New Conditions for Testing the Asymptotic Behavior of Solutions of Odd-Order Neutral Differential Equations with Multiple Delays

Abstract

The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in order to determine the oscillation terms of the studied equation. In order to contribute to this subject, we create new and significant relationships and inequalities. We use these relationships to create conditions in which positive and N-Kneser solutions of the considered equation are excluded. To obtain these terms, we employ the comparison method and the Riccati technique. Furthermore, we use the relationships obtained to create new criteria, so expanding the existing literature on the field. Finally, we provide an example from the general case to demonstrate the results’ significance. The findings given in this work provide light on the behavior of odd-order neutral differential equation solutions with multiple delays.

1. Introduction

Differential equations (DEs) are a powerful and useful mathematical tool for understanding and analyzing many natural and technological processes. Differential equations are made up of functions and their derivatives. These equations are important tools in engineering, physics, computer science, natural sciences, economics, and other domains. It is used to research the motion and dynamics of things, to analyze biological growth and disease spread, and to advance technology in sectors like electrical engineering, mechanics, and others. Differential equations aid in the study of natural occurrences, as well as the development of scientific and technical models and forecasts. Differential equations are crucial in scientific research and practical applications, see [1,2,3,4].

Neutral differential equations are an important branch of differential equations because they contain time delays, functions, and their derivatives. These equations are crucial in understanding and analyzing occurrences and processes in various domains, from engineering and physics to medical and economic sciences. Using neutral differential equations, we may develop a scientific understanding in the field of dynamic analysis and control, as well as better technological and economic systems and processes (see [5,6,7]).

The oscillation theorem is a mathematical branch that studies the behavior of oscillating solutions in differential equations. This theorem helps to comprehend the pattern and form that the solution to the equation takes, such as regular vibration and oscillation. Oscillations are a phenomenon that occurs in many domains, including physics, engineering, and economics, and they play a significant part in integrated systems in electronic systems. Understanding oscillation theory allows us to collaborate, stabilize, and control it more efficiently (see [8,9,10]).

The stability of a time-delay force feedback teleoperation system based on a scattering matrix is crucial. It ensures reliable and robust communication between the operator and the remote robot, preventing delays or disruptions that could compromise control. By analyzing the system’s scattering matrix, key stability properties can be assessed, such as stability margins and robustness to external disturbances, enabling the development of more efficient and dependable teleoperation systems (see [11,12]).

The main objective of this investigation is to investigate the oscillatory characteristics exhibited by solutions of a non-linear DE of odd order, represented by the following expression

$${\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m{\mathrm{q}}_i\left(\mathrm{s}\right)x^{\alpha }\left({\upsilon }_i\left(\mathrm{s}\right)\right))=0,\mathrm{s}\ge {\mathrm{s}}_0,$$

(1)

where $\mathcal{U}\left(\mathrm{s}\right)=x\left(\mathrm{s}\right)+$p$\left(\mathrm{s}\right)x\left(\mathfrak{y}\left(\mathrm{s}\right)\right).$ In this paper, we make the following assumptions:

(H₁)

n is an odd natural number, while $\alpha$ represents the ratio of two positive odd integers;

(H₂)

$b,\mathfrak{y},{\upsilon }_i\in C^1\left(\left[{\mathrm{s}}_0,\infty \right),{\mathbb{R}}^{+}\right),$ $b^{\prime }\left(\mathrm{s}\right)\ge 0,$ and $0\le$ p$\left(\mathrm{s}\right)<$ p${}_0<\infty ;$

(H₃)

$\mathfrak{y}\left(\mathrm{s}\right)<\mathrm{s},$ ${\upsilon }_i\left(\mathrm{s}\right)\le \mathrm{s},$ ${\upsilon }_i^{\prime }\left(\mathrm{s}\right)>0,$ ${\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)}^{\prime }\ge {\upsilon }_0>0,$ ${\mathfrak{y}}^{\prime }\left(\mathrm{s}\right)\ge {\mathfrak{y}}_0>0,$ and${lim}_{\mathrm{s}\to \infty }\mathfrak{y}\left(\mathrm{s}\right)={lim}_{\mathrm{s}\to \infty }{\upsilon }_i\left(\mathrm{s}\right)=\infty ;$

(H₄)

$\mathfrak{y}\circ {\upsilon }_i={\upsilon }_i\circ \mathfrak{y},$ for $i=1,2,\dots ,m;$

(H₅)

${\mathrm{q}}_i\in C\left(\left[{\mathrm{s}}_0,\infty \right),\left[0,\infty \right)\right),$ for $i=1,2,\dots ,m.$

Moreover, we consider the canonical case, i.e.,

$${\int }_{{\mathrm{s}}_0}^{\infty }\frac{1}{b^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu =\infty .$$

(2)

A function $x\in C^{n-1}\left([{\mathrm{S}}_x,\infty )\right)$, ${\mathrm{S}}_x⩾{\mathrm{s}}_0,$ is considered a solution of (1) which has the property $b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\in C^1[{\mathrm{s}}_x,\infty )$, and it satisfies the Equation (1) for all $x\in [{\mathrm{S}}_x,\infty )$. We examine, exclusively, the solutions x from (1) that are present on a half-line $[{\mathrm{S}}_x,\infty )$ and fulfill the requirement:

$$sup\left\{\right|x\left(\mathrm{s}\right)|:\mathrm{s}⩾\mathrm{S}\}>0,\mathrm{for}\mathrm{all}\mathrm{S}\ge {\mathrm{S}}_x.$$

A solution is referred to as oscillatory if it does not eventually positive or negative. Otherwise, it is considered non-oscillatory. Equation (1) is considered oscillatory when all of its solutions exhibit oscillatory behavior.

Previous works on the subject of neutral DEs opened the path for advances in our understanding of delayed systems. Researchers have extensively researched neutral DEs of various orders, making substantial contributions to the theoretical foundations and practical applications of these equations.

In the canonical case, multiple studies have explored the oscillation behavior of even-order quasilinear neutral functional differential equations. Baculikova et al. [13], Dzurina [14], Graef et al. [15], and Bohner et al. [16] have specifically delved into this topic, shedding light on the properties of these equations. In the non-canonical case, several studies have concentrated on analyzing the oscillation criteria for even-order neutral delay differential equations. These studies include the works of Moaaz et al. [17], Almari et al. [18], another study by Bohner et al. [19], and Jadlovska [20]. The investigations conducted have provided valuable insights regarding the oscillatory behavior and dynamics exhibited by these equations.

Dzurina and Baculıkova [21] introduced a generalized version of Philos and Staikos lemmas, which aimed to examine the oscillatory behavior and asymptotic characteristics of a higher-order DDE

$${\left(b\left(\mathrm{s}\right){\left(x^{\prime }\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}+q\left(\mathrm{s}\right)x^{\alpha }\left(\upsilon \left(\mathrm{s}\right)\right)=0.$$

They employed a comparison theory to obtain their results.

Karpuz et al. [22] focused on higher-order neutral DEs of the form:

$${\left(x\left(\mathrm{s}\right)+\mathrm{p}\left(\mathrm{s}\right)x\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\left(n\right)}+q\left(\mathrm{s}\right)x\left(\upsilon \left(\mathrm{s}\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0,$$

The authors compared the asymptotic and oscillatory behaviors of all solutions of higher-order neutral DEs with those of first-order delay DEs.

Sun et al. [23] investigated the oscillatory behavior of neutral DEs of the form

$${\left(b\left(\mathrm{s}\right){\left(x\left(\mathrm{s}\right)+\mathrm{p}\left(\mathrm{s}\right)x\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\left(n-1\right)}\right)}^{\prime }+q\left(\mathrm{s}\right)K\left(x\left(\upsilon \left(\mathrm{s}\right)\right)\right)=0,$$

under two conditions, the canonical condition (2) and non-canonical condition

$${\int }_{{\mathrm{s}}_0}^{\infty }\frac{1}{b^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu <\infty ,$$

where $K\left(x\right)/x\ge k>0.$

Baculková et al. [24] examined the oscillation behavior and asymptotic properties of the equation

$${\left(b\left(\mathrm{s}\right){\left(x^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+q\left(\mathrm{s}\right)K\left(x\left(\upsilon \left(\mathrm{s}\right)\right)\right)=0,$$

when K is a non-decreasing function satisfying:

$$-K\left(-{\mathrm{s}}_1{\mathrm{s}}_2\right)\ge K\left({\mathrm{s}}_1{\mathrm{s}}_2\right)\ge K\left({\mathrm{s}}_1\right)K\left({\mathrm{s}}_2\right)\mathrm{for}{\mathrm{s}}_1{\mathrm{s}}_2>0.$$

Xing et al. [25] developed several oscillation criteria for a particular higher-order quasi-linear NDE

$${\left(b\left(\mathrm{s}\right){\left({\left(x\left(\mathrm{s}\right)+\mathrm{p}\left(\mathrm{s}\right)x\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+q\left(\mathrm{s}\right)x^{\alpha }\left(\upsilon \left(\mathrm{s}\right)\right)=0.$$

(3)

Moaaz et al. [26] focused on the oscillatory characteristics of the neutral DE (3) in the non-canonical case.

This work aims to look into the oscillatory behavior of solutions in odd-order neutral DEs with multiple delays. The study aims to establish new relationships and terms within this field. Additionally, a novel approach is employed to derive new criteria that guarantee the oscillatory nature of the solutions for the considered equation.

2. Preliminary Results

This section will introduce several essential lemmas that will be utilized to demonstrate the main results. To simplify our notation, let us denote the following:

$${\rho }_{+}^{\prime }\left(\mathrm{s}\right):=max\left\{0,{\rho }^{\prime }\left(\mathrm{s}\right)\right\},$$

$$\tilde{\upsilon }\left(\mathrm{s}\right):=min\left\{{\upsilon }_i\left(\mathrm{s}\right),i=1,2,\dots ,m\right\},\upsilon \left(\mathrm{s}\right):=max\left\{{\upsilon }_i\left(\mathrm{s}\right),i=1,2,\dots ,m\right\},$$

$${\pi }_0\left(\varsigma ,\varrho \right):={\int }_{\varrho }^{\varsigma }{b}^{-1/\alpha }\left(\nu \right)\mathrm{d}\nu ,{\pi }_i\left(\varsigma ,\varrho \right):={\int }_{\varrho }^{\varsigma }{\pi }_{i-1}\left(\varsigma ,\nu \right)\mathrm{d}\nu ,$$

$${\pi }_0\left(\mathrm{s}\right):={\int }_{{\mathrm{s}}_0}^{\mathrm{s}}{b}^{-1/\alpha }\left(\nu \right)\mathrm{d}\nu ,{\pi }_i\left(\mathrm{s}\right):={\int }_{{\mathrm{s}}_0}^{\mathrm{s}}{\pi }_{i-1}\left(\nu \right)\mathrm{d}\nu ,i=1,2,\dots ,n-2,$$

$${\mathrm{Q}}_0\left(\mathrm{s}\right):=\sum _{i=1}^m{\mathrm{q}}_i\left(\mathrm{s}\right){\left(1-\mathrm{p}\left({\upsilon }_i\left(\mathrm{s}\right)\right)\right)}^{\alpha },$$

$$\mathrm{Q}\left(\mathrm{s}\right):=\sum _{i=1}^m{\tilde{\mathrm{q}}}_i\left(\mathrm{s}\right),\widehat{\mathrm{Q}}\left(\mathrm{s}\right):=\sum _{i=1}^m{\widehat{\mathrm{q}}}_i\left(\mathrm{s}\right),$$

and

$${\tilde{\mathrm{q}}}_i\left(\mathrm{s}\right):=min\{{\mathrm{q}}_i\left(\mathrm{s}\right),{\mathrm{q}}_i\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\},{\widehat{\mathrm{q}}}_i\left(\mathrm{s}\right):=min\{{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right),{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\}.$$

Lemma 1

([27]). Assume that $x_1,x_2$ $\in \left[0,\infty \right)$. Then,

$${\left(x_1+x_2\right)}^{\alpha }\le \mu \left(x_1^{\alpha }+x_2^{\alpha }\right),$$

and

$$\mu =\left\{\begin{array}{cc}{2}^{\alpha -1}\hfill & \mathit{for}\alpha >1;\hfill \\ 1\hfill & \mathit{for}0<\alpha \le 1.\hfill \end{array}\right.$$

Lemma 2

([28]). For $A>0$ and B is any real number, the following inequality holds:

$$B\psi -A{\psi }^{\left(\alpha +1\right)/\alpha }\le \frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{B^{\alpha +1}}{A^{\alpha }}.$$

(4)

Lemma 3

([29]). Assume that $\psi \in C^n\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right).$ Additionally, ${\psi }^{\left(n\right)}\left(\mathrm{s}\right)$ has a fixed sign and is not equal to zero throughout $\left[{\mathrm{s}}_0,\infty \right).$ Furthermore, there exists ${\mathrm{s}}_1\ge {\mathrm{s}}_0$ satisfying the condition ${\psi }^{\left(n-1\right)}\left(\mathrm{s}\right){\psi }^{\left(n\right)}\left(\mathrm{s}\right)\le 0$ for all $\mathrm{s}\ge {\mathrm{s}}_1$. If ${lim}_{\mathrm{s}\to \infty }\psi \left(\mathrm{s}\right)\ne 0,$ then for any $\lambda \in \left(0,1\right),$ there exists ${\mathrm{s}}_{\mu }\ge {\mathrm{s}}_1$ satisfying the inequality:

$$\psi \left(\mathrm{s}\right)\ge \frac{\lambda }{\left(n-1\right)!}{\mathrm{s}}^{n-1}\left|{\psi }^{\left(n-1\right)}\left(\mathrm{s}\right)\right|for\mathrm{s}\ge {\mathrm{s}}_{\mu }.$$

Lemma 4

([30]). Consider $x\left(\mathrm{s}\right)$ as a positive solution of (1). Consequently, $b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }$ is a decreasing function. Furthermore, all derivatives ${\mathcal{U}}^{\left(i\right)}\left(\mathrm{s}\right),$ $1\le i\le n-1$ have constant signs. Additionally, $\mathcal{U}\left(\mathrm{s}\right)$ satisfies one of the following cases:

$$\begin{array}{cc}{\mathrm{C}}_1:\hfill & \mathcal{U}\left(\mathrm{s}\right)>0,{\mathcal{U}}^{\prime }\left(\mathrm{s}\right)>0,{\mathcal{U}}^{″}\left(\mathrm{s}\right)>0,{\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)>0,{\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }<0;\hfill \\ {\mathrm{C}}_2:\hfill & {\left(-1\right)}^k{\mathcal{U}}^{\left(k\right)}\left(\mathrm{s}\right)>0,fork=0,1,2,\dots ,n.\hfill \end{array}$$

Notation 1.

The symbols ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$ represent sets of solutions that are eventually positive and satisfy the corresponding function conditions (${\mathrm{C}}_1$) and (${\mathrm{C}}_2$), respectively.

Definition 1

([31]). We define a Kneser solution for Equation (1) as a solution x that satisfies the following condition, there exists a ${\mathrm{s}}_{*}\in \left[{\mathrm{s}}_0,\infty \right)$ such that $\mathcal{U}\left(\mathrm{s}\right){\mathcal{U}}^{\prime }\left(\mathrm{s}\right)<0$ for all $\mathrm{s}\in \left[{\mathrm{s}}_{*},\infty \right)$.

3. Criteria for Non-Existence of N-Kneser Solutions

In this section, we introduce specific criteria that ensure the non-existence of N-Kneser solutions that satisfy condition (${\mathrm{C}}_2$).

Theorem 1.

If $\zeta \in C\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right)$ fulfilling $\upsilon \left(\mathrm{s}\right)<\zeta \left(\mathrm{s}\right)$and ${\mathfrak{y}}^{-1}\left(\zeta \left(\mathrm{s}\right)\right)<\mathrm{s}$, such that the DE

$$G^{\prime }\left(\mathrm{s}\right)+\frac{1}{\mu }\frac{{\mathfrak{y}}_0}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}{\pi }_{n-2}^{\alpha }\left(\zeta \left(\mathrm{s}\right),\upsilon \left(\mathrm{s}\right)\right)\mathrm{Q}\left(\mathrm{s}\right)G\left({\mathfrak{y}}^{-1}\left(\zeta \left(\mathrm{s}\right)\right)\right)=0,$$

(5)

is oscillatory, then ${\mathrm{K}}_2=⌀$.

Proof. 

Let $x\in {\mathrm{K}}_2$, say $x\left(\mathrm{s}\right)>0$ and $x\left({\upsilon }_i\left(\mathrm{s}\right)\right)>0$ for $\mathrm{s}\ge {\mathrm{s}}_1\ge {\mathrm{s}}_0$. This implies that

$${\left(-1\right)}^k{\mathcal{U}}^{\left(k\right)}\left(\mathrm{s}\right)>0,\mathrm{for}k=0,1,2,\dots ,n.$$

(6)

From (1), we see that:

$$\begin{array}{ccc}\hfill 0& \ge \hfill & \hfill \frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}^{\prime }}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+{\mathrm{p}}_0^{\alpha }\sum _{i=1}^m{\mathrm{q}}_i\left(\mathfrak{y}\right)x^{\alpha }\left({\upsilon }_i\left(\mathfrak{y}\right)\right)\\ \hfill & \ge \hfill & \hfill \frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+{\mathrm{p}}_0^{\alpha }\sum _{i=1}^m{\mathrm{q}}_i\left(\mathfrak{y}\right)x^{\alpha }\left({\upsilon }_i\left(\mathfrak{y}\right)\right)\\ \hfill & =\hfill & \hfill \frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+{\mathrm{p}}_0^{\alpha }\sum _{i=1}^m{\mathrm{q}}_i\left(\mathfrak{y}\right)x^{\alpha }\left(\mathfrak{y}\left({\upsilon }_i\right)\right).\end{array}$$

(7)

When we combine (1) with (7), we obtain:

$$\begin{array}{cc}\hfill 0& \ge {\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m{\mathrm{q}}_i{x}^{\alpha }\left({\upsilon }_i\right)+{\mathrm{p}}_0^{\alpha }\sum _{i=1}^n{\mathrm{q}}_i\left(\mathfrak{y}\right)x^{\alpha }\left(\mathfrak{y}\left({\upsilon }_i\right)\right)\hfill \\ \hfill & ={\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m\left[{\mathrm{q}}_i{x}^{\alpha }\left({\upsilon }_i\right)+{\mathrm{p}}_0^{\alpha }{\mathrm{q}}_i\left(\mathfrak{y}\right)x^{\alpha }\left(\mathfrak{y}\left({\upsilon }_i\right)\right)\right]\hfill \\ \hfill & \ge {\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m{\tilde{\mathrm{q}}}_i\left[x^{\alpha }\left({\upsilon }_i\right)+{\mathrm{p}}_0^{\alpha }{x}^{\alpha }\left(\mathfrak{y}\left({\upsilon }_i\right)\right)\right].\hfill \end{array}$$

Using Lemma 1, we have:

$$\begin{array}{ccc}\hfill 0& \ge & {\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\frac{1}{\mu }\sum _{i=1}^m{\tilde{\mathrm{q}}}_i{\left(x\left({\upsilon }_i\right)+{\mathrm{p}}_{0}x\left(\mathfrak{y}\left({\upsilon }_i\right)\right)\right)}^{\alpha }.\hfill \end{array}$$

(8)

From definition of $\mathcal{U}$, we have:

$$\mathcal{U}\left({\upsilon }_i\right)=x\left({\upsilon }_i\right)+\mathrm{p}\left({\upsilon }_i\right)x\left(\mathfrak{y}\left({\upsilon }_i\right)\right)\le x\left({\upsilon }_i\right)+{\mathrm{p}}_{0}x\left(\mathfrak{y}\left({\upsilon }_i\right)\right).$$

From (8), we obtain:

$$0\ge {\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{1}{\mu }\sum _{i=1}^m{\tilde{\mathrm{q}}}_i{\mathcal{U}}^{\alpha }\left({\upsilon }_i\right).$$

Since $\mathcal{U}$ is decreasing, then:

$$\begin{array}{ccc}\hfill 0& \ge & {\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{1}{\mu }{\mathcal{U}}^{\alpha }\left(\upsilon \right)\sum _{i=1}^m{\tilde{\mathrm{q}}}_i\hfill \\ & =& {\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}{\left(b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{1}{\mu }\mathrm{Q}{\mathcal{U}}^{\alpha }\left(\upsilon \right).\hfill \end{array}$$

That is

$${\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{1}{\mu }\mathrm{Q}{\mathcal{U}}^{\alpha }\left(\upsilon \right)\le 0.$$

(9)

It follows from ${\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }\le 0$ that

$$-{\mathcal{U}}^{\left(n-2\right)}\left(\varrho \right)\ge {\int }_{\varrho }^{\varsigma }\frac{b^{1/\alpha }\left(\nu \right){\mathcal{U}}^{\left(n-1\right)}\left(\nu \right)}{b^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu \ge b^{1/\alpha }\left(\varsigma \right){\mathcal{U}}^{\left(n-1\right)}\left(\varsigma \right){\pi }_0\left(\varsigma ,\varrho \right).$$

(10)

By integrating (10) over $\left(\varrho ,\varsigma \right),$we arrive at

$$-{\mathcal{U}}^{\left(n-3\right)}\left(\varrho \right)\le b^{1/\alpha }\left(\varsigma \right){\mathcal{U}}^{\left(n-1\right)}\left(\varsigma \right){\pi }_1\left(\varsigma ,\varrho \right).$$

(11)

By applying the integration process $n-3$ times to (11) over $\left(\varrho ,\varsigma \right)$ and then using (6), we obtain

$$\mathcal{U}\left(\varrho \right)\ge b^{1/\alpha }\left(\varsigma \right){\mathcal{U}}^{\left(n-1\right)}\left(\varsigma \right){\pi }_{n-2}\left(\varsigma ,\varrho \right).$$

(12)

Therefore, we obtain:

$$\mathcal{U}\left(\upsilon \right)\ge b^{1/\alpha }\left(\zeta \right){\mathcal{U}}^{\left(n-1\right)}\left(\zeta \right){\pi }_{n-2}\left(\zeta ,\upsilon \right).$$

By virtue of (9), it follows that

$$\begin{array}{ccc}\hfill 0& \ge & {\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\frac{1}{\mu }\mathrm{Q}b\left(\zeta \right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\zeta \right)\right)}^{\alpha }{\pi }_{n-2}^{\alpha }\left(\zeta ,\upsilon \right).\hfill \end{array}$$

(13)

Now, set

$$G=b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }>0.$$

From ${\left(b{\left({\mathcal{U}}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }\le 0$, we have

$$G\le b\left(\mathfrak{y}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\right)\right)}^{\alpha }\left(1+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}\right),$$

or, equivalently,

$$b\left(\zeta \right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\zeta \right)\right)}^{\alpha }\ge \frac{{\mathfrak{y}}_0}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}G\left({\mathfrak{y}}^{-1}\left(\zeta \right)\right).$$

(14)

By applying (14) within (13), it becomes evident that G represents a positive solution of the differential inequality

$$G^{\prime }+\frac{1}{\mu }\frac{{\mathfrak{y}}_0}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}{\pi }_{n-2}^{\alpha }\left(\zeta ,\upsilon \right)\mathrm{Q}G\left({\mathfrak{y}}^{-1}\left(\zeta \right)\right)\le 0.$$

Considering [32] (Theorem 1), it can be inferred that (5) also possesses a positive solution, which contradicts the previous inequality. As a result, the proof is concluded. □

Corollary 1.

If $\zeta \in C\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right)$ fulfilling $\upsilon \left(\mathrm{s}\right)<\zeta \left(\mathrm{s}\right)$and ${\mathfrak{y}}^{-1}\left(\zeta \left(\mathrm{s}\right)\right)<\mathrm{s},$ such that

$$\underset{\mathrm{s}\to \infty }{liminf}{\int }_{{\mathfrak{y}}^{-1}\left(\zeta \left(\mathrm{s}\right)\right)}^{\mathrm{s}}{\pi }_{n-2}^{\alpha }\left(\zeta \left(\nu \right),\upsilon \left(\nu \right)\right)\mathrm{Q}\left(\nu \right)\mathrm{d}\nu >\frac{\mu \left({\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }\right)}{{\mathfrak{y}}_0\mathrm{e}},$$

(15)

then ${\mathrm{K}}_2=⌀$.

Theorem 2.

If $\delta \left(\mathrm{s}\right)\in C\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right)$ fulfilling $\delta \left(\mathrm{s}\right)<\mathrm{s}$ and $\upsilon \left(\mathrm{s}\right)<\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}\frac{{\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right),\upsilon \left(\mathrm{s}\right)\right)}{b\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right)}{\int }_{\delta \left(\mathrm{s}\right)}^{\mathrm{s}}\mathrm{Q}\left(\nu \right)\mathrm{d}\nu >\frac{\mu \left({\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }\right)}{{\mathfrak{y}}_0},$$

(16)

then ${\mathrm{K}}_2=⌀$.

Proof. 

Using the same procedure as in the proof of Theorem 1, we obtain

$$0\ge {\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}b\left(\mathfrak{y}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{1}{\mu }\mathrm{Q}\left(\mathrm{s}\right){\mathcal{U}}^{\alpha }\left(\upsilon \left(\mathrm{s}\right)\right).$$

Integrating the previous inequality over $\left(\delta \left(\mathrm{s}\right),\mathrm{s}\right)$ and utilizing the property that $\mathcal{U}$ is decreasing, we have

$$\begin{array}{ccc}& & b\left(\delta \left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\delta \left(\mathrm{s}\right)\right)\right)}^{\alpha }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}b\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\hfill \\ & \ge & b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}b\left(\mathfrak{y}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\alpha }+\frac{1}{\mu }{\mathcal{U}}^{\alpha }\left(\upsilon \left(\mathrm{s}\right)\right){\int }_{\delta \left(\mathrm{s}\right)}^{\mathrm{s}}\mathrm{Q}\left(\nu \right)\mathrm{d}\nu \hfill \\ & \ge & \frac{1}{\mu }{\mathcal{U}}^{\alpha }\left(\upsilon \left(\mathrm{s}\right)\right){\int }_{\delta \left(\mathrm{s}\right)}^{\mathrm{s}}\mathrm{Q}\left(\nu \right)\mathrm{d}\nu .\hfill \end{array}$$

Since $\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)<\mathfrak{y}\left(\mathrm{s}\right)$ and ${\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }\le 0$, we obtain

$$b\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\left(1+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}\right)\ge \frac{1}{\mu }{\mathcal{U}}^{\alpha }\left(\upsilon \left(\mathrm{s}\right)\right){\int }_{\delta \left(\mathrm{s}\right)}^{\mathrm{s}}\mathrm{Q}\left(\nu \right)\mathrm{d}\nu .$$

(17)

By utilizing (12) with $\varsigma =\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)$ and $\varrho =\upsilon \left(\mathrm{s}\right)$ into (17), we derive the following inequality

$$\begin{array}{ccc}& & b\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\left(1+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}\right)\hfill \\ & \ge & \frac{1}{\mu }{\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }{\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right),\upsilon \left(\mathrm{s}\right)\right){\int }_{\delta \left(\mathrm{s}\right)}^{\mathrm{s}}\mathrm{Q}\left(\nu \right)\mathrm{d}\nu .\hfill \end{array}$$

That is

$$\frac{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}\ge \frac{1}{\mu }\frac{{\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right),\upsilon \left(\mathrm{s}\right)\right)}{b\left(\mathfrak{y}\left(\delta \left(\mathrm{s}\right)\right)\right)}{\int }_{\delta \left(\mathrm{s}\right)}^{\mathrm{s}}\mathrm{Q}\left(\nu \right)\mathrm{d}\nu .$$

By considering the $limsup$ of both sides of the aforementioned inequality, it becomes apparent that it contradicts (16). As a result, we can conclude the proof. □

Theorem 3.

Assume that ${\upsilon }_i\left(\mathfrak{y}\left(\mathrm{s}\right)\right)<\mathrm{s},$ $i=1,2,\dots ,n$holds. If the DE

$${\mathsf{\Psi }}^{\prime }\left(\mathrm{s}\right)+\widehat{\mathrm{Q}}\left(\mathrm{s}\right){\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right),\mathrm{s}\right)\left(\frac{{\upsilon }_0{\mathfrak{y}}_0}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}\right)\mathsf{\Psi }\left(\upsilon \left(\mathrm{s}\right)\right)=0,$$

(18)

is oscillatory, then ${\mathrm{K}}_2=⌀$.

Proof. 

Suppose $u\in {\mathrm{K}}_2,$ with $x\left(\mathrm{s}\right)>0,x\left(\mathfrak{y}\left(\mathrm{s}\right)\right)>0$and $x\left({\upsilon }_i\left(\mathrm{s}\right)\right)>0$ for $\mathrm{s}\ge {\mathrm{s}}_1\ge {\mathrm{s}}_0$. Consequently, it follows that

$${\left(-1\right)}^k{\mathcal{U}}^{\left(k\right)}\left(\mathrm{s}\right)>0,\mathrm{for}k=0,1,2,\dots ,n.$$

Utilising (1), we can be observed that

$$\begin{array}{ccc}\hfill 0& \ge & \frac{1}{{\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)}^{\prime }}{\left(b\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)x^{\alpha }\left(\mathrm{s}\right)\hfill \\ & \ge & \frac{1}{{\upsilon }_0}{\left(b\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)x^{\alpha }\left(\mathrm{s}\right).\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill 0& \ge & \frac{{\mathrm{p}}_0^{\alpha }}{{\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\prime }}{\left(b\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +{\mathrm{p}}_0^{\alpha }\sum _{i=1}^m{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)x^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\hfill \\ & \ge & \frac{{\mathrm{p}}_0^{\alpha }}{{\upsilon }_0{\mathfrak{y}}_0}{\left(b\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +{\mathrm{p}}_0^{\alpha }\sum _{i=1}^m{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)x^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right)\right).\hfill \end{array}$$

Combining the above inequalities yields that

$$\begin{array}{ccc}\hfill 0& \ge & \frac{1}{{\upsilon }_0}{\left(b\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\frac{{\mathrm{p}}_0^{\alpha }}{{\upsilon }_0{\mathfrak{y}}_0}{\left(b\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\sum _{i=1}^m\left[{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)x^{\alpha }\left(\mathrm{s}\right)+{\mathrm{p}}_0^{\alpha }{\mathrm{q}}_i\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)x^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right]\hfill \\ & \ge & \frac{1}{{\upsilon }_0}{\left(b\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\frac{{\mathrm{p}}_0^{\alpha }}{{\upsilon }_0{\mathfrak{y}}_0}{\left(b\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\sum _{i=1}^m{\widehat{\mathrm{q}}}_i\left(\mathrm{s}\right)\left[x^{\alpha }\left(\mathrm{s}\right)+{\mathrm{p}}_0^{\alpha }{x}^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right].\hfill \end{array}$$

That is

$$\begin{array}{ccc}\hfill 0& \ge & \left[\left(\frac{1}{{\upsilon }_0}b\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\right)\right.\hfill \\ & & {\left.+\frac{{\mathrm{p}}_0^{\alpha }}{{\upsilon }_0{\mathfrak{y}}_0}b\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }\right]}^{\prime }+\widehat{\mathrm{Q}}\left(\mathrm{s}\right){\mathcal{U}}^{\alpha }\left(\mathrm{s}\right).\hfill \end{array}$$

(19)

Now, we set

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(\mathrm{s}\right)& =& \frac{1}{{\upsilon }_0}b\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\hfill \\ & & +\frac{{\mathrm{p}}_0^{\alpha }}{{\upsilon }_0{\mathfrak{y}}_0}b\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }.\hfill \end{array}$$

(20)

Since ${\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }\le 0$, it is clear that

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(\mathrm{s}\right)& \le & \frac{b\left(\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }_i^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }}{{\upsilon }_0}\left(1+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}\right)\hfill \\ & \le & \frac{b\left(\left({\upsilon }^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left({\upsilon }^{-1}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)\right)}^{\alpha }}{{\upsilon }_0}\left(1+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}\right).\hfill \end{array}$$

(21)

By using (12) with $\varsigma =\mathfrak{y}\left(\mathrm{s}\right)$ and $\varrho =\mathrm{s}$ and (21), we have $\mathcal{U}\left(\varrho \right)\ge b^{1/\alpha }\left(\varsigma \right){\mathcal{U}}^{\left(n-1\right)}\left(\varsigma \right){\pi }_{n-2}\left(\varsigma ,\varrho \right)$, and

$${\mathcal{U}}^{\alpha }\left(\mathrm{s}\right)\ge b\left(\mathfrak{y}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\alpha }{\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right),\mathrm{s}\right)\ge \mathsf{\Psi }\left(\upsilon \left(\mathrm{s}\right)\right){\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right),\mathrm{s}\right)\left(\frac{{\upsilon }_0{\mathfrak{y}}_0}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}\right).$$

Using the preceding inequality and the definition of $\mathsf{\Psi }$ in (19), we obtain

$${\mathsf{\Psi }}^{\prime }\left(\mathrm{s}\right)+\widehat{\mathrm{Q}}\left(\mathrm{s}\right){\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\mathrm{s}\right),\mathrm{s}\right)\left(\frac{{\upsilon }_0{\mathfrak{y}}_0}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}\right)\mathsf{\Psi }\left(\upsilon \left(\mathrm{s}\right)\right)\le 0.$$

Based on [32] (Theorem 1), it can be inferred that Equation (18) also has a positive solution, which contradicts the previous inequality. As a result, the proof is concluded. □

Corollary 2.

Suppose that ${\upsilon }_i\left(\mathfrak{y}\left(\mathrm{s}\right)\right)<\mathrm{s},$ $i=1,2,\dots ,n$holds. If

$$\underset{\mathrm{s}\to \infty }{liminf}{\int }_{\upsilon \left(\mathrm{s}\right)}^{\mathrm{s}}{\pi }_{n-2}^{\alpha }\left(\mathfrak{y}\left(\nu \right),\nu \right)\widehat{\mathrm{Q}}\left(\nu \right)\mathrm{d}\nu >\frac{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}{{\upsilon }_0{\mathfrak{y}}_0\mathrm{e}},$$

(22)

then ${\mathrm{K}}_2=⌀.$

4. Non-Existence of Solutions from the Class ${\mathrm{C}}_{\mathbf{1}}$

The main objective of this section is to analyze the asymptotic and monotonic properties displayed by the positive solutions of the examined equation. Additionally, we present limitations to guarantee that none of the positive solutions meet the criteria stated as condition (${\mathrm{C}}_1$).

Lemma 5.

Assume that $x\in {\mathrm{K}}_1$. Then, eventually

$$x\left(\mathrm{s}\right)>\left(1-\mathrm{p}\left(\mathrm{s}\right)\right)\mathcal{U}\left(\mathrm{s}\right),$$

and (1) becomes

$${\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+{\mathrm{Q}}_0\left(\mathrm{s}\right){\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\le 0,$$

(23)

eventually.

Proof. 

Since

$$\mathcal{U}\left(\mathrm{s}\right)=x\left(\mathrm{s}\right)+\mathrm{p}\left(\mathrm{s}\right)x\left(\mathfrak{y}\left(\mathrm{s}\right)\right),$$

then $\mathcal{U}\left(\mathrm{s}\right)\ge x\left(\mathrm{s}\right)$ and

$$x\left(\mathrm{s}\right)=\mathcal{U}\left(\mathrm{s}\right)-\mathrm{p}\left(\mathrm{s}\right)x\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\ge \mathcal{U}\left(\mathrm{s}\right)-\mathrm{p}\left(\mathrm{s}\right)\mathcal{U}\left(\mathfrak{y}\left(\mathrm{s}\right)\right).$$

Since $\mathcal{U}\left(\mathrm{s}\right)$ is increasing, then

$$x\left(\mathrm{s}\right)\ge \left(1-\mathrm{p}\left(\mathrm{s}\right)\right)\mathcal{U}\left(\mathrm{s}\right).$$

(24)

From (1), we have

$$\begin{array}{ccc}\hfill 0& =& {\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m{\mathrm{q}}_i\left(\mathrm{s}\right)x^{\alpha }\left({\upsilon }_i\left(\mathrm{s}\right)\right)\hfill \\ & \ge & {\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^n{\mathrm{q}}_i\left(\mathrm{s}\right){\left(1-\mathrm{p}\left({\upsilon }_i\left(\mathrm{s}\right)\right)\right)}^{\alpha }{\mathcal{U}}^{\alpha }\left({\upsilon }_i\left(\mathrm{s}\right)\right)\hfill \\ & \ge & {\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+{\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\sum _{i=1}^n{\mathrm{q}}_i\left(\mathrm{s}\right){\left(1-\mathrm{p}\left({\upsilon }_i\left(\mathrm{s}\right)\right)\right)}^{\alpha }\hfill \\ & \ge & {\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+{\mathrm{Q}}_0\left(\mathrm{s}\right){\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right).\hfill \end{array}$$

Therefore, the proof is concluded. □

Theorem 4.

Assume that there is a $\rho \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right)$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\left(\rho \left(\nu \right){\mathrm{Q}}_0\left(\nu \right)-\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{b\left(\tilde{\upsilon }\left(\nu \right)\right){\left({\rho }^{\prime }\left(\nu \right)\right)}^{\alpha +1}}{{\left(\lambda \rho \left(\nu \right){\tilde{\upsilon }}^{n-2}\left(\nu \right){\tilde{\upsilon }}^{\prime }\left(\nu \right)\right)}^{\alpha }}\right)\mathrm{d}\nu =\infty .$$

(25)

Then ${\mathrm{K}}_1=⌀.$

Proof. 

Suppose the opposite that $x\in {\mathrm{K}}_1$. We introduce w defined as

$$w\left(\mathrm{s}\right)=\rho \left(\mathrm{s}\right)\frac{b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }}{{\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}.$$

(26)

Then $w\left(\mathrm{s}\right)>0$. Differentiating (26), we have

$$\begin{array}{ccc}\hfill w^{\prime }\left(\mathrm{s}\right)& =& {\rho }^{\prime }\left(\mathrm{s}\right)\frac{b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }}{{\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}+\rho \left(\mathrm{s}\right)\frac{{\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }}{{\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}\hfill \\ & & -\alpha {\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right)\rho \left(\mathrm{s}\right)\frac{b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }{\mathcal{U}}^{\prime }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}{{\mathcal{U}}^{\alpha +1}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}\hfill \\ & \le & \frac{{\rho }^{\prime }\left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}w\left(\mathrm{s}\right)-\rho \left(\mathrm{s}\right){\mathrm{Q}}_0\left(\mathrm{s}\right)\frac{{\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}{{\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}-\alpha {\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right)w\left(\mathrm{s}\right)\frac{{\mathcal{U}}^{\prime }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}{\mathcal{U}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}.\hfill \end{array}$$

(27)

Using Lemma 3 with $\psi \left(\mathrm{s}\right)={\mathcal{U}}^{\prime }\left(\mathrm{s}\right)$, we see that

$${\mathcal{U}}^{\prime }\left(\mathrm{s}\right)\ge \frac{\lambda }{\left(n-2\right)!}{\mathrm{s}}^{n-2}{\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right),\mathrm{for}\mathrm{all}\lambda \in \left(0,1\right),$$

and

$${\mathcal{U}}^{\prime }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\ge \frac{\lambda }{\left(n-2\right)!}{\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right){\mathcal{U}}^{\left(n-1\right)}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right).$$

Putting the last inequality into (27), we obtain

$$w^{\prime }\left(\mathrm{s}\right)\le \frac{{\rho }^{\prime }\left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}w\left(\mathrm{s}\right)-\rho \left(\mathrm{s}\right){\mathrm{Q}}_0\left(\mathrm{s}\right)-\frac{\alpha \lambda }{\left(n-2\right)!}\frac{{\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right){\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right)}{b^{1/\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}\frac{b^{1/\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right){\mathcal{U}}^{\left(n-1\right)}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)w\left(\mathrm{s}\right)}{\mathcal{U}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}.$$

(28)

Since $b^{1/\alpha }\left(\mathrm{s}\right){\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)$ is decreasing, then

$$b^{1/\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right){\mathcal{U}}^{\left(n-1\right)}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\ge b^{1/\alpha }\left(\mathrm{s}\right){\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right).$$

Therefore, (28) can be expressed as

$$\begin{array}{ccc}\hfill w^{\prime }\left(\mathrm{s}\right)& \le & \frac{{\rho }^{\prime }\left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}w\left(\mathrm{s}\right)-\rho \left(\mathrm{s}\right){\mathrm{Q}}_0\left(\mathrm{s}\right)-\frac{\alpha \lambda }{\left(n-2\right)!}\frac{{\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right){\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right)}{b^{1/\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}\frac{b^{1/\alpha }\left(\mathrm{s}\right){\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)w\left(\mathrm{s}\right)}{\mathcal{U}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}\hfill \\ & =& \frac{{\rho }^{\prime }\left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}w\left(\mathrm{s}\right)-\rho \left(\mathrm{s}\right){\mathrm{Q}}_0\left(\mathrm{s}\right)-\frac{\alpha \lambda }{\left(n-2\right)!}\frac{{\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right){\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right)}{{\left(\rho \left(\mathrm{s}\right)b\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\right)}^{1/\alpha }}{w}^{\left(\alpha +1\right)/\alpha }\left(\mathrm{s}\right).\hfill \end{array}$$

(29)

Using Lemma 2 where $B={\rho }^{\prime }\left(\mathrm{s}\right)/\rho \left(\mathrm{s}\right),$ $B=\alpha \lambda {\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right){\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right)/{\left(\rho \left(\mathrm{s}\right)b\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\right)}^{1/\alpha },$ and $\psi =w,$ we obtain

$$\begin{array}{ccc}& & \frac{{\rho }^{\prime }\left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}w\left(\mathrm{s}\right)-\frac{\alpha \lambda }{\left(n-2\right)!}\frac{{\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right){\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right)}{{\left(\rho \left(\mathrm{s}\right)b\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\right)}^{1/\alpha }}{w}^{\left(\alpha +1\right)/\alpha }\left(\mathrm{s}\right)\hfill \\ & \le & \frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\left({\rho }^{\prime }\left(\mathrm{s}\right)\right)}^{\alpha +1}b\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}{{\left(\lambda \rho \left(\mathrm{s}\right){\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right){\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right)\right)}^{\alpha }}.\hfill \end{array}$$

Substituting the previous inequality into (29), we obtain

$$w^{\prime }\left(\mathrm{s}\right)\le -\rho \left(\mathrm{s}\right){\mathrm{Q}}_0\left(\mathrm{s}\right)+\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\left({\rho }^{\prime }\left(\mathrm{s}\right)\right)}^{\alpha +1}b\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}{{\left(\lambda \rho \left(\mathrm{s}\right){\tilde{\upsilon }}^{\prime }\left(\mathrm{s}\right){\tilde{\upsilon }}^{n-2}\left(\mathrm{s}\right)\right)}^{\alpha }}.$$

(30)

Integrating (30) from ${\mathrm{s}}_1$ to $\mathrm{s}$, we have

$${\int }_{{\mathrm{s}}_1}^{\mathrm{s}}\left(\rho \left(\nu \right){\mathrm{Q}}_0\left(\nu \right)-\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{b\left(\tilde{\upsilon }\left(\nu \right)\right){\left({\rho }^{\prime }\left(\nu \right)\right)}^{\alpha +1}}{{\left(\lambda \rho \left(\nu \right){\tilde{\upsilon }}^{n-2}\left(\nu \right){\tilde{\upsilon }}^{\prime }\left(\nu \right)\right)}^{\alpha }}\right)\mathrm{d}\nu \le w\left({\mathrm{s}}_1\right),$$

which contradicts (25). □

Theorem 5.

If

$$\underset{\mathrm{s}\to \infty }{liminf}{\int }_{\tilde{\upsilon }\left(\mathrm{s}\right)}^{\mathrm{s}}{\mathrm{Q}}_0\left(\nu \right){\left({\tilde{\upsilon }}^{n-1}\left(\nu \right)\right)}^{\alpha }\mathrm{d}\nu >\frac{{\left(\left(n-1\right)!\right)}^{\alpha }}{\mathrm{e}},$$

(31)

then ${\mathrm{K}}_1=⌀.$

Proof. 

Let us assume the opposite, that $x\in {\mathrm{K}}_1.$ It is clear from the use of the Lemma 3 that

$$\mathcal{U}\left(\mathrm{s}\right)\ge \frac{\lambda }{\left(n-1\right)!}{\mathrm{s}}^{n-1}{\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right),\mathrm{for}\mathrm{all}\lambda \in \left(0,1\right).$$

(32)

Substituting from (32) into (23), we conclude that

$${\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+{\left(\frac{\lambda }{\left(n-1\right)!}{\tilde{\upsilon }}^{n-1}\left(\mathrm{s}\right)\right)}^{\alpha }{\mathrm{Q}}_0\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\right)}^{\alpha }\le 0.$$

Let us define $\phi \left(\mathrm{s}\right)=b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }>0$, It follows that $\phi$ is a positive solution to the inequality

$${\phi }^{\prime }\left(\mathrm{s}\right)+\frac{{\lambda }^{\alpha }}{{\left(\left(n-1\right)!\right)}^{\alpha }}{\left({\tilde{\upsilon }}^{n-1}\left(\mathrm{s}\right)\right)}^{\alpha }{\mathrm{Q}}_0\left(\mathrm{s}\right)\phi \left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\le 0.$$

However, from Theorem 2.1.1 in [33], condition (31) confirms the oscillatory nature of all solutions to Equation (32). This contradicts the previous inequality. □

Theorem 6.

If the DE

$${\theta }^{\prime }\left(\mathrm{s}\right)+\frac{{\mathfrak{y}}_0{\lambda }^{\alpha }}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}\frac{{\left({\tilde{\upsilon }}^{n-1}\left(\mathrm{s}\right)\right)}^{\alpha }\mathrm{Q}\left(\mathrm{s}\right)}{{\left(\left(n-1\right)!\right)}^{\alpha }b\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}\theta \left({\mathfrak{y}}^{-1}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\right)=0$$

(33)

is oscillatory, then ${\mathrm{K}}_1=⌀$.

Proof. 

Assume that $x\in {\mathrm{K}}_1$. Similar to the proof of Theorem 1, we observe that (9) can be expressed as

$$0\ge {\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}b\left(\mathfrak{y}\left(\mathrm{s}\right)\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{1}{\mu }\mathrm{Q}\left(\mathrm{s}\right){\mathcal{U}}^{\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right).$$

(34)

Applying Lemma 3, we obtain

$$\mathcal{U}\left(\mathrm{s}\right)\ge \frac{\lambda }{\left(n-1\right)!b^{1/\alpha }\left(\mathrm{s}\right)}{\mathrm{s}}^{n-1}{b}^{1/\alpha }\left(\mathrm{s}\right){\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right).$$

(35)

Therefore, by setting $w\left(\mathrm{s}\right)=b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }$in (34) and employing (35), it becomes evident that w is a positive solution of the equation

$${\left(w\left(\mathrm{s}\right)+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}w\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\right)}^{\prime }+{\left(\frac{\lambda }{\left(n-1\right)!b^{1/\alpha }\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}{\tilde{\upsilon }}^{n-1}\left(\mathrm{s}\right)\right)}^{\alpha }\mathrm{Q}\left(\mathrm{s}\right)w\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)=0.$$

(36)

Since $w\left(\mathrm{s}\right)=b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }$is decreasing and it satisfies (36). Let us denote

$$\theta \left(\mathrm{s}\right)=w\left(\mathrm{s}\right)+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}w\left(\mathfrak{y}\left(\mathrm{s}\right)\right).$$

It follows from $\mathfrak{y}\left(\mathrm{s}\right)<\mathrm{s},$

$$\theta \left(\mathrm{s}\right)\le w\left(\mathfrak{y}\left(\mathrm{s}\right)\right)\left(1+\frac{{\mathrm{p}}_0^{\alpha }}{{\mathfrak{y}}_0}\right).$$

By replacing these expressions into (36), we discover that $\theta$ is a positive solution of

$${\theta }^{\prime }\left(\mathrm{s}\right)+\frac{{\mathfrak{y}}_0{\lambda }^{\alpha }}{{\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }}\frac{{\left({\tilde{\upsilon }}^{n-1}\left(\mathrm{s}\right)\right)}^{\alpha }\mathrm{Q}\left(\mathrm{s}\right)}{{\left(\left(n-1\right)!\right)}^{\alpha }b\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}\theta \left({\mathfrak{y}}^{-1}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)\right)\le 0.$$

According to [32] (Theorem 1), it implies that (33) has a positive solution as well, resulting in a contradiction with (33). □

Corollary 3.

If

$$\underset{\mathrm{s}\to \infty }{liminf}{\int }_{{\mathfrak{y}}^{-1}\left(\tilde{\upsilon }\left(\mathrm{s}\right)\right)}^{\mathrm{s}}\frac{{\left({\tilde{\upsilon }}^{n-1}\left(\nu \right)\right)}^{\alpha }\mathrm{Q}\left(\nu \right)}{b\left(\tilde{\upsilon }\left(\nu \right)\right)}\mathrm{d}\nu >\frac{\left({\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }\right){\left(\left(n-1\right)!\right)}^{\alpha }}{{\lambda }^{\alpha }{\mathfrak{y}}_0\mathrm{e}},$$

(37)

then ${\mathrm{K}}_1=⌀.$

5. Oscillation Theorem

This section builds on the preceding section’s findings to establish new criteria for investigating the oscillatory behavior of all solutions in (1). By merging the established conditions that eliminate positive solutions for both cases (C${}_1$) and (C${}_2$), we can formulate criteria outlined in the next theorem to determine the oscillation characteristics of the studied equation.

Theorem 7.

One of conditions (15), (16), or (22), together with one of conditions (25), (31), or (37), ensure that all solutions of Equation (1) oscillate.

Proof. 

Let us assume the opposite scenario, that x is a solution to Equation (1). that eventually becomes positive. Based on Lemma 4, we can deduce that there are two potential situations for the behavior of z and its derivatives. Applying Corollary 1 and Theorem 4, it can be determined that conditions (15) and (25) guarantee that there are no solutions to (1) satisfy $\left({\mathrm{C}}_1\right)$ and $\left({\mathrm{C}}_2\right)$, respectively. The same approach is used for the remaining conditions mentioned in the theorem. As a result, we can conclude that the proof is finished. □

6. Application

In this section, we will utilize the derived findings to address a specific case of the (1). Let us examine the non-linear differential equation (NDE) given by:

$${\left({\left({\left(x\left(\mathrm{s}\right)+{\mathrm{p}}_{0}x\left({\mathfrak{y}}_0\mathrm{s}\right)\right)}^{″}\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m\frac{{\mathrm{q}}_i}{{\mathrm{s}}^{2\alpha +1}}{x}^{\alpha }\left({\upsilon }_i\mathrm{s}\right)=0,\mathrm{s}\ge 1.$$

(38)

From (38) we have $n=3,$ $b\left(\mathrm{s}\right)=1,$ p$\left(\mathrm{s}\right)=$ p${}_0,\mathfrak{y}\left(\mathrm{s}\right)={\mathfrak{y}}_0\mathrm{s},$ $\upsilon \left(\mathrm{s}\right)={\upsilon }_0\mathrm{s}=max\left\{{\upsilon }_i\left(\mathrm{s}\right),i=1,2,\dots ,m\right\},$ $\tilde{\upsilon }\left(\mathrm{s}\right)={\tilde{\upsilon }}_0\mathrm{s}=min\left\{{\upsilon }_i\mathrm{s},i=1,2,\dots ,m\right\}$, ${\mathrm{q}}_i\left(\mathrm{s}\right)={\mathrm{q}}_i/{\mathrm{s}}^{2\alpha +1},$

$${\pi }_0\left(\mathrm{s}\right)=\mathrm{s},{\pi }_1\left(\mathrm{s}\right)=\frac{{\mathrm{s}}^2}{2},$$

$${\pi }_0\left(\mathfrak{y}\left(\mathrm{s}\right),\upsilon \left(\mathrm{s}\right)\right)=\left({\mathfrak{y}}_0-{\upsilon }_0\right)\mathrm{s},{\pi }_1\left(\mathfrak{y}\left(\mathrm{s}\right),\upsilon \left(\mathrm{s}\right)\right)={\left({\mathfrak{y}}_0-{\upsilon }_0\right)}^3\frac{{\mathrm{s}}^2}{2},$$

and

$${\mathrm{Q}}_0\left(\mathrm{s}\right)=\frac{{\left(1-{\mathrm{p}}_0\right)}^{\alpha }}{{\mathrm{s}}^{3\alpha +1}}\sum _{i=1}^m{\mathrm{q}}_i.$$

The condition described in (15) is fulfilled when:

$${\left({\zeta }_0-{\upsilon }_0\right)}^{3\alpha }{2}^{-\alpha }\left(\sum _{i=1}^n{\mathrm{q}}_i\right)ln\frac{{\mathfrak{y}}_0}{{\zeta }_0}>\frac{\mu \left({\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }\right)}{{\mathfrak{y}}_0\mathrm{e}}.$$

(39)

The condition stated in (16) is satisfied when:

$${\left({\mathfrak{y}}_0{\delta }_0-{\upsilon }_0\right)}^{3\alpha }{2}^{-\alpha }\left(1-{\delta }_0^{2\alpha }\right)\sum _{i=1}^n{\mathrm{q}}_i>\frac{2\alpha \mu {\delta }_0^{2\alpha }\left({\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }\right)}{{\mathfrak{y}}_0}.$$

(40)

The condition presented in (22) is met when

$${\left(1-{\mathfrak{y}}_0\right)}^{3\alpha }\sum _{i=1}^m{q}_{i}ln\frac{1}{{\upsilon }_0}>\frac{2^{\alpha }\left({\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }\right)}{{\upsilon }_0{\mathfrak{y}}_0\mathrm{e}}.$$

(41)

The condition described in (25) is satisfied when

$${\left(1-{\mathrm{p}}_0\right)}^{\alpha }\sum _{i=1}^m{\mathrm{q}}_i>{\left(\frac{2\alpha }{\alpha +1}\right)}^{\alpha +1}{\left(\frac{1}{\lambda {\tilde{\upsilon }}_0^2}\right)}^{\alpha },\rho \left(\mathrm{s}\right)={\mathrm{s}}^{2\alpha }.$$

(42)

The condition given in (31) is met when

$${\tilde{\upsilon }}^{2\alpha }{\left(1-{\mathrm{p}}_0\right)}^{\alpha }\left(\sum _{i=1}^m{\mathrm{q}}_i\right)ln\frac{1}{{\tilde{\upsilon }}_0}>\frac{2^{\alpha }}{\mathrm{e}}.$$

(43)

The condition presented in (31) is satisfied when

$${\tilde{\upsilon }}^{2\alpha }\left(\sum _{i=1}^m{\mathrm{q}}_i\right)ln\frac{{\mathfrak{y}}_0}{{\tilde{\upsilon }}_0}>\frac{2^{\alpha }\left({\mathfrak{y}}_0+{\mathrm{p}}_0^{\alpha }\right)}{{\lambda }^{\alpha }{\mathfrak{y}}_0\mathrm{e}}.$$

(44)

Now, by applying conditions (39)–(44), we observe that the Theorem 7 demonstrates oscillatory behavior. This can be verified by assigning specific values to Equation (38).

Remark 1.

If we substitute $p=0,$ $\alpha =1,$ and $m=1$ into (38), we obtain a third-order Euler-type equation in the given form:

$$x^{‴}\left(\mathrm{s}\right)+\frac{{\mathrm{q}}_1}{{\mathrm{s}}^3}x\left({\upsilon }_1\mathrm{s}\right)=0,\mathrm{s}\ge 1.$$

7. Conclusions

This research aims to investigate the oscillatory and asymptotic characteristics of solutions to odd-order NDEs with multiple delays. By comprehending the relationship between the solution, its derivatives of various orders, and its corresponding function, we have significantly advanced the research of oscillation conditions in neutral differential equations. We derived criteria that eliminate N-Kneser solutions and positive solutions of the studied equation by deducing novel relationships and inequalities. These inferred relationships and variances also enable the development of additional criteria that contribute to the expansion of the literature and provide a better understanding of the behavior of NDE solutions containing multiple delays. To demonstrate the importance of our findings, We provided a general example. The obtained results provide useful insights into the behavior of solutions in odd-order NDEs and emphasize the necessity for future study to investigate the state when the equation:

$${\left(b\left(\mathrm{s}\right){\left({\mathcal{U}}^{\left(n-1\right)}\left(\mathrm{s}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{i=1}^m{\mathrm{q}}_i\left(\mathrm{s}\right)x^{\beta }\left({\upsilon }_i\left(\mathrm{s}\right)\right))=0,$$

Moreover, one of the interesting open research points is obtaining oscillation criteria for all solutions of Equation (1) without the need for constraint (H${}_4$).