(𝑘))exp_𝑇^𝑘+1₁𝑝(𝑠)𝑅₂𝑠,𝑇₁𝑟₁(𝑠)𝑑𝑠=Ψ((𝑘+1)⁻). (3.14) Thus Ψ(𝑘+𝑚)=Ψ(𝑘)+^{𝑛=𝑘+𝑚}_𝑛=𝑘+1[Ψ(𝑛)((𝑘)+−Ψ𝑛−1)]≤Ψ^{𝑛=𝑘+𝑚}_𝑛=𝑘+1[Ψ(𝑛⁻)((𝑘)−−Ψ𝑛−1)]≤Ψ_𝑘^𝑘+𝑚𝑞(𝑡)exp^𝑡_𝑇1𝑝(𝑠)𝑅₂𝑠,𝑇₁𝑟₁(𝑠)𝑑𝑠𝑑𝑡, (3.15) where 𝑚=1,2,…. For fixed 𝑘, by letting 𝑚→∞, we get from (3.2) that Ψ(𝑗)<0 for large integer 𝑗, which is contrary to the fact that Ψ(𝑡)>0,𝑡≥𝑇₁.
Therefore, we only need to consider Case 2: 𝑥(𝑡)>0, 𝑥^′(𝑡)<0, 𝑡≥𝑇. We assert that lim_𝑡→∞𝑥(𝑡)=𝜆=0. (3.16) Suppose to the contrary that 𝜆>0. For any arbitrary constant 𝛿>1, it follows from Lemma 2.7 that, there exists an integer 𝑘₀≥𝑇 such that 𝜆<𝑥(𝑡)<𝛿𝜆 (𝑡≥𝑘₀) and 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)≤𝐿₂𝑥(𝑠)+𝑝(𝑠)𝑥(𝑠)−𝜆^𝑡_𝑠𝐾𝑞(𝑢)−𝑝^′(𝑢)𝑑𝑢, (3.17) where 𝑡>𝑠≥𝑘₀.
Now assume that (3.3) holds. Letting 𝑠=𝑘₀ in (3.17), we have 𝐿₂𝑥(𝑡)≤𝑀−𝜆^𝑡_𝑘0𝐾𝑞(𝑢)−𝑝^′(𝑢)𝑑𝑢, (3.18) where 𝑀 is a constant. By this inequality and (3.3), there exists a negative constant 𝜇 and a positive constant 𝑇₁≥𝑘₀ such that 𝐿₂𝑥(𝑡)<𝜇 for 𝑡≥𝑇₁. Integrating (𝑟₁(𝑡)𝑥′(𝑡))′<𝜇𝑟₂⁻¹(𝑡) from 𝑇₁ to 𝑡 twice, we have 𝑥(𝑡)≤𝑥𝑇₁+𝐿₁𝑥𝑇₁𝑅₁𝑡,𝑇₁+𝜇^𝑡_𝑇1𝑅₂𝑠,𝑇₁𝑟₁(𝑠)𝑑𝑠. (3.19) Thus we get from (2.5) that 𝑥(𝑡)<0 for large 𝑡, a contradiction.
Next, assume that (3.4) holds. We distinguish the following three subcases: (i) 𝐿₂𝑥(𝑡)≤0 for all large 𝑡, (ii) 𝐿₂𝑥(𝑡)≥0 for all large 𝑡, (iii) 𝐿₂𝑥(𝑡) changes sign for arbitrarily large 𝑡.
Case (i) is equivalent to that 𝑥^′′(𝑡)≤0 for all large 𝑡. Since 𝑥^′(𝑡)<0, we conclude that 𝑥(𝑡) is eventually negative, a contradiction.
In case (ii), we get from (2.22) that 𝐿₂𝑥(𝑠)+𝑝(𝑠)𝑥(𝑠)≥𝜆^𝑡_𝑠𝐾𝑞(𝑢)−𝑝^′(𝑢)𝑑𝑢,𝑡>𝑠≥𝑇₂≥𝑘₀, (3.20) where 𝑇₂ is a constant such that 𝐿₂𝑥(𝑡)≥0 on [𝑇₂,∞). Letting 𝑡→∞, we obtain that 𝐿₂𝑥(𝑠)+𝛿𝜆𝑝(𝑠)≥𝜆Φ(𝑠). (3.21) Integrating (3.21) from 𝑠 to 𝑡, using 𝐿₁𝑥(𝑡)≤0, it turns out that 𝐿₁𝑥(𝑠)≤𝛿𝜆^𝑡_𝑠𝑝(𝑢)𝑟₂(𝑢)𝑑𝑢−𝜆^𝑡_𝑠Φ(𝑢)𝑟₂(𝑢)𝑑𝑢. (3.22)
Integrating again from 𝑇₂ to 𝑡, we find 𝑥(𝑡)≤𝑥𝑇₂+^𝑡_𝑇2𝛿𝜆𝑟₁(𝑠)^𝑡_𝑠𝑝(𝑢)𝑟₂(𝑢)𝑑𝑢𝑑𝑠−^𝑡_𝑇2𝜆𝑟₁(𝑠)^𝑡_𝑠Φ(𝑢)𝑟₂(𝑢)d𝑢𝑑𝑠. (3.23) By the condition (3.4), we have that 𝑥(𝑡)<0 for all large 𝑡. This is a contradiction.
Finally, in case (iii), we let {𝑡_𝑛} be the sequence of zeros of 𝐿₂𝑥(𝑡) such that lim_𝑛→∞𝑡_𝑛=∞. By choosing 𝑡=𝑡_𝑛 in (2.22), we get 𝐿₂𝑥(𝑠)+𝑝(𝑠)𝑥(𝑠)≥𝜆^𝑡𝑛_𝑠𝐾𝑞(𝑢)−𝑝^′(𝑢)𝑑𝑢. (3.24) Let 𝑛→∞, it follows that 𝐿₂𝑥(𝑠)+𝛿𝜆𝑝(𝑠)≥𝜆Φ(𝑠). (3.25)
Integrating this inequality twice yields that 𝑥𝑡_𝑛≤𝑥𝑇₀+^𝑡𝑛_𝑇0𝛿𝜆𝑟₁(𝑠)^𝑡𝑛_𝑠𝑝(𝑢)𝑟₂(𝑢)𝑑𝑢𝑑𝑠−^𝑡𝑛_𝑇0𝜆𝑟₁(𝑠)^𝑡𝑛_𝑠Φ(𝑢)𝑟₂(𝑢)𝑑𝑢𝑑𝑠, (3.26) which also leads to a contradiction.
Therefore, in Case 2 we conclude that (3.16) holds. This completes the proof of the theorem.

From the conclusion of Theorem 3.1, we get the following corollary.

Corollary 3.2. Suppose that (2.5), (2.6) hold and that (2.7) is nonoscillatory. Assume further that 𝑝′(𝑡)≤0 for large 𝑡. If ∫^∞_𝑇𝑞(𝑡)𝑑𝑡=∞, then any solution 𝑥 of (1.1) is oscillatory or satisfies 𝑥(𝑡)→0 as 𝑡→∞.

Proof. Since 𝑝^′(𝑡)≤0, it follows that Φ(𝑇)=^∞_𝑇𝐾𝑞(𝜏)−𝑝^′(𝜏)𝑑𝜏≥𝐾^∞_𝑇𝑞(𝑠)𝑑𝑠=∞. (3.27) Thus (3.3) is satisfied.
On the other hand, with the condition ∫^∞_𝑇𝑞(𝑡)𝑑𝑡=∞ we get (3.2) immediately. Consequently, the expected conclusion follows from Theorem 3.1 directly.

We next consider the following equation which is different from (1.1): 𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′′+𝑝(𝑡)𝑥^′(𝑡)+𝑞(𝑡)𝑥^𝛼([𝑡])=0,𝑡≥0, (3.28) or the more general equation 𝑟₂(𝑡)𝑟₁(𝑡)𝑥^′(𝑡)^′′+𝑝(𝑡)𝑥^′(𝑡)+𝑓(𝑡,𝑥([𝑡]))=0,𝑡≥0, (3.29) with 𝑓(𝑡,𝑥)≥𝑞(𝑡)𝜙(𝑥),𝜙(𝑥)𝑥^𝛼≥𝐾>0,𝑞(𝑡)>0, (3.30) where 𝛼∈(0,1)∪(1,∞) is a quotient of odd integers. We note that the result of Theorem 3.1 is not applicable. In what follows we give an oscillation criteria for (3.29).

Let Φ₁(𝑡)=^∞_𝑡𝐾₁𝑞(𝜏)−𝑝^′(𝜏)𝑑𝜏,𝑡≥0. (3.31) Suppose that 𝐾₁𝑞(𝑡)−𝑝′(𝑡)≥0 and not identically zero in any subinterval of [0,∞) for any 𝐾₁>0. Moreover, we need the following conditions: Φ₁(𝑇)=∞,forsome𝑇≥0, (3.32) or forevery𝑡≥0,Φ₁(𝑡)<∞,andthereexistsa𝛿>1,suchthatlim_𝑡→∞^𝑡_𝑇1𝑟₁(𝑠)^𝑡_𝑠Φ₁(𝑢)−𝛿𝑝(𝑢)𝑟₂(𝑢)𝑑𝑢𝑑𝑠=∞. (3.33)

Theorem 3.3. Suppose that(i)(2.7) is nonoscillatory,(ii)(2.5) and (2.6) are satisfied, and for every 𝑇≥0, lim_𝑡→∞^𝑡_𝑇𝑞(𝑠)exp^𝑠_𝑇𝑝(𝑢)𝑅₂(𝑢,𝑇)𝑟₁(𝑢)𝑑𝑢𝑑𝑠=∞, (3.34)(iii)for any 𝐾₁>0, one of (3.32) and (3.33) is satisfied.
Then any solution 𝑥 of (3.29) is oscillatory or satisfies 𝑥(𝑡)→0 as 𝑡→∞.

Proof. From the proof of Theorem 3.1, it suffices to verify in Case 2 that there exists a positive constant 𝐾₁ such that 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)≤𝐿₂𝑥(𝑠)+𝑝(𝑠)𝑥(𝑠)−𝜆^𝑡_𝑠𝐾₁𝑞(𝑢)−𝑝^′(𝑢)𝑑𝑢, (3.35) for any 𝑡>𝑠≥𝑘₀, where 𝑘₀≥𝑇 is the integer such that 𝜆<𝑥(𝑡)<𝛿𝜆, 𝑡≥𝑘₀.
By (3.29), we obtain that 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)^′=𝑝^′(𝑡)𝑥(𝑡)−𝑓(𝑡(𝑡𝑝𝑡,𝑥([]))≤𝑥)^′(𝑡)−𝑞(𝑡)𝜙(𝑥([𝑡]))𝑥𝑡(𝑡)𝑝([])≤𝑥^′(𝑡)−𝐾𝑞(𝑡)𝑥^𝛼−1([𝑡]), (3.36)𝑡∈[𝑘,𝑘+1), where 𝑘=𝑘₀,𝑘₀+1,𝑘₀+2,…. If 𝛼>1, then it follows from this inequality that 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)^′≤𝜆𝑝^′(𝑡)−𝐾𝜆^𝛼−1𝑞(𝑡), (3.37) while 𝛼<1, we have 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)^′≤𝜆𝑝^′(𝑡)−𝐾(𝛿𝜆)^𝛼−1𝑞(𝑡), (3.38) Let 𝐾₁=𝐾𝜆^𝛼−1 if 𝛼>1 and 𝐾₁=𝐾(𝛿𝜆)^𝛼−1 if 𝛼<1, it turns out for the both cases that 𝐿₂𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡)^′≤𝜆𝑝^′(𝑡)−𝐾₁𝑞(𝑡),𝑡∈[𝑘,𝑘+1). (3.39) The rest of the proof is exactly the same as in Theorem 3.1 and hence is omitted.

The following result follows from Theorem 3.3 directly.

Corollary 3.4. Suppose that (2.5), (2.6) hold and that (2.7) is nonoscillatory. Assume further that 𝑝^′(𝑡)≤0 for large 𝑡. If ∫^∞_𝑇𝑞(𝑡)𝑑𝑡=∞, then any solution 𝑥(𝑡) of (3.29) is oscillatory or satisfies 𝑥(𝑡)→0 as 𝑡→∞.

Remark 3.5. In the literature dealing with the third-order delay differential equation, the Riccati transformation 𝑤(𝑡)=𝜌(𝑡)𝐿₂𝑥(𝑡)/𝑥(𝑔(𝑡)), where 𝑔(𝑡) is the delay and 𝜌(𝑡) is a differentiable positive function, is used widely, see [3, 4] and the references therein. However, in our paper we find that the transformation 𝑤(𝑡)=𝐿₂𝑥(𝑡)/𝜙(𝑥([𝑡])) (see (3.7)) plays the same role as the more general one 𝑤(𝑡)=𝜌(𝑡)𝐿₂𝑥(𝑡)𝜙(𝑥𝑡.([])) (3.40) In fact, if we replace (3.7) by (3.40) in the proof of Theorem 3.1, then it yields that 𝑤^′(𝑡)(𝑡)𝑞(𝑡)+𝜌≤−𝜌^′(𝑡)𝜌(𝑡)(𝑡)𝑅−𝑝²(𝑡,𝑇)𝑟₁(𝑡)𝑤(𝑡). (3.41) Similarly to Theorem 3.1, we need the following condition (instead of (3.2)) lim_𝑡→∞^𝑡_𝑇𝜌(𝑠)𝑞(𝑠)exp^𝑠_𝑇−𝜌^′(𝑢)𝜌(𝑢)(𝑢)R+𝑝²(𝑢,𝑇)𝑟₁(𝑡)𝑑𝑢𝑑𝑠=∞ (3.42) to obtain a contradiction. Noting that 𝜌(𝑠)exp−^𝑠_𝑇𝜌^′(𝑢)𝜌(𝑢)𝑑𝑢=𝜌(𝑠)exp−_𝜌(𝑇)^𝜌(𝑠)𝑑𝜌𝜌=𝜌(𝑇), (3.43) we conclude that (3.42) is equivalent to (3.2).
Therefore, we cannot get a more general result by using the “more general” transformation (3.40). This is different from the theory of function differential equation with the continuous delay.

4. Examples

In this section, we give several examples to illustrate our main results. For the convenience of readers, let us firstly recall the famous lemma of Kneser [13]. Consider the following second-order ordinary differential equation: 𝑦^′′(𝑡)+𝑎(𝑡)𝑦(𝑡)=0, (4.1) where 𝑎(𝑡) is a locally integrable function of 𝑡. Kneser [13] shows that (4.1) is nonoscillatory if 𝑎(𝑡)≤(1/4𝑡²) and is oscillatory if 𝑎(𝑡)≥(1+𝜀)/4𝑡², where 𝜀 is an any arbitrary positive constant.

Example 4.1. Consider the third-order delay differential equation with piecewise constant argument 𝑥^′′′(𝑡)+3√𝑡𝑒^𝑡+17𝑡²𝑥^′(𝑡)+1+𝑡²𝑥^𝛼([𝑡])𝑒^{1+𝑡𝑥([𝑡])}=0,𝑡≥0, (4.2) where 𝛼>0 is a quotient of odd integers. Clearly, 𝑝(𝑡)=3√𝑡/(𝑒^𝑡+17𝑡²),𝑓(𝑡,𝑥)=𝑡²𝑥^𝛼𝑒^1+𝑡𝑥. Let 𝑞(𝑡)=1+𝑡², 𝜙(𝑥)=𝑥^𝛼, then |𝑓(𝑡,𝑥)|≥𝑞(𝑡)|𝜙(𝑥)|. By the results of [13], 𝑦^′′(𝑡)+𝑝(𝑡)𝑦(𝑡)=0 is nonoscillatory. A simple calculation shows that 𝑝′(𝑡)<0 for 𝑡>1. Therefore, it is easy to see from Corollary 3.4 that any solution of (4.2) is either oscillatory or converges to zero.

Example 4.2. Consider the third-order delay differential equation with piecewise constant argument (𝑡+1)𝑥^′(𝑡)^′′+1𝑥5𝑡+5^′(𝑡)+(𝑡+1)^−6/5𝑥([𝑡])1+𝑒^𝑥([𝑡])=0,𝑡≥0, (4.3) where 𝑟₁(𝑡)=𝑡+1,𝑟₂(𝑡)=1,𝑝(𝑡)=(5𝑡+5)⁻¹,and𝑓(𝑡,𝑥)=(𝑡+1)^−6/5𝑥(1+𝑒^𝑥). Let 𝑞(𝑡)=(𝑡+1)^−6/5,𝜙(𝑥)=𝑥, then |𝑓(𝑡,𝑥)|≥𝑞(𝑡)|𝜙(𝑥)|. It is easy to see that the Euler equation 5𝜏²𝑧^′′(𝜏)+𝑧(𝜏)=0 is nonoscillatory, hence 5(𝑡+1)²𝑦^′′(𝑡)+𝑦(𝑡)=0 is nonoscillatory. A simple calculation leads to Φ(𝑢)=^∞_𝑢𝑞(𝜏)−𝑝^′(𝜏5)𝑑𝜏=(𝑢+1)^1/5+1,Φ(𝑢)(𝑢)≥45𝑢+5−2𝑝5(𝑢+1)^1/5,andhence^𝑡_𝑠Φ(𝑢)−2𝑝(𝑢)𝑟₂(𝑢)𝑑𝑢≥(1+𝑡)^4/5−(1+𝑠)^4/5. (4.4) Thus ^𝑡_𝑇1𝑟₁(𝑠)^𝑡_𝑠Φ(𝑢)−2𝑝(𝑢)𝑟₂(𝑢)()−5𝑑𝑢𝑑𝑠≥ln1+𝑡4−ln(1+𝑇)(1+𝑡)^4/5+54(1+𝑇)^4/5→∞, (4.5) as 𝑡→∞.
On the other hand, exp^𝑡_𝑇𝑝(𝑢)𝑅₂(𝑢,𝑇)𝑟₁(𝑢)=1𝑑𝑢5^𝑡_𝑇(𝑠−𝑇)(𝑠+1)²1𝑑𝑠=51+𝑇()(,1+𝑡+ln1+𝑡−1−ln1+𝑇) (4.6) which yields that exp^𝑡_𝑇𝑝(𝑢)𝑅₂(𝑢,𝑇)𝑟₁(𝑢)𝑑𝑢=𝑎𝑒^{𝑏(1+𝑡)−1}(1+𝑡)^1/5≥𝑎(1+𝑡)^1/5, (4.7) where 𝑎,𝑏 are positive constants. Hence ^𝑡_𝑇𝑞(𝑠)exp^𝑠_𝑇𝑝(𝑢)𝑅₂(𝑢,𝑇)𝑟₁(𝑢)𝑑𝑢𝑑𝑠≥𝑎^𝑡_𝑇11+𝑠𝑑𝑠→∞as𝑡→∞. (4.8)
Obviously, the other conditions of Theorem 3.1 are also satisfied. Hence we conclude that any solution of (4.3) is either oscillatory or converges to zero.

Example 4.3. Consider the third-order delay differential equation with piecewise constant argument 𝑒^−𝑡𝑒^−𝑡𝑥^′′′+𝑒^−2𝑡4𝑥^′(𝑡)+3𝑡2+sin2𝑡ln𝑡+1√2𝑥³([𝑡])=0,𝑡≥0. (4.9) Here 𝑝(𝑡)=𝑒^−2𝑡/4,𝑓(𝑡,𝑥)=(3𝑡/(2+sin2𝑡))ln((𝑡+1)/√2)𝑥³. Let 𝑞(𝑡)=(3𝑡/(2+sin2𝑡))ln((𝑡+1)/√2),𝜙(𝑥)=𝑥³, then |𝑓(𝑡,𝑥)|=𝑞(𝑡)|𝜙(𝑥)|. Since the equation 𝑒^−𝑡(𝑒^−𝑡𝑦^′)^′+𝑝(𝑡)𝑦^′(𝑡)=0 can be reduced to 𝑦^′′−𝑦^′+(1/4)𝑦=0, we conclude that the former is nonoscillatory. It is easy to see that 𝑝^′(𝑡)<0 and ∫^∞₀𝑞(𝑡)𝑑𝑡=∞. Therefore, By Corollary 3.4, any solution of (4.9) is either oscillatory or converges to zero.

Example 4.4. Consider the third-order delay differential equation with piecewise constant argument 1()()𝑥ln1+𝑡2𝑡+3^′′′+1𝑡²(2+𝑡)²𝑥^′(𝑡)+𝑔(𝑡)−4(1+𝑡)𝑡³(2+𝑡)³𝑥^𝛼([𝑡])=0,𝑡≥0, (4.10) where 𝛼>0 is a quotient of odd integers 𝑔(𝑡) is a integrable function such that ^∞𝑔(𝑡)(𝑡)>4()𝑑𝑡=∞,𝑔1+𝑡𝑡³(2+𝑡)³(𝑡>0). (4.11) Let 𝑟₁(𝑡)=2𝑡+3,𝑟₂(𝑡)=1/ln(1+𝑡), 𝑝(𝑡)=1/𝑡²(2+𝑡)²,and𝑓(𝑡,𝑥)=(𝑔(𝑡)−4(1+𝑡)/𝑡³(2+𝑡)³)𝑥^𝛼. We will show that 𝑟₂(𝑡)𝑦^′(𝑡)^′+𝑝(𝑡)𝑟₁(𝑡)𝑦(𝑡)=0 (4.12) is nonoscillatory. We introduce the change of variables: 𝑠=𝑠(𝑡)=^𝑡₀ln(1+𝜏)𝑑𝜏⁻¹,𝑦(𝑡)=𝑠⁻¹𝑧(𝑠), (4.13) which transforms (4.12) into 𝑧^′′(𝑠)+𝑝(𝑡)𝑟₂(𝑡)𝑠⁴𝑟₁(𝑡)(𝑠)𝑧(𝑠)𝑡=𝑡=0. (4.14) Note that 𝑝(𝑡)𝑟₂(𝑡)𝑠⁴𝑟₁(𝑡)(𝑡)=∫𝑠=𝑠^𝑡0ln(1+𝜏)𝑑𝜏⁴(2𝑡+3)𝑡²(2+𝑡)²(),ln1+𝑡 (4.15) and that 𝑡²(2+𝑡)²𝑡=4²2+𝑡²=4^𝑡₀(1+𝜏)𝑑𝜏²≥4^𝑡₀ln(1+𝜏)𝑑𝜏²,(2𝑡+3)ln(1+𝑡)>1,𝑡≥2, (4.16) we obtain that 𝑝(𝑡)𝑟₂(𝑡)𝑠⁴𝑟₁(𝑡)(𝑡)≤∫𝑠=𝑠^𝑡0ln(1+𝜏)𝑑𝜏⁴4∫^𝑡₀ln(1+𝜏)𝑑𝜏²=14𝑠². (4.17) Thus by Kneser [13], it follows that (4.14) (and hence (4.12)) is nonoscillatory.
Let 𝑞(𝑡)=𝑔(𝑡)−4(1+𝑡)/𝑡³(2+𝑡)³, then it is easy to find that ∫^∞𝑞(𝑡)𝑑𝑡=∞. Therefore, by Corollary 3.4, any solution of (4.10) is either oscillatory or converges to zero.

Acknowledgments

The authors would like to thank the referee very much for his valuable comments and suggestions. Liang was supported by the NSF of China (no. 11201086) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant 2012LYM_0087).