Investigation of the Oscillatory Behavior of the Solutions of a Class of Third-Order Delay Differential Equations with Several Terms

Abstract

In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing positive solutions and decreasing positive solutions so that the resulting criteria are easier to apply than other criteria that have appeared in the literature. We have obtained new oscillation criteria that hold up more robustly upon application. Some examples are presented to illustrate the significance of our main findings.

1. Introduction

Studying the oscillation of solutions of third-order delay differential equations is extremely important because it describes complex dynamical systems where future states depend not only on the current state but also on previous states. By studying oscillations, researchers can better understand the long-term behavior of these systems, including stability and cycle. These equations frequently appear in engineering, physics, biology, and economics. For example, they can model phenomena like population dynamics in biology, control systems in engineering, and economic cycles. Understanding oscillations can lead to improved system design and control. The study of oscillations provides insights into the qualitative properties of differential equations, contributing to the broader field of dynamical systems. It involves techniques from various mathematical disciplines, including functional analysis, complex analysis, and numerical methods. By analysing the oscillatory solutions, computational models can be refined to more accurately simulate real-world systems. This can lead to better predictions and optimizations in applied settings [1,2].

There are instances where third-order differential equations are used to describe complex dynamic systems in biology. For instance, in regard to the Lotka–Volterra Predator–Prey Model with Delays, this model can be extended to include delays that can be described by third-order differential equations to account for more complex dynamics and interactions between species. Also, third-order delay differential equations can model the dynamics of glucose and insulin levels in the bloodstream, incorporating delays and feedback mechanisms to better reflect physiological processes.Third-order differential equations are used to model the electrical activity and mechanical responses of the heart, taking into account the propagation of electrical signals and the resulting muscle contractions. Furthermore, some models of neural networks, especially those involving the detailed dynamics of individual neurons and synapses, can involve third-order differential equations to describe the membrane potential changes with higher accuracy [3,4,5,6,7,8,9,10].

Oscillation theory is a fundamental area of study in mathematics and physics that deals with the behavior of systems that exhibit periodic motion. Oscillatory phenomena are ubiquitous in nature, appearing in various forms, such as the swinging of a pendulum, electrical circuits, population dynamics, and even the rhythmic beating of the human heart. The study of oscillations is not only vital for understanding these natural systems but also for the development of technological applications such as signal processing, control systems, and mechanical engineering.

At its core, oscillation theory explores the conditions under which the solutions of differential equations exhibit repetitive behaviors. This includes both linear and nonlinear systems, spanning from simple harmonic motions to complex, chaotic oscillations. By examining the properties of these solutions, such as amplitude, frequency, and stability, researchers can predict and control the behavior of oscillatory systems. Recent developments in oscillation theory have expanded its applicability to more complex domains, including delay differential equations, fractional calculus, and quantum mechanics. As modern challenges such as climate modeling, biological rhythms, and advanced communication systems emerge, the need to understand oscillatory behavior in these contexts becomes increasingly important [11,12,13].

Some third-order delay differential equations are extremely complex and cannot be solved directly. In these cases, understanding the behavior of solutions (such as how they change over time or under the influence of different conditions) can be more realistic and beneficial. The absence of closed-form solutions for some of these equations, especially the nonlinear ones, implies that they may not have an exact analytical solution. In such cases, studying the behavior aids in estimating solutions and understanding the general properties of their solutions (such as the existence of limits, stability, oscillation, etc.) without the need for an exact solution [14,15,16,17,18,19,20,21,22,23,24].

In this paper, we are concerned with the oscillatory behavior of the third-order delay differential equations of the form

$${\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }+\sum _{i=1}^k{q}_i\left(\top \right)h\left({\delta }_i\left(\top \right)\right)=0,$$

(1)

where $k\ge 1$ is an integer and $\top \in [{\top }_0,\infty ),$${\top }_0>0.$ Moreover, throughout this work it is assumed that

(C₁) 

$b_1,b_2\in C([{\top }_0,\infty ),\mathbb{R})$ are positive functions, and

$${\int }_{{\top }_0}^{\infty }{\displaystyle \frac{1}{b_1\left(s\right)}}\mathrm{d}s={\int }_{{\top }_0}^{\infty }{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s=\infty ;$$

(2)

(C₂) 

$q_i\in C([{\top }_0,\infty ),\mathbb{R}),\delta ,{\delta }_i\in C^1([{\top }_0,\infty ),\mathbb{R}),q_i\left(\top \right)\ge 0,$$\delta \left(\top \right)\le {\delta }_i\left(\top \right)<\top ,$${\delta }^{\prime }\left(\top \right)\ge 0$ and ${lim}_{\top \to \infty }\delta \left(\top \right)={lim}_{\top \to \infty }{\delta }_i\left(\top \right)=\infty .$

We consider only solution $h(\top )$ of (1) that belong to $C^1([{\top }_h,\infty ),\mathbb{R}),$ verifying $b_1(\top )h^{\prime }(\top )$$\in C^1[{\top }_h,\infty ),\mathbb{R})$ and $b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\in C^1[{\top }_h,\infty ),\mathbb{R}),$ which satisfy (1) on $[{\top }_h,\infty )$ for every $\top \ge {\top }_h\ge {\top }_0.$ Furthermore, we are interested in those solutions of (1) that exist on $[{\top }_h,\infty )$ and satisfy

$$sup\left\{\left|h(\top )\right|:\top \ge {\top }_1\right\}>0$$

for any ${\top }_h\le {\top }_1.$

Definition 1 

([25]). A solution $h(\top )$ of (1) is called oscillatory if it is neither eventually negative nor eventually positive; otherwise, it is called a nonoscillatory solution.

Definition 2 

([25]). An equation of the type in (1) is called an oscillatory equation if all its solutions are oscillatory. Otherwise, it is called a nonoscillatory equation.

Definition 3 

([26]). We say that Equation (1) is almost oscillatory if its solutions are either oscillatory or tend to zero.

The authors in [27,28] studied new oscillation criteria for the equations

$${\left(b_2(\top ){\left(h^{″}(\top )\right)}^{\varrho }\right)}^{\prime }+q(\top )h^{\rho }\left(\delta (\top )\right))=0,$$

(3)

where $\varrho$ and $\rho$ are quotients of two odd positive integers. They concluded that every solution of (3) is oscillatory or satisfies (5).

By using the principle of comparison, Zhang et al. [29] established some conditions that guarantee that the solutions of the equation

$${\left(b_2(\top ){\left(h^{\left(n-1\right)}(\top )\right)}^{\varrho }\right)}^{\prime }+q(\top )h^{\rho }\left(\delta (\top )\right)=0,\mathrm{where}\rho \le \varrho .$$

are oscillatory.

On the other hand, the oscillation of the nonlinear delay differential equations

$${\left(b_2(\top ){\left(h^{″}(\top )\right)}^{\varrho }\right)}^{\prime }+q(\top )f\left(h\left(\delta (\top )\right)\right)=0,$$

in their canonical and non-canonical forms, where $\varrho$ is a quotient of two odd positive integers, was discussed in [30,31].

Some of the asymptotic and oscillatory properties of Equation (1) have been studied in the previous literature; for more details, see [27,29,32,33,34] and the references cited therein. Below, we present two important results related to the topic under study.

Li et al. [34] studied equation

$${\left(b_2(\top ){\left(b_1(\top )\left(h(\top )+\mathsf{\Omega }\left(\top \right)h\left(\sigma \left(\top \right)\right)\right)\right)}^{\prime }\right)}^{\prime }+q\left(\top \right)h\left(\delta \left(\top \right)\right)=0$$

(4)

and obtained sufficient conditions that ensure that the solutions of (4) are oscillatory or satisfy

$$\underset{\top \to \infty }{lim}h(\top )=0.$$

(5)

By setting $\mathsf{\Omega }\left(\top \right)=0$, the authors arrived at the following theorem:

Theorem 1 

(Theorem 2.1, [34]). Assume that (2) holds. If there exists a function $w\left(\top \right)\in C^1\left([{\top }_0,\infty ),(0,\infty )\right),$ for ${\top }_1\ge {\top }_0$ and for ${\top }_3\ge {\top }_2\ge {\top }_1$ such that

$${\int }_{{\top }_0}^{\infty }{\displaystyle \frac{1}{b_1\left(v\right)}}\left({\int }_v^{\infty }{\displaystyle \frac{1}{b_2\left(\varsigma \right)}}\left({\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)\mathrm{d}\varsigma \right)\mathrm{d}v=\infty$$

(6)

and

$$\underset{\top \to \infty }{lim\; sup}{\int }_{{\top }_3}^{\top }\left(w\left(s\right)q\left(s\right){\displaystyle \frac{{\int }_{{\top }_2}^{\delta \left(s\right)}\left({\int }_{{\top }_1}^{v}1/b_2\left(u\right)\mathrm{d}u/b_1\left(v\right)\right)\mathrm{d}v}{{\int }_{{\top }_1}^v\left(1/b_2\left(u\right)\right)\mathrm{d}u}}-{\displaystyle \frac{b_2\left(s\right){\left(w^{\prime }\left(s\right)\right)}^2}{4w\left(s\right)}}\right)\mathrm{d}s=\infty ,$$

then (4) is almost oscillatory.

Theorem 2 

(Theorem 3, [35]). Assume that $q\in C([{\top }_0,\infty ),\mathbb{R})$ and there exists a function $\zeta \left(\top \right)\in C^1\left([{\top }_0,\infty ),\mathbb{R}\right)$ such that ${\zeta }^{\prime }\left(\top \right)\ge 0,\zeta \left(\top \right)>\top ,\eta \left(\top \right)=\delta \left(\zeta \left(\zeta \left(\top \right)\right)\right)<\top .$ If both the equations

$$x^{\prime }\left(\top \right)+q\left(\top \right){\int }_{{\top }_0}^{\delta \left(\top \right)}\left({\displaystyle \frac{1}{b_2\left(\varsigma \right)}}\left(\delta \left(\top \right)-\varsigma \right)\right)\mathrm{d}\varsigma *x\left(\delta \left(\top \right)\right)=0$$

(7)

and

$$z^{\prime }\left(\top \right)+{\int }_{\top }^{\zeta \left(\top \right)}{\displaystyle \frac{1}{b_2\left(v\right)}}\left({\int }_v^{\zeta \left(v\right)}q\left(\varsigma \right)\mathrm{d}\varsigma \right)\mathrm{d}v*z\left(\eta \left(\top \right)\right)=0$$

are oscillatory, then (1) with $b_1(\top )=1$ is also oscillatory.

Theorem 3 

(Theorem 2, [35]). Assume that

$${\int }_{{\top }_0}^{\infty }\left({\int }_v^{\infty }{\displaystyle \frac{1}{b_2\left(\varsigma \right)}}\left({\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)\mathrm{d}\varsigma \right)\mathrm{d}v=\infty .$$

If Equation (7) is oscillatory, then every solution of Equation (1) with $b_1(\top )=1$ is almost oscillatory.

The main purpose of this paper is to study the asymptotic and oscillatory behavior of the solutions of Equation (1). The relationships linking the solutions of this equation with its derivatives, which have been studied in the previous literature, are considered routine and may sometimes fail under certain conditions (see [5,18,20,21,22,24,27,35]). Through this study, we provide new relationships characterised by flexibility and greater stability upon applications. Using these relations, we provide some new conditions that guarantee the oscillation of the solutions of the considered equation. Our results are characterized by their novelty and generality compared to previous results, as well as the improvement of previous techniques. Therefore, they represent a new contribution to the oscillation theory. The rest of the article is as follows: the next section reviews some previous results necessary for further development. Section 3 collects the main results that we have obtained on the oscillatory behavior of Equation (1). Section 4 is dedicated to presenting the conclusions of the work.

2. Some Preliminary Lemmas

The use of first-order delay differential equations dates back to the second half of the 19th century and the early 20th century. They were used to model a variety of problems in fields such as engineering, physics, economics, and biology. For instance, in 1925, the French physicist Louis Bachelier applied delay differential equations in his study of stock market movements and financial markets. In the field of biology, these equations were used to study the growth cycles of grasses, trees, and other biological phenomena that necessitate consideration of time factors. The utilisation of this type of equation in research and applications became more widespread in the 20th century, especially with the advancement of technology and computing, which aided in the analysis and solution of these equations more efficiently and accurately [36,37,38].

One of the most important practical applications of first-order delay differential equations is the regression model, which is used to estimate changes in states over time in many fields and adopts the form

$$h^{\prime }\left(\top \right)=-\lambda h\left(\top -\delta \right),$$

where $\delta$ is the delay and $\lambda$ is the regression coefficient. This model is utilized in various domains, such as in biology to study population growth, in economics to analyze time series, in engineering to study the response of dynamic systems, and many other applications.

We now present some essential relationships that will help us to obtain the main results.

Lemma 1. 

([39]). Assume that

$$\delta \in C^1[{\top }_0,\infty ),\delta \left(\top \right)<\top ,\sigma \left(\top \right)<\top ,{\delta }^{\prime }\left(\top \right)\ge 0$$

(8)

and

$$g\in C[{\top }_0,\infty ),g\left(\top \right)>0,\underset{\top \to \infty }{lim}g\left(\top \right)=\infty .$$

hold, and the first-order delay differential inequality

$$h^{\prime }\left(\top \right)+g\left(\top \right)h\left(\delta \left(\top \right)\right)\le 0$$

has a positive solution, then the delay differential equation

$$h^{\prime }\left(\top \right)+g\left(\top \right)h\left(\delta \left(t\right)\right)=0,$$

(9)

also has a positive solution.

Lemma 2. 

Suppose that

$${\int }_{\top }^{\top +\delta }g\left(s\right)ds>0\mathrm{for}\top \ge {\top }_0>0,$$

and

$${\int }_{{\top }_0}^{\infty }g\left(\top \right)ln\left(e{\int }_{\top }^{\top +\delta }g\left(s\right)ds\right)d\top =\infty .$$

(10)

Then, every solution of (9) oscillates.

Proof. 

Assume that $h\left(\top \right)>0$ is a solution of (9). It is clear that

$$h^{\prime }\left(\top \right)=-g\left(\top \right)h\left(\delta \left(t\right)\right).$$

That is $h^{\prime }\left(\top \right)<0.$ Let

$$\xi \left(\top \right)={\displaystyle \frac{-h^{\prime }\left(\top \right)}{h\left(\top \right)}}.$$

Not that the function $\xi$ is continuous and $\xi \left(\top \right)\ge 0$ for large ⊤, and

$${\displaystyle \frac{h\left(\top \right)}{h\left({\top }_1\right)}}=exp\left(-{\int }_{{\top }_1}^{\top }\xi \left(s\right)\mathrm{d}s\right),$$

where $h\left({\top }_1\right)>0$ for some ${\top }_1\ge {\top }_0$. Moreover, $\xi \left(\top \right)$ satisfies the generalized characteristic equation

$${\displaystyle \frac{\xi \left(\top \right)}{g\left(\top \right)}}=exp\left({\int }_{\top -\delta }^{\top }\xi \left(s\right)\mathrm{d}s\right).$$

That is

$$e^{Ah}-h\ge {\displaystyle \frac{ln\left(eA\right)}{A}}\mathrm{for}A>0.$$

Define

$$H\left(\top \right)={\int }_{\top }^{\top +\delta }g\left(\top \right)\mathrm{d}s,$$

and thus,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\xi \left(\top \right)}{g\left(\top \right)}}& =& exp\left(H\left(\top \right){\displaystyle \frac{{\int }_{\top -\delta }^{\top }\xi \left(s\right)\mathrm{d}s}{H\left(\top \right)}}\right)\hfill \\ & \ge & \left({\displaystyle \frac{{\int }_{\top -\delta }^{\top }\xi \left(s\right)\mathrm{d}s}{H\left(\top \right)}}+{\displaystyle \frac{ln\left(eH\left(\top \right)\right)}{H\left(\top \right)}}\right).\hfill \end{array}$$

It follows that

$${\displaystyle \frac{\xi \left(\top \right){\int }_{\top }^{\top +\delta }g\left(s\right)-g\left(\top \right){\int }_{\top -\delta }^{\top }\xi \left(s\right)\mathrm{d}s}{g\left(\top \right)}}\ge ln\left(e{\int }_{\top }^{\top +\delta }g\left(s\right)\mathrm{d}s\right),$$

and for $y>y_{*},$ we have

$$\begin{array}{ccc}& & {\int }_{y_{*}}^y\xi \left(\top \right)\left({\int }_{\top }^{\top +\delta }g\left(s\right)\right)\mathrm{d}s\mathrm{d}\top \hfill \\ & \ge & {\int }_{y_{*}}^{y}g\left(\top \right)\left({\int }_{\top -\delta }^{\top }\xi \left(s\right)\mathrm{d}s\right)\mathrm{d}\top +{\int }_{y_{*}}^{y}g\left(\top \right)ln\left(e{\int }_{\top }^{\top +\delta }g\left(s\right)\mathrm{d}s\right)\mathrm{d}\top .\hfill \end{array}$$

Now, changing the order of integration

$$\begin{array}{ccc}\hfill {\int }_{y_{*}}^{y}g\left(\top \right)\left({\int }_{\top -\delta }^{\top }\xi \left(s\right)\mathrm{d}s\right)\mathrm{d}\top & \ge & {\int }_{y_{*}}^{y-y_{*}}\left({\int }_s^{s+\delta }g\left(\top \right)\xi \left(\top \right)\mathrm{d}\top \right)\mathrm{d}s\hfill \\ & =& {\int }_{y_{*}}^{y-\delta }\xi \left(s\right)\left({\int }_s^{s+\delta }g\left(\top \right)\mathrm{d}\top \right)\mathrm{ds}\hfill \\ & =& {\int }_{y_{*}}^{y-\delta }\xi \left(s\right)\left({\int }_{\top }^{\top +\delta }g\left(s\right)\mathrm{d}s\right)\mathrm{d}\top .\hfill \end{array}$$

It follows that

$${\int }_{y-\delta }^y\xi \left(s\right){\int }_{\top }^{\top +\delta }g\left(\top \right)\mathrm{d}s\mathrm{d}\top \ge {\int }_{y_{*}}^{y}g\left(\top \right)ln\left(e{\int }_{\top }^{\top +\delta }g\left(s\right)\mathrm{d}s\right)\mathrm{d}\top .$$

(11)

In view of (Lemma 2, [40]), we see that

$${\int }_{\top }^{\top +\delta }g\left(s\right)\mathrm{d}s-1\le 0\mathrm{eventually}$$

(12)

From (11) and (12), we obtain

$${\int }_{y-\delta }^y\xi \left(\top \right)\mathrm{d}\top -{\int }_{y_{*}}^{y}g\left(\top \right)ln\left(e{\int }_{\top }^{\top +\delta }g\left(s\right)\mathrm{d}s\right)\mathrm{d}\top \ge 0,$$

that is

$$ln{\displaystyle \frac{h\left(y-\delta \right)}{h\left(y\right)}}-{\int }_{y_{*}}^{y}g\left(\top \right)ln\left(e{\int }_{\top }^{\top +\delta }g\left(s\right)\mathrm{d}s\right)\mathrm{d}\top \ge 0.$$

By

$$lim{\displaystyle \frac{h\left(y-\delta \right)}{h\left(y\right)}}=\infty .$$

According to (10), there exists a sequence $\left\{{\top }_f\right\}$ with ${\top }_f\to \infty$ as $f\to \infty$ such that

$${\int }_{{\top }_f}^{{\top }_f+\delta }g\left(s\right)\mathrm{d}s-{\displaystyle \frac{1}{e}}\ge 0\mathrm{for}\mathrm{all}f.$$

Thus, by (Lemma 1, [40]), we have $lim\; inf{\displaystyle \frac{h\left(y-\delta \right)}{h\left(y\right)}}<\infty$. The proof is complete. □

Lemma 3. 

Assume that $h(\top )>0$ is a solution of (1). Then

$$\begin{array}{ccc}\hfill {\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }& <& 0,\hfill \\ \hfill b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }& >& 0,\hfill \end{array}$$

and one of the following cases is satisfied:

$$h^{\prime }(\top )>0$$

(13)

or

$$h^{\prime }(\top )<0,$$

(14)

eventually on $[{\top }_0,\infty )$.

Proof. 

Suppose that $h(\top )>0.$ Then, since ${lim}_{\top \to \infty }{\delta }_i\left(\top \right)=\infty$ for $i=1,2,\cdots ,k,$ there exists a ${\top }_1\ge {\top }_0$ such that $h\left({\delta }_i\left(\top \right)\right)>0$ for $\top \ge {\top }_1.$ Using (1), we find

$${\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }=-\sum _{i=1}^k{q}_i\left(\top \right)h\left({\delta }_i\left(\top \right)\right).$$

Therefore, since $h\left({\delta }_i\left(\top \right)\right)>0$ and $q_i\left(\top \right)\ge 0$ for $\top \ge {\top }_1,$ we have

$${\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }<0.$$

That is, ${\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }$ is nonincreasing and of one sign. If we admit that

$$b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\le 0.$$

Then, there exists a constant $J<0$ such that

$$b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\le J<0,$$

(15)

that is

$${\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\le {\displaystyle \frac{J}{b_2(\top )}}.$$

Integrating (15), we obtain

$$h^{\prime }(\top )\le {\displaystyle \frac{b_1\left({\top }_3\right)}{b_1(\top )}}{h}^{\prime }\left({\top }_3\right)+J{\displaystyle \frac{1}{b_1(\top )}}{\int }_{{\top }_3}^{\top }{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s.$$

Letting $\top \to \infty$, from (2), we obtain $h^{\prime }(\top )\to -\infty$. Thus, $h^{\prime }(\top )<0$ eventually, this implies $h(\top )\to -\infty$ as $\top \to \infty ,$ which is a contradiction. Therefore $b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\ge 0$. The proof is complete. □

Lemma 4. 

Suppose that $h(\top )>0$ is a solution of (1) and holds property (14). If

$${\int }_{{\top }_1}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}\left({\int }_v^{\infty }{\displaystyle \frac{1}{b_2\left(\varsigma \right)}}\left({\int }_{\varsigma }^{\infty }\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s\right)\mathrm{d}\varsigma \right)\mathrm{d}v=\infty ,$$

(16)

then $h(\top )$ tends to zero as $\top \to \infty$.

Proof. 

It is clear that there exists a finite ${lim}_{\top \to \infty }h(\top )=L$. Assume that $L>0$.

Integrating (1) from ⊤ to ∞ and using the property $h\left({\delta }_i\left(\top \right)\right)>L$, we obtain

$$\begin{array}{ccc}\hfill b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }& \ge & {\int }_{\top }^{\infty }\sum _{i=1}^k{q}_i\left(s\right)h\left({\delta }_i\left(s\right)\right)\mathrm{d}s\hfill \\ & \ge & L{\int }_{\top }^{\infty }\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s\hfill \end{array}$$

that is,

$${\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\ge {\displaystyle \frac{L}{b_2(\top )}}{\int }_{\top }^{\infty }\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s.$$

Integrating the above inequality from ⊤ to ∞, we obtain

$$-b_1(\top )h^{\prime }(\top )\ge {\int }_{\top }^{\infty }{\displaystyle \frac{L}{b_2(\top )}}\left({\int }_{\top }^{\infty }\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s\right)\mathrm{d}u,$$

which implies

$$-h^{\prime }(\top )\ge {\displaystyle \frac{1}{b_1(\top )}}{\int }_{\top }^{\infty }{\displaystyle \frac{L}{b_2(\top )}}\left({\int }_{\top }^{\infty }\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s\right)\mathrm{d}u.$$

Integrating this inequality ${\top }_1$ to ⊤, we have

$$h\left({\top }_1\right)\ge L{\int }_{{\top }_1}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}\left({\int }_v^{\infty }{\displaystyle \frac{1}{b_2\left(\varsigma \right)}}\left({\int }_{\varsigma }^{\infty }\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s\right)\mathrm{d}\varsigma \right)\mathrm{d}v.$$

Letting $\top \to \infty ,$ this contradicts (16) and so we have that ${lim}_{\top \to \infty }h(\top )=0.$ □

Now, we state the following useful property concerning the solutions verifying (13). Let us iteratively define the following functions:

$$\begin{array}{ccc}\hfill R_1(\top ,{\top }_{*})& =& {\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}{\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s\mathrm{d}v,\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill R_{m+1}(\top ,{\top }_{*})& =& {\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}\left({\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left(\tilde{R}\left(\top \right)\right)\mathrm{d}s\right)\mathrm{d}v,m\in \mathbb{N},\hfill \end{array}$$

(18)

and from these,

$${\beta }_1\left(\top ,{\top }_{*}\right)={\int }_s^{\top }{R}_m(\delta \left(\varsigma \right),{\top }_{*})\sum _{i=1}^k{q}_i\left(\varsigma \right)\mathrm{d}\varsigma ,$$

for $\top \ge {\top }_{*},$${\top }_{*}\in [{\top }_0,\infty ).$

Lemma 5. 

Assume that $h(\top )>0$ is a solution of (1) satisfying (13). Then, for $\top \ge {\top }_{*}$, is large enough, with ${\top }_{*}\in [{\top }_0,\infty )$, it holds

$$R_m(\delta (\top ),{\top }_{*})\le {\displaystyle \frac{h\left(\delta \right(\top \left)\right)}{b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }}}.$$

(19)

Proof. 

We proceed by induction. Assume that $h(\top )>0$ is a solution of (1) and (13) holds on $[{\top }_1,\infty ),$${\top }_1\in [{\top }_0,\infty ).$ According to

$${\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }\le 0,$$

we have that

$$\begin{array}{ccc}\hfill b_1(\top )h^{\prime }(\top )& \ge & {\int }_{{\top }_1}^{\top }{\displaystyle \frac{1}{b_2\left(s\right)}}{b}_2\left(s\right){\left(b_1\left(s\right)h^{\prime }\left(s\right)\right)}^{\prime }\mathrm{d}s\hfill \\ & \ge & b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }{\int }_{{\top }_1}^{\top }{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s,\hfill \end{array}$$

which means that

$$h^{\prime }(\top )\ge {\displaystyle \frac{b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }}{b_1(\top )}}{\int }_{{\top }_1}^{\top }{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s,\top \ge {\top }_1.$$

(20)

Integrating (20) and using (17), we have

$$\begin{array}{ccc}\hfill h(\top )& \ge & {\int }_{{\top }_1}^{\top }{\displaystyle \frac{b_2\left(v\right){\left(b_1\left(v\right){\left(h\left(v\right)\right)}^{\prime }\right)}^{\prime }}{b_1\left(v\right)}}{\int }_{{\top }_1}^v{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s\mathrm{d}v\hfill \\ & \ge & b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }{R}_1(\top ,{\top }_1).\hfill \end{array}$$

Then, by using ${lim}_{\top \to \infty }\delta \left(\top \right)=\infty ,$ we arrive at

$${\displaystyle \frac{h\left(\delta \right(\top \left)\right)}{b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }}}\ge R_1(\delta (\top ),{\top }_2),\mathrm{for}\mathrm{all}\top \ge {\top }_2.$$

That is, (19) is satisfied when $m=1.$

Now, suppose that (19) holds for some $m>1$. Then we see that

$${\displaystyle \frac{h\left(\delta \right(\top \left)\right)}{b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }}}\ge R_m(\delta (\top ),{\top }_{*}),\top \ge {\top }_{*}\ge {\top }_{2.}.$$

(21)

Substituting (21) in (1) and using $\delta \left(\top \right)\le {\delta }_i\left(\top \right)<\top$, we receive

$${\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }\le -R_m(\delta (\top ),{\top }_{*})b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }\sum _{i=1}^k{q}_i\left(\top \right).$$

(22)

Setting $x(\top ):=b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime },$ (22) implies that

$$x^{\prime }(\top )\le -R_m(\delta (\top ),{\top }_{*})\sum _{i=1}^k{q}_i\left(\top \right)*x(\top ).$$

(23)

That is, $x(\top )$ is nonincreasing, and using the fact that $\delta (\top )<\top ,$ we obtain

$$x\left(\delta (\top )\right)/x(\top )\ge 1.$$

Applying the Gronwall inequality to (23), we receive

$${\displaystyle \frac{b_2\left(\delta \left(s\right)\right){\left(b_1\left(\delta \left(s\right)\right)h^{\prime }\left(\delta \left(s\right)\right)\right)}^{\prime }}{b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }}}\ge exp\left({\int }_s^{\top }\left(R_m(\delta \left(\varsigma \right),{\top }_{*})\sum _{i=1}^k{q}_i\left(\varsigma \right)\right)\mathrm{d}\varsigma \right),$$

(24)

$\top \ge s\ge {\top }_{*},$ and thus,

$$\begin{array}{ccc}\hfill b_1(\top )h^{\prime }(\top )& \ge & {\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{b_2\left(\delta \left(s\right)\right){\left(b_1\left(\delta \left(s\right)\right)h^{\prime }\left(\delta \left(s\right)\right)\right)}^{\prime }}{b_2\left(s\right)}}\mathrm{d}s\hfill \\ & \ge & b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }{\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left({\beta }_1\left(s,{\top }_{*}\right)\right)\mathrm{d}s,\hfill \end{array}$$

which gives,

$$\begin{array}{ccc}\hfill h^{\prime }(\top )& \ge & {\displaystyle \frac{b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }}{b_1(\top )}}\hfill \\ & & \times {\int }_{{\top }_{*}}^{\top }\left({\displaystyle \frac{1}{b_2\left(s\right)}}exp\left({\beta }_1\left(s,{\top }_{*}\right)\right)\right)\mathrm{d}s.\hfill \end{array}$$

After integrating the above inequality from ${\top }_{*}$ to $\top ,$ due to (24), we arrive at

$$\begin{array}{ccc}\hfill h(\top )& \ge & {\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{b_2\left(\delta \left(v\right)\right){\left(b_1\left(\delta \left(v\right)\right)h^{\prime }\left(\delta \left(v\right)\right)\right)}^{\prime }}{b_1\left(v\right)}}{\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left({\beta }_1\left(v,{\top }_{*}\right)\right)\mathrm{d}s\mathrm{d}v\hfill \\ & \ge & b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }{\int }_{{\top }_{*}}^{\top }exp\left({\beta }_1\left(v,{\top }_{*}\right)\right){\displaystyle \frac{1}{b_1\left(v\right)}}{\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left({\beta }_1\left(v,{\top }_{*}\right)\right)\mathrm{d}s\mathrm{d}v\hfill \\ & =& b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\hfill \\ & & \times {\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}{\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left({\beta }_1\left(\top ,{\top }_{*}\right)\right)\mathrm{d}s\mathrm{d}v.\hfill \end{array}$$

That is, there exists ${\top }_{**}\ge {\top }_{*}$ such that

$${\displaystyle \frac{h\left(\delta \right(\top )}{b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }}}\ge {\int }_{{\top }_{**}}^{\delta (\top )}{\displaystyle \frac{1}{b_1\left(v\right)}}{\int }_{{\top }_{**}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left({\int }_s^{\delta (\top )}\left(R_m(\delta \left(\varsigma \right),{\top }_{**})\sum _{i=1}^k{q}_i\left(\varsigma \right)\right)\mathrm{d}\varsigma \right)\mathrm{d}s\mathrm{d}v,$$

(25)

for all $\top \ge {\top }_{**},$ which according to (18) gives

$${\displaystyle \frac{h\left(\delta \right(\top \left)\right)}{b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }}}\ge R_{m+1}(\delta (\top ),{\top }_{**}),\top \ge {\top }_{**}.$$

The proof is complete. □

Remark 1. 

By setting $m=1$ in (19) we will recover a relationship that appeared in the previous literature; for example in [5,22,24].

Now, we state a useful property related to the solutions of (1) that verify condition (14).

We define the following functions iteratively

$$\begin{array}{ccc}\hfill {\tilde{R}}_1(v,\varsigma )& =& {\int }_{\varsigma }^v{\displaystyle \frac{1}{b_1\left(x\right)}}{\int }_x^v{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s\mathrm{d}x,\hfill \\ \hfill {\tilde{R}}_{n+1}(v,\varsigma )& =& {\int }_{\varsigma }^v{\displaystyle \frac{1}{b_1\left(x\right)}}{\int }_x^v{\displaystyle \frac{1}{b_2\left(s\right)}}\hfill \\ & & \times exp\left(\left({\int }_s^v{\tilde{R}}_n(z,\delta \left(z\right))\sum _{i=1}^k{q}_i\left(z\right)\right)\mathrm{d}z\right)\mathrm{d}s\mathrm{d}x,n\in \mathbb{N}\hfill \end{array}$$

and from these,

$${\beta }_2(v,\varsigma )=\left({\int }_s^v{\tilde{R}}_n(z,\delta \left(z\right))\sum _{i=1}^k{q}_i\left(z\right)\right)\mathrm{d}z$$

for $v\ge \varsigma \ge {\top }_{*},$${\top }_{*}\in [{\top }_0,\infty ).$

Lemma 6. 

Assume that $h(\top )>0$ is a solution of (1), which satisfies property (14). Then

$${\tilde{R}}_n(v,\varsigma )\le {\displaystyle \frac{h\left(\varsigma \right)}{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}},v\ge \varsigma \ge {\top }_{*},{\top }_{*}\in [{\top }_0,\infty )\mathit{is}\mathit{large}\mathit{enough}.$$

(26)

Proof. 

We proceed by induction. Let $h(\top )>0$ be a solution of (1) and assume that (14) holds on $[{\top }_1,\infty ),$${\top }_1\in [{\top }_0,\infty ).$ According to ${\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }\le 0$, we see that

$$\begin{array}{ccc}\hfill -b_1\left(\varsigma \right)h^{\prime }\left(\varsigma \right)& \ge & {\int }_{\varsigma }^v{\displaystyle \frac{b_2\left(s\right){\left(b_1\left(s\right)h^{\prime }\left(s\right)\right)}^{\prime }}{b_2\left(s\right)}}\mathrm{d}s\hfill \\ & \ge & b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }{\int }_{\varsigma }^v{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s,\hfill \end{array}$$

(27)

for some $v\ge \varsigma \ge {\top }_1,$ that is

$$-h^{\prime }\left(\varsigma \right)\ge {\displaystyle \frac{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}{b_1\left(\varsigma \right)}}{\int }_{\varsigma }^v{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s.$$

(28)

Integrating the above inequality from $\varsigma$ to v, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{h\left(\varsigma \right)}{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}}& \ge & {\int }_{\varsigma }^v{\displaystyle \frac{1}{b_1\left(x\right)}}\left({\int }_x^v{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s\right)\mathrm{d}x\hfill \\ & =& {\tilde{R}}_1(v,\varsigma ).\hfill \end{array}$$

(29)

Thus, (27) holds when $n=1.$

Now, assume that (27) is satisfied for some $n>1$. Then, we have

$${\displaystyle \frac{h\left(\delta \left(\top \right)\right)}{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}}\ge {\tilde{R}}_n(v,\delta \left(\top \right)),v\ge \varsigma \ge {\top }_{*}\ge \top .$$

(30)

Using (30) with $v=\top$ and $\varsigma =\delta (\top )$ in (1), we obtain

$${\left(b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }\right)}^{\prime }+b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }{\tilde{R}}_n(v,\delta (\top ))\sum _{i=1}^k{q}_i\left(\top \right)\le 0.$$

Similar to what is seen in the proof of Lemma 5, we find that

$${\displaystyle \frac{b_2\left(s\right){\left(b_1\left(s\right)h^{\prime }\left(s\right)\right)}^{\prime }}{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}}\ge exp\left(\left({\int }_s^v{\tilde{R}}_n(z,z\left(\delta \right))\sum _{i=1}^k{q}_i\left(z\right)\right)\mathrm{d}z\right),v\ge s\ge {\top }_{*}.$$

(31)

Using (27) in (31), we have

$${\displaystyle \frac{-b_1\left(\varsigma \right)h^{\prime }\left(\varsigma \right)}{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}}\ge {\int }_{\varsigma }^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left(\left({\int }_s^v{\tilde{R}}_n(z,z\left(\delta \right))\sum _{i=1}^k{q}_i\left(z\right)\right)\mathrm{d}z\right)\mathrm{d}s,$$

from which we receive

$${\displaystyle \frac{-h^{\prime }\left(\varsigma \right)}{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}}\ge {\displaystyle \frac{1}{b_1\left(\varsigma \right)}}{\int }_{\varsigma }^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left(\left({\int }_s^v{\tilde{R}}_n(z,z\left(\delta \right))\sum _{i=1}^k{q}_i\left(z\right)\right)\mathrm{d}z\right)\mathrm{d}s.$$

(32)

Integrating (32) from $\varsigma$ to $v\ge s\ge {\top }_{*},$ it follows that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{h\left(\varsigma \right)}{b_2\left(v\right){\left(b_1\left(v\right)h^{\prime }\left(v\right)\right)}^{\prime }}}& \ge & {\int }_{\varsigma }^v{\displaystyle \frac{1}{b_1\left(x\right)}}{\int }_x^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left(\left({\int }_s^v{\tilde{R}}_n(z,z\left(\delta \right))\sum _{i=1}^k{q}_i\left(z\right)\right)\mathrm{d}z\right)\mathrm{d}s\mathrm{d}x\hfill \\ & =& {\tilde{R}}_{n+1}(\varsigma ,v).\hfill \end{array}$$

This completes the proof. □

3. Oscillatory Results

In this section, we first establish sufficient criteria to ensure that any nonoscillatory solution to Equation (1) approaches zero when $\top \to \infty$. Second, we provide new criteria to ensure that Equation (1) is oscillatory.

Theorem 4. 

Assume that (6) holds. If

$$\underset{\top \to \infty }{lim\; inf}{\int }_{\delta \left(\top \right)}^{\top }{R}_m(\delta \left(s\right),{\top }_{*})\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s>{\displaystyle \frac{1}{e}},\mathit{for}\mathit{some}{\top }_{*}\in [{\top }_0,\infty )\mathit{and}m\in \mathbb{N},$$

(33)

then Equation (1) is almost oscillatory.

Proof. 

We proceed by contradiction. Let us suppose that $h(\top )>0$ is a solution of (1) and assume that it is $h\left(\delta (\top )\right)>0$ for $\top \ge {\top }_1\ge {\top }_0.$ Set

$$P(\top ):=b_2(\top ){\left(b_1(\top )h^{\prime }(\top )\right)}^{\prime }>0.$$

According to Lemma 3, there are two exclusive cases. First, we will consider that case (13) holds for $\top \ge {\top }_1$. Using (19) and (1), we have

$$P^{\prime }(\top )+P\left(\delta (\top )\right)R_m(\delta (\top ),{\top }_{*})\sum _{i=1}^k{q}_i\left(\top \right)\le 0.$$

(34)

Since $P(\top )$ is a solution of inequality (34), in view of Lemma 1, the associated delay differential equation

$$P^{\prime }(\top )+P\left(\delta (\top )\right)R_m(\delta (\top ),{\top }_{*})\sum _{i=1}^k{q}_i\left(\top \right)=0$$

(35)

also has a positive solution. From (Theorem 2.1.1, [41]), (33) leads to the oscillation of (35), which is a contradiction.

Now, we will consider case (14). This follows directly, since by Lemma 4 we see that ${lim}_{\top \to \infty }h(\top )=0.$ The proof is complete. □

Example 1. 

Setting in (1)

$${\delta }_1\left(\top \right)={\displaystyle \frac{1}{2}}\top \mathit{and}{\delta }_2\left(\top \right)={\displaystyle \frac{4}{5}}\top ,\mathit{with}\delta \left(\top \right)={\displaystyle \frac{1}{2}}\top ,$$

and

$$b_2(\top )=b_1(\top )=1,q_1\left(\top \right)={\displaystyle \frac{3}{{\top }^3}},\mathit{with}\top \in [1,\infty ),$$

(36)

we see that taking ${\top }_{*}=0,$ we obtain

$$R_1\left(\top ,{\top }_{*}\right)={\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}{\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}ds\mathrm{d}v={\displaystyle \frac{{\top }^2}{2}},$$

and

$$R_2\left(\top ,{\top }_{*}\right)={\top }^{3/8}{\int }_{{\top }_{*}}^{\top }{\int }_{{\top }_{*}}^v{s}^{-3/8}\mathrm{d}s\mathrm{d}v={\displaystyle \frac{64}{65}}{\top }^2.$$

Applying Condition (33) when $m=1$ and $m=2$, we obtain

$$0.25993<1/e$$

and

$$0.511863>1/e,$$

(37)

respectively.

Therefore, (33) is satisfied and (35) holds for $m=2$. That is, from Theorem 4, every solution of (1) with the settings in (36) is almost oscillatory.

Remark 2. 

When we substitute $m=1$ in Theorem 4, we receive the same results as in references (Theorem 1, [21]), (Theorem 2, [35]), (Theorem 12, [42]), and (Theorem 2.9, [43]). Note that all of these results fail, since $0.25993<1/e$, while our results held up when $m=2.$

Example 2. 

Setting in (1)

$${\delta }_1\left(\top \right)={\displaystyle \frac{1}{4}}\top \mathit{and}{\delta }_2\left(\top \right){\displaystyle \frac{1}{2}}\top ,\mathit{with}\delta \left(\top \right)={\displaystyle \frac{1}{4}}\top ,$$

and

$$b_2(\top )=b_1(\top )=1,q_1\left(\top \right)={\displaystyle \frac{8}{{\top }^3}},\mathit{such}\mathit{that}\top \in [1,\infty ),$$

(38)

we see that, by taking ${\top }_{*}=0$, we obtain

$$R_1\left(\top ,{\top }_{*}\right)={\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}{\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}\mathrm{d}s\mathrm{d}v={\displaystyle \frac{{\top }^2}{2}},$$

and

$$\begin{array}{ccc}\hfill R_2\left(\top ,{\top }_{*}\right)& =& {\int }_{{\top }_{*}}^{\top }{\displaystyle \frac{1}{b_1\left(v\right)}}{\int }_{{\top }_{*}}^v{\displaystyle \frac{1}{b_2\left(s\right)}}exp\left({\int }_s^{\top }\sum _{i=1}^k{q}_i\left(\varsigma \right)R_1\left(\delta \left(\varsigma \right),{\top }_{*}\right)\mathrm{d}\varsigma \right)\mathrm{d}s\mathrm{d}v\hfill \\ & =& {\displaystyle \frac{16}{21}}{\top }^2.\hfill \end{array}$$

Applying Condition (33) when $m=1$ and $m=2,$ we obtain

$$0.34657<{\displaystyle \frac{1}{\mathrm{e}}}$$

and

$$2.11245>{\displaystyle \frac{1}{\mathrm{e}}},$$

respectively. Note that (33) holds and (35) is satisfied when $m=2$. That is, from Theorem 4, every solution of (1) with the settings in (38) is almost oscillatory.

Theorem 5. 

Assume that (33) holds. If

$$\underset{\top \to \infty }{lim\; sup}{\int }_{\delta \left(\top \right)}^{\top }{\tilde{R}}_n(\delta (\top ),\delta \left(s\right))\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s>1,\mathrm{for}\mathrm{some}{\top }_{*}\in [{\top }_0,\infty )\mathrm{and}n\in \mathbb{N},$$

(39)

then all solutions of (1) are oscillatory.

Proof. 

We proceed again by contradiction. Let us suppose that $h(\top )>0$ is a solution of (1) and assume that it is $h\left(\delta (\top )\right)>0$ for $\top \ge {\top }_1$, ${\top }_1\ge {\top }_0.$

According to Lemma 3 there are two exclusive cases. If case (13) holds for $\top \ge {\top }_1$, proceeding similarly as in the proof of Theorem 4, we arrive at a contradiction.

Now, assume that (14) holds for $\top \ge {\top }_1$. Integrating (1) from $\delta (\top )$ to ⊤, it follows that

$$b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }\ge {\int }_{\delta \left(\top \right)}^{\top }h\left(\delta \left(s\right)\right)\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s.$$

(40)

By virtue of (26) and (40) with $\varsigma =\delta \left(s\right)$ and $v=\delta (\top ),$ we have

$$\begin{array}{ccc}& & b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }\hfill \\ & \ge & b_2\left(\delta (\top )\right){\left(b_1\left(\delta (\top )\right)h^{\prime }\left(\delta (\top )\right)\right)}^{\prime }\times {\int }_{\delta \left(\top \right)}^{\top }{\tilde{R}}_n(\delta (\top ),\delta \left(s\right))\sum _{i=1}^k{q}_i\left(s\right)\mathrm{d}s.\hfill \end{array}$$

Therefore, this leads to a contradiction with (39). The proof is complete. □

Remark 3. 

In Theorem 5, we provided the conditions leading to the exclusion of both nonoscillatory (positive) solutions of type (13) and type (14), thereby ensuring the oscillation of all solutions of Equation (1), while Theorem 1 did not provide conditions for excluding nonoscillatory solutions of type (14) and merely stated condition (6), resulting in the conclusion that the solutions of Equation (1) are either oscillatory or tend to zero. Note also that our results are less restrictive compared to the result in [34].

Example 3. 

Setting in (1)

$${\delta }_i\left(\top \right)={\delta }_{1,2}\left(\top \right)=\{{\displaystyle \frac{1}{2}}\top ,{\displaystyle \frac{3}{5}}\top \},\mathit{with}\delta \left(\top \right)={\displaystyle \frac{1}{2}}\top$$

and

$$b_2(\top )=b_1(\top )=1,q_1\left(\top \right)={\displaystyle \frac{18}{{\top }^3}},\mathit{such}\mathit{that}\top \in [1,\infty ).$$

(41)

Applying Condition (39) when $n=1,n=2$ and $n=3$ gives $0.643702<1,0.874346<1$ and $1.364297>1$, respectively. Thus, (39) is satisfied when $n=3$. That is, from Theorem 5, we see that every solution of (1) with settings in (41) is oscillatory.

4. Conclusions

In this paper, we present some new criteria to determine the oscillation of a class of third-order delay differential equations. The results of this study are characterised by their stability over an infinite interval and their ease of application. Furthermore,

1.

The idea that our conditions can be applied more than once makes them significantly more valuable compared to previous results, as it opens up opportunities for broader applications, as in example 1, where the results in (Theorem 1, [21]), (Theorem 2, [35]), (Theorem 12, [42]), and (Theorem 2.9, [43]) fail, while the criterion (33) holds and leads to (37).

2.

The current results provide criteria to ensure the oscillation of all solutions of the studied equation. In contrast, most previous findings showed criteria that guarantee the exclusion of positive solutions of type (13), while no similar conditions were provided to exclude the other type (14). Therefore, these results ensure the oscillation of the solutions of Equation (1) or their convergence to zero [29,30,31].

3.

Our results do not require additional constraints, although most findings in the previous literature necessitate further assumptions (see [22,24,27,31,43,44]).

Exploring (1) in its non-canonical form is an intriguing topic that will open up new and inspiring avenues for future research.