Investigating Oscillatory Behavior in Third-Order Neutral Differential Equations with Canonical Operators

Abstract

In this study, we aim to set new criteria regarding the asymptotic behavior of the neutral differential equation of the third order. These criteria are designed to ensure that this equation is oscillatory using comparisons with first-order differential equations and Riccati substitution. The results we obtained improve some of the results found in the literature. Some examples are provided to illustrate the applicability of our results and compare them with results found in some previous studies.

1. Introduction

In this study, we are interested in studying the oscillatory behavior of solutions of differential equations (DEs),

$${\left(r\left(\mathrm{s}\right){\left({\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)\right)}^{″}\right)}^{\lambda }\right)}^{\prime }+q\left(\mathrm{s}\right){\varkappa }^{\lambda }\left(\alpha \left(\mathrm{s}\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0,$$

(1)

where the corresponding function to the solution $\varkappa$ is defined by

$$y\left(\mathrm{s}\right):=\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right).$$

(2)

Now, we present some hypotheses to use in this study:

(I) 

$\lambda$ is the ratio of positive odd integers, $r\in C^1\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right),r^{\prime }\left(\mathrm{s}\right)\ge 0,$and

$$\mathrm{\Phi }\left(l,\mathrm{s}\right)={\int }_l^{\mathrm{s}}{\displaystyle \frac{1}{r^{1/\lambda }\left(\nu \right)}}\mathrm{d}\nu =\infty \mathrm{as}\mathrm{s}\to \infty ;$$

(3)

(II) 

$g,q\in C\left(\left[{\mathrm{s}}_0,\infty \right),\left[0,\infty \right)\right)$with $g\left(\mathrm{s}\right)\ge 1,g\left(\mathrm{s}\right)\ne 1$ for large $\mathrm{s}$ values, and $q\left(\mathrm{s}\right)$ is not identical to zero for large $\mathrm{s}$ values;

(III) 

$\beta ,\alpha \in C\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right),$$\beta \left(\mathrm{s}\right)\le \mathrm{s}$, $\alpha \left(\mathrm{s}\right)\le \mathrm{s},$$\beta \left(\mathrm{s}\right)$ is strictly increasing and ${lim}_{\mathrm{s}\to \infty }\beta \left(\mathrm{s}\right)={lim}_{\mathrm{s}\to \infty }\alpha \left(\mathrm{s}\right)=\infty .$

By a solution of (1), we mean a function $\varkappa \in C\left(\left[{\mathrm{s}}_{\varkappa },\infty \right),\mathbb{R}\right)$ for some${\mathrm{s}}_{\varkappa }\ge {\mathrm{s}}_0$ such that $y\in C^2\left(\left[{\mathrm{s}}_{\varkappa },\infty \right),\mathbb{R}\right),$$r{\left(y^{″}\right)}^{\lambda }\in C^1\left(\left[{\mathrm{s}}_{\varkappa },\infty \right),\mathbb{R}\right),$ and $\varkappa$ satisfies (1) on $\left[{\mathrm{s}}_{\varkappa },\infty \right).$ We consider only those proper solutions of (1) that exist on some half-line $\left[{\mathrm{s}}_{\varkappa },\infty \right)$ and satisfy

$$sup\left\{\left|\varkappa \left(\mathrm{s}\right)\right|:\mathrm{s}\ge {\mathrm{s}}_{*}\right\}>0\mathrm{for}\mathrm{all}{\mathrm{s}}_{*}\ge {\mathrm{s}}_{\varkappa }.$$

Qualitative theory plays an important role in DEs, providing essential insights into their behavior across various contexts. One of its primary branches is dedicated to exploring the qualitative properties demonstrated by solutions to these equations.

Oscillation theory as a part of the qualitative theory of differential and difference equations has been developed rapidly in recent years. It has concerned itself largely with the oscillatory and nonoscillatory properties of solutions. Overall, oscillation theory provides valuable tools for predicting and understanding the behavior of solutions to DEs, particularly in terms of their tendency to oscillate or not.

In the past few years, there has been a significant surge in research dedicated to studying the oscillation of solutions of delay/neutral DEs of different orders (see, for example, [1,2,3,4,5,6,7,8,9,10]). This increased interest largely stems from recognizing the crucial role such equations play in various real-world scenarios. We are witnessing a growing number of applications across physics, biology, ecology, and physiology that rely on these equations for modeling diverse phenomena.

Many studies have focused on studying the qualitative properties of DEs of the form

$${\left(r\left(\mathrm{s}\right){\left({\varkappa }^{″}\left(\mathrm{s}\right)\right)}^{\lambda }\right)}^{\prime }+q\left(\mathrm{s}\right){\varkappa }^{\lambda }\left(\alpha \left(\mathrm{s}\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0,$$

(4)

and their special cases or generalizations (see, for example, [11,12,13,14]). Mostly, the previous equation has been studied under the condition

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

Chatzarakis et al. [15] focused their attention on studying (4) under the following condition:

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

They set new sufficient conditions such that these conditions ensure that all solutions are oscillatory or almost oscillatory.

Astashova [16,17] studied the behavior of singular solutions of Emden–Fowler-type equations:

$${\varkappa }^{\left(n\right)}=P\left(\mathrm{s},\varkappa ,{\varkappa }^{\prime },\dots ,{\varkappa }^{\left(n-1\right)}\right){\left|\varkappa \right|}^{k}sgn\varkappa ,$$

where $n>2$ with a regular ($k>1$) or singular ($0<k<1$) nonlinearity. The author discussed the existence and behavior of “blown” solutions. The results concerning the asymptotic behavior of oscillating solutions were formulated. For third- and fourth-order equations, an asymptotic classification of all solutions was presented.

Regarding the oscillation of neutral DEs of the third order, we refer the reader to studies [18,19,20,21,22]. We mention the following studies in some detail. Jiang and Li [23] studied the oscillation behavior of DEs with variable delay arguments:

$${\left(r\left(\mathrm{s}\right){\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\mathrm{s}-\beta \left(\mathrm{s}\right)\right)\right)}^{″}\right)}^{\prime }+\sum _{i=1}^m{q}_i\left(\mathrm{s}\right)f_i\left(\varkappa \left(\mathrm{s}-{\alpha }_i\left(\mathrm{s}\right)\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0,$$

where $0\le g\left(\mathrm{s}\right)\le g_0<1,m\ge 1$ is an integer, $f_i\left(\zeta \right)/\zeta \ge j_i,$ for $\zeta \ne 0,j_i>0$ and

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

Using the integral averaging technique and generalized Riccati substitution, they proved that every solution is either oscillatory or convergent to zero using a new criterion of the Philos type.

Graef et al. [24] investigated the asymptotic properties of solutions of DEs:

$${\left({\left({\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)\right)}^{″}\right)}^{\lambda }\right)}^{\prime }+q\left(\mathrm{s}\right){\varkappa }^{\lambda }\left(\alpha \left(\mathrm{s}\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0.$$

(5)

They proved that the solution is either oscillatory or converges to zero by deducing some new conditions. We display one of the results obtained in [24] at the end of this study for the purpose of comparison with the results we obtained.

Theorem 1. 

Assume that $\beta \left(\mathrm{s}\right)\ge \alpha \left(\mathrm{s}\right),$

$${\int }_{{\mathrm{s}}_0}^{\infty }{\int }_v^{\infty }{\left({\int }_{\zeta }^{\infty }q\left(\nu \right){\left(p^{*}\left(\alpha \left(\nu \right)\right)\right)}^{\lambda }\mathrm{d}\nu \right)}^{1/\lambda }\mathrm{d}\zeta \mathrm{d}v=\infty ,$$

(6)

and $\mu \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right)$such that $2\mu \left(\mathrm{s}\right)-\mathrm{s}{\mu }^{\prime }\left(\mathrm{s}\right)\le 0$ holds, $p_{*}\left(\mathrm{s}\right)>0,$and $p^{*}\left(\mathrm{s}\right)>0.$ If $\xi \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right)$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\left(\xi \left(\nu \right)q\left(\nu \right){\left(p_{*}\left(\alpha \left(\nu \right)\right)\right)}^{\lambda }{\left({\displaystyle \frac{{\beta }^{-1}\left(\alpha \left(\nu \right)\right)}{\nu }}\right)}^{2\lambda }-{\xi }_{+}^{\prime }\left(\nu \right){\left({\displaystyle \frac{8}{{\nu }^2}}\right)}^{\lambda }\right)\mathrm{d}\nu =\infty ,$$

(7)

then any solution of (5) is either oscillatory or tends to zero as $\mathrm{s}\to \infty ,$ where

$$p^{*}\left(\mathrm{s}\right)={\displaystyle \frac{1}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}\left(1-{\displaystyle \frac{1}{g\left({\beta }^{-1}\left({\beta }^{-1}\left(\mathrm{s}\right)\right)\right)}}\right),$$

$$p_{*}\left(\mathrm{s}\right)={\displaystyle \frac{1}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}\left(1-{\displaystyle \frac{1}{g\left({\beta }^{-1}\left({\beta }^{-1}\left(\mathrm{s}\right)\right)\right)}}{\displaystyle \frac{\mu \left({\beta }^{-1}\left({\beta }^{-1}\left(\mathrm{s}\right)\right)\right)}{\mu \left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}\right),$$

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

and the function μ can be considered, for example, for $\mu \left(\mathrm{s}\right)={\mathrm{e}}^{\mathrm{s}},\mu \left(\mathrm{s}\right)={\mathrm{s}}^{\tau }{\mathrm{e}}^{ϵ\mathrm{s}},$$\mu \left(\mathrm{s}\right)={\mathrm{s}}^2,$ and $\mu \left(\mathrm{s}\right)={\mathrm{s}}^3,$ with $\tau \ge 2$ and $ϵ\ge 0,$ etc.

Qaraad et al. [25] established some oscillation criteria of

$${\left(r\left(\mathrm{s}\right){\left(y^{″}\left(\mathrm{s}\right)\right)}^{\lambda }\right)}^{\prime }+q\left(\mathrm{s}\right){\varkappa }^{\lambda }\left(\alpha \left(\mathrm{s}\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0,$$

(8)

where $g_{*},g\in C\left(\left[{\mathrm{s}}_0,\infty \right),\left[0,\infty \right)\right),$ $\alpha \circ \beta =\beta \circ \alpha ,$ $\alpha \circ {\beta }_{*}={\beta }_{*}\circ \alpha ,{\beta }_{*}\left(\mathrm{s}\right)>\mathrm{s}$ and

$$y\left(\mathrm{s}\right):=\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)+g_{*}\left(\mathrm{s}\right)\varkappa \left({\beta }_{*}\left(\mathrm{s}\right)\right).$$

They presented new criteria for the oscillation of Equation (8) by using the Riccati transformation and the comparison technique with first-order differential inequalities. We display one of the results obtained in [24] at the end of this study for the purpose of comparison with the results we obtained.

Theorem 2. 

Assume that $\alpha \left(\mathrm{s}\right)<\beta \left(\mathrm{s}\right)$. If ${\mu }_{*},{\delta }_{*}\in C\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right)$ and satisfying ${\beta }^{-1}\left({\mu }_{*}\left(\mathrm{s}\right)\right)<\mathrm{s}$ and $\alpha \left(\mathrm{s}\right)<{\mu }_{*}\left(\mathrm{s}\right)$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\left({\displaystyle \frac{k^{\lambda }{\delta }_{*}\left(\nu \right)\widehat{q}\left(\nu \right){\alpha }^{\lambda }\left(\nu \right)}{2^{\lambda }{\left({\kappa }_{*}\right)}^2}}-{\displaystyle \frac{1}{{\left(\lambda +1\right)}^{\lambda +1}}}\left({\displaystyle \frac{1}{{\kappa }_{*}}}+{\displaystyle \frac{g^{\lambda }}{\beta }}+{\displaystyle \frac{g_{*}^{\lambda }}{{\beta }_{*}}}\right){\displaystyle \frac{{\left({\delta }_{*}^{\prime }\left(\nu \right)\right)}^{\lambda +1}r\left(\alpha \left(\nu \right)\right)}{{\left({\delta }_{*}\left(\nu \right){\alpha }^{\prime }\left(\nu \right)\right)}^{\lambda }}}\right)\mathrm{d}\nu =\infty$$

(9)

and

$$\underset{\mathrm{s}\to \infty }{liminf}{\int }_{{\beta }^{-1}\left({\mu }_{*}\left(\mathrm{s}\right)\right)}^{\mathrm{s}}\widehat{q}\left(\nu \right){\widehat{g}}^{\lambda }\left({\mu }_{*}\left(\nu \right),\alpha \left(\nu \right)\right)\mathrm{d}\nu >{\displaystyle \frac{\beta {\beta }_{*}+{\kappa }_{*}{\beta }_{*}{g}^{\lambda }+{\kappa }_{*}\beta g_{*}^{\lambda }}{{\kappa }_{*}\beta {\beta }_{*}}}$$

(10)

hold, then (8) is oscillatory, where $k\in \left(0,1\right),$

$${\kappa }_{*}=\left\{\begin{array}{cc}1& \mathit{if}\lambda \le 1,\\ 2^{\lambda -1}& \mathit{if}\lambda >1,\end{array}\right.$$

$$\widehat{q}\left(\mathrm{s}\right)=min\left\{q\left(\mathrm{s}\right).q\left(\beta \left(\mathrm{s}\right)\right),q\left({\beta }_{*}\left(\mathrm{s}\right)\right)\right\}$$

and

$$\widehat{g}\left(v,w\right)={\int }_w^v{\int }_h^v{\displaystyle \frac{1}{r^{1/\lambda }\left(\nu \right)}}\mathrm{d}\nu \mathrm{d}h.$$

This study aims to reach new oscillatory results for (1), which would improve the results mentioned above. This study is organized as follows. In Section 2, we present some basic results and notations that will be used to obtain our main results. In Section 3, we present the results we obtained, and at the end of this section we present some examples and some remarks that illustrate the importance of the results we obtained. Section 4 concerns the conclusion.

2. Some Lemmas and Auxiliary Notations

In this section, we provide the necessary lemmas and notations of our results.

First, we present three basic lemmas.

Lemma 1 

([26]). Assume that $\mathcal{H}\in C^{m+1}\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right)$, ${\mathcal{H}}^{\left(\vartheta \right)}\left(\mathrm{s}\right)>0,$for $\vartheta =0,1,\dots ,m,$ and ${\mathcal{H}}^{\left(m+1\right)}\left(\mathrm{s}\right)\le 0$ eventually. Then, for every $k_2\in \left(0,1\right)$,

$${\displaystyle \frac{\mathcal{H}\left(\mathrm{s}\right)}{{\mathcal{H}}^{\prime }\left(\mathrm{s}\right)}}\ge {\displaystyle \frac{k_2}{m}}\mathrm{s},$$

eventually.

Lemma 2 

([27], Lemma 3). Assume that $\mathcal{H}\in C^2\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right).$If $\mathcal{H}\left(\mathrm{s}\right)>0,$${\mathcal{H}}^{\prime }\left(\mathrm{s}\right)\ge 0$, and ${\mathcal{H}}^{″}\left(\mathrm{s}\right)\le 0,$ then, for every $k_1\in \left(0,1\right)$,

$${\displaystyle \frac{\mathcal{H}\left(\alpha \left(\mathrm{s}\right)\right)}{\mathcal{H}\left(\mathrm{s}\right)}}\ge k_1{\displaystyle \frac{\alpha \left(\mathrm{s}\right)}{\mathrm{s}}},$$

eventually.

Lemma 3 

([27], Lemma 1). Assume that ϰ is an eventually positive solution of (1). Then, y eventually satisfies one of the following cases:

(i)

$y>0,y^{\prime }>0,$$y^{″}>0$and ${\left(r{\left(y^{″}\right)}^{\lambda }\right)}^{\prime }\le 0;$

(ii)

$y>0,y^{\prime }<0,$$y^{″}>0$and ${\left(r{\left(y^{″}\right)}^{\lambda }\right)}^{\prime }\le 0.$

Now, we consider the following notations:

$$u\left(\mathrm{s}\right)={\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\mathrm{\Phi }\left({\mathrm{s}}_0,\zeta \right)\mathrm{d}\zeta ,$$

$$\hslash \left(\mathrm{s}\right)={\left(1-{\displaystyle \frac{1}{g\left({\beta }^{-1}\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)\right)}}\right)}^{\lambda }{\left({\int }_{{\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)}^{{\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)}{\int }_{\upsilon }^{{\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)}{\displaystyle \frac{1}{r^{1/\lambda }\left(\zeta \right)}}\mathrm{d}\zeta \mathrm{d}\upsilon \right)}^{\lambda },$$

$$\mathcal{T}=\sum _{\vartheta =0}^{\lambda +1}\left[\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +1}{\lambda -\vartheta +1}}\right){\left(-1\right)}^{\lambda -\vartheta +1}{\epsilon }^{\lambda -\vartheta +1}{\kappa }^{\vartheta }{\int }_0^1{\left(1-\sigma \right)}^{\epsilon +\vartheta -\lambda -1}{\sigma }^{\kappa -\vartheta }\mathrm{d}\sigma \right],\mathrm{for}\lambda \in {\mathbb{Z}}^{+},$$

(11)

where $\epsilon ,$$\kappa \in \left(\lambda ,\infty \right),$

$${\int }_0^1{\sigma }^{\epsilon -1}{\left(1-\sigma \right)}^{\kappa -1}\mathrm{d}\sigma ={\displaystyle \frac{\mathrm{\Gamma }\left(\epsilon \right)\mathrm{\Gamma }\left(\kappa \right)}{\mathrm{\Gamma }\left(\epsilon +\kappa \right)}},$$

(12)

$$\mathrm{\Gamma }\left(\nu \right)={\int }_0^{+\infty }{\varkappa }^{\nu -1}{e}^{-\varkappa }\mathrm{d}\varkappa ,\nu >0,$$

$$\mathrm{\Psi }\left(\mathrm{s},l,\nu \right):={\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }\gamma \left(\nu \right){\displaystyle \frac{r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}\mho \left(\mathrm{s},l,\nu \right),$$

$$\mho \left(\mathrm{s},l,\nu \right)={\left({\displaystyle \frac{\kappa \mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\left(\epsilon +\kappa \right)\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)+\epsilon \mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)}{r^{1/\lambda }\left(\nu \right)\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}}+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda +1},$$

$$𝘍\left(\mathrm{s},l,\nu \right):={\left({\displaystyle \frac{\kappa \mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\left(\epsilon +\kappa \right)\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)+\epsilon \mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)}{r^{1/\lambda }\left(\nu \right)\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}}\right)}^{\lambda +1},$$

$$\left(\mathrm{s}\right)=\left({\displaystyle \frac{1}{g\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}-{\displaystyle \frac{u\left({\beta }^{-2}\left(\alpha \left(\mathrm{s}\right)\right)\right)}{g\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)g\left({\beta }^{-2}\left(\alpha \left(\mathrm{s}\right)\right)\right)u\left(\alpha \left(\mathrm{s}\right)\right)}}\right)$$

and

$$\mathsf{\Omega }\left(\mathrm{s}\right)={{\displaystyle \frac{{\alpha }^{2\lambda }\left(\nu \right)}{{\left(2\nu \right)}^{\lambda }}}}^{\lambda }\left(\mathrm{s}\right)$$

for $g\left({\beta }^{-2}\left(\alpha \left(\mathrm{s}\right)\right)\right)>u\left({\beta }^{-2}\left(\alpha \left(\mathrm{s}\right)\right)\right)/u\left(\alpha \left(\mathrm{s}\right)\right),$where ${\beta }^{-2}\left(\mathrm{s}\right)={\beta }^{-1}\left({\beta }^{-1}\left(\mathrm{s}\right)\right).$

3. Main Results

This section contains the results for the oscillation of the studied DE.

First, we present two lemmas that will be used in oscillation theorems.

Lemma 4. 

Assume that $\varkappa \left(\mathrm{s}\right)$ is an eventually positive solution of (1) and Case (i) satisfies. Then,

$${\left(r\left(\mathrm{s}\right){\left({\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)\right)}^{″}\right)}^{\lambda }\right)}^{\prime }\le -q\left(\mathrm{s}\right)y^{\lambda }{\left(\alpha \left(\mathrm{s}\right)\right)}^{\lambda }\left(\mathrm{s}\right)$$

(13)

holds.

Proof. 

Assume that $\varkappa \left(\mathrm{s}\right)$ is an eventually positive solution of (1) satisfying $\varkappa \left(\mathrm{s}\right)>0,$$\varkappa \left(\beta \left(\mathrm{s}\right)\right)>0,$and $\varkappa \left(\alpha \left(\mathrm{s}\right)\right)>0$ for $\mathrm{s}\ge {\mathrm{s}}_1$. Clearly, $y\left(\mathrm{s}\right)>0$. From (2), we obtain

$$\begin{array}{ccc}\hfill \varkappa \left(\mathrm{s}\right)& =& {\displaystyle \frac{1}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}\left(y\left({\beta }^{-1}\left(\mathrm{s}\right)\right)-\varkappa \left({\beta }^{-1}\left(\mathrm{s}\right)\right)\right)\hfill \\ & \ge & {\displaystyle \frac{y\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}-{\displaystyle \frac{y\left({\beta }^{-2}\left(\mathrm{s}\right)\right)}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)g\left({\beta }^{-2}\left(\mathrm{s}\right)\right)}}.\hfill \end{array}$$

(14)

Since ${\beta }^{-1}\left(\mathrm{s}\right)\ge \mathrm{s},y^{\prime }>0,$ we see that

$$y\left({\beta }^{-1}\left(\mathrm{s}\right)\right)\ge y\left(\mathrm{s}\right),\mathrm{s}\ge {\mathrm{s}}_1$$

(15)

and so (14) becomes

$$\varkappa \left(\mathrm{s}\right)\ge {\displaystyle \frac{y\left(\mathrm{s}\right)}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}-{\displaystyle \frac{y\left({\beta }^{-2}\left(\mathrm{s}\right)\right)}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)g\left({\beta }^{-2}\left(\mathrm{s}\right)\right)}}.$$

(16)

Since

$$\begin{array}{ccc}\hfill y^{\prime }\left(\mathrm{s}\right)& \ge & {\int }_{{\mathrm{s}}_0}^{\mathrm{s}}{\displaystyle \frac{r^{1/\lambda }\left(\zeta \right)y^{″}\left(\zeta \right)}{r^{1/\lambda }\left(\zeta \right)}}\mathrm{d}\zeta \ge r^{1/\lambda }\left(\mathrm{s}\right)y^{″}\left(\mathrm{s}\right){\int }_{{\mathrm{s}}_0}^{\mathrm{s}}{\displaystyle \frac{1}{r^{1/\lambda }\left(\zeta \right)}}\mathrm{d}\zeta \hfill \\ & \ge & r^{1/\lambda }\left(\mathrm{s}\right)y^{″}\left(\mathrm{s}\right)\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right),\mathrm{s}\ge {\mathrm{s}}_1,\hfill \end{array}$$

we find

$${\left({\displaystyle \frac{y^{\prime }\left(\mathrm{s}\right)}{\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)}}\right)}^{\prime }={\displaystyle \frac{\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)y^{″}\left(\mathrm{s}\right)-y^{\prime }\left(\mathrm{s}\right)r^{-1/\lambda }\left(\mathrm{s}\right)}{{\mathrm{\Phi }}^2\left({\mathrm{s}}_0,\mathrm{s}\right)}}={\displaystyle \frac{r^{1/\lambda }\left(\mathrm{s}\right)\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)y^{″}\left(\mathrm{s}\right)-y^{\prime }\left(\mathrm{s}\right)}{r^{1/\lambda }\left(\mathrm{s}\right){\mathrm{\Phi }}^2\left({\mathrm{s}}_0,\mathrm{s}\right)}}\le 0,\mathrm{s}\ge {\mathrm{s}}_1.$$

(17)

Since

$$y\left(\mathrm{s}\right)\ge {\int }_{{\mathrm{s}}_0}^{\mathrm{s}}{\displaystyle \frac{\mathrm{\Phi }\left({\mathrm{s}}_0,\zeta \right)y^{\prime }\left(\zeta \right)}{\mathrm{\Phi }\left({\mathrm{s}}_0,\zeta \right)}}\mathrm{d}\zeta ,\mathrm{for}\mathrm{s}\ge {\mathrm{s}}_1,$$

by using (17), we see that

$$y\left(\mathrm{s}\right)\ge {\displaystyle \frac{y^{\prime }\left(\mathrm{s}\right)}{\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)}}u\left(\mathrm{s}\right)$$

and so

$${\left({\displaystyle \frac{y\left(\mathrm{s}\right)}{u\left(\mathrm{s}\right)}}\right)}^{\prime }={\displaystyle \frac{{\mathrm{\Phi }}^{-1}\left({\mathrm{s}}_0,\mathrm{s}\right)u\left(\mathrm{s}\right)y^{\prime }\left(\mathrm{s}\right)-y\left(\mathrm{s}\right)}{{\mathrm{\Phi }}^{-1}\left({\mathrm{s}}_0,\mathrm{s}\right)u^2\left(\mathrm{s}\right)}}\le 0,\mathrm{s}\ge {\mathrm{s}}_1.$$

(18)

Also, since ${\beta }^{-2}\left(\mathrm{s}\right)\ge {\beta }^{-1}\left(\mathrm{s}\right)\ge \mathrm{s},$ and using (18), we obtain

$$y\left({\beta }^{-2}\left(\mathrm{s}\right)\right)\le {\displaystyle \frac{u\left({\beta }^{-2}\left(\mathrm{s}\right)\right)y\left(\mathrm{s}\right)}{u\left(\mathrm{s}\right)}},\mathrm{s}\ge {\mathrm{s}}_1.$$

(19)

Substituting (19) into (16) and replacing each $\mathrm{s}$ with $\alpha \left(\mathrm{s}\right)$, we obtain

$$\varkappa \left(\alpha \left(\mathrm{s}\right)\right)\ge y\left(\alpha \left(\mathrm{s}\right)\right)\left({\displaystyle \frac{1}{g\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}-{\displaystyle \frac{u\left({\beta }^{-2}\left(\alpha \left(\mathrm{s}\right)\right)\right)}{g\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)g\left({\beta }^{-2}\left(\alpha \left(\mathrm{s}\right)\right)\right)u\left(\alpha \left(\mathrm{s}\right)\right)}}\right),\mathrm{s}\ge {\mathrm{s}}_1.$$

(20)

Using (20) and (1), we arrive at

$${\left(r\left(\mathrm{s}\right){\left({\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)\right)}^{″}\right)}^{\lambda }\right)}^{\prime }\le -q\left(\mathrm{s}\right)y^{\lambda }{\left(\alpha \left(\mathrm{s}\right)\right)}^{\lambda }\left(\mathrm{s}\right),$$

which gives (13). The proof is complete. □

Lemma 5. 

Assume that $\varkappa \left(\mathrm{s}\right)$ is an eventually positive solution of (1) and Case (ii) satisfies. Then,

$${\left(r\left(\mathrm{s}\right){\left({\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)\right)}^{″}\right)}^{\lambda }\right)}^{\prime }\le -q\left(\mathrm{s}\right){\displaystyle \frac{y^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}{g^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}{\left(1-{\displaystyle \frac{1}{g\left({\beta }^{-1}\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)\right)}}\right)}^{\lambda }$$

(21)

holds.

Proof. 

Assume that $\varkappa \left(\mathrm{s}\right)$ is an eventually positive solution of (1) satisfying $\varkappa \left(\mathrm{s}\right)>0,$$\varkappa \left(\beta \left(\mathrm{s}\right)\right)>0,$and $\varkappa \left(\alpha \left(\mathrm{s}\right)\right)>0$ for $\mathrm{s}\ge {\mathrm{s}}_1$. Clearly, $y\left(\mathrm{s}\right)>0$. Since $\beta \left(\mathrm{s}\right)\le \mathrm{s}$ is a strictly increasing function, we have

$$\mathrm{s}\le {\beta }^{-1}\left(\mathrm{s}\right)$$

and so

$$y\left(\mathrm{s}\right)\ge y\left({\beta }^{-1}\left(\mathrm{s}\right)\right).$$

(22)

From (2), we obtain

$$\begin{array}{ccc}\hfill \varkappa \left(\beta \left(\mathrm{s}\right)\right)& =& {\displaystyle \frac{y\left(\mathrm{s}\right)}{g\left(\mathrm{s}\right)}}-{\displaystyle \frac{\varkappa \left(\mathrm{s}\right)}{g\left(\mathrm{s}\right)}}\hfill \\ & \ge & {\displaystyle \frac{y\left(\mathrm{s}\right)}{g\left(\mathrm{s}\right)}}-{\displaystyle \frac{y\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}{g\left(\mathrm{s}\right)g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}.\hfill \end{array}$$

(23)

Using (22) in (23), we obtain

$$\varkappa \left(\beta \left(\mathrm{s}\right)\right)\ge {\displaystyle \frac{y\left(\mathrm{s}\right)}{g\left(\mathrm{s}\right)}}\left(1-{\displaystyle \frac{1}{g\left({\beta }^{-1}\left(\mathrm{s}\right)\right)}}\right),$$

which implies that

$$\varkappa \left(\alpha \left(\mathrm{s}\right)\right)\ge {\displaystyle \frac{y\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}{g\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}\left(1-{\displaystyle \frac{1}{g\left({\beta }^{-1}\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)\right)}}\right).$$

(24)

Using (24) in Equation (1), we conclude that

$${\left(r\left(\mathrm{s}\right){\left({\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)\right)}^{″}\right)}^{\lambda }\right)}^{\prime }\le -q\left(\mathrm{s}\right){\displaystyle \frac{y^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}{g^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}{\left(1-{\displaystyle \frac{1}{g\left({\beta }^{-1}\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)\right)}}\right)}^{\lambda },$$

which gives (21). The proof is complete. □

Now, in order to introduce some new oscillation criteria for (1), we introduce a class of functions ℜ. Let

$$H=\left\{\left(\mathrm{s},\nu ,\phi \right):{\mathrm{s}}_0\le \phi \le \nu \le \mathrm{s}\le \infty \right\}.$$

The function $\mathcal{N}\in C\left(H,\mathbb{R}\right)$ is said to belong to the class ℜ if

(1)

$\mathcal{N}\left(\mathrm{s},\mathrm{s},\phi \right)=0,\mathcal{N}\left(\mathrm{s},\phi ,\phi \right)=0,and\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)\ne 0$for $\phi <\nu <\mathrm{s};$

(2)

$\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)$ has the partial derivative $\partial \mathcal{N}/\partial \nu$ on H such that $\partial \mathcal{N}/\partial \nu$ is locally integrable with respect to $\nu$ in H and

$${\displaystyle \frac{\partial \mathcal{N}\left(\mathrm{s},\nu ,\phi \right)}{\partial \nu }}=h\left(\mathrm{s},\nu ,\phi \right)\mathcal{N}\left(\mathrm{s},\nu ,\phi \right),$$

(25)

where $h\in C\left(H,\mathbb{R}\right)$ (see Philos [28]).

Theorem 3. 

Assume that the function $\mathcal{N}\in \Re .$ Suppose that there exists a functions $\varpi \in C\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right)$ and $\gamma \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),{\mathbb{R}}^{+}\right)$ such that $\alpha \left(\mathrm{s}\right)\le \varpi \left(\mathrm{s}\right)\le \beta \left(\mathrm{s}\right)$ and ${\gamma }^{\prime }\left(\mathrm{s}\right)\ge 0$. If

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)\gamma \left(\nu \right)\left(k_1^{\lambda }{k}_2^{\lambda }q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)-{\displaystyle \frac{r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}{\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda +1}\right)\mathrm{d}\nu >0$$

(26)

for any $k_1,k_2\in \left(0,1\right)$, and the delay DE

$${\mathrm{Y}}^{\prime }\left(\mathrm{s}\right)+q\left(\mathrm{s}\right){\displaystyle \frac{\hslash \left(\mathrm{s}\right)}{g^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}\mathrm{Y}\left({\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)\right)=0$$

(27)

is oscillatory, then (1) is oscillatory.

Proof. 

Assume that $\varkappa \left(\mathrm{s}\right)$ is an eventually positive solution of (1) satisfying $\varkappa \left(\mathrm{s}\right)>0,$$\varkappa \left(\beta \left(\mathrm{s}\right)\right)>0,$and $\varkappa \left(\alpha \left(\mathrm{s}\right)\right)>0$ for $\mathrm{s}\ge {\mathrm{s}}_1$. Clearly, $y\left(\mathrm{s}\right)>0$. From Lemma 3, we find that Case (i) or Case (ii) is satisfied.

First, suppose that Case (i) holds. We assume that the function $\varphi \left(\mathrm{s}\right)$ is defined as follows:

$$\varphi \left(\mathrm{s}\right)=\gamma \left(\mathrm{s}\right){\displaystyle \frac{r\left(\mathrm{s}\right){\left(y^{″}\left(\mathrm{s}\right)\right)}^{\lambda }}{{\left(y^{\prime }\left(\mathrm{s}\right)\right)}^{\lambda }}}>0,\mathrm{s}\ge {\mathrm{s}}_2.$$

(28)

Differentiating (28), we obtain

$$\begin{array}{ccc}\hfill {\varphi }^{\prime }\left(\mathrm{s}\right)& =& {\gamma }^{\prime }\left(\mathrm{s}\right){\displaystyle \frac{r\left(\mathrm{s}\right){\left(y^{″}\left(\mathrm{s}\right)\right)}^{\lambda }}{{\left(y^{\prime }\left(\mathrm{s}\right)\right)}^{\lambda }}}\hfill \\ & & +{\displaystyle \frac{\gamma \left(\mathrm{s}\right){\left(y^{\prime }\left(\mathrm{s}\right)\right)}^{\lambda }{\left(r\left(\mathrm{s}\right){\left(y^{\prime \prime }\left(\mathrm{s}\right)\right)}^{\lambda }\right)}^{\prime }-\lambda \gamma \left(\mathrm{s}\right)r\left(\mathrm{s}\right){\left(y^{″}\left(\mathrm{s}\right)\right)}^{\lambda }{\left(y^{\prime }\left(\mathrm{s}\right)\right)}^{\lambda -1}{y}^{″}\left(\mathrm{s}\right)}{{\left(y^{\prime }\left(\mathrm{s}\right)\right)}^{2\lambda }}},\mathrm{s}\ge {\mathrm{s}}_2.\hfill \end{array}$$

(29)

Using Lemma 4 and (28), we see that (29) becomes

$${\varphi }^{\prime }\left(\mathrm{s}\right)\le {\gamma }^{\prime }\left(\mathrm{s}\right){\displaystyle \frac{\varphi \left(\mathrm{s}\right)}{\gamma \left(\mathrm{s}\right)}}-\gamma \left(\mathrm{s}\right){\displaystyle \frac{q\left(\mathrm{s}\right)y^{\lambda }{\left(\alpha \left(\mathrm{s}\right)\right)}^{\lambda }\left(\mathrm{s}\right)}{{\left(y^{\prime }\left(\mathrm{s}\right)\right)}^{\lambda }}}-\lambda {\displaystyle \frac{{\varphi }^{\left(\lambda +1\right)/\lambda }\left(\mathrm{s}\right)}{r^{1/\lambda }\left(\mathrm{s}\right){\gamma }^{1/\lambda }\left(\mathrm{s}\right)}},\mathrm{s}\ge {\mathrm{s}}_2.$$

(30)

By using Lemma 2 with $\mathcal{H}\left(\mathrm{s}\right)=y^{\prime }\left(\mathrm{s}\right),$ there exists an ${\mathrm{s}}_3\ge {\mathrm{s}}_2$ such that

$${\displaystyle \frac{y^{\prime }\left(\alpha \left(\mathrm{s}\right)\right)}{y^{\prime }\left(\mathrm{s}\right)}}\ge k_1{\displaystyle \frac{\alpha \left(\mathrm{s}\right)}{\mathrm{s}}},\mathrm{s}\ge {\mathrm{s}}_3\ge {\mathrm{s}}_2.$$

(31)

From Lemma 1, we have

$${\displaystyle \frac{y\left(\mathrm{s}\right)}{y^{\prime }\left(\mathrm{s}\right)}}\ge {\displaystyle \frac{1}{2}}{k}_2\mathrm{s},\mathrm{s}\ge {\mathrm{s}}_3.$$

(32)

Using (31) and (32), we obtain

$${\displaystyle \frac{1}{y^{\prime }\left(\mathrm{s}\right)}}\ge k_1{\displaystyle \frac{\alpha \left(\mathrm{s}\right)}{\mathrm{s}{y}^{\prime }\left(\alpha \left(\mathrm{s}\right)\right)}}\ge k_1{k}_2{\displaystyle \frac{{\alpha }^2\left(\mathrm{s}\right)}{2\mathrm{s}}}{\displaystyle \frac{1}{y\left(\alpha \left(\mathrm{s}\right)\right)}},\mathrm{s}\ge {\mathrm{s}}_3.$$

(33)

From (30) and (33), we obtain

$${\varphi }^{\prime }\left(\mathrm{s}\right)\le {\gamma }^{\prime }\left(\mathrm{s}\right){\displaystyle \frac{\varphi \left(\mathrm{s}\right)}{\gamma \left(\mathrm{s}\right)}}-k_1^{\lambda }{k}_2^{\lambda }\mathsf{\Omega }\left(\mathrm{s}\right)\gamma \left(\mathrm{s}\right)q\left(\mathrm{s}\right)-\lambda {\displaystyle \frac{{\varphi }^{\left(\lambda +1\right)/\lambda }\left(\mathrm{s}\right)}{r^{1/\lambda }\left(\mathrm{s}\right){\gamma }^{1/\lambda }\left(\mathrm{s}\right)}},\mathrm{s}\ge {\mathrm{s}}_3$$

and so

$$k_1^{\lambda }{k}_2^{\lambda }\gamma \left(\mathrm{s}\right)q\left(\mathrm{s}\right)\mathsf{\Omega }\left(\mathrm{s}\right)\le -{\varphi }^{\prime }\left(\mathrm{s}\right)+{\gamma }^{\prime }\left(\mathrm{s}\right){\displaystyle \frac{\varphi \left(\mathrm{s}\right)}{\gamma \left(\mathrm{s}\right)}}-\lambda {\displaystyle \frac{{\varphi }^{\left(\lambda +1\right)/\lambda }\left(\mathrm{s}\right)}{r^{1/\lambda }\left(\mathrm{s}\right){\gamma }^{1/\lambda }\left(\mathrm{s}\right)}},\mathrm{s}\ge {\mathrm{s}}_3.$$

(34)

Multiplying (34) by $\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)$ and integrating from $\phi \ge {\mathrm{s}}_3$ to $\mathrm{s}$, we obtain

$$\begin{array}{ccc}& & {\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)k_1^{\lambda }{k}_2^{\lambda }\gamma \left(\nu \right)q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)\mathrm{d}\nu \hfill \\ & & \le -{\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right){\varphi }^{\prime }\left(\nu \right)\mathrm{d}\nu +{\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right){\gamma }^{\prime }\left(\nu \right){\displaystyle \frac{\varphi \left(\nu \right)}{\gamma \left(\nu \right)}}\mathrm{d}\nu \hfill \\ & & -\lambda {\int }_{\phi }^{\mathrm{s}}{\displaystyle \frac{\mathcal{N}\left(\mathrm{s},\nu ,\phi \right){\varphi }^{\left(\lambda +1\right)/\lambda }\left(\nu \right)}{r^{1/\lambda }\left(\nu \right){\gamma }^{1/\lambda }\left(\nu \right)}}\mathrm{d}\nu ,\phi \ge {\mathrm{s}}_3.\hfill \end{array}$$

(35)

By using (25), for all $\mathrm{s}\ge \phi ,$we have

$$\begin{array}{ccc}& & {\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)k_1^{\lambda }{k}_2^{\lambda }\gamma \left(\nu \right)q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)\mathrm{d}\nu \hfill \\ & & \le {\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)\left(\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)\varphi \left(\nu \right)-\lambda {\displaystyle \frac{{\varphi }^{\left(\lambda +1\right)/\lambda }\left(\nu \right)}{r^{1/\lambda }\left(\nu \right){\gamma }^{1/\lambda }\left(\nu \right)}}\right)\mathrm{d}\nu ,\mathrm{s}\ge \phi .\hfill \end{array}$$

(36)

Set

$$F\left(\eta \right)=\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)\eta -\lambda {\displaystyle \frac{{\eta }^{\left(\lambda +1\right)/\lambda }}{r^{1/\lambda }\left(\nu \right){\gamma }^{1/\lambda }\left(\nu \right)}},\mathrm{s}\ge \phi .$$

A simple calculation implies that when

$$\eta ={\displaystyle \frac{\gamma \left(\nu \right)r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}{\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda },\mathrm{s}\ge \phi$$

$F\left(\eta \right)$ has the maximum

$${\displaystyle \frac{\gamma \left(\nu \right)r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}{\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda +1},\mathrm{s}\ge \phi ,$$

that is,

$$F\left(\eta \right)\le F_{max}={\displaystyle \frac{\gamma \left(\nu \right)r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}{\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda +1},\mathrm{s}\ge \phi .$$

(37)

Using (36) and (37), we have

$$\begin{array}{ccc}& & 0\ge {\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)k_1^{\lambda }{k}_2^{\lambda }\gamma \left(\nu \right)q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)\mathrm{d}\nu \hfill \\ & & -{\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right){\displaystyle \frac{\gamma \left(\nu \right)r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}{\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda +1}\mathrm{d}\nu ,\mathrm{s}\ge \phi \hfill \end{array}$$

and so

$${\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)\gamma \left(\nu \right)\left(k_1^{\lambda }{k}_2^{\lambda }q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)-{\displaystyle \frac{r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}{\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda +1}\right)\mathrm{d}\nu \le 0,\mathrm{s}\ge \phi .$$

Taking the super limit, we obtain

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\phi }^{\mathrm{s}}\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)\gamma \left(\nu \right)\left(k_1^{\lambda }{k}_2^{\lambda }q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)-{\displaystyle \frac{r\left(\nu \right)}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}{\left(h\left(\mathrm{s},\nu ,\phi \right)+{\displaystyle \frac{{\gamma }^{\prime }\left(\nu \right)}{\gamma \left(\nu \right)}}\right)}^{\lambda +1}\right)\mathrm{d}\nu \le 0,\mathrm{s}\ge \phi .$$

From this inequality, we obtain a contradiction with (26).

Now, assume that Case (ii) holds. We can write

$$y^{\prime }\left(\mathrm{s}\right)-y^{\prime }\left(\varsigma \right)={\int }_{\varsigma }^{\mathrm{s}}{\displaystyle \frac{r^{1/\lambda }\left(\zeta \right)y^{″}\left(\zeta \right)}{r^{1/\lambda }\left(\zeta \right)}}\mathrm{d}\zeta \mathrm{for}\mathrm{s}\ge \varsigma \ge {\mathrm{s}}_2\ge {\mathrm{s}}_1$$

and so

$$-y^{\prime }\left(\varsigma \right)\ge r^{1/\lambda }\left(\mathrm{s}\right)y^{″}\left(\mathrm{s}\right){\int }_{\varsigma }^{\mathrm{s}}{\displaystyle \frac{1}{r^{1/\lambda }\left(\zeta \right)}}\mathrm{d}\zeta .$$

(38)

Integrating (38) from $\varsigma$ to $\mathrm{s}$, we have

$$y\left(\varsigma \right)\ge r^{1/\lambda }\left(\mathrm{s}\right)y^{″}\left(\mathrm{s}\right){\int }_{\varsigma }^{\mathrm{s}}{\int }_{\upsilon }^{\mathrm{s}}{\displaystyle \frac{1}{r^{1/\lambda }\left(\zeta \right)}}\mathrm{d}\zeta \mathrm{d}\upsilon .$$

(39)

Since $\alpha \left(\mathrm{s}\right)\le \varpi \left(\mathrm{s}\right)<\beta \left(\mathrm{s}\right)$ for all $\mathrm{s}\ge {\mathrm{s}}_0$ and $\beta \left(\mathrm{s}\right)$ is strictly increasing, then ${\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\le {\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)$. Now, if we set $\varsigma ={\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)$ and $\mathrm{s}={\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)$ in (39), we have

$$y\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)\ge r^{1/\lambda }\left({\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)\right)y^{″}\left({\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)\right){\int }_{{\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)}^{{\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)}{\int }_{\upsilon }^{{\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)}{\displaystyle \frac{1}{r^{1/\lambda }\left(\zeta \right)}}\mathrm{d}\zeta \mathrm{d}\upsilon .$$

(40)

From Lemma 5 and (40), we obtain

$${\left(r\left(\mathrm{s}\right){\left(y^{″}\left(\mathrm{s}\right)\right)}^{\lambda }\right)}^{\prime }+q\left(\mathrm{s}\right){\displaystyle \frac{\hslash \left(\mathrm{s}\right)}{g^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}r\left({\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)\right){\left(y^{″}\left({\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)\right)\right)}^{\lambda }\le 0.$$

If we set $\mathrm{Y}\left(\mathrm{s}\right)=r\left(\mathrm{s}\right){\left(y^{″}\left(\mathrm{s}\right)\right)}^{\lambda }$, we find that $\mathrm{Y}\left(\mathrm{s}\right)$ is a positive solution of

$${\mathrm{Y}}^{\prime }\left(\mathrm{s}\right)+q\left(\mathrm{s}\right){\displaystyle \frac{\hslash \left(\mathrm{s}\right)}{g^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\mathrm{s}\right)\right)\right)}}\mathrm{Y}\left({\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)\right)\le 0.$$

Hence, according to Corollary 1 in [29], it follows that Equation (27) also possesses a positive solution, leading to a contradiction. The proof is complete. □

Corollary 1. 

Assume that the function $\mathcal{N}\in \Re .$ Suppose that there exists a functions $\varpi \in C\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right)$ and $\gamma \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),{\mathbb{R}}^{+}\right)$ such that $\alpha \left(\mathrm{s}\right)\le \varpi \left(\mathrm{s}\right)\le \beta \left(\mathrm{s}\right)$ and ${\gamma }^{\prime }\left(\mathrm{s}\right)\ge 0$. If (26) and

$$\underset{\mathrm{s}\to \infty }{liminf}{\int }_{{\beta }^{-1}\left(\varpi \left(\mathrm{s}\right)\right)}^{\mathrm{s}}q\left(\zeta \right){\displaystyle \frac{\hslash \left(\zeta \right)}{g^{\lambda }\left({\beta }^{-1}\left(\alpha \left(\zeta \right)\right)\right)}}\mathrm{d}\zeta >{\displaystyle \frac{1}{\mathrm{e}}}$$

(41)

hold for any $k_1,k_2\in \left(0,1\right)$, then Equation (1) is oscillatory.

Below, we give a form of the function $\mathcal{N}$ and present some results.

Assume that

$$\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)={\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa },$$

(42)

where $\epsilon ,\kappa >\lambda .$ Then,

$$\begin{array}{ccc}\hfill h\left(\mathrm{s},\nu ,\phi \right)\mathcal{N}\left(\mathrm{s},\nu ,\phi \right)& =& {\displaystyle \frac{\partial \mathcal{N}\left(\mathrm{s},\nu ,\phi \right)}{\partial \nu }}\hfill \\ & =& \epsilon {\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon -1}\left(-{\mathrm{\Phi }}^{\prime }\left({\mathrm{s}}_0,\nu \right)\right){\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }\hfill \\ & & +{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }\kappa {\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa -1}\left({\mathrm{\Phi }}^{\prime }\left({\mathrm{s}}_0,\nu \right)\right).\hfill \end{array}$$

Using Theorem 3, we obtain the following:

Theorem 4. 

Assume that (41) holds and $\mathcal{N}\in \Re .$ Suppose that there exists functions $\varpi \in C\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right)$ and $\gamma \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),{\mathbb{R}}^{+}\right)$ such that $\alpha \left(\mathrm{s}\right)\le \varpi \left(\mathrm{s}\right)\le \beta \left(\mathrm{s}\right)$ and ${\gamma }^{\prime }\left(\mathrm{s}\right)\ge 0$. If

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\phi }^{\mathrm{s}}\left[{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }{k}_1^{\lambda }{k}_2^{\lambda }\gamma \left(\nu \right)q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)-\mathrm{\Psi }\left(\mathrm{s},l,\nu \right)\right]\mathrm{d}\nu >0$$

(43)

for any $k_1,k_2\in \left(0,1\right),$then (1) is oscillatory, where $\epsilon ,\kappa >\lambda .$

Corollary 2. 

Assume that (41) holds and that $\gamma \left(\mathrm{s}\right)=1$ and$\lambda$ is an odd positive integer. If

$$\underset{\mathrm{s}\to \infty }{limsup}{\displaystyle \frac{{\int }_{\phi }^{\mathrm{s}}{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }{k}_1^{\lambda }{k}_2^{\lambda }q\left(\nu \right)\mathsf{\Omega }\left(\nu \right)\mathrm{d}\nu }{{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\epsilon +\kappa -\lambda }}}>{\displaystyle \frac{\mathcal{T}}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}$$

(44)

for any $k_1,k_2\in \left(0,1\right)$, then (1) is oscillatory, where $\epsilon ,\kappa >\lambda .$

Proof. 

In view of Theorem 4 with $\gamma \left(\mathrm{s}\right)=1$, it is sufficient to show that (44) implies that (43) holds. By putting $\sigma =\mathsf{\ell }/\delta$ in (12), we obtain

$$\begin{array}{ccc}\hfill {\int }_0^{\delta }{\left(\delta -\mathsf{\ell }\right)}^{\epsilon +\vartheta -\lambda -1}{\mathsf{\ell }}^{\kappa -\vartheta }\mathrm{d}\mathsf{\ell }& =& {\int }_0^1{\delta }^{\epsilon +\kappa -\lambda }{\left(1-\sigma \right)}^{\epsilon +\vartheta -\lambda -1}{\sigma }^{\kappa -\vartheta }\mathrm{d}\sigma \hfill \\ & =& {\delta }^{\epsilon +\kappa -\lambda }{\int }_0^1{\left(1-\sigma \right)}^{\epsilon +\vartheta -\lambda -1}{\sigma }^{\kappa -\vartheta }\mathrm{d}\sigma .\hfill \end{array}$$

(45)

Assume that $\mathsf{\ell }=\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)$ and$\delta =\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right).$Then, by (26), we obtain

$$\begin{array}{ccc}& & {\int }_{\phi }^{\mathrm{s}}r\left(\nu \right){\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }𝘍\left(\mathrm{s},l,\nu \right)\mathrm{d}\nu \hfill \\ & & ={\int }_0^{\delta }{\left(\delta -\mathsf{\ell }\right)}^{\epsilon -\lambda -1}{\mathsf{\ell }}^{\kappa -\lambda -1}{\left(\kappa \left(\delta -\mathsf{\ell }\right)-\epsilon \mathsf{\ell }\right)}^{\lambda +1}\mathrm{d}\mathsf{\ell }\hfill \\ & & ={\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\epsilon +\kappa -\lambda }\mathcal{T},\mathrm{s}\ge \phi ,\hfill \end{array}$$

(46)

where

$${\left(\kappa \left(\delta -\mathsf{\ell }\right)-\epsilon \mathsf{\ell }\right)}^{\lambda +1}=\sum _{\vartheta =0}^{\lambda +1}{\left(-1\right)}^{\vartheta }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +1}{\vartheta }}\right){\left(\kappa \left(\delta -\mathsf{\ell }\right)\right)}^{\vartheta }{\left(\epsilon \mathsf{\ell }\right)}^{\lambda +1-\vartheta }.$$

From (44) and (46), we satisfy (43). The proof is complete. □

Corollary 3. 

Suppose that (41) holds, $\gamma \left(\mathrm{s}\right)=1$, and$\lambda$ is an odd positive integer. If

$$\underset{\mathrm{s}\to \infty }{limsup}{\displaystyle \frac{{\int }_{\phi }^{\mathrm{s}}{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }\frac{{\alpha }^{2\lambda }\left(\nu \right)}{{\nu }^{\lambda }}q{\left(\nu \right)}^{\lambda }\left(\nu \right)\mathrm{d}\nu }{{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\epsilon +\kappa -\lambda }}}>2^{\lambda }{\displaystyle \frac{\mathcal{T}}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}},$$

(47)

then (1) is oscillatory, where $\epsilon ,\kappa >\lambda .$

Proof. 

We shall show that (47) implies (44). Note that

$${\left({\displaystyle \frac{k_1{k}_2}{2}}\right)}^{\lambda }q{\left(\mathrm{s}\right)}^{\lambda }\left(\mathrm{s}\right){\left({\displaystyle \frac{{\alpha }^2\left(\mathrm{s}\right)}{\mathrm{s}}}\right)}^{\lambda }={\left({\displaystyle \frac{k}{2}}\right)}^{\lambda }q{\left(\mathrm{s}\right)}^{\lambda }\left(\mathrm{s}\right){\left({\displaystyle \frac{{\alpha }^2\left(\mathrm{s}\right)}{\mathrm{s}}}\right)}^{\lambda },$$

(48)

where $k=k_1{k}_2$. On the other hand, (47) implies

$$\underset{\mathrm{s}\to \infty }{limsup}{\displaystyle \frac{{\int }_{\phi }^{\mathrm{s}}{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }\frac{{\alpha }^{2\lambda }\left(\nu \right)}{{\nu }^{\lambda }}q{\left(\nu \right)}^{\lambda }\left(\nu \right)\mathrm{d}\nu }{{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\epsilon +\kappa -\lambda }}}>2^{\lambda }{\displaystyle \frac{\mathcal{T}}{k^{\lambda }{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}},$$

(49)

for $k\in \left(0,1\right)$. Combining (48) and (49), we obtain that (44) holds. Hence, by Corollary 2, we complete the proof. □

Example 1. 

Consider a neutral DE,

$${\left(\varkappa \left(\mathrm{s}\right)+g_0\varkappa \left({\beta }_0\mathrm{s}\right)\right)}^{″\prime }+{\displaystyle \frac{q_0}{{\mathrm{s}}^3}}\varkappa \left({\alpha }_0\mathrm{s}\right)=0,$$

(50)

where $\lambda =1$and $r\left(\mathrm{s}\right)=1.$Now, we note that $g\left(\mathrm{s}\right)=g_0>1,$ $\beta \left(\mathrm{s}\right)={\beta }_0\mathrm{s},$ $q\left(\mathrm{s}\right)=q_0/{\mathrm{s}}^3,$ $\alpha \left(\mathrm{s}\right)={\alpha }_0\mathrm{s},$ ${\beta }_0,{\alpha }_0\in \left(0,1\right),$ and

$$\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)=\left(\mathrm{s}-{\mathrm{s}}_0\right).$$

Let $\varpi \left(\mathrm{s}\right)={\varpi }_0\mathrm{s}$ such that $\alpha \left(\mathrm{s}\right)\le \varpi \left(\mathrm{s}\right)\le \beta \left(\mathrm{s}\right);$ then, it is easy to see that

$$\hslash \left(\mathrm{s}\right)=\left({\displaystyle \frac{g_0-1}{g_0}}\right)\left({\displaystyle \frac{{\varpi }_0^2}{2{\beta }_0^2}}-{\displaystyle \frac{{\varpi }_0{\alpha }_0}{{\beta }_0^2}}+{\displaystyle \frac{{\alpha }_0^2}{2{\beta }_0^2}}\right){\mathrm{s}}^2,$$

and condition (41) becomes

$${\displaystyle \frac{q_0}{g_0}}\left({\displaystyle \frac{g_0-1}{g_0}}\right)\left({\displaystyle \frac{{\varpi }_0^2}{2{\beta }_0^2}}-{\displaystyle \frac{{\varpi }_0{\alpha }_0}{{\beta }_0^2}}+{\displaystyle \frac{{\alpha }_0^2}{2{\beta }_0^2}}\right)\left(ln{\displaystyle \frac{{\beta }_0}{{\varpi }_0}}\right)>{\displaystyle \frac{1}{\mathrm{e}}},$$

Therefore, condition (41) is satisfied if

$$q_0>{\displaystyle \frac{g_0}{\mathrm{e}\left(\frac{g_0-1}{g_0}\right)\left(\frac{{\varpi }_0^2}{2{\beta }_0^2}-\frac{{\varpi }_0{\alpha }_0}{{\beta }_0^2}+\frac{{\alpha }_0^2}{2{\beta }_0^2}\right)\left(ln\frac{{\beta }_0}{{\varpi }_0}\right)}}.$$

(51)

If we choose $\epsilon =4and\kappa =5$, then

$$\mathcal{T}=4.1664\times {10}^{-2}$$

and so

$$2^{\lambda }{\displaystyle \frac{\mathcal{T}}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}=2.0832\times {10}^{-2}.$$

Now, we see that

$$u\left(\alpha \left(\mathrm{s}\right)\right)={\displaystyle \frac{1}{2}}{\left({\alpha }_0\mathrm{s}\right)}^2,u\left({\beta }^{-2}\left(\alpha \left(\mathrm{s}\right)\right)\right)={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\alpha }_0\mathrm{s}}{{\beta }_0^2}}\right)}^2$$

and

$$\left(\mathrm{s}\right)=\left({\displaystyle \frac{1}{g_0}}-{\displaystyle \frac{1}{g_0^2{\beta }_0^4}}\right)={\varrho }_0,$$

and, for $\mathrm{s}>\phi >1$, the left side of (47) becomes

$$\underset{\mathrm{s}\to \infty }{limsup}{\displaystyle \frac{{\int }_{\phi }^{\mathrm{s}}{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)\right)}^{\epsilon }{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\nu \right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\kappa }\frac{{\alpha }^{2\lambda }\left(\nu \right)}{{\nu }^{\lambda }}q{\left(\nu \right)}^{\lambda }\left(\nu \right)\mathrm{d}\nu }{{\left(\mathrm{\Phi }\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathrm{\Phi }\left({\mathrm{s}}_0,\phi \right)\right)}^{\epsilon +\kappa -\lambda }}}={\displaystyle \frac{{\alpha }_0^2{q}_0{\varrho }_0}{280}}$$

So, condition (47) is satisfied if

$$q_0>{\displaystyle \frac{\left(280\right)\left(2.0832\times {10}^{-2}\right)}{{\varrho }_0{\alpha }_0^2}}.$$

(52)

Hence, by Corollary 3, we find that Equation (50) is oscillatory if

$$q_0>max\left\{{\displaystyle \frac{g_0}{\mathrm{e}\left(\frac{g_0-1}{g_0}\right)\left(\frac{{\varpi }_0^2}{2{\beta }_0^2}-\frac{{\varpi }_0{\alpha }_0}{{\beta }_0^2}+\frac{{\alpha }_0^2}{2{\beta }_0^2}\right)\left(ln\frac{{\beta }_0}{{\varpi }_0}\right)}},{\displaystyle \frac{\left(280\right)\left(2.0832\times {10}^{-2}\right)}{{\varrho }_0{\alpha }_0^2}}\right\}.$$

Example 2. 

Consider the third-order neutral DE

$${\left(\varkappa \left(\mathrm{s}\right)+5\varkappa \left(\left(0.8\right)\mathrm{s}\right)\right)}^{‴}+{\displaystyle \frac{q_0}{{\mathrm{s}}^3}}\varkappa \left(\left(0.3\right)\mathrm{s}\right)=0,$$

(53)

where $g_0=5>1,$ ${\beta }_0=\left(0.8\right),$ $q\left(\mathrm{s}\right)=q_0/{\mathrm{s}}^3,$ and ${\alpha }_0=\left(0.3\right).$ Let $\varpi \left(\mathrm{s}\right)={\varpi }_0\mathrm{s}=\left(0.5\right)\mathrm{s};$ then, it is easy to see that

$$\begin{array}{ccc}\hfill \hslash \left(\mathrm{s}\right)& =& \left({\displaystyle \frac{5-1}{5}}\right)\left({\displaystyle \frac{{\left(0.5\right)}^2}{2{\left(0.8\right)}^2}}-{\displaystyle \frac{\left(0.5\right)\left(0.3\right)}{{\left(0.8\right)}^2}}+{\displaystyle \frac{{\left(0.3\right)}^2}{2{\left(0.8\right)}^2}}\right){\mathrm{s}}^2\hfill \\ & =& 0.025{\mathrm{s}}^2,\hfill \end{array}$$

Therefore, condition (41) is satisfied if

$$q_0>{\displaystyle \frac{5}{\mathrm{e}\left(\frac{5-1}{5}\right)\left(\frac{{\left(0.5\right)}^2}{2{\left(0.8\right)}^2}-\frac{\left(0.5\right)\left(0.3\right)}{{\left(0.8\right)}^2}+\frac{{\left(0.3\right)}^2}{2{\left(0.8\right)}^2}\right)\left(ln\frac{0.8}{0.5}\right)}}=156.54.$$

(54)

Let $\epsilon =4and\kappa =5$; then,

$$2^{\lambda }{\displaystyle \frac{\mathcal{T}}{{\left(\lambda +1\right)}^{\left(\lambda +1\right)}}}=2.0832\times {10}^{-2}.$$

Now, we see that

$$\left(\mathrm{s}\right)=0.10234={\varrho }_0$$

and so condition (47) is satisfied if

$$q_0>{\displaystyle \frac{\left(280\right)\left(2.0832\times {10}^{-2}\right)}{\left(0.10234\right){\left(0.3\right)}^2}}=633.29.$$

(55)

Hence, by Corollary 3, we find that Equation (53) is oscillatory if $q_0>633.29.$

Next, by using Theorem 1, where $\mu \left(\mathrm{s}\right)={\mathrm{s}}^3$and $\xi \left(\mathrm{s}\right)={\mathrm{s}}^2,$ we see that condition (6) is satisfied, and condition (7) is satisfied if $q_0>933.56;$ therefore, we find that any solution of (53) is either oscillatory or tends to zero as $\mathrm{s}\to \infty .$

Finally, by using Theorem 2, where $g_{*}\left(\mathrm{s}\right)=0,$ ${\mu }_{*}\left(\mathrm{s}\right)=3\mathrm{s}/4,$ and${\delta }_{*}\left(\mathrm{s}\right)=\mathrm{s}$, we see that condition (9) is satisfied if $q_0>40.278,$ and condition (10) is satisfied if $q_0>1109.5.$ Therefore, we find that Equation (53) is oscillatory if $q_0>1109.5$.

Remark 1. 

We will mention below the most important observations that distinguish the results we obtained compared to some previous studies.

(1)

We note that the approach used in [25] requires the condition $\alpha \circ \beta =\beta \circ \alpha$, which is a harsh condition on the delay functions. In our results, we do not require this condition.

(2)

We also note that condition (7) in Theorem 1 regarding the exclusion of Case (ii) is satisfied if $q_0>933.56$, while, using the results we obtained, we find that condition (47) in Corollary 3 is satisfied if $q_0>633.29$.

(3)

We find that, using Theorem 2, Equation (53) is oscillatory if $q_0>1109.5$, while, using Corollary 3, Equation (53) is oscillatory if $q_0>633.29$.

Therefore, the results we obtained improve the results of [24,25].

Remark 2. 

Let

$${\left(\varkappa \left(\mathrm{s}\right)+5\varkappa \left(\left(0.8\right)\mathrm{s}\right)\right)}^{″\prime }+{\left({\displaystyle \frac{9}{\mathrm{s}}}\right)}^3\varkappa \left(\left(0.3\right)\mathrm{s}\right)=0$$

(56)

be a special case of Equation (53). We find that Theorem 1 and Theorem 2 fail to study the oscillation of (56) because $q_0=9^3$. While using the results we obtained, we see that (56) is oscillatory. Therefore, our results improve the results in [24,25].

4. Conclusions

It is known that there are many papers studying the oscillatory behavior of solutions of third-order DEs in canonical form,

$${\int }_l^{\mathrm{s}}{\displaystyle \frac{1}{r^{1/\lambda }\left(\nu \right)}}\mathrm{d}\nu =\infty ,$$

(57)

and in noncanonical form,

$${\int }_l^{\mathrm{s}}{\displaystyle \frac{1}{r^{1/\lambda }\left(\nu \right)}}\mathrm{d}\nu <\infty ,$$

(58)

but most of them explain that every solution is either oscillatory or tends to zero. In this study, the oscillatory behavior of DE (1) was studied in the canonical form and, using the method of comparison with first-order equations and the Riccati substitution method, we concluded that all solutions to the studied DE are oscillatory. In addition, we provided examples showing that our results improve some of the results in previous studies. It is interesting to use the same approach to study the following cases:

(1)

The oscillation behavior of DE (1) in noncanonical form (58).

(2)

The oscillation behavior of the higher-order DE

$${\left(r\left(\mathrm{s}\right){\left({\left(\varkappa \left(\mathrm{s}\right)+g\left(\mathrm{s}\right)\varkappa \left(\beta \left(\mathrm{s}\right)\right)\right)}^{\left(n-1\right)}\right)}^{\lambda }\right)}^{\prime }+q\left(\mathrm{s}\right){\varkappa }^{\lambda }\left(\alpha \left(\mathrm{s}\right)\right)=0,$$

where $n>3$ is an odd natural number, in canonical form (57).