More Effective Conditions for Testing the Oscillatory Behavior of Solutions to a Class of Fourth-Order Functional Differential Equations

Abstract

This paper presents an investigation into the qualitative behavior of solutions for a specific class of fourth-order half-linear neutral differential equations. The main objective of this study is to improve the relationship between the solution and its corresponding function. By developing improved relationships, a novel criterion is proposed to determine the oscillatory behavior of the studied equation. The exclusion of positive solutions is achieved through a comparative approach in which the examined equation is compared to second-order equations. Additionally, the significance of the obtained results is demonstrated by applying them to various illustrative examples.

1. Introduction

Differential equations are fundamental mathematical tools that find extensive applications in numerous scientific disciplines. Among the various types of differential equations, neutral differential equations hold a special place due to their ability to capture dynamic systems influenced by past behaviors. This paper focuses on a specific class of neutral differential equations, namely, fourth-order half-linear equations, which exhibit a combination of linearity and nonlinearity. Understanding the oscillatory nature of solutions to these equations is of paramount importance in light of their practical relevance in modeling complex dynamic systems encountered in engineering, physics, biology, and economics. The analysis of oscillations in such equations offers invaluable insights into system stability and dynamics, ultimately guiding the design and control of real-world systems, including mechanical structures, electrical circuits, biological processes, and economic models. Thus, a comprehensive grasp of oscillatory behavior in these equations serves as a cornerstone for optimizing system performance and ensuring reliability in practical applications; see [1,2,3].

The qualitative behavior of solutions to differential equations plays a crucial role in understanding the dynamics and stability properties of the underlying systems. Oscillation is an essential aspect that characterizes the periodic and repetitive nature of solutions. The investigation of oscillation criteria has received significant attention in the field of differential equations, aiming to establish the conditions under which solutions exhibit oscillatory behavior. Such criteria are valuable in predicting the presence or absence of oscillations in real-world phenomena, see [4,5,6,7].

The practical importance of understanding the oscillatory behavior of solutions to fourth-order functional differential equations lies in its applicability to various real-world phenomena. These equations often model complex dynamic systems encountered in engineering, physics, biology, and economics. Analyzing their oscillatory behavior provides crucial insights into the stability and dynamics of these systems. This knowledge, in turn, guides the design and control of practical systems, such as mechanical structures, electrical circuits, biological processes, and economic models. Therefore, a comprehensive understanding of oscillations in such equations is fundamental for optimizing system performance and ensuring their reliability in practical applications.

In recent years, there has been growing interest in the study of neutral differential equations due to their oscillatory behavior. This literature review aims to highlight notable research studies that have contributed to understanding the oscillatory properties of these types of equations.

One area of focus has been the investigation of oscillation in second-order neutral differential equations. Several studies have explored this topic, including [8,9,10]. Additionally, the oscillatory behavior of fourth-order neutral differential equations has been examined in studies such as [11,12,13]. Furthermore, the oscillation properties of even-order neutral differential equations have been investigated in [14,15,16].

Baculíková [17] focused on the oscillatory and asymptotic properties of the differential equation, as follows:

$${\left(\zeta (\ell ){\upsilon }^{\prime }(\ell )\right)}^{\prime }+q(\ell )f\left(\upsilon \left(\mu \right(\ell \left)\right)\right)=0,$$

(1)

which, in the noncanonical case, is

$${\int }_{{\ell }_0}^{\infty }\frac{1}{{\zeta }^{1/\varrho }\left(\varsigma \right)}\mathrm{d}\varsigma <\infty .$$

(2)

The function f in Equation (1) is defined as follows: $f\in C\left(\mathbb{R}\right)$, $vf\left(v\right)>0$ for $v\ne 0,$ $f\left(v_1{v}_2\right)\ge f\left(v_1\right)f\left(v_2\right)$ for $v_1{v}_2>0$, and f is non-decreasing.

El-Nabulsi et al. [18] conducted a study on the oscillation properties of solutions to a nonlinear fourth-order differential equation. The equation is represented as follows:

$${\left(\zeta (\ell ){\left({\upsilon }^{‴}(\ell )\right)}^{\varrho }\right)}^{\prime }+q(\ell )f\left(\upsilon \left(\mu \right(\ell \left)\right)\right)=0.$$

(3)

In this equation, the function $f\left(\theta \right)/{\theta }^{\varrho }\ge k>0$ for $\theta \ne 0$ and condition (5) is satisfied. Zhang et al. [19] conducted an investigation into the oscillatory patterns of (3) while taking into account condition (2).

Karpuz et al. [20] investigated higher-order neutral differential equations of the following form:

$${\kappa }^{\left(n\right)}\left(\ell \right)+q(\ell )\upsilon \left(\mu (\ell )\right)=0.$$

They conducted a comparison between the oscillatory and asymptotic characteristics of the solutions for higher-order neutral differential equations and first-order delay differential equations.

Li and Rogovchenko [21] and Zhang et al. [22,23] discussed oscillation results for higher-order half-linear delay differential equations. These equations were of the following form:

$${\left(\zeta (\ell ){\left({\kappa }^{\left(n-1\right)}(\ell )\right)}^{\varrho }\right)}^{\prime }+q(\ell ){\upsilon }^{\beta }\left(\mu (\ell )\right)=0.$$

Their results were obtained under condition (2), and they employed the Riccati technique in their analysis.

Alnafisah et al. [24] introduced augmented inequalities in order to enhance the characteristics of solutions to neutral differential equations of even order. These equations can be represented as follows:

$${\left(\zeta (\ell ){\left({\kappa }^{\left(n-1\right)}(\ell )\right)}^{\varrho }\right)}^{\prime }+q(\ell ){\upsilon }^{\varrho }\left(\mu (\ell )\right)=0.$$

These improved inequalities were established under condition (2).

This study focuses on investigating the oscillatory behavior exhibited by the solutions of a fourth-order quasi-linear neutral differential equation, provided by

$${\left(\zeta (\ell ){\left({\kappa }^{‴}(\ell )\right)}^{\varrho }\right)}^{\prime }+q(\ell ){\upsilon }^{\varrho }\left(\mu (\ell )\right)=0,\ell \ge {\ell }_0,$$

(4)

where $\kappa \left(\ell \right)=\upsilon \left(\ell \right)+\mathrm{p}\left(\ell \right)\upsilon \left(\tau \left(\ell \right)\right).$ Note that throughout this paper we consistently make the following assumptions:

(H₁)

$\varrho \ge 1$ is expressed as the ratio of two positive odd integers.

(H₂)

$\tau , \mu , \zeta \in C^1\left([{\ell }_0, \infty )\right)$ and $q\left(\ell \right)\in C\left(\left[{\ell }_0, \infty \right)\right)$.

(H₃)

$\tau (\ell )\le \ell$, $\mu \left(\ell \right)\le \ell$, ${\mu }^{\prime }(\ell )>0,$ and ${lim}_{\ell \to \infty }\tau \left(\ell \right) = {lim}_{\ell \to \infty }\mu \left(\ell \right) = \infty$.

(H₄)

$\zeta \left(\ell \right)>0$, ${\zeta }^{\prime }\left(\ell \right)\ge 0$, $0\le \mathrm{p}\left(\ell \right)<{\mathrm{p}}_0$, $q\left(\ell \right)\ge 0,$ and

$${\int }_{{\ell }_0}^{\ell }\frac{1}{{\zeta }^{1/\varrho }\left(\varsigma \right)}\mathrm{d}\varsigma \to \infty \mathrm{as}\ell \to \infty .$$

(5)

A function $\upsilon \in C^3([{\mathrm{L}}_{\upsilon },\infty ),\mathbb{R})$, ${\mathrm{L}}_{\upsilon }⩾{\ell }_0$ is said to be a solution of (4) which has the property $\zeta {\left({\kappa }^{‴}\right)}^{\varrho }\in C^1[{\mathrm{L}}_{\upsilon },\infty )$ and satisfies Equation (4) for all $\upsilon \in [{\ell }_{\upsilon },\infty )$. We consider only those solutions $\upsilon$ of (4) which exist on some half-line $[{\mathrm{L}}_{\upsilon },\infty )$ and satisfy the condition

$$sup\left\{\right|\upsilon (\ell )|:\ell ⩾\mathrm{L}\}>0,\mathrm{for}\mathrm{all}\mathrm{L}\ge {\mathrm{L}}_{\upsilon }.$$

A solution of (4) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be non-oscillatory. Equation (4) is said to be oscillatory if all of its solutions are oscillatory.

In this paper, we present an investigation of fourth-order half-linear neutral differential equations, with a focus on enhancing the relationship between variables and introducing an improved oscillation criterion. This study contributes to the existing body of knowledge in the field of differential equations and offers valuable insights into the qualitative behavior of solutions for this specific class of equations.

2. Preliminary Considerations

We start by introducing several helpful lemmas that pertain to the monotonic characteristics of the non-oscillatory solutions of the examined equations. In order to make our notation more concise, we define the following expressions:

$${\phi }_0\left(\ell ,{\ell }_0\right):={\int }_{{\ell }_0}^{\ell }\frac{1}{{\zeta }^{1/\varrho }\left(\varsigma \right)}\mathrm{d}\varsigma ,{\phi }_i\left(\ell ,{\ell }_0\right):={\int }_{{\ell }_0}^{\ell }{\phi }_{i-1}\left(\varsigma ,{\ell }_0\right)\mathrm{d}\varsigma ,i=1,2,$$

$$G^{\left[0\right]}\left(\ell \right):=G\left(\ell \right)\mathrm{and}{G}^{\left[j\right]}\left(\ell \right):=G\left(G^{\left[j-1\right]}\left(\ell \right)\right),\mathrm{for}j=1,2,\cdots ,n,$$

$${\mathrm{p}}_1\left(\ell ;n\right):=\sum _{k=0}^n\left(\prod _{i=0}^{2k}\mathrm{p}\left({\tau }^{\left[i\right]}\left(\ell \right)\right)\right)\left[\frac{1}{\mathrm{p}\left({\tau }^{\left[2k\right]}\left(\ell \right)\right)}-1\right]\frac{{\phi }_2\left({\tau }^{\left[2k\right]}\left(\ell \right),{\ell }_1\right)}{{\phi }_2\left(\ell ,{\ell }_1\right)},$$

$${\widehat{\mathrm{p}}}_1\left(\ell ,n\right):=\sum _{k=1}^n\left(\prod _{i=1}^{2k-1}\frac{1}{\mathrm{p}\left({\tau }^{\left[-i\right]}\left(\ell \right)\right)}\right)\left[\frac{{\phi }_2\left({\tau }^{\left[-2k+1\right]}\left(\ell \right),{\ell }_1\right)}{{\phi }_2\left({\tau }^{\left[-2k\right]}\left(\ell \right),{\ell }_1\right)}-\frac{1}{\mathrm{p}\left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)}\right],$$

$${\mathrm{B}}_0\left(\ell ,n\right):=\left\{\begin{array}{cc}\hfill p_1\left(\ell ;n\right)& \mathrm{f}\mathrm{o}\mathrm{r} p_0<1,\\ \hfill {\widehat{\mathrm{p}}}_1\left(\ell ;n\right)& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} p_0>{\phi }_2\left(\ell , {\ell }_1\right)/{\phi }_2\left(\tau \left(\ell \right),{\ell }_1\right),\end{array}\right.$$

and

$$\mathrm{L}:=max\left\{m{\left(1-m\right)}^{\varrho }{\lambda }^{-\varrho m}:m\in \left(0,1\right)\right\}.$$

In studying the asymptotic properties of the positive solutions of Equation (4), it is easy to verify, as demonstrated in [25] (Lemma 2.2.1), that the function $\kappa$ exhibits the following two distinct possible cases.

Lemma 1 

([25]). Assume that υ is an eventually positive solution of (4); then, κ eventually satisfies the following cases:

$$\begin{array}{ccc}\hfill C_1& :& \kappa >0,{\kappa }^{\prime }>0,{\kappa }^{″}>0,{\kappa }^{‴}>0,{\left(\zeta ·{\left({\kappa }^{‴}\right)}^{\varrho }\right)}^{\prime }<0,\hfill \\ \hfill C_2& :& \kappa >0,{\kappa }^{\prime }>0,{\kappa }^{″}<0,{\kappa }^{‴}>0,\hfill \end{array}$$

for $\ell ⩾{\ell }_1⩾{\ell }_0.$

Notation 1. 

The symbol ${\mathsf{\Psi }}_i$ (Category ${\mathsf{\Psi }}_i$) represents the collection of all solutions that eventually become positive for which the corresponding function fulfills condition ($C_i$) for $i=1,2.$

In the oscillation theory of neutral differential equations, the relationship between the solution and its corresponding function holds significant importance. Therefore, our work focuses on improving these relationships through the utilization of the following lemma.

Lemma 2 

(see [26], Lemma 1). Suppose that υ represents an eventually positive solution to Equation (4). If $p<1$, then eventually

$$\upsilon \left(\ell \right)>\sum _{k=0}^n\left(\prod _{i=0}^{2k}\mathrm{p}\left({\tau }^{\left[i\right]}\left(\ell \right)\right)\right)\left[\frac{\kappa \left({\tau }^{\left[2k\right]}\left(\ell \right)\right)}{\mathrm{p}\left({\tau }^{\left[2k\right]}\left(\ell \right)\right)}-\kappa \left({\tau }^{\left[2k+1\right]}\left(\ell \right)\right)\right]$$

for any integer $n\ge 0.$

Lemma 3. 

Suppose that υ represents an eventually positive solution to Equation (4). If $p_0>1$, then

$$\upsilon \left(\ell \right)>\sum _{k=1}^n\left(\prod _{i=1}^{2k-1}\frac{1}{\mathrm{p}\left({\tau }^{\left[-k\right]}\left(\ell \right)\right)}\right)\left[\kappa \left({\tau }^{\left[-2k+1\right]}\left(\ell \right)\right)-\frac{1}{\mathrm{p}\left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)}\kappa \left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)\right].$$

Proof. 

From

$$\kappa \left(\ell \right)=\upsilon \left(\ell \right)+\mathrm{p}\left(\ell \right)\upsilon \left(\tau \left(\ell \right)\right),$$

we can deduce that

$$\begin{array}{ccc}\hfill \upsilon \left(\ell \right)& =& \frac{1}{\mathrm{p}\left({\tau }^{-1}\left(\ell \right)\right)}\left[\kappa \left({\tau }^{-1}\left(\ell \right)\right)-\upsilon \left({\tau }^{-1}\left(\ell \right)\right)\right]\hfill \\ & =& \frac{1}{\mathrm{p}\left({\tau }^{\left[-1\right]}\left(\ell \right)\right)}\kappa \left({\tau }^{\left[-1\right]}\left(\ell \right)\right)\hfill \\ & & -\frac{1}{\mathrm{p}\left({\tau }^{\left[-1\right]}\left(\ell \right)\right)}\frac{1}{\mathrm{p}\left({\tau }^{\left[-2\right]}\left(\ell \right)\right)}\left[\kappa \left({\tau }^{\left[-2\right]}\left(\ell \right)\right)-\upsilon \left({\tau }^{\left[-2\right]}\left(\ell \right)\right)\right].\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \upsilon \left(\ell \right)& =& \frac{1}{\mathrm{p}\left({\tau }^{\left[-1\right]}\left(\ell \right)\right)}\kappa \left({\tau }^{\left[-1\right]}\left(\ell \right)\right)-{\displaystyle \prod _{i=1}^2}\frac{1}{\mathrm{p}\left({\tau }^{\left[-i\right]}\left(\ell \right)\right)}\kappa \left({\tau }^{\left[-2\right]}\left(\ell \right)\right)\hfill \\ & & +{\displaystyle \prod _{i=1}^3}\frac{1}{\mathrm{p}\left({\tau }^{\left[-i\right]}\left(\ell \right)\right)}\left[\kappa \left({\tau }^{\left[-3\right]}\left(\ell \right)\right)-\upsilon \left({\tau }^{\left[-3\right]}\left(\ell \right)\right)\right].\hfill \end{array}$$

By employing the same method repeatedly, we achieve

$$\upsilon \left(\ell \right)>\sum _{k=1}^n\left(\prod _{i=1}^{2k-1}\frac{1}{\mathrm{p}\left({\tau }^{\left[i\right]}\left(\ell \right)\right)}\right)\left[\kappa \left({\tau }^{\left[-2k+1\right]}\left(\ell \right)\right)-\frac{1}{\mathrm{p}\left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)}\kappa \left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)\right].$$

Hence, we have successfully demonstrated the proof of the lemma. □

3. Asymptotic and Monotonic Properties

This section discusses the characteristics of positive solutions of the studied equation in terms of their asymptotic behavior and monotonic properties. It is further categorized into two distinct subtopics, outlined below.

3.1. Category ${\mathsf{\Psi }}_1$

Lemma 4. 

Suppose that $\upsilon \in {\mathsf{\Psi }}_1.$ Then, for sufficiently large $\ell ⩾{\ell }_1$:

$\left({\mathrm{A}}_{1,1}\right)$

$\kappa (\ell )⩾{\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}(\ell ){\phi }_2(\ell ,{\ell }_1)$.

$\left({\mathrm{A}}_{1,2}\right)$

${\kappa }^{″}(\ell ,{\ell }_1)/{\phi }_0(\ell ,{\ell }_1)$, ${\kappa }^{\prime }(\ell )/{\phi }_1(\ell ,{\ell }_1)$ and $\kappa (\ell )/{\phi }_2(\ell ,{\ell }_1)$ are decreasing.

$\left({\mathrm{A}}_{1,3}\right)$

${\phi }_0(\ell ,{\ell }_1)\kappa (\ell )⩾{\phi }_2(\ell ,{\ell }_1){\kappa }^{″}(\ell )$.

Proof. 

$\left({\mathrm{A}}_{1,1}\right)$ The monotonicity of ${\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}(\ell )$ implies that

$$\begin{array}{ccc}\hfill {\kappa }^{″}\left(\ell \right)& \ge & {\int }_{{\ell }_1}^{\ell }{\zeta }^{1/\varrho }\left(\varsigma \right){\kappa }^{‴}\left(\varsigma \right)\frac{1}{{\zeta }^{1/\varrho }\left(\varsigma \right)}\mathrm{d}\varsigma \hfill \\ & \ge & {\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}\left(\ell \right){\int }_{{\ell }_1}^{\ell }\frac{1}{{\zeta }^{1/\varrho }\left(\varsigma \right)}\mathrm{d}\varsigma \hfill \\ & \ge & {\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}\left(\ell \right){\phi }_0\left(\ell ,{\ell }_1\right).\hfill \end{array}$$

(6)

Integrating twice more from ${\ell }_1$ to ℓ, we obtain

$${\kappa }^{\prime }\left(\ell \right)\ge {\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}\left(\ell \right){\phi }_1\left(\ell ,{\ell }_1\right)$$

(7)

and

$$\kappa \left(\ell \right)\ge {\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}\left(\ell \right){\phi }_2\left(\ell ,{\ell }_1\right).$$

$\left({\mathrm{A}}_{1,2}\right)$ From (6), we obtain

$${\left(\frac{{\kappa }^{″}(\ell )}{{\phi }_0(\ell ,{\ell }_1)}\right)}^{\prime }=\frac{{\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}(\ell ){\phi }_0(\ell ,{\ell }_1)-{\kappa }^{″}(\ell )}{{\zeta }^{1/\varrho }(\ell ){\phi }_0^2(\ell ,{\ell }_1)}\le 0.$$

Because ${\kappa }^{″}\left(\ell \right)/{\phi }_0(\ell ,{\ell }_1)$ is decreasing,

$${\kappa }^{\prime }\left(\ell \right)\ge {\int }_{{\ell }_1}^{\ell }\frac{{\kappa }^{″}\left(\varsigma \right)}{{\phi }_0(\varsigma ,{\ell }_1)}{\phi }_0\left(\varsigma ,{\ell }_1\right)\mathrm{d}\varsigma \ge \frac{{\kappa }^{″}(\ell )}{{\phi }_0(\ell ,{\ell }_1)}{\phi }_1(\ell ,{\ell }_1).$$

(8)

From this, we can deduce that

$${\left(\frac{{\kappa }^{\prime }(\ell )}{{\phi }_1(\ell ,{\ell }_1)}\right)}^{\prime }=\frac{{\kappa }^{″}(\ell ){\phi }_1(\ell ,{\ell }_1)-{\phi }_0\left(\ell ,{\ell }_1\right){\kappa }^{\prime }(\ell )}{{\phi }_1^2(\ell ,{\ell }_1)}\le 0.$$

Because ${\kappa }^{\prime }\left(\ell \right)/{\phi }_1(\ell ,{\ell }_1)$ is decreasing,

$$\kappa \left(\ell \right)\ge {\int }_{{\ell }_1}^{\ell }\frac{{\kappa }^{\prime }\left(\varsigma \right)}{{\phi }_1(\varsigma ,{\ell }_1)}{\phi }_1\left(\varsigma ,{\ell }_1\right)\mathrm{d}\varsigma \ge \frac{{\kappa }^{\prime }(\ell )}{{\phi }_1(\ell ,{\ell }_1)}{\phi }_2(\ell ,{\ell }_1).$$

(9)

Consequently,

$${\left(\frac{\kappa (\ell )}{{\phi }_2(\ell ,{\ell }_1)}\right)}^{\prime }=\frac{{\kappa }^{\prime }(\ell ){\phi }_2(\ell ,{\ell }_1)-{\phi }_1\left(\ell ,{\ell }_1\right)\kappa (\ell )}{{\phi }_2^2(\ell ,{\ell }_1)}\le 0.$$

$\left({\mathrm{A}}_{1,3}\right)$ From (8) and (9), we find that

$$\kappa \left(\ell \right)\ge \frac{{\phi }_2(\ell ,{\ell }_1)}{{\phi }_0(\ell ,{\ell }_1)}{\kappa }^{″}(\ell ).$$

Therefore, we have successfully illustrated the lemma’s validity. □

Lemma 5. 

Assume that $\upsilon \in {\mathsf{\Psi }}_1$. Then,

$\left({\mathrm{A}}_{2,1}\right)$

$\upsilon \left(\ell \right)>{\mathrm{B}}_0\left(\ell ,n\right)\kappa \left(\ell \right)$.

$\left({\mathrm{A}}_{2,2}\right)$

${\left(\zeta (\ell ){\left({\kappa }^{‴}(\ell )\right)}^{\varrho }\right)}^{\prime }\le -q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\kappa }^{\varrho }\left(\mu (\ell )\right)$.

Proof. 

$\left({\mathrm{A}}_{2,1}\right)$ If p${}_0<1$, due to the fact that $\kappa \left(\ell \right)$ is increasing and ${\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\ge {\tau }^{{}_{\left[2k+1\right]}}\left(\ell \right)$, we have

$$\kappa \left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right)\ge \kappa \left({\tau }^{{}_{\left[2k+1\right]}}\left(\ell \right)\right),$$

which, along with Lemma 2, implies that

$$\begin{array}{ccc}\hfill \upsilon \left(\ell \right)& >& {\displaystyle \sum _{k=0}^n}\left({\displaystyle \prod _{i=0}^{2k}}\mathrm{p}\left({\tau }^{\left[i\right]}\left(\ell \right)\right)\right)\left[\frac{\kappa \left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right)}{\mathrm{p}\left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right)}-\kappa \left({\tau }^{{}_{\left[2k+1\right]}}\left(\ell \right)\right)\right]\hfill \\ & \ge & {\displaystyle \sum _{k=0}^n}\left({\displaystyle \prod _{i=0}^{2k}}\mathrm{p}\left({\tau }^{\left[i\right]}\left(\ell \right)\right)\right)\left[\frac{1}{\mathrm{p}\left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right)}-1\right]\kappa \left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right).\hfill \end{array}$$

(10)

Moreover, as $\kappa \left(\ell \right)/{\phi }_2\left(\ell ,{\ell }_1\right)$ is decreasing and ${\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\le \ell ,$ we have

$$\frac{\kappa \left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right)}{{\phi }_2\left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right),{\ell }_1\right)}\ge \frac{\kappa \left(\ell \right)}{{\phi }_2\left(\ell ,{\ell }_1\right)}$$

and

$$\kappa \left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right)\ge \frac{{\phi }_2\left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right),{\ell }_1\right)}{{\phi }_2\left(\ell ,{\ell }_1\right)}\kappa \left(\ell \right).$$

Thus, using the above inequality and substituting in (10), we obtain

$$\begin{array}{ccc}\hfill \upsilon \left(\ell \right)& >& {\displaystyle \sum _{k=0}^n}\left({\displaystyle \prod _{i=0}^{2k}}\mathrm{p}\left({\tau }^{\left[i\right]}\left(\ell \right)\right)\right)\left[\frac{1}{\mathrm{p}\left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right)\right)}-1\right]\frac{{\phi }_2\left({\tau }^{{}_{\left[2k\right]}}\left(\ell \right),{\ell }_1\right)}{{\phi }_2\left(\ell ,{\ell }_1\right)}\kappa \left(\ell \right)\hfill \\ & =& {\mathrm{p}}_1\left(\ell ;n\right)\kappa \left(\ell \right).\hfill \end{array}$$

On the other hand, if p${}_0>1,$ then $\kappa \left(\ell \right)/{\phi }_2\left(\ell ,{\ell }_1\right)$ is decreasing and ${\tau }^{\left[-2k\right]}\left(\ell \right)\ge {\tau }^{\left[-2k+1\right]}\left(\ell \right)$, implying that

$$\frac{\kappa \left({\tau }^{\left[-2k+1\right]}\left(\ell \right)\right)}{{\phi }_2\left({\tau }^{\left[-2k+1\right]}\left(\ell \right),{\ell }_1\right)}\ge \frac{\kappa \left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)}{{\phi }_2\left({\tau }^{\left[-2k\right]}\left(\ell \right),{\ell }_1\right)}$$

and

$$\kappa \left({\tau }^{\left[-2k+1\right]}\left(\ell \right)\right)\ge \frac{{\phi }_2\left({\tau }^{\left[-2k+1\right]}\left(\ell \right),{\ell }_1\right)}{{\phi }_2\left({\tau }^{\left[-2k\right]}\left(\ell \right),{\ell }_1\right)}\kappa \left({\tau }^{\left[-2k\right]}\left(\ell \right)\right).$$

Using Lemma 3, we can conclude that

$$\upsilon \left(\ell \right)>\sum _{k=1}^n\left(\prod _{i=1}^{2k-1}\frac{1}{\mathrm{p}\left({\tau }^{\left[-i\right]}\left(\ell \right)\right)}\right)\left[\frac{{\phi }_2\left({\tau }^{\left[-2k+1\right]}\left(\ell \right),{\ell }_1\right)}{{\phi }_2\left({\tau }^{\left[-2k\right]}\left(\ell \right),{\ell }_1\right)}-\frac{1}{\mathrm{p}\left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)}\right]\kappa \left({\tau }^{\left[-2k\right]}\left(\ell \right)\right).$$

As $\kappa \left(\ell \right)$ is increasing and ${\tau }^{\left[-2k\right]}\left(\ell \right)\ge \ell ,$ we have

$$\begin{array}{ccc}\hfill \upsilon \left(\ell \right)& >& {\displaystyle \sum _{k=1}^n}\left({\displaystyle \prod _{i=1}^{2k-1}}\frac{1}{\mathrm{p}\left({\tau }^{\left[-i\right]}\left(\ell \right)\right)}\right)\left[\frac{{\phi }_2\left({\tau }^{\left[-2k+1\right]}\left(\ell \right),{\ell }_1\right)}{{\phi }_2\left({\tau }^{\left[-2k\right]}\left(\ell \right),{\ell }_1\right)}-\frac{1}{\mathrm{p}\left({\tau }^{\left[-2k\right]}\left(\ell \right)\right)}\right]\kappa \left(\ell \right)\hfill \\ & =& {\widehat{\mathrm{p}}}_1\left(\ell ,n\right)\kappa \left(\ell \right).\hfill \end{array}$$

$\left({\mathrm{A}}_{2,2}\right)$ From (4), we have

$${\left(\zeta (\ell ){\left({\kappa }^{‴}(\ell )\right)}^{\varrho }\right)}^{\prime }=-q(\ell ){\upsilon }^{\varrho }\left(\mu (\ell )\right).$$

Using $\left({\mathrm{A}}_{2,1}\right),$ we obtain

$${\left(\zeta (\ell ){\left({\kappa }^{‴}(\ell )\right)}^{\varrho }\right)}^{\prime }\le -q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\kappa }^{\varrho }\left(\mu (\ell )\right).$$

Therefore, we have successfully illustrated the lemma’s validity. □

Lemma 6. 

Assume that $\upsilon \in {\mathsf{\Psi }}_1.$ Then,

$${\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{\prime }+\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\phi }_2^{\varrho -1}(\mu \left(\ell \right),{\ell }_1)\kappa \left(\mu (\ell )\right)\le 0.$$

Proof. 

Assume that $\upsilon \in {\mathsf{\Psi }}_1$ for $\ell \ge {\ell }_1\ge {\ell }_0.$ Therefore, we obtain

$${\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{\prime }=\frac{1}{\varrho }{\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{1-\varrho }{\left(\zeta \left(\ell \right){\left({\kappa }^{‴}\left(\ell \right)\right)}^{\varrho }\right)}^{\prime }.$$

From $\left({\mathrm{A}}_{2,2}\right),$ we obtain

$${\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{\prime }\le -\frac{1}{\varrho }{\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{1-\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\kappa }^{\varrho }\left(\mu (\ell )\right).$$

(11)

From Lemma 4, we can deduce that

$${\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}(\ell )\le \frac{\kappa \left(\ell \right)}{{\phi }_2(\ell ,{\ell }_1)}\le \frac{\kappa \left(\mu \left(\ell \right)\right)}{{\phi }_2(\mu \left(\ell \right),{\ell }_1)}.$$

Because $\varrho \ge 1$,

$${\left({\zeta }^{1/\varrho }(\ell ){\kappa }^{‴}(\ell )\right)}^{1-\varrho }\ge {\left(\frac{\kappa \left(\mu \left(\ell \right)\right)}{{\phi }_2(\mu \left(\ell \right),{\ell }_1)}\right)}^{1-\varrho },$$

which, with (11) provides

$$\begin{array}{ccc}\hfill {\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{\prime }& \le & -\frac{1}{\varrho }{\left(\frac{\kappa \left(\mu \left(\ell \right)\right)}{{\phi }_2(\mu \left(\ell \right),{\ell }_1)}\right)}^{1-\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\kappa }^{\varrho }\left(\mu (\ell )\right)\hfill \\ & =& -\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\phi }_2^{\varrho -1}(\mu \left(\ell \right),{\ell }_1)\kappa \left(\mu (\ell )\right).\hfill \end{array}$$

□

Theorem 1. 

Assume that $\upsilon \in {\mathsf{\Psi }}_1.$ Then, the DE

$${\left({\zeta }^{1/\varrho }\left(\ell \right){\omega }^{\prime }\left(\ell \right)\right)}^{\prime }+\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right)\frac{{\phi }_2^{\varrho }(\mu \left(\ell \right),{\ell }_1)}{{\phi }_0(\mu (\ell ),{\ell }_1)}\omega \left(\mu (\ell )\right)=0$$

(12)

has a positive solution.

Proof. 

Assume that $\upsilon \in {\mathsf{\Psi }}_1$ for $\ell \ge {\ell }_1\ge {\ell }_0.$ From Lemma 4, $\left({\mathrm{A}}_{1,3}\right)$ holds. Hence, it follows that from Lemma 6 we can obtain

$$\begin{array}{ccc}\hfill 0& \ge & {\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{\prime }+\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\phi }_2^{\varrho -1}(\mu \left(\ell \right),{\ell }_1)\kappa \left(\mu (\ell )\right)\hfill \\ & \ge & {\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{\prime }+\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right){\phi }_2^{\varrho -1}(\mu \left(\ell \right),{\ell }_1)\frac{{\phi }_2(\mu (\ell ),{\ell }_1)}{{\phi }_0(\mu (\ell ),{\ell }_1)}{\kappa }^{″}\left(\mu (\ell )\right)\hfill \\ & =& {\left({\zeta }^{1/\varrho }\left(\ell \right){\kappa }^{‴}\left(\ell \right)\right)}^{\prime }+\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right)\frac{{\phi }_2^{\varrho }(\mu \left(\ell \right),{\ell }_1)}{{\phi }_0(\mu (\ell ),{\ell }_1)}{\kappa }^{″}\left(\mu (\ell )\right).\hfill \end{array}$$

(13)

Now, let $\omega \left(\ell \right)={\kappa }^{″}\left(\ell \right)>0$; then, (13) reduces to

$${\left({\zeta }^{1/\varrho }\left(\ell \right){\omega }^{\prime }\left(\ell \right)\right)}^{\prime }+\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right)\frac{{\phi }_2^{\varrho }(\mu \left(\ell \right),{\ell }_1)}{{\phi }_0(\mu (\ell ),{\ell }_1)}\omega \left(\mu (\ell )\right)\le 0.$$

Using Corollary 1 in [27], the corresponding DE (12) has a positive solution as well. Therefore, we have completed the proof of the Lemma. □

3.2. Category ${\mathsf{\Psi }}_2$

Because

$$\kappa \left(\ell \right)=\upsilon \left(\ell \right)+\mathrm{p}\left(\ell \right)\upsilon \left(\tau \left(\ell \right)\right),$$

$\kappa \left(\ell \right)\ge \upsilon \left(\ell \right)$ and

$$\upsilon \left(\ell \right)=\kappa \left(\ell \right)-\mathrm{p}\left(\ell \right)\upsilon \left(\tau \left(\ell \right)\right)\ge \kappa \left(\ell \right)-\mathrm{p}\left(\ell \right)\kappa \left(\tau \left(\ell \right)\right).$$

Because $\kappa \left(\ell \right)$ is increasing, $\kappa \left(\ell \right)\ge \kappa \left(\tau \left(\ell \right)\right)$ and

$$\upsilon \left(\ell \right)\ge \left(1-\mathrm{p}\left(\ell \right)\right)\kappa \left(\ell \right).$$

(14)

Theorem 2. 

Assume that $\upsilon \in {\mathsf{\Psi }}_2$; then, the DE

$${\kappa }^{″}\left(\ell \right)+\kappa \left(\mu (\ell )\right){\int }_{\ell }^{\infty }{\left(\frac{1}{\zeta \left(u\right)}{\int }_u^{\infty }q\left(\varsigma \right){\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)}^{\varrho }\mathrm{d}\varsigma \right)}^{1/\varrho }\mathrm{d}u=0$$

(15)

has a positive solution.

Proof. 

Assume that $\upsilon \in {\mathsf{\Psi }}_2$ for $\ell \ge {\ell }_1\ge {\ell }_0.$ Integrating (4) from ℓ to ∞, we have

$$\begin{array}{ccc}\hfill \zeta (\ell ){\left({\kappa }^{‴}(\ell )\right)}^{\varrho }& \ge & {\int }_{\ell }^{\infty }q\left(\varsigma \right){\upsilon }^{\varrho }\left(\mu \left(\varsigma \right)\right)\mathrm{d}\varsigma \hfill \\ & \ge & {\int }_{\ell }^{\infty }q\left(\varsigma \right){\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)}^{\varrho }{\kappa }^{\varrho }\left(\mu \left(\varsigma \right)\right)\mathrm{d}\varsigma \hfill \\ & \ge & {\kappa }^{\varrho }\left(\mu (\ell )\right){\int }_{\ell }^{\infty }q\left(\varsigma \right){\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)}^{\varrho }\mathrm{d}\varsigma ,\hfill \end{array}$$

or

$${\kappa }^{‴}\left(\ell \right)\ge \kappa \left(\mu (\ell )\right){\left(\frac{1}{\zeta \left(\ell \right)}{\int }_{\ell }^{\infty }q\left(\varsigma \right){\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)}^{\varrho }\mathrm{d}\varsigma \right)}^{1/\varrho }.$$

Integrating once again from ℓ to ∞, we obtain

$${\kappa }^{″}\left(\ell \right)\le -\kappa \left(\mu (\ell )\right){\int }_{\ell }^{\infty }{\left(\frac{1}{\zeta \left(u\right)}{\int }_u^{\infty }q\left(\varsigma \right){\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)}^{\varrho }\mathrm{d}\varsigma \right)}^{1/\varrho }\mathrm{d}u.$$

(16)

Therefore, $\kappa$ is a positive solution of differential inequality (16). Using Corollary 1 in [27], the corresponding DE (15) has a positive solution as well. This ends the proof. □

4. Oscillation Theorem and Examples

In this section, we establish a condition that ensures the occurrence of oscillation in the differential Equation (4). Furthermore, we investigate specific instances of the studied equation by employing this novel criterion.

Theorem 3. 

Assume that

$$\lambda :=\underset{\ell \to \infty }{liminf}\frac{{\phi }_0\left(\ell ,{\ell }_0\right)}{{\phi }_0\left(\mu \left(\ell \right),{\ell }_0\right)}<\infty .$$

(17)

If

$$\underset{\ell \to \infty }{liminf}{\zeta }^{1/\varrho }\left(\ell \right){\phi }_0\left(\ell ,{\ell }_0\right){\phi }_2^{\varrho }\left(\mu \left(\ell \right),{\ell }_0\right)q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right)>\varrho \mathrm{L}$$

(18)

and

$$\underset{\ell \to \infty }{liminf}\ell \mu \left(\ell \right){\int }_{\ell }^{\infty }{\left(\frac{1}{\zeta \left(u\right)}{\int }_u^{\infty }q\left(\varsigma \right){\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)}^{\varrho }\mathrm{d}\varsigma \right)}^{1/\varrho }\mathrm{d}u>\mathrm{L},$$

(19)

then (4) is oscillatory.

Proof. 

Let us consider the opposite scenario, where $\upsilon$ is assumed to be a positive solution of (4). Then, $\upsilon$ satisfies one of the two cases ($C_1$) and ($C_2$) for $\varsigma \ge {\varsigma }_1\ge {\varsigma }_0$. From Theorems 1 and 2, we know that the two DEs

$${\left({\zeta }^{1/\varrho }\left(\ell \right){\omega }^{\prime }\left(\ell \right)\right)}^{\prime }+\frac{1}{\varrho }q(\ell ){\mathrm{B}}_0^{\varrho }\left(\mu (\ell ),n\right)\frac{{\phi }_2^{\varrho }(\mu \left(\ell \right),{\ell }_1)}{{\phi }_0(\mu (\ell ),{\ell }_1)}\omega \left(\mu (\ell )\right)=0$$

and

$${\kappa }^{″}\left(\ell \right)+\left({\int }_{\ell }^{\infty }{\left(\frac{1}{\zeta \left(u\right)}{\int }_u^{\infty }q\left(\varsigma \right){\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)}^{\varrho }\mathrm{d}\varsigma \right)}^{1/\varrho }\mathrm{d}u\right)\kappa \left(\mu (\ell )\right)=0$$

have positive solutions. However, according to Theorem 2 in [28], conditions (18) and (19) confirm the oscillation of all solutions of these equations. □

Corollary 1. 

Assume that (17) holds. If

$$\underset{\ell \to \infty }{liminf}\ell {\mu }^3\left(\ell \right)q(\ell ){\mathrm{B}}_0\left(\mu (\ell ),n\right)>6\mathrm{L}$$

and

$$\underset{\ell \to \infty }{liminf}\ell \mu \left(\ell \right){\int }_{\ell }^{\infty }{\int }_u^{\infty }q\left(\varsigma \right)\left(1-\mathrm{p}\left(\mu \left(\varsigma \right)\right)\right)\mathrm{d}\varsigma \mathrm{d}u>\mathrm{L},$$

then the linear NDE

$${\left(\upsilon \left(\ell \right)+\mathrm{p}\left(\ell \right)\upsilon \left(\tau \left(\ell \right)\right)\right)}^{\left(4\right)}+q\left(\ell \right)\upsilon \left(\mu \left(\ell \right)\right)=0$$

is oscillatory.

Example 1. 

Consider the neutral differential equation

$${\left({\ell }^{\varrho -1}{\left({\left(\upsilon \left(\ell \right)+{\mathrm{p}}_0\upsilon \left({\tau }_0\ell \right)\right)}^{‴}\right)}^{\varrho }\right)}^{\prime }+\frac{q_0}{{\ell }^{2\varrho +2}}\upsilon \left({\mu }_0\ell \right)=0,\ell >0.$$

(20)

Here, ${\tau }_0,{\mu }_0\in \left(0,1\right)$ and $q_0>0$. We can verify that $\zeta \left(\ell \right)={\ell }^{\varrho -1}$, $\tau \left(\ell \right)={\tau }_0\ell$, ${\mu }_0\left(\ell \right)={\mu }_0\ell ,$ $q\left(\ell \right)=q_0/{\ell }^{2\varrho +2},$

$${\phi }_0\left(\ell \right)=\varrho {\ell }^{1/\varrho },{\phi }_1\left(\ell \right)=\frac{{\varrho }^2}{\varrho +1}{\ell }^{1+1/\varrho },{\phi }_2\left(\ell \right)=\frac{{\varrho }^3}{\left(\varrho +1\right)\left(2\varrho +1\right)}{\ell }^{2+1/\varrho },$$

$$\lambda =\frac{1}{{\mu }_0^{1/\varrho }},$$

$${\mathrm{p}}_1\left(\ell ;n\right)=\left[1-{\mathrm{p}}_0\right]\sum _{k=0}^n{\mathrm{p}}_0^{2k}{\tau }_0^{2k\left(2+1/\varrho \right)},$$

$${\widehat{\mathrm{p}}}_1\left(\ell ,n\right)=\left[{\mathrm{p}}_0{\tau }_0^{2+1/\varrho }-1\right]\sum _{k=1}^n\frac{1}{{\mathrm{p}}_0^{2k+1}},$$

and

$${\mathrm{B}}_0\left(\ell ,n\right)=\left\{\begin{array}{cc}\hfill p_1\left(\ell ;n\right)& \mathit{f}\mathit{o}\mathit{r} p_0<1,\\ \hfill {\widehat{\mathrm{p}}}_1\left(\ell ;n\right)& \hspace{1em}\hspace{1em}\hspace{1em}\mathit{f}\mathit{o}\mathit{r} p_0>1/{\tau }^{2+1/\varrho }.\end{array}\right.$$

Using Theorem 3, we can establish the conditions for the oscillation of all solutions of Equation (20). These conditions are provided by

$$\frac{{\varrho }^{3\varrho }}{{\left(2{\varrho }^2+3\varrho +1\right)}^{\varrho }}{\mu }_0^{2\varrho +1}{q}_0{\mathrm{B}}_0^{\varrho }>\mathrm{L}$$

and

$${\mu }_0\frac{\left(1-{\mathrm{p}}_0\right)q_0^{1/\varrho }}{2{\left(1+2\varrho \right)}^{1/\varrho }}>\mathrm{L},$$

where

$$\mathrm{L}=max\left\{m{\left(1-m\right)}^{\varrho }{\mu }_0^m:m\in \left(0,1\right)\right\}.$$

By satisfying these conditions, the oscillation of all solutions of Equation (20) is confirmed.

Example 2. 

Consider the neutral differential equation

$${\left({\ell }^2{\left({\left(\upsilon \left(\ell \right)+0.5\upsilon \left(0.8\ell \right)\right)}^{‴}\right)}^3\right)}^{\prime }+\frac{q_0}{{\ell }^8}\upsilon \left(0.5\ell \right)=0,\ell >0.$$

(21)

We can verify that $\zeta \left(\ell \right)={\ell }^2$, $\tau \left(\ell \right)=0.8\ell$, ${\mu }_0\left(\ell \right)=0.5\ell$, $q\left(\ell \right)=q_0/{\ell }^8,$

$${\phi }_0\left(\ell \right)=3{\ell }^{1/3},{\phi }_1\left(\ell \right)=\frac{9}{4}{\ell }^{4/3},{\phi }_2\left(\ell \right)=\frac{27}{28}{\ell }^{7/3},$$

$$\lambda =2^{1/3},$$

$${\mathrm{p}}_1\left(\ell ;10\right)=\sum _{k=0}^{10}{\left(0.5\right)}^{2k+1}{\left(0.8\right)}^{14k/3}=0.54839,$$

and

$${\mathrm{B}}_0\left(\ell ,10\right)={\mathrm{p}}_1\left(\ell ;10\right)=0.54839.$$

Using Theorem 3, we can establish the conditions for the oscillation of all solutions of Equation (21). We find that

$$\mathrm{L}=0.08964\mathit{at}m=0.22024.$$

Therefore, conditions (18) and (19) respectively reduce to

$$q_0>34.081\mathrm{L}=3.05469,$$

and

$$q_0>3584{\mathrm{L}}^3=2.5815.$$

For $q_0>3.05469$, the oscillation of all solutions of Equation (21) is confirmed.

Example 3. 

Consider the neutral differential equation

$${\left(\upsilon \left(\ell \right)+{\mathrm{p}}_0\upsilon \left({\tau }_0\ell \right)\right)}^{\left(4\right)}+\frac{q_0}{{\ell }^4}\upsilon \left({\mu }_0\ell \right)=0,\ell >0,$$

(22)

where ${\tau }_0,{\mu }_0\in \left(0,1\right)$ and $q_0>0$. We can easily verify that $\zeta \left(\ell \right)=1$, $\tau \left(\ell \right)={\tau }_0\ell$, ${\mu }_0\left(\ell \right)={\mu }_0\ell$, $q\left(\ell \right)=q_0/{\ell }^4,$

$${\phi }_0\left(\ell \right)=\ell ,{\phi }_1\left(\ell \right)=\frac{1}{2}{\ell }^2,{\phi }_2\left(\ell \right)=\frac{1}{6}{\ell }^3,$$

$$\lambda =\frac{1}{{\mu }_0}$$

$${\mathrm{p}}_1\left(\ell ;n\right)=\left[1-{\mathrm{p}}_0\right]\sum _{k=0}^n{\mathrm{p}}_0^{2k}{\tau }_0^{6k},$$

$${\widehat{\mathrm{p}}}_1\left(\ell ,n\right)=\left[{\mathrm{p}}_0{\tau }_0^3-1\right]\sum _{k=1}^n\frac{1}{{\mathrm{p}}_0^{2k+1}},$$

and

$${\mathrm{B}}_0\left(\ell ,n\right)=\left\{\begin{array}{cc}\hfill p_1\left(\ell ;n\right)& \mathit{f}\mathit{o}\mathit{r} p_0<1,\\ \hfill {\widehat{\mathrm{p}}}_1\left(\ell ;n\right)& \hspace{1em}\mathit{f}\mathit{o}\mathit{r} p_0>1/{\tau }^3.\end{array}\right.$$

Using Corollary 1, the conditions

$$q_0>\frac{6\mathrm{L}}{{\mu }_0^3{\mathrm{B}}_0}$$

and

$$q_0>\frac{6\mathrm{L}}{{\mu }_0\left(1-{\mathrm{p}}_0\right)}$$

confirm the oscillation of all solutions of (22), where

$$\mathrm{L}=max\left\{m\left(1-m\right){\mu }_0^m:m\in \left(0,1\right)\right\}.$$

Example 4. 

Consider Equation (22) with $\tau \left(\ell \right)=0.8\ell$, ${\mu }_0\left(\ell \right)=0.5\ell$, and $q\left(\ell \right)=16/{\ell }^4.$ It is straightforward to observe that

$${\phi }_0\left(\ell \right)=\ell ,{\phi }_1\left(\ell \right)=\frac{1}{2}{\ell }^2,{\phi }_2\left(\ell \right)=\frac{1}{6}{\ell }^3,$$

$$\lambda =2,$$

$${\mathrm{p}}_1\left(\ell ;10\right)=\sum _{k=0}^{10}{\left(0.5\right)}^{2k+1}{\left(0.9\right)}^{6k}=0.57661,$$

and

$${\mathrm{B}}_0\left(\ell ,10\right)={\mathrm{p}}_1\left(\ell ;10\right)=0.57661.$$

Using Corollary 1, we can deduce that

$$\mathrm{L}=0.18209\mathit{at}m=0.41581,$$

and that conditions

$$q_0=16>\frac{6\mathrm{L}}{{\mu }_0^3{\mathrm{B}}_0}=15.158$$

and

$$q_0=16>\frac{6\mathrm{L}}{{\mu }_0\left(1-{\mathrm{p}}_0\right)}=4.3702$$

are satisfied. Therefore, every solution of (22) is oscillatory.

5. Conclusions

In this paper, we have made significant contributions to the study of fourth-order half-linear neutral differential equations by improving the link between the solution and its corresponding function. The identification of these improved links leads to the development of a novel criterion for assessing the oscillatory characteristics of the examined equation. Positive solutions are efficiently excluded using the comparison approach with second-order equations, yielding useful insights into the nature of the solutions. We demonstrate the practical significance and use of the suggested oscillatory standard by applying our findings to several cases.

Overall, our research represents an advance in this field and improves previous findings regarding the qualitative behavior of solutions of fourth-order half-linear neutral differential equations. To further expand the scope of this study, it would be interesting to extend the research to higher-order nonlinear neutral differential equations with $n\ge 4$. Such an extension could deepen our understanding of these equations and contribute to the development of more comprehensive analytical tools.