Oscillatory Features of Fourth-Order Emden–Fowler Differential Equations with Sublinear Neutral Terms

Abstract

This article examines the oscillatory characteristics of a fourth-order Emden–Fowler differential equation, specifically when it includes a sublinear neutral term. Our methodology centers on establishing multiple theorems that introduce innovative conditions to guarantee that there are no positive solutions to the examined equation. Due to the symmetry between non-oscillatory solutions, we obtain oscillation conditions by excluding only positive solutions. We employ the Riccati technique in various ways to achieve this objective. The criteria presented in this study complement and generalize many findings published in the literature. We support the efficiency of our findings by applying them to an example.

1. Introduction

In this study, we aim to investigate the fourth-order Emden–Fowler delay differential equations (DDE) with a sublinear neutral term, represented by

$${\left(\mathfrak{a}\left(\nu \right){\left({\left(x\left(\nu \right)+\mathrm{g}\left(\nu \right)x^{\alpha }\left(\xi \left(\nu \right)\right)\right)}^{‴}\right)}^{\gamma }\right)}^{\prime }+\mathrm{p}\left(\nu \right)x^{\beta }\left(\lambda \left(\nu \right)\right)=0,$$

(1)

where $\nu \ge {\nu }_0$. We assume the following throughout this paper:

(H₁)

$0<\alpha \le 1,$ $\beta$ and $\gamma$ are ratios of two positive odd integers;

(H₂)

$\xi ,$ $\lambda \in C^1([{\nu }_0,\infty ),\mathbb{R}),$ $\xi \left(\nu \right)\le \nu ,$ $\lambda \left(\nu \right)\le \nu ,$ ${\lambda }^{\prime }\left(\nu \right)>0,$ and ${lim}_{\nu \to \infty }\xi \left(\nu \right)={lim}_{\nu \to \infty }\lambda \left(\nu \right)$ $=\infty ;$

(H₃)

$\mathfrak{a}\in C^1([{\nu }_0,\infty ),\left(0,\infty \right)),$ ${\mathfrak{a}}^{\prime }\left(\nu \right)\ge 0,$ $\mathrm{g},$ $\mathrm{p}\in C([{\nu }_0,\infty ),\left(0,\infty \right)),$ ${lim}_{\nu \to \infty }$g$\left(\nu \right)=0,$ and

$${\int }_{{\nu }_0}^{\nu }{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\gamma }\left(s\right)}}\mathrm{d}s\to \infty \mathrm{as}\nu \to \infty .$$

(2)

A solution of (1) is defined as a function $x\in C\left([{\nu }_x,\infty )\right)$, ${\nu }_x⩾{\nu }_0,$ which satisfies $\left(x+\mathrm{g}·\left(x^{\alpha }\circ \xi \right)\right)\in C^3\left([{\nu }_x,\infty )\right)$, $\left[\mathfrak{a}·{\left(x+\mathrm{g}·\left(x^{\alpha }\circ \xi \right)\right)}^{\gamma }\right]\in C^1\left([{\nu }_x,\infty )\right)$, and satisfies (1) on $[{\nu }_x,\infty )$. Only solutions x of (1) that meet the condition

$$sup\left\{\right|x\left(\nu \right)|:\nu ⩾{\nu }_{*}\}>0,\mathrm{for}\mathrm{all}{\nu }_{*}\ge {\nu }_x$$

are taken into consideration.

A solution of (1) is said to be non-oscillatory if it is eventually positive or eventually negative; otherwise, it is said to be oscillatory. If every solution to Equation (1) is oscillatory, then the equation is considered oscillatory.

Differential equations (DEs) have long been a milestone of mathematics and its application to the sciences. These equations serve as incredible tools for modeling a wide range of natural phenomena, ranging from population growth to the behavior of electrical circuits, disease spread, and celestial movement. Their importance is appraised by the fact that they offer a systematic approach to understanding and predicting change, making them indispensable in physics, engineering, biology, and numerous other disciplines; see [1,2,3,4,5].

The qualitative theory of differential inequalities focuses on the study of solution characteristics such as stability, periodicity, symmetry, and oscillation. Research into oscillatory phenomena in differential equations is a crucial and captivating area in both mathematics and applied sciences. Oscillations, characterized by periodic and repetitive patterns, are essential for understanding dynamic processes in various natural and engineered systems. These patterns appear in fields like physics, engineering, biology, and economics, offering valuable insights into system behaviors over time. This study delves into the intricate realm of oscillations in differential equations, aiming to uncover their fundamental principles, explore their applications, and contribute to the growing body of knowledge in this intriguing field (see [6,7,8,9,10]).

Oscillation criteria play a pivotal role in the study of nonlinear dynamics by providing sufficient criteria for the oscillatory features of solutions to DEs. In various applications, the ability to predict whether a system’s solutions will oscillate or remain steady is crucial. For instance, in mechanical and structural engineering, understanding oscillation helps in designing structures that can withstand periodic forces without leading to resonance, which could result in structural failure. In biological systems, oscillation criteria are essential for modeling phenomena such as cardiac rhythms or neural activity, where oscillatory patterns are indicative of healthy function or pathological conditions. In the realm of electrical engineering, oscillation criteria are fundamental for the design of oscillators and filters in circuits, ensuring that they perform their intended functions efficiently. Furthermore, in control systems engineering, these criteria aid in the design of controllers that manage the dynamic behavior of systems, preventing unwanted oscillations that can cause instability. Thus, oscillation criteria are indispensable tools in nonlinear dynamics, providing insights that help in the prediction, analysis, and control of complex systems across a wide range of scientific and engineering disciplines; see [11,12].

The Emden–Fowler equations, named after Robert Emden and Arthur Fowler, are central to the study of fourth-order differential models, particularly in mathematical physics. These equations are essential for modeling the structure and behavior of stellar objects such as white dwarfs, as they describe the complex relationships between pressure, density, and temperature. This modeling significantly contributes to the understanding of physical processes inside stars and their evolution. The applications of the Emden–Fowler equations extend to various fields. In thermodynamics, these equations describe temperature equilibria in complex thermal systems such as furnaces or geothermal models. In biological sciences and mathematical biology, they are used to model population dynamics and the spread of diseases, such as bacterial growth or virus transmission in different environments. In mechanical engineering and applied physics, they aid in modeling physical processes like fluid flow through porous media or material behavior under specific conditions. In geology, they assist in studying subsurface fluid movement and thermal distribution within the Earth’s crust, aiding in understanding geological formations and optimizing resource extraction; see [13,14,15,16].

The prior in-depth literature has enhanced our understanding of oscillatory properties of Emden–Fowler differential equations, with numerous influential authors shaping this area of study. Their significant contributions have left a lasting impact on the field, continuing to influence its direction and outcomes and attracting prominent attention. Below, we highlight some of these significant contributions:

Agarwal et al. [17] explored the oscillatory characteristics of DEs with a neutral term, described by

$${\left(\mathfrak{a}\left(\nu \right){\left(x\left(\nu \right)+\mathrm{g}\left(\nu \right)x^{\alpha }\left(\xi \left(\nu \right)\right)\right)}^{\prime }\right)}^{\prime }+\mathrm{p}\left(\nu \right)x\left(\lambda \left(\nu \right)\right)=0.$$

They established several sufficient conditions for the oscillations of these equations, considering the cases

$${\int }_{{\nu }_0}^{\infty }{\displaystyle \frac{1}{\mathfrak{a}\left(s\right)}}\mathrm{d}s=\infty ,$$

(3)

and

$${\int }_{{\nu }_0}^{\infty }{\displaystyle \frac{1}{\mathfrak{a}\left(s\right)}}\mathrm{d}s<\infty .$$

Baculíková et al. [18] examined the oscillatory properties of neutral differential equations (NDEs), represented by

$${\left(\mathfrak{a}\left(\nu \right){\left(x\left(\nu \right)-\mathrm{g}\left(\nu \right)x^{\alpha }\left(\xi \left(\nu \right)\right)\right)}^{\prime }\right)}^{\prime }+\mathrm{p}\left(\nu \right)x^{\beta }\left(\lambda \left(\nu \right)\right)=0.$$

They introduced new oscillatory criteria under the condition (3).

Zhang et al. [19] studied the oscillatory properties of a specific class of NDEs, given by

$${\left(\mathfrak{a}\left(\nu \right){\left|{\kappa }^{\prime }\left(\nu \right)\right|}^{\alpha -1}{\kappa }^{\prime }\left(\nu \right)\right)}^{\prime }+\mathrm{p}\left(\nu \right){\left|x\left(\lambda \left(\nu \right)\right)\right|}^{\alpha -1}x\left(\lambda \left(\nu \right)\right)=0,$$

where $\kappa \left(\nu \right)=x\left(\nu \right)+{\sum }_{i=1}^m{\mathrm{g}}_i\left(\nu \right)x\left({\xi }_i\left(\nu \right)\right).$ Their results simplify the analysis of these equations.

Tamilvanan et al. [20] investigated the oscillatory characteristics of the Emden–Fowler DEs, expressed as

$${\left(\mathfrak{a}\left(\nu \right){\left(x\left(\nu \right)+\mathrm{g}\left(\nu \right)x^{\alpha }\left(\xi \left(\nu \right)\right)\right)}^{\prime }\right)}^{\prime }+\mathrm{p}\left(\nu \right)x^{\beta }\left(\lambda \left(\nu \right)\right)=0.$$

(4)

Wu et al. [21] investigated the oscillatory characteristics of Emden–Fowler DDEs, described by

$${\left(\mathfrak{a}\left(\nu \right){\left({\left(x\left(\nu \right)+\mathrm{g}\left(\nu \right)x^{\alpha }\left(\xi \left(\nu \right)\right)\right)}^{\prime }\right)}^{\gamma }\right)}^{\prime }+\mathrm{p}\left(\nu \right)x^{\beta }\left(\lambda \left(\nu \right)\right)=0.$$

They introduced new criteria for determining oscillatory behavior under the conditions

$${\int }_{{\nu }_0}^{\infty }{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\gamma }\left(s\right)}}\mathrm{d}s=\infty ,$$

and

$${\int }_{{\nu }_0}^{\infty }{\displaystyle \frac{1}{{\mathfrak{a}}^{1/\gamma }\left(s\right)}}\mathrm{d}s<\infty .$$

El-Nabulsi et al. [22] studied the oscillation of solutions to DEs characterized by

$${\left(\mathfrak{a}\left(\nu \right){\left(x^{‴}\left(\nu \right)\right)}^{\alpha }\right)}^{\prime }+\mathrm{p}\left(\nu \right)f\left(x\left(\lambda \right(\nu \left)\right)\right)=0,$$

(5)

where $f\left(y\right)/y^{\alpha }\ge k>0$ for $y\ne 0$, and condition (2) holds. Related studies by Zhang et al. [10] and Moaaz et al. [23] further explored the oscillatory properties of (5) under condition (3).

Li and Rogovchenko [24] investigated the asymptotic behavior of solutions to a specific class of higher-order Emden–Fowler DDEs of the form

$${\left(\mathfrak{a}\left(\nu \right){\left(x^{\left(n\right)}\left(\nu \right)\right)}^{\alpha }\right)}^{\prime }+\mathrm{p}\left(\nu \right)x^{\beta }\left(\lambda \left(\nu \right)\right)=0.$$

They introduced new theorems that refined and advanced the understanding of these equations.

Graef et al. [25] introduced a new technique for linearizing even-order nonlinear NDEs of the form

$${\left(x\left(\nu \right)+\mathrm{g}\left(\nu \right)x^{\alpha }\left(\xi \left(\nu \right)\right)\right)}^{\left(n\right)}+\mathrm{p}\left(\nu \right)x^{\beta }\left(\lambda \left(\nu \right)\right)=0,n\ge 2.$$

They developed new oscillation criteria by comparing these equations with first-order linear DDEs.

Baculikova et al. [26] and Xing [27] established oscillation criteria for a specific class of higher-order quasi-linear NDEs:

$${\left(\mathfrak{a}\left(\nu \right){\left({\left(x\left(\nu \right)+\mathrm{g}\left(\nu \right)x\left(\xi \left(\nu \right)\right)\right)}^{\left(n-1\right)}\right)}^{\gamma }\right)}^{\prime }+\mathrm{p}\left(\nu \right)x^{\gamma }\left(\lambda \left(\nu \right)\right)=0,$$

(6)

subject to condition (2). Alnafisah et al. [28] introduced refined properties that optimize the oscillatory characteristics of solutions to Equation (6) under condition (3).

Our research aims to extend the investigation of oscillatory behavior to a specific class of fourth-order NDEs, particularly those with sublinear neutral terms. The motivation of our work is to extend and generalize previous studies (for example, [21]) that explored the asymptotic properties and oscillatory nature of second-order NDEs.

2. Preliminaries

This section introduces some important preliminary results related to the monotonic properties of non-oscillatory solutions to the equation under study, which contribute significantly to obtaining our results. We start with the following notations:

$$\kappa \left(\nu \right):=x\left(\nu \right)+\mathrm{g}\left(\nu \right)x^{\alpha }\left(\xi \left(\nu \right)\right),$$

(7)

$$\tilde{\mathrm{g}}\left(\nu \right):=1-{\displaystyle \frac{\mathrm{g}\left(\nu \right)}{c_1^{1-\alpha }}},$$

and

$$\tilde{\mathrm{p}}\left(\nu \right):=\mathrm{p}\left(\nu \right){\tilde{\mathrm{g}}}^{\beta }\left(\lambda \left(\nu \right)\right).$$

Lemma 1

([29]). Suppose that $\varrho \in C^n\left([{\nu }_0,\infty ),{\mathbb{R}}^{+}\right)$, where ${\varrho }^{\left(n\right)}\left(\nu \right)$ has a constant sign and is non-zero on $[{\nu }_0,\infty )$. Additionally, suppose there is ${\nu }_1\ge {\nu }_0$ such that ${\varrho }^{(n-1)}\left(\nu \right){\varrho }^{\left(n\right)}\left(\nu \right)\le 0$ for every $\nu \ge {\nu }_1$. If ${lim}_{\nu \to \infty }\varrho \left(\nu \right)\ne 0$, then for any $\delta \in (0,1)$, there is ${\nu }_{ϵ}\in [{\nu }_1,\infty )$ such that

$$\varrho \left(\nu \right)\ge {\displaystyle \frac{ϵ}{(n-1)!}}{\nu }^{n-1}\left|{\varrho }^{(n-1)}\left(\nu \right)\right|,$$

for $\varrho \in [{\nu }_{ϵ},\infty )$.

Lemma 2

([30]). Let γ be a ratio of two odd positive integers and A and B be constants. Then,

$$Bu-A{u}^{\left(\gamma +1\right)/\gamma }\le {\displaystyle \frac{{\gamma }^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{B^{\gamma +1}}{A^{\gamma }}},\mathfrak{a}>0.$$

(8)

Lemma 3

([31]). Let $\varrho \in {\mathbf{C}}^n\left([{\nu }_0,\infty ),\left(0,\infty \right)\right)$, ${\varrho }^{\left(i\right)}\left(\nu \right)>0$ for $i=1,2,\dots ,n$, and ${\varrho }^{\left(n+1\right)}\left(\nu \right)\le 0$, eventually. Then, eventually, $\varrho \left(\nu \right)/{\varrho }^{\prime }\left(\nu \right)\ge ϵ\nu /n$ for every $ϵ\in \left(0,1\right)$.

Lemma 4

([32]). Suppose that x represents an eventually positive solution of (1). In such a case, x will eventually fulfill the conditions of the following scenarios:

$$\begin{array}{ccc}\hfill {\mathrm{C}}_1& :& \kappa >0,{\kappa }^{\prime }>0,{\kappa }^{″}>0,{\kappa }^{‴}>0,{\left(\mathfrak{a}{\left({\kappa }^{‴}\right)}^{\gamma }\right)}^{\prime }<0,\hfill \\ \hfill {\mathrm{C}}_2& :& \kappa >0,{\kappa }^{\prime }>0,{\kappa }^{″}<0,{\kappa }^{‴}>0,{\left(\mathfrak{a}{\left({\kappa }^{‴}\right)}^{\gamma }\right)}^{\prime }<0,\hfill \end{array}$$

for $\nu ⩾{\nu }_1⩾{\nu }_0$.

Notation 1.

The category ${\Omega }_i$ denotes the set of all solutions that eventually become positive, with the corresponding function satisfying ${\mathrm{C}}_i$ for $i=1,2$. We note that all eventually positive solutions belong to ${\Omega }_1\cup {\Omega }_2$.

3. Auxiliary Lemmas

In the following discussion, we will establish several novel conditions that are adequate for excluding positive solutions that satisfy Equation (1). We will then combine these conditions to derive oscillation criteria.

Lemma 5.

Let us consider $x\in {\Omega }_1\cup {\Omega }_2$. Then, we have

$$x\left(\nu \right)>\tilde{\mathrm{g}}\left(\nu \right)\kappa \left(\nu \right),$$

(9)

and

$${\left(\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\right)}^{\prime }\le -\tilde{\mathrm{p}}\left(\nu \right){\kappa }^{\beta }\left(\lambda \left(\nu \right)\right).$$

(10)

Proof. 

From (7), we have $\kappa \left(\nu \right)\ge x\left(\nu \right)$. Using Lemma 4 for ${\nu }_1\ge {\nu }_0,$ we obtain

$${\left(\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\right)}^{\prime }<0.$$

(11)

Since $\lambda \left(\nu \right)\le \nu$, we can derive from (11) that

$$\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\le \mathfrak{a}\left(\lambda \left(\nu \right)\right){\left({\kappa }^{‴}\left(\lambda \left(\nu \right)\right)\right)}^{\gamma },\nu \ge {\nu }_1.$$

(12)

Notably, ${\kappa }^{\prime }\left(\nu \right)>0$. Therefore, there exists $c_1>0$ such that $\kappa \left(\nu \right)\ge c_1$ for sufficiently large $\nu$. By using (7), we conclude that

$$\begin{array}{ccc}\hfill x\left(v\right)& =& \kappa \left(v\right)-\mathrm{g}\left(v\right)x^{\alpha }\left(\xi \left(v\right)\right)\hfill \\ & \ge & \kappa \left(v\right)-\mathrm{g}\left(v\right){\kappa }^{\alpha }\left(\xi \left(v\right)\right)\hfill \\ & \ge & \kappa \left(v\right)-\mathrm{g}\left(v\right){\kappa }^{\alpha }\left(v\right)\hfill \\ & =& \left(1-\mathrm{g}\left(v\right){\kappa }^{\alpha -1}\left(v\right)\right)\kappa \left(v\right).\hfill \end{array}$$

(13)

Given that $0<\alpha \le 1,$ and $\kappa \left(\nu \right)\ge c_1>0,$ we have

$${\kappa }^{\alpha -1}\left(\nu \right)\le c_1^{\alpha -1}={\displaystyle \frac{1}{c_1^{1-\alpha }}}.$$

Substituting this inequality into (13), we obtain

$$x\left(\nu \right)\ge \left(1-{\displaystyle \frac{\mathrm{g}\left(\nu \right)}{c_1^{1-\alpha }}}\right)\kappa \left(\nu \right)=\tilde{\mathrm{g}}\left(\nu \right)\kappa \left(\nu \right).$$

(14)

Finally, based on (1), we can establish

$$\begin{array}{ccc}\hfill {\left(\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\right)}^{\prime }& =& -\mathrm{p}\left(\nu \right)x^{\beta }\left(\lambda \left(\nu \right)\right)\hfill \\ & \le & -\mathrm{p}\left(\nu \right){\left(1-{\displaystyle \frac{\mathrm{g}\left(\lambda \left(\nu \right)\right)}{c_1^{1-\alpha }}}\right)}^{\beta }{\kappa }^{\beta }\left(\lambda \left(\nu \right)\right)\hfill \\ & =& -\tilde{\mathrm{p}}\left(\nu \right){\kappa }^{\beta }\left(\lambda \left(\nu \right)\right).\hfill \end{array}$$

 □

Lemma 6.

Assume that (2) holds. If $\beta \ge \gamma$ and there is a nondecreasing function $\mu \in C^1([{\nu }_0,\infty ),$ $\left(0,\infty \right))$ such that

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(\mu \left(s\right)\tilde{\mathrm{p}}\left(s\right)-{\displaystyle \frac{2^{\gamma }\mathfrak{a}\left(\lambda \left(s\right)\right){\left[{\mu }^{\prime }\left(s\right)\right]}^{\gamma +1}}{{\left(\gamma +1\right)}^{\gamma +1}{c}_2^{\beta -\gamma }{\left(ϵ{\lambda }^2\left(s\right){\lambda }^{\prime }\left(s\right)\mu \left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s=\infty ,$$

(15)

holds for every $c_2>0,$ $ϵ\in \left(0,1\right)$, then ${\Omega }_1=⌀$.

Proof. 

We assume, for the sake of contradiction, that $x\left(\nu \right)\in {\Omega }_1$. Now, define the function $w\left(\nu \right)$ by

$$w\left(\nu \right):=\mu \left(\nu \right){\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }}{{\kappa }^{\beta }\left(\lambda \left(\nu \right)\right)}},$$

(16)

which yields $w\left(\nu \right)>0,$ $\nu \ge {\nu }_1,$

$$w^{\prime }\left(\nu \right)={\mu }^{\prime }\left(\nu \right){\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }}{{\kappa }^{\beta }\left(\lambda \left(\nu \right)\right)}}+\mu \left(\nu \right){\displaystyle \frac{{\left(\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\right)}^{\prime }}{{\kappa }^{\beta }\left(\lambda \left(\nu \right)\right)}}-\beta \mu \left(\nu \right){\lambda }^{\prime }\left(\nu \right){\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }{\kappa }^{\prime }\left(\lambda \left(\nu \right)\right)}{{\kappa }^{\beta +1}\left(\lambda \left(\nu \right)\right)}}.$$

(17)

We see from (9), (10), (16), and (17) that

$$w^{\prime }\left(\nu \right)\le {\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)-\beta {\lambda }^{\prime }\left(\nu \right)w\left(\nu \right){\displaystyle \frac{{\kappa }^{\prime }\left(\lambda \left(\nu \right)\right)}{\kappa \left(\lambda \left(\nu \right)\right)}}.$$

(18)

Since $\kappa ,$ ${\kappa }^{\prime },$ ${\kappa }^{″},$ and ${\kappa }^{‴}$ are positive, and ${\left(\mathfrak{a}{\left({\kappa }^{‴}\right)}^{\gamma }\right)}^{\prime }<0,$ according to Lemma 1, we can deduce that

$${\kappa }^{\prime }\left(\nu \right)\ge {\displaystyle \frac{ϵ}{2}}{\nu }^2{\kappa }^{‴}\left(\nu \right),$$

and

$${\kappa }^{\prime }\left(\lambda \left(\nu \right)\right)\ge {\displaystyle \frac{ϵ}{2}}{\lambda }^2\left(\nu \right){\kappa }^{‴}\left(\lambda \left(\nu \right)\right),$$

(19)

for all $ϵ\in \left(0,1\right)$ and every sufficiently large $\nu$. Substituting (19) into (18), we obtain

$$\begin{array}{ccc}\hfill w^{\prime }\left(\nu \right)& \le & {\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)-{\displaystyle \frac{ϵ}{2}}\beta {\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right){\displaystyle \frac{{\kappa }^{‴}\left(\lambda \left(\nu \right)\right)}{\kappa \left(\lambda \left(\nu \right)\right)}}w\left(\nu \right)\hfill \\ & =& -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-{\displaystyle \frac{ϵ}{2}}\beta {\displaystyle \frac{{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)}{{\mathfrak{a}}^{1/\gamma }\left(\lambda \left(\nu \right)\right)}}{\displaystyle \frac{{\mathfrak{a}}^{1/\gamma }\left(\lambda \left(\nu \right)\right){\kappa }^{‴}\left(\lambda \left(\nu \right)\right)}{\kappa \left(\lambda \left(\nu \right)\right)}}w\left(\nu \right).\hfill \end{array}$$

Since ${\left(\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\right)}^{\prime }<0,$ we conclude that

$${\mathfrak{a}}^{1/\gamma }\left(\nu \right){\kappa }^{‴}\left(\nu \right)\le {\mathfrak{a}}^{1/\gamma }\left(\lambda \left(\nu \right)\right){\kappa }^{‴}\left(\lambda \left(\nu \right)\right).$$

Then,

$$\begin{array}{ccc}\hfill w^{\prime }\left(\nu \right)& \le & -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-{\displaystyle \frac{ϵ\beta }{2}}{\displaystyle \frac{{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)}{{\mathfrak{a}}^{1/\gamma }\left(\lambda \left(\nu \right)\right)}}w\left(\nu \right){\displaystyle \frac{{\mathfrak{a}}^{1/\gamma }\left(\nu \right){\kappa }^{‴}\left(\nu \right)}{\kappa \left(\lambda \left(\nu \right)\right)}}\hfill \\ & =& -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-{\displaystyle \frac{ϵ\beta }{2}}{\displaystyle \frac{{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)}{{\left[\mu \left(\nu \right)\mathfrak{a}\left(\lambda \left(\nu \right)\right)\right]}^{1/\gamma }}}{\left[\kappa \left(\lambda \left(\nu \right)\right)\right]}^{{\displaystyle \frac{\beta -\gamma }{\gamma }}}{w}^{{\displaystyle \frac{\gamma +1}{\gamma }}}\left(\nu \right).\hfill \end{array}$$

(20)

Because ${\kappa }^{\prime }>0$ and $\beta \ge \gamma$, there exist constants $c_2>0$ and ${\nu }_2\ge {\nu }_1$ such that

$$\kappa \left(\lambda \left(\nu \right)\right)\ge c_2,$$

and

$${\kappa }^{\beta /\gamma -1}\left(\lambda \left(\nu \right)\right)\ge c_2^{\beta /\gamma -1},\nu \ge {\nu }_2.$$

(21)

Thus, inequality (20) gives

$$w^{\prime }\left(\nu \right)\le -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-{\displaystyle \frac{ϵ\gamma c_2^{\beta /\gamma -1}}{2}}{\displaystyle \frac{{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)}{{\left[\mu \left(\nu \right)\mathfrak{a}\left(\lambda \left(\nu \right)\right)\right]}^{1/\gamma }}}{w}^{{\displaystyle \frac{\gamma +1}{\gamma }}}\left(\nu \right).$$

(22)

Using Lemma 2 with

$$B={\mu }^{\prime }\left(\nu \right)/\mu \left(\nu \right),A=ϵ\gamma c_2^{\beta /\gamma -1}{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)/2{\left[\mu \left(\nu \right)\mathfrak{a}\left(\lambda \left(\nu \right)\right)\right]}^{1/\gamma }$$

and $u\left(\nu \right)=w\left(\nu \right),$ we can derive the following inequality:

$$w^{\prime }\left(\nu \right)\le -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{2^{\gamma }\mathfrak{a}\left(\lambda \left(\nu \right)\right){\left[{\mu }^{\prime }\left(\nu \right)\right]}^{\gamma +1}}{{\left(\gamma +1\right)}^{\gamma +1}{c}_2^{\beta -\gamma }{\left(ϵ{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)\mu \left(\nu \right)\right)}^{\gamma }}}.$$

(23)

Integrating (23) from ${\nu }_3\ge {\nu }_2$ to $\nu ,$ one arrives at

$${\int }_{{\nu }_3}^{\nu }\left(\mu \left(s\right)\tilde{\mathrm{p}}\left(s\right)-{\displaystyle \frac{2^{\gamma }\mathfrak{a}\left(\lambda \left(s\right)\right){\left[{\mu }^{\prime }\left(s\right)\right]}^{\gamma +1}}{{\left(\gamma +1\right)}^{\gamma +1}{c}_2^{\beta -\gamma }{\left(ϵ{\lambda }^2\left(s\right){\lambda }^{\prime }\left(s\right)\mu \left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s\le w\left({\nu }_3\right),$$

which contradicts (15) as $\nu \to \infty$. This completes the proof. □

Lemma 7.

Suppose that (2) holds. If $\beta \ge \gamma$ and there is a ${\mu }_1\in C^1([{\nu }_0,\infty ),\left(0,\infty \right))$ such that ${\mu }_1^{\prime }\left(\nu \right)\ge 0$

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(M_1^{\beta -\gamma }{\mu }_1\left(s\right)\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{3\beta /\widetilde{ϵ}}-{\displaystyle \frac{2^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left({\mu }_1^{\prime }\left(s\right)\right)}^{\gamma +1}}{{\left(ϵs^2{\mu }_1\left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s=\infty ,$$

(24)

holds for every $ϵ,\widetilde{ϵ}\in \left(0,1\right)$, then ${\Omega }_1=⌀$.

Proof. 

We assume, for the sake of contradiction, that $x\left(\nu \right)\in {\Omega }_1$. Now, define a function $w_1\left(\nu \right)$ by

$$w_1\left(\nu \right):={\mu }_1\left(\nu \right){\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }}{{\kappa }^{\gamma }\left(\nu \right)}},$$

(25)

which yields $w_1\left(\nu \right)>0.$ The derivative $w_1^{\prime }\left(\nu \right)$ is given by

$$w_1^{\prime }\left(\nu \right)={\mu }_1^{\prime }\left(\nu \right){\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }}{{\kappa }^{\gamma }\left(\nu \right)}}+{\mu }_1\left(\nu \right){\displaystyle \frac{{\left(\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\right)}^{\prime }}{{\kappa }^{\gamma }\left(\nu \right)}}-\gamma {\mu }_1\left(\nu \right){\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }{\kappa }^{\prime }\left(\nu \right)}{{\kappa }^{\gamma +1}\left(\nu \right)}}.$$

(26)

We see from (10), (25), and (26) that

$$w_1^{\prime }\left(\nu \right)\le -{\mu }_1\left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right){\displaystyle \frac{{\kappa }^{\beta }\left(\lambda \left(\nu \right)\right)}{{\kappa }^{\gamma }\left(\nu \right)}}+{\displaystyle \frac{{\mu }_1^{\prime }\left(\nu \right)}{{\mu }_1\left(\nu \right)}}{w}_1\left(\nu \right)-\gamma {\displaystyle \frac{{\kappa }^{\prime }\left(\nu \right)}{\kappa \left(\nu \right)}}{w}_1\left(\nu \right).$$

(27)

From Lemma 3, we deduce that

$$\kappa \left(\nu \right)\ge {\displaystyle \frac{\widetilde{ϵ}}{3}}\nu {\kappa }^{\prime }\left(\nu \right),$$

and hence,

$${\displaystyle \frac{\kappa \left(\lambda \left(\nu \right)\right)}{\kappa \left(\nu \right)}}\ge {\left({\displaystyle \frac{{\lambda }^3\left(\nu \right)}{\nu }}\right)}^{3/\widetilde{ϵ}}.$$

(28)

From Lemma 1, we conclude that

$${\kappa }^{\prime }\left(\nu \right)\ge {\displaystyle \frac{ϵ}{2}}{\nu }^2{\kappa }^{‴}\left(\nu \right),$$

(29)

for all $ϵ\in \left(0,1\right)$. Thus, by (27)–(29), we have

$$\begin{array}{ccc}\hfill w_1^{\prime }\left(\nu \right)& \le & -{\mu }_1\left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right){\kappa }^{\beta -\gamma }\left(\nu \right){\displaystyle \frac{{\kappa }^{\beta }\left(\lambda \left(\nu \right)\right)}{{\kappa }^{\beta }\left(\nu \right)}}+{\displaystyle \frac{{\mu }_1^{\prime }\left(\nu \right)}{{\mu }_1\left(\nu \right)}}{w}_1\left(\nu \right)-{\displaystyle \frac{ϵ\gamma }{2}}{\nu }^2{\displaystyle \frac{{\kappa }^{‴}\left(\nu \right)}{\kappa \left(\nu \right)}}{w}_1\left(\nu \right)\hfill \\ & =& -{\mu }_1\left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right){\kappa }^{\beta -\gamma }\left(\nu \right){\left({\displaystyle \frac{{\lambda }^3\left(\nu \right)}{\nu }}\right)}^{3\beta /\widetilde{ϵ}}\hfill \\ & & +{\displaystyle \frac{{\mu }_1^{\prime }\left(\nu \right)}{{\mu }_1\left(\nu \right)}}{w}_1\left(\nu \right)-{\displaystyle \frac{ϵ\gamma }{2}}{\displaystyle \frac{{\nu }^2}{{\left(\mathfrak{a}\left(\nu \right){\mu }_1\left(\nu \right)\right)}^{1/\gamma }}}{w}_1^{\left(1+\gamma \right)/\gamma }\left(\nu \right).\hfill \end{array}$$

(30)

Since ${\kappa }^{\prime }\left(\nu \right)>0$, there exist a ${\nu }_3\ge {\nu }_2$ and a constant $M_1>0$ such that

$$\kappa \left(\nu \right)>M_1.$$

Since $\beta \ge \gamma ,$ then

$${\kappa }^{\beta -\gamma }\left(\nu \right)>M_1^{\beta -\gamma }.$$

(31)

Thus, inequality (30) gives

$$\begin{array}{ccc}\hfill w_1^{\prime }\left(\nu \right)& \le & -M_1^{\beta -\gamma }{\mu }_1\left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right){\left({\displaystyle \frac{\lambda \left(\nu \right)}{\nu }}\right)}^{3\beta /\widetilde{ϵ}}\hfill \\ & & +{\displaystyle \frac{{\mu }_1^{\prime }\left(\nu \right)}{{\mu }_1\left(\nu \right)}}{w}_1\left(\nu \right)-{\displaystyle \frac{ϵ\gamma {\nu }^2}{2{\left(\mathfrak{a}\left(\nu \right){\mu }_1\left(\nu \right)\right)}^{1/\gamma }}}{w}_1^{\left(1+\gamma \right)/\gamma }\left(\nu \right).\hfill \end{array}$$

(32)

Using Lemma 2, with

$$B={\mu }_1^{\prime }\left(\nu \right)/{\mu }_1\left(\nu \right),A=ϵ\gamma {\nu }^2/2{\left(\mathfrak{a}\left(\nu \right){\mu }_1\left(\nu \right)\right)}^{1/\gamma },$$

and $u\left(\nu \right)=w_1\left(\nu \right),$ we can derive the following inequality:

$$w_1^{\prime }\left(\nu \right)\le -M_1^{\beta -\gamma }{\mu }_1\left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right){\left({\displaystyle \frac{\lambda \left(\nu \right)}{\nu }}\right)}^{3\beta /\widetilde{ϵ}}+{\displaystyle \frac{2^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left({\mu }_1^{\prime }\left(\nu \right)\right)}^{\gamma +1}}{{\left(ϵ{\nu }^2{\mu }_1\left(\nu \right)\right)}^{\gamma }}}.$$

(33)

Integrating (33) from ${\nu }_4\ge {\nu }_3$ to $\nu ,$ one arrives at

$${\int }_{{\nu }_4}^{\nu }\left(M_1^{\beta -\gamma }{\mu }_1\left(s\right)\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{3\beta /\widetilde{ϵ}}-{\displaystyle \frac{2^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left({\mu }_1^{\prime }\left(s\right)\right)}^{\gamma +1}}{{\left(ϵs^2{\mu }_1\left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s\le w_1\left({\nu }_4\right),$$

which contradicts (15) as $\nu \to \infty$. This completes the proof. □

Lemma 8.

Assume that (2) holds. If $0<\beta <\gamma$ and there is a nondecreasing function $\mu \in C^1([{\nu }_0,\infty ),\left(0,\infty \right))$ such that

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(\mu \left(s\right)\tilde{\mathrm{p}}\left(s\right)-{\displaystyle \frac{2^{\beta }}{{\left(\beta +1\right)}^{\beta +1}{c}_3^{\beta -\gamma }}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left[{\mu }^{\prime }\left(s\right)\right]}^{\beta +1}}{{\left(ϵ\mu \left(s\right){\lambda }^2\left(s\right){\lambda }^{\prime }\left(s\right)\right)}^{\beta }}}\right)\mathrm{d}s=\infty ,$$

(34)

holds for every $c_3>0,$ $ϵ\in \left(0,1\right)$, then ${\Omega }_1=⌀$.

Proof. 

We assume, for the sake of contradiction, that $x\left(\nu \right)\in {\Omega }_1$. As in the proof of Lemma 5, we have (9) and (10). The function $w\left(\nu \right)$ is defined in (16), and then (17) holds. By (1), (16), and (17), we conclude that

$$w^{\prime }\left(\nu \right)\le -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-\beta {\lambda }^{\prime }\left(\nu \right)w\left(\nu \right){\displaystyle \frac{{\kappa }^{\prime }\left(\lambda \left(\nu \right)\right)}{\kappa \left(\lambda \left(\nu \right)\right)}}.$$

By using (19), we observe that

$$\begin{array}{ccc}\hfill w^{\prime }\left(\nu \right)& \le & -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-{\displaystyle \frac{ϵ\beta }{2}}{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right){\left[{\kappa }^{‴}\left(\nu \right)\right]}^{\left(\beta -\gamma \right)/\beta }{\displaystyle \frac{{\left[{\kappa }^{‴}\left(\nu \right)\right]}^{\gamma /\beta }}{\kappa \left(\lambda \left(\nu \right)\right)}}w\left(\nu \right)\hfill \\ & =& -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-{\displaystyle \frac{ϵ\beta }{2}}{\displaystyle \frac{{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)}{{\left(\mu \left(\nu \right)\mathfrak{a}\left(\nu \right)\right)}^{1/\beta }}}{\left[{\kappa }^{‴}\left(\nu \right)\right]}^{\left(\beta -\gamma \right)/\beta }{w}^{\left(\beta +1\right)/\beta }\left(\nu \right).\hfill \end{array}$$

(35)

Note that $0<\beta <\gamma$ and (${\mathrm{C}}_1$) hold. Since ${\mathfrak{a}}^{\prime }\left(\nu \right)\ge 0$, we can conclude that ${\kappa }^{\left(4\right)}\left(\nu \right)\le 0$. This straightforwardly implies that ${\kappa }^{‴}\left(\nu \right)$ is nonincreasing. Consequently, there exist constants $c_3>0$ and ${\nu }_3\ge {\nu }_2$ such that

$${\kappa }^{‴}\left(\nu \right)\le c_3,$$

and

$${\left({\kappa }^{‴}\left(\nu \right)\right)}^{\left(\beta -\gamma \right)/\beta }\ge c_3^{\left(\beta -\gamma \right)/\beta },\nu \ge {\nu }_3.$$

(36)

From (35) and (36), it follows that

$$w\left(\nu \right)\le -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{{\mu }^{\prime }\left(\nu \right)}{\mu \left(\nu \right)}}w\left(\nu \right)-{\displaystyle \frac{ϵ\beta c_3^{\left(\beta -\gamma \right)/\beta }}{2}}{\displaystyle \frac{{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)}{{\left(\mu \left(\nu \right)\mathfrak{a}\left(\nu \right)\right)}^{1/\beta }}}{w}^{\left(\beta +1\right)/\beta }\left(\nu \right).$$

(37)

Using Lemma 2 with

$$B={\mu }^{\prime }\left(\nu \right)/\mu \left(\nu \right),A=ϵ\beta c_3^{\left(\beta -\gamma \right)/\beta }{\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)/2{\left[\mu \left(\nu \right)\mathfrak{a}\left(\nu \right)\right]}^{1/\beta },$$

and $u\left(\nu \right)=w\left(\nu \right),$ It can be deduced from (37) that

$$w\left(\nu \right)\le -\mu \left(\nu \right)\tilde{\mathrm{p}}\left(\nu \right)+{\displaystyle \frac{2^{\beta }}{{\left(\beta +1\right)}^{\beta +1}{c}_3^{\beta -\gamma }}}{\displaystyle \frac{\mathfrak{a}\left(\nu \right){\left[{\mu }^{\prime }\left(\nu \right)\right]}^{\beta +1}}{{\left(ϵ\mu \left(\nu \right){\lambda }^2\left(\nu \right){\lambda }^{\prime }\left(\nu \right)\right)}^{\beta }}}.$$

(38)

By performing the integration of (38) over the interval $\left[{\nu }_5,\nu \right]$, we can deduce that

$${\int }_{{\nu }_5}^{\nu }\left(\mu \left(s\right)\tilde{\mathrm{p}}\left(s\right)-{\displaystyle \frac{2^{\beta }}{{\left(\beta +1\right)}^{\beta +1}{c}_3^{\beta -\gamma }}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left[{\mu }^{\prime }\left(s\right)\right]}^{\beta +1}}{{\left(ϵ\mu \left(s\right){\lambda }^2\left(s\right){\lambda }^{\prime }\left(s\right)\right)}^{\beta }}}\right)\mathrm{d}s\le w\left({\nu }_5\right),$$

which contradicts (34) as $\nu \to \infty$. This completes the proof. □

Lemma 9.

If $\beta \ge \gamma$ and there is a $\tilde{\mu }\in C\left(\left[{\nu }_0,\infty \right),\left(0,\infty \right)\right)$ such that

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(M^{\beta /\gamma -1}\tilde{\mu }\left(s\right){\int }_s^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(v\right){\left({\displaystyle \frac{\lambda \left(v\right)}{v}}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}}\right)\mathrm{d}s=\infty ,$$

(39)

hold for some ${ϵ}_1\in \left(0,1\right)$, then ${\Omega }_2=⌀.$

Proof. 

We suppose for contradiction that $x\left(\nu \right)\in {\Omega }_2.$ By integrating (10) from $\nu$ to ∞ and utilizing the property that ${\left(r{\left({\kappa }^{‴}\right)}^{\gamma }\right)}^{\prime }\le 0$, we derive

$$\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\ge {\int }_{\nu }^{\infty }\mathrm{p}\left(s\right){\tilde{\mathrm{g}}}^{\beta }\left(\lambda \left(s\right)\right){\kappa }^{\beta }\left(\lambda \left(s\right)\right)\mathrm{d}s.$$

(40)

As $\kappa >0,$ ${\kappa }^{\prime }>0,$ and ${\kappa }^{″}<0,$ Lemma 3 implies that $\kappa \ge {ϵ}_1\nu {\kappa }^{\prime }$ for all ${ϵ}_1\in \left(0,1\right)$. Upon integrating this inequality from $\lambda \left(\nu \right)$ to $\nu ,$ we find

$${\displaystyle \frac{\kappa \left(\lambda \left(\nu \right)\right)}{\kappa \left(\nu \right)}}\ge {\left({\displaystyle \frac{\lambda \left(\nu \right)}{\nu }}\right)}^{1/{ϵ}_1}.$$

Therefore, (40) becomes

$$\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\ge {\int }_{\nu }^{\infty }\mathrm{p}\left(s\right){\tilde{\mathrm{g}}}^{\beta }\left(\lambda \left(s\right)\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}{\kappa }^{\beta }\left(s\right)\mathrm{d}s.$$

Since ${\kappa }^{\prime }\left(\nu \right)>0,$ then

$$\mathfrak{a}\left(\nu \right){\left({\kappa }^{‴}\left(\nu \right)\right)}^{\gamma }\ge {\kappa }^{\beta }\left(\nu \right){\int }_{\nu }^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s,$$

or equivalently,

$${\kappa }^{‴}\left(\nu \right)\ge {\kappa }^{\beta /\gamma }\left(\nu \right){\left({\displaystyle \frac{1}{\mathfrak{a}\left(\nu \right)}}{\int }_{\nu }^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }.$$

Upon integrating this inequality from $\nu$ to ∞, the result is obtained as

$${\kappa }^{″}\left(\nu \right)\le -{\kappa }^{\beta /\gamma }{\int }_{\nu }^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }\mathrm{d}u.$$

(41)

Now, define

$$G\left(\nu \right):=\tilde{\mu }\left(\nu \right){\displaystyle \frac{{\kappa }^{\prime }\left(\nu \right)}{\kappa \left(\nu \right)}}.$$

(42)

Then, $G\left(\nu \right)\ge 0$, and

$$\begin{array}{ccc}\hfill G^{\prime }& =& {\tilde{\mu }}^{\prime }\left(\nu \right){\displaystyle \frac{{\kappa }^{\prime }\left(\nu \right)}{\kappa \left(\nu \right)}}+\tilde{\mu }\left(\nu \right){\displaystyle \frac{{\kappa }^{″}\left(\nu \right)}{\kappa \left(\nu \right)}}-\tilde{\mu }\left(\nu \right){\displaystyle \frac{{\left({\kappa }^{\prime }\left(\nu \right)\right)}^2}{{\kappa }^2\left(\nu \right)}}\hfill \\ & =& \tilde{\mu }\left(\nu \right){\displaystyle \frac{{\kappa }^{″}\left(\nu \right)}{\kappa \left(\nu \right)}}+{\displaystyle \frac{{\tilde{\mu }}^{\prime }\left(\nu \right)}{\tilde{\mu }\left(\nu \right)}}G\left(\nu \right)-{\displaystyle \frac{1}{\tilde{\mu }\left(\nu \right)}}{G}^2\left(\nu \right).\hfill \end{array}$$

Hence, by (41), we obtain

$$\begin{array}{ccc}\hfill G^{\prime }\left(\nu \right)& \le & -\tilde{\mu }\left(\nu \right){\kappa }^{\beta /\gamma -1}{\int }_{\nu }^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }\mathrm{d}u\hfill \\ & & +{\displaystyle \frac{{\tilde{\mu }}_{+}^{\prime }\left(\nu \right)}{\tilde{\mu }\left(\nu \right)}}G\left(\nu \right)-{\displaystyle \frac{1}{\tilde{\mu }\left(\nu \right)}}{G}^2\left(\nu \right).\hfill \end{array}$$

(43)

Because ${\kappa }^{\prime }\left(\nu \right)>0$ and $\beta \ge \gamma$, there exist constants $M>0$ and ${\nu }_2\ge {\nu }_1$ such that

$$\kappa \left(\nu \right)\ge M,\nu \ge {\nu }_2,$$

and

$${\kappa }^{\beta /\gamma -1}\left(\nu \right)\ge M^{\beta /\gamma -1}.$$

(44)

Substituting (44) into (43), we have

$$\begin{array}{ccc}\hfill G^{\prime }\left(\nu \right)& \le & -M^{\beta /\gamma -1}\tilde{\mu }\left(\nu \right){\int }_{\nu }^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }\mathrm{d}u\hfill \\ & & +{\displaystyle \frac{{\tilde{\mu }}_{+}^{\prime }\left(\nu \right)}{\tilde{\mu }\left(\nu \right)}}G\left(\nu \right)-{\displaystyle \frac{1}{\tilde{\mu }\left(\nu \right)}}{G}^2\left(\nu \right).\hfill \end{array}$$

Using Lemma 2 with $B={\tilde{\mu }}_{+}^{\prime }\left(\nu \right)/\tilde{\mu }\left(\nu \right),$ $A=1/\tilde{\mu }\left(\nu \right)$, and $u\left(\nu \right)=G\left(\nu \right),$ we obtain

$${\displaystyle \frac{{\tilde{\mu }}_{+}^{\prime }\left(\nu \right)}{\tilde{\mu }\left(\nu \right)}}G\left(\nu \right)-{\displaystyle \frac{1}{\tilde{\mu }\left(\nu \right)}}{G}^2\left(\nu \right)\le {\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(\nu \right)\right)}^2}{4\tilde{\mu }\left(\nu \right)}}.$$

Consequently, (44) leads to

$$G^{\prime }\left(\nu \right)\le -M^{\beta /\gamma -1}\tilde{\mu }\left(\nu \right){\int }_{\nu }^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }\mathrm{d}u+{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(\nu \right)\right)}^2}{4\tilde{\mu }\left(\nu \right)}}.$$

(45)

Integrating (45) from ${\nu }_1$ to $\nu$, we have

$${\int }_{{\nu }_1}^{\nu }\left(M^{\beta /\gamma -1}\tilde{\mu }\left(s\right){\int }_s^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(v\right){\left({\displaystyle \frac{\lambda \left(v\right)}{v}}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}}\right)\mathrm{d}s\le G\left({\nu }_1\right),$$

which contradicts (39) as $\nu \to \infty$. This completes the proof. □

Lemma 10.

If $0<\beta <\gamma$ and there is a $\tilde{\mu }\in C\left(\left[{\nu }_0,\infty \right),\left(0,\infty \right)\right)$ such that

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(K^{\frac{\beta -\gamma }{\gamma }}{s}^{\widehat{ϵ}}\tilde{\mu }\left(s\right){\int }_s^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(v\right){\left({\displaystyle \frac{\lambda \left(v\right)}{v}}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}}\right)\mathrm{d}s=\infty ,$$

(46)

hold for some ${ϵ}_1\in \left(0,1\right),$ $\widehat{ϵ}=\left(\beta -\gamma \right)/{ϵ}_2\gamma$, then ${\Omega }_2=⌀.$

Proof. 

We begin by assuming, for the sake of contradiction, that $x\left(\nu \right)\in {\Omega }_2$. As demonstrated in the proof of Lemma 9, we have (9) and (10). The function $G\left(\nu \right)$ is defined in (42), and consequently, (43) holds, which can be expressed as

$$\begin{array}{ccc}\hfill G^{\prime }\left(\nu \right)& \le & -\tilde{\mu }\left(\nu \right){\kappa }^{\beta /\gamma -1}{\int }_{\nu }^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }\mathrm{d}u\hfill \\ & & +{\displaystyle \frac{{\tilde{\mu }}_{+}^{\prime }\left(\nu \right)}{\tilde{\mu }\left(\nu \right)}}G\left(\nu \right)-{\displaystyle \frac{1}{\tilde{\mu }\left(\nu \right)}}{G}^2\left(\nu \right).\hfill \end{array}$$

(47)

Since $\kappa >0,$ ${\kappa }^{\prime }>0,$ and ${\kappa }^{″}<0,$ Lemma 3 implies that

$$\kappa \left(\nu \right)\ge {ϵ}_2\nu {\kappa }^{\prime }\left(\nu \right),$$

for all ${ϵ}_2\in \left(0,1\right),$ and

$${\displaystyle \frac{1}{{ϵ}_2\nu }}\ge {\displaystyle \frac{{\kappa }^{\prime }\left(\nu \right)}{\kappa \left(\nu \right)}}.$$

(48)

Integrating (48) from ${\nu }_1$ to $\nu ,$ we obtain

$$ln{\left({\displaystyle \frac{\nu }{{\nu }_1}}\right)}^{1/{ϵ}_2}\ge ln{\displaystyle \frac{\kappa \left(\nu \right)}{\kappa \left({\nu }_1\right)}},$$

which implies

$$\kappa \left(\nu \right)\le \kappa \left({\nu }_1\right){\left({\displaystyle \frac{\nu }{{\nu }_1}}\right)}^{1/{ϵ}_2}={\displaystyle \frac{\kappa \left({\nu }_1\right)}{{\nu }_1^{1/{ϵ}_2}}}{\nu }^{1/{ϵ}_2}=K{\nu }^{1/{ϵ}_2},$$

(49)

where $K={\displaystyle \frac{\kappa \left({\nu }_1\right)}{{\nu }_1^{1/{ϵ}_2}}}$. As $0<\beta <\gamma ,$ it follows that $0<\beta /\gamma <1,$ which, combined with (49), yields

$${\kappa }^{\beta /\gamma -1}\left(\nu \right)\ge K^{\beta /\gamma -1}{\nu }^{\left(\beta /\gamma -1\right)/{ϵ}_2}.$$

(50)

Hence, inequality (47) becomes

$$\begin{array}{ccc}\hfill G^{\prime }\left(\nu \right)& \le & -K^{\beta /\gamma -1}\tilde{\mu }\left(\nu \right){\nu }^{\left(\beta /\gamma -1\right)/{ϵ}_2}{\int }_{\nu }^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }\mathrm{d}u\hfill \\ & & +{\displaystyle \frac{{\tilde{\mu }}_{+}^{\prime }\left(\nu \right)}{\tilde{\mu }\left(\nu \right)}}G\left(\nu \right)-{\displaystyle \frac{1}{\tilde{\mu }\left(\nu \right)}}{G}^2\left(\nu \right).\hfill \end{array}$$

(51)

Using Lemma 2 with $B={\tilde{\mu }}_{+}^{\prime }\left(\nu \right)/\tilde{\mu }\left(\nu \right),$ $\mathfrak{a}=1/\tilde{\mu }\left(\nu \right)$, and $u\left(\nu \right)=G\left(\nu \right),$ we conclude that

$${\displaystyle \frac{{\tilde{\mu }}_{+}^{\prime }\left(\nu \right)}{\tilde{\mu }\left(\nu \right)}}G\left(\nu \right)-{\displaystyle \frac{1}{\tilde{\mu }\left(\nu \right)}}{G}^2\left(\nu \right)\le {\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(\nu \right)\right)}^2}{4\tilde{\mu }\left(\nu \right)}}.$$

Consequently, (51) leads to

$$G^{\prime }\left(\nu \right)\le -K^{\beta /\gamma -1}\tilde{\mu }\left(\nu \right){\nu }^{\left(\beta /\gamma -1\right)/{ϵ}_2}{\int }_{\nu }^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{\beta /{ϵ}_1}\mathrm{d}s\right)}^{1/\gamma }\mathrm{d}u+{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(\nu \right)\right)}^2}{4\tilde{\mu }\left(\nu \right)}}.$$

Integrating this inequality from ${\nu }_2$ to $\nu$, we infer that

$${\int }_{{\nu }_2}^{\nu }\left(K^{\beta /\gamma -1}\tilde{\mu }\left(s\right)s^{\left(\beta /\gamma -1\right)/{ϵ}_2}{\int }_s^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(v\right)}{v}}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}}\right)\mathrm{d}s\le G\left({\nu }_1\right),$$

which contradicts (39) as $\nu \to \infty$. Thus, we have completed the proof. □

4. Oscillation Criteria

This section builds on the conclusions from our earlier results to establish new conditions for analyzing the oscillatory properties of all solutions to Equation (1). By merging the criteria previously concluded to rule out positive solutions in both scenarios $\left({\mathrm{C}}_1\right)$ and $\left({\mathrm{C}}_2\right)$, we can formulate the criteria presented in the following theorems. These criteria will help us ascertain the oscillatory features of the examined equation.

Theorem 1.

Suppose that

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(\mu \left(s\right)\tilde{\mathrm{p}}\left(s\right)-{\displaystyle \frac{2^{\gamma }\mathfrak{a}\left(\lambda \left(s\right)\right){\left[{\mu }^{\prime }\left(s\right)\right]}^{\gamma +1}}{{\left(\gamma +1\right)}^{\gamma +1}{c}_2^{\beta -\gamma }{\left(ϵ{\lambda }^2\left(s\right){\lambda }^{\prime }\left(s\right)\mu \left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s=\infty ,$$

and

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(M^{\beta /\gamma -1}\tilde{\mu }\left(s\right){\int }_s^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(v\right){\left({\displaystyle \frac{\lambda \left(v\right)}{v}}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}}\right)\mathrm{d}s=\infty ,$$

hold. Then, (1) oscillates.

Proof. 

Assume that $x\in {\Omega }_1\cup {\Omega }_2$. Lemma 4 presents two possible cases for $\kappa$ and its derivatives. Using Lemmas 6 and 9, it is easy to see that cases $\left({\mathrm{C}}_1\right)$ and $\left({\mathrm{C}}_2\right)$ are excluded by conditions (15) and (39), respectively. This ends the proof. □

Theorem 2.

Suppose that

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(M_1^{\beta -\gamma }{\mu }_1\left(s\right)\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{3\beta /\widetilde{ϵ}}-{\displaystyle \frac{2^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left({\mu }_1^{\prime }\left(s\right)\right)}^{\gamma +1}}{{\left(ϵs^2{\mu }_1\left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s=\infty ,$$

and

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(M^{\beta /\gamma -1}\tilde{\mu }\left(s\right){\int }_s^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(v\right){\left({\displaystyle \frac{\lambda \left(v\right)}{v}}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}}\right)\mathrm{d}s=\infty ,$$

hold. Then, (1) oscillates.

Proof. 

This proof follows a method analogous to that of Theorem 1 and thus was omitted. □

Theorem 3.

Assume that

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(\mu \left(s\right)\tilde{\mathrm{p}}\left(s\right)-{\displaystyle \frac{2^{\beta }}{{\left(\beta +1\right)}^{\beta +1}{c}_3^{\beta -\gamma }}}\frac{\mathfrak{a}\left(s\right){\left[{\mu }^{\prime }\left(s\right)\right]}^{\beta +1}}{{\left(ϵ\mu \left(s\right){\lambda }^2\left(s\right){\lambda }^{\prime }\left(s\right)\right)}^{\beta }}\right)\mathrm{d}s=\infty ,$$

and

$$\underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(K^{\frac{\beta -\gamma }{\gamma }}{s}^{\widehat{ϵ}}\tilde{\mu }\left(s\right){\int }_s^{\infty }{\left(\frac{1}{\mathfrak{a}\left(u\right)}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(v\right){\left(\frac{\lambda \left(v\right)}{v}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-\frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}\right)\mathrm{d}s=\infty ,$$

hold. Then, (1) is oscillatory.

Proof. 

This proof follows a method analogous to that of Theorem 1 and thus was omitted. □

Example 1.

Consider the NDE given by

$${\left(\frac{1}{v}\left({\left(x\left(\nu \right)+\frac{{\mathrm{g}}_0}{v}{x}^{1/3}\left({\xi }_0\nu \right)\right)}^{‴}\right)\right)}^{\prime }+{\displaystyle \frac{{\mathrm{p}}_0}{{\nu }^5}}x\left({\lambda }_0\nu \right)=0,\nu \ge 1,$$

(52)

where $\beta \ge \gamma ,$ ${\xi }_0,$ ${\lambda }_0\in \left(0,1\right)$, and ${\mathrm{p}}_0>0.$ By comparing (1) and (52), we observe that

$$\mathfrak{a}\left(\nu \right)=1/v,\xi \left(\nu \right)={\xi }_0\nu ,\mathrm{and}\lambda \left(\nu \right)={\lambda }_0\nu .$$

As a result, we have

$$\tilde{\mathrm{g}}\left(\nu \right)=1-{\displaystyle \frac{{\mathrm{g}}_0}{c_1^{2/3}v}},$$

and

$$\tilde{\mathrm{p}}\left(\nu \right):=\left(1-{\displaystyle \frac{{\mathrm{g}}_0}{v{c}_1^{2/3}}}\right){\displaystyle \frac{{\mathrm{p}}_0}{{\nu }^5}}.$$

Let us consider the function $\mu \left(\nu \right)={\nu }^4.$ Condition (15) can then be expressed as

$$\begin{array}{ccc}& & \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(\mu \left(s\right)\tilde{\mathrm{p}}\left(s\right)-{\displaystyle \frac{2^{\gamma }\mathfrak{a}\left(\lambda \left(s\right)\right){\left[{\mu }^{\prime }\left(s\right)\right]}^{\gamma +1}}{{\left(\gamma +1\right)}^{\gamma +1}{c}_2^{\beta -\gamma }{\left(ϵ{\lambda }^2\left(s\right){\lambda }^{\prime }\left(s\right)\mu \left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(s^4\left(1-{\displaystyle \frac{{\mathrm{g}}_0}{s{c}_1^{2/3}}}\right){\displaystyle \frac{{\mathrm{p}}_0}{s^5}}-{\displaystyle \frac{2}{2^2}}{\displaystyle \frac{1}{ϵ{\lambda }_{0}s}}{\displaystyle \frac{4^2{s}^6}{{\lambda }_0^2{s}^2{\lambda }_0{s}^4}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left({\displaystyle \frac{{\mathrm{p}}_0}{s}}-{\displaystyle \frac{{\mathrm{g}}_0{\mathrm{p}}_0}{s^6{c}_1^{2/3}}}-{\displaystyle \frac{8}{ϵ{\lambda }_0^4}}{\displaystyle \frac{1}{s}}\right)\mathrm{d}s\hfill \\ & =& \left({\mathrm{p}}_0-{\displaystyle \frac{8}{ϵ{\lambda }_0^4}}\right)\underset{\nu \to \infty }{limsup}ln{\displaystyle \frac{v}{v_0}}+{\displaystyle \frac{{\mathrm{g}}_0{\mathrm{p}}_0}{5c_1^{2/3}}}\underset{\nu \to \infty }{limsup}{\displaystyle \frac{1}{s^5}}-{\displaystyle \frac{{\mathrm{g}}_0{\mathrm{p}}_0}{5c_1^{2/3}{v}_0^5}}\hfill \\ & =& \infty ,\hfill \end{array}$$

which is satisfied when

$${\mathrm{p}}_0>{\displaystyle \frac{8}{ϵ{\lambda }_0^4}}.$$

(53)

Similarly, if we consider the function ${\mu }_1\left(\nu \right)={\nu }^4,$ condition (24) becomes

$$\begin{array}{ccc}& & \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(M_1^{\beta -\gamma }{\mu }_1\left(s\right)\tilde{\mathrm{p}}\left(s\right){\left({\displaystyle \frac{\lambda \left(s\right)}{s}}\right)}^{3\beta /\widetilde{ϵ}}-{\displaystyle \frac{2^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}}{\displaystyle \frac{\mathfrak{a}\left(s\right){\left({\mu }_1^{\prime }\left(s\right)\right)}^{\gamma +1}}{{\left(ϵs^2{\mu }_1\left(s\right)\right)}^{\gamma }}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(s^4\left(1-{\displaystyle \frac{{\mathrm{g}}_0}{s{c}_1^{2/3}}}\right){\displaystyle \frac{{\mathrm{p}}_0}{s^5}}{\lambda }_0^{3/\widetilde{ϵ}}-{\displaystyle \frac{2}{2^2}}{\displaystyle \frac{1}{s}}{\displaystyle \frac{4^2{s}^6}{ϵs^2{s}^4}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left({\displaystyle \frac{{\mathrm{p}}_0{\lambda }_0^{3/\widetilde{ϵ}}}{s}}-{\displaystyle \frac{{\mathrm{g}}_0{\mathrm{p}}_0{\lambda }_0^{3/\widetilde{ϵ}}}{s^6{c}_1^{2/3}}}-{\displaystyle \frac{8}{ϵ}}{\displaystyle \frac{1}{s}}\right)\mathrm{d}s\hfill \\ & =& \left({\mathrm{p}}_0{\lambda }_0^{3/\widetilde{ϵ}}-{\displaystyle \frac{8}{ϵ}}\right)\underset{\nu \to \infty }{limsup}ln{\displaystyle \frac{\nu }{{\nu }_0}}+{\displaystyle \frac{{\mathrm{g}}_0{\mathrm{p}}_0{\lambda }_0^{3/\widetilde{ϵ}}}{5c_1^{2/3}}}\underset{\nu \to \infty }{limsup}{\displaystyle \frac{1}{v^5}}-{\displaystyle \frac{{\mathrm{g}}_0{\mathrm{p}}_0{\lambda }_0^{3/\widetilde{ϵ}}}{5c_1^{2/3}{v}_0^5}}\hfill \\ & =& \infty ,\hfill \end{array}$$

which is satisfied when

$${\mathrm{p}}_0>{\displaystyle \frac{8}{ϵ{\lambda }_0^{3/\widetilde{ϵ}}}}.$$

(54)

If we consider the function $\tilde{\mu }\left(\nu \right)=\nu ,$ then condition (39) can be expressed as

$$\begin{array}{ccc}& & \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(M^{\beta /\gamma -1}s{\int }_s^{\infty }{\left({\displaystyle \frac{1}{\mathfrak{a}\left(u\right)}}{\int }_u^{\infty }\tilde{\mathrm{p}}\left(v\right){\left({\displaystyle \frac{\lambda \left(v\right)}{v}}\right)}^{\beta /{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{{\left({\tilde{\mu }}_{+}^{\prime }\left(s\right)\right)}^2}{4\tilde{\mu }\left(s\right)}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(s{\int }_s^{\infty }{\left(u{\int }_u^{\infty }\left(1-{\displaystyle \frac{{\mathrm{g}}_0}{v{c}_1^{2/3}}}\right){\displaystyle \frac{{\mathrm{p}}_0}{{\nu }^5}}{\lambda }_0^{1/{ϵ}_1}\mathrm{d}v\right)}^{1/\gamma }\mathrm{d}u-{\displaystyle \frac{1}{4s}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(s{\int }_s^{\infty }\left(u{\int }_u^{\infty }\left({\displaystyle \frac{{\mathrm{p}}_0}{{\nu }^5}}-{\displaystyle \frac{{\mathrm{p}}_0{\mathrm{g}}_0}{v^6{c}_1^{2/3}}}\right){\lambda }_0^{1/{ϵ}_1}\mathrm{d}v\right)\mathrm{d}u-{\displaystyle \frac{1}{4s}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left(s{\int }_s^{\infty }\left({\displaystyle \frac{{\mathrm{p}}_0}{4u^3}}-{\displaystyle \frac{{\mathrm{p}}_0{\mathrm{g}}_0}{5u^4{c}_1^{2/3}}}\right){\lambda }_0^{1/{ϵ}_1}\mathrm{d}u-{\displaystyle \frac{1}{4s}}\right)\mathrm{d}s\hfill \\ & =& \underset{\nu \to \infty }{limsup}{\int }_{{\nu }_0}^{\nu }\left({\displaystyle \frac{{\mathrm{p}}_0{\lambda }_0^{1/{ϵ}_1}}{8s}}-{\displaystyle \frac{{\mathrm{p}}_0{\mathrm{g}}_0{\lambda }_0^{1/{ϵ}_1}}{15s^2{c}_1^{2/3}}}-{\displaystyle \frac{1}{4s}}\right)\mathrm{d}s\hfill \\ & =& \left({\displaystyle \frac{{\mathrm{p}}_0{\lambda }_0^{1/{ϵ}_1}}{8}}-{\displaystyle \frac{1}{4}}\right)\underset{v\to \infty }{limsup}ln{\displaystyle \frac{v}{v_0}}+{\displaystyle \frac{{\mathrm{p}}_0{\mathrm{g}}_0{\lambda }_0^{1/{ϵ}_1}}{15s^2{c}_1^{2/3}}}\underset{v\to \infty }{limsup}{\displaystyle \frac{1}{v^2}}-{\displaystyle \frac{{\mathrm{p}}_0{\mathrm{g}}_0{\lambda }_0^{1/{ϵ}_1}}{15v_0^2{c}_1^{2/3}}}=\infty ,\hfill \end{array}$$

which is satisfied when

$${\mathrm{p}}_0>{\displaystyle \frac{2}{{\lambda }_0^{1/{ϵ}_1}}}.$$

(55)

Now, according to Theorem 1, if conditions (53) and (55) are satisfied, then (52) exhibits oscillatory behavior. Similarly, Theorem 2 asserts that if conditions (54) and (55) are satisfied, then (54) exhibits oscillatory behavior.

Remark 1.

Consider the NDE given by

$${\left({\displaystyle \frac{1}{v}}\left({\left(x\left(\nu \right)+{\displaystyle \frac{1}{2v}}{x}^{1/3}\left(0.3\nu \right)\right)}^{‴}\right)\right)}^{\prime }+{\displaystyle \frac{{\mathrm{p}}_0}{{\nu }^5}}x\left(0.9\nu \right)=0,\nu \ge 1,$$

(56)

where ${\mathrm{p}}_0>0.$ Now, conditions (53)–(55) are satisfied when ${\mathrm{p}}_0>24.387,$ ${\mathrm{p}}_0>30.107,$ and ${\mathrm{p}}_0>8,$ respectively. Consequently, based on Theorem 1, we can deduce that Equation (56) exhibits oscillatory behavior if ${\mathrm{p}}_0>24.387$.

5. Conclusions

This work discussed the oscillatory features of solutions to a class of fourth-order Emden–Fowler DDEs with a sublinear neutral term. We were first able to obtain criteria that excluded all cases of the corresponding function of the positive solutions. Our approach, leveraging the Riccati technique in various ways, has allowed us to derive comprehensive oscillation criteria for the studied equation. As is clear from the remarks, the new findings are a development and extension of previous relevant findings.

Studying the oscillation of DDEs with a sublinear neutral term has many analytical problems. So, our study imposes some restrictions that limit the application of the results to a wider area of equations of this type. Therefore, we propose—as future work—obtaining an oscillation criterion that does not require constraints $\alpha \in \left(0,1\right]$ and ${lim}_{t\to \infty }\mathrm{g}\left(t\right)=0$ and is also considered the noncanonical case.