Kamenev-Type Criteria for Testing the Asymptotic Behavior of Solutions of Third-Order Quasi-Linear Neutral Differential Equations

Abstract

This paper aims to study the asymptotic properties of nonoscillatory solutions (eventually positive or negative) of a class of third-order canonical neutral differential equations. We use Riccati substitution to reduce the order of the considered equation, and then we use the Philos function class to obtain new criteria of the Kamenev type, which guarantees that all nonoscillatory solutions converge to zero. This approach is characterized by the possibility of applying its conditions to a wider area of equations. This is not the only aspect that distinguishes our results; we also use improved relationships between the solution and the corresponding function, which in turn is reflected in a direct improvement of the criteria. The findings in this article extend and generalize previous findings in the literature and also improve some of these findings.

1. Introduction

One type of functional differential equation (FDE) that accounts for the temporal memory of phenomena is the delay differential equation (DDE). Thus, it is simple to understand how these equations are applied in a wide spectrum of fields, including as biological, engineering, and physical models, as well as in other sciences [1,2].

A variety of inquiries concerning oscillatory behavior and asymptotic features of DDE solutions are addressed by oscillation theory, a subfield of qualitative theory. The basic task of oscillation theory is to identify the criteria that eliminate the nonoscillatory solutions. A variety of findings, techniques, and strategies for examining the oscillation of DDEs were gathered in monographs [3,4,5,6].

The investigation of oscillation for solutions of ordinary, partial, and fractional FDEs with delay, neutral delay (NDDE), mixed delay, and damping is a recent, significant expansion and enhancement of the oscillation theory. It is known that differential equations with delay have received the most attention, particularly for non-canonical cases. For instance, refer to [7,8,9,10,11,12,13,14,15] for delay, advanced, and neutral equations, respectively. Furthermore, Refs. [16,17,18,19,20,21] show how investigations of odd-order equations have evolved. Moreover, one may trace the variation of fractional DDEs in Survey [22]. Whereas [23,24,25] dealt with damping equations, and [26,27,28,29] studied mixed equations. Over the past 20 years, functional dynamic equations have also drawn a lot of attention; see, for instance, [30,31,32].

In this paper, we present new criteria for the oscillation of quasi-linear third-order neutral DDEs:

$${\left(a\left(\mathrm{s}\right){\left({\left(x\left(\mathrm{s}\right)+\eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right)\right)}^{″}\right)}^r\right)}^{\prime }+q\left(\mathrm{s}\right)x^r\left(\tau \left(\mathrm{s}\right)\right)=0,$$

(1)

where $\mathrm{s}\ge {\mathrm{s}}_{0,}$ and r is the ratio of any two positive odd integers. Here, in this work, the following assumptions are satisfied:

(I) 

$a\in C^1\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right),a^{\prime }\left(\mathrm{s}\right)\ge 0,$and $\mathcal{T}\left({\mathrm{s}}_0,\infty \right)=\infty ,$ where

$$\mathcal{T}\left(l,\mathrm{s}\right)={\int }_l^{\mathrm{s}}\frac{1}{a^{1/r}\left(\theta \right)}\mathrm{d}\theta ;$$

(2)

(II) 

$\eta ,q\in C\left(\left[{\mathrm{s}}_0,\infty \right),\left[0,\infty \right)\right)$with $0\le \eta \left(\mathrm{s}\right)\le {\eta }_0<\infty$ and $q\left(\mathrm{s}\right)$ does not vanish eventually;

(III) 

$g,\tau \in C\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right),$ $g\left(\mathrm{s}\right)\le \mathrm{s}$, $\tau \left(\mathrm{s}\right)\le \mathrm{s},$ and ${lim}_{\mathrm{s}\to \infty }g\left(\mathrm{s}\right)={lim}_{\mathrm{s}\to \infty }\tau \left(\mathrm{s}\right)=\infty .$

For the solution of (1) on $\left[{\mathrm{s}}_x,\infty \right)$, we refer to a real-valued function $x\in C\left(\left[{\mathrm{s}}_x,\infty \right),\mathbb{R}\right),$ ${\mathrm{s}}_x\ge {\mathrm{s}}_0,$ which satisfies (1) on $\left[{\mathrm{s}}_x,\infty \right)$, and has the properties $\left(x+\eta ·\left(x\circ g\right)\right)\in C^2\left(\left[{\mathrm{s}}_x,\infty \right),\mathbb{R}\right)$ and $\left(a·{\left({\left(x+\eta ·\left(x\circ g\right)\right)}^{″}\right)}^r\right)\in C\left(\left[{\mathrm{s}}_x,\infty \right),\mathbb{R}\right)$. We only consider those solutions $x\left(\mathrm{s}\right)$ of (1) satisfying $sup\left\{\left|x\left(\mathrm{s}\right)\right|:\mathrm{s}\ge \mathrm{S}\right\}>0$ for all $\mathrm{S}\ge {\mathrm{s}}_x$, and we assume that (1) has such solutions. A solution of (1) is said to be oscillatory if it has arbitrarily large zeros in $\left[{\mathrm{s}}_0,\infty \right)$, and is called nonoscillatory otherwise. Equation (1) is said to be oscillatory if all of its solutions are oscillatory.

In the study of neutral equations, the corresponding function z to the solution x, defined as

$$z\left(\mathrm{s}\right):=x\left(\mathrm{s}\right)+\eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right),$$

(3)

is vital.

Numerous studies have been conducted on third-order functional differential equations and the oscillation behavior of solutions; see [33,34,35,36,37,38,39,40]. There exists a theoretical and applicable interest in the problem of oscillatory properties of neutral DDEs; see Hale [1] for some important applications in various applied sciences.

In what follows, we survey some of the most important research that handles the study of third-order NDDEs using different techniques and some different restrictions to obtain conditions that ensure that the solution is oscillatory or tends to zero to cover the the largest area when applied to special cases.

Baculikova and Dzurina [41] tested the asymptotic features of a pair of third-order NDDEs,

$${\left(a\left(\mathrm{s}\right){\left({\left(x\left(\mathrm{s}\right)\pm \eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right)\right)}^{″}\right)}^r\right)}^{\prime }+q\left(\mathrm{s}\right)x^r\left(\tau \left(\mathrm{s}\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0,$$

(4)

where $0\le \eta \left(\mathrm{s}\right)\le \eta <1.$They established novel sufficient conditions that all confirm that all nonoscillatory solutions of (4) converge to zero.

Thandapani and Li [42] studied the oscillatory features of the third-order NDDE (1), where $g^{\prime }\left(\mathrm{s}\right)\ge g_0>0,\tau \circ g=g\circ \tau$ and $0\le \eta \left(\mathrm{s}\right)\le \eta <\infty .$By using the Riccati transformation, they established some sufficient criteria, which confirm that any solution of (1) is oscillatory or tends to zero.

Graef et al. [43] discussed the oscillatory properties of a class of solutions of third-order nonlinear NDDEs:

$${\left({\left({\left(x\left(\mathrm{s}\right)+\eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right)\right)}^{″}\right)}^r\right)}^{\prime }+q\left(\mathrm{s}\right)x^r\left(\tau \left(\mathrm{s}\right)\right)=0$$

where $a=1$and $\eta \left(\mathrm{s}\right)\ge 1$. They presented novel sufficient criteria for any solution of the studied equation to be either oscillating or converging to zero.

Kumar and Ganesan [44] discussed the third-order nonlinear NDDE in the form

$${\left(a\left(\mathrm{s}\right)\phi \left(z^{″}\left(\mathrm{s}\right)\right)\right)}^{\prime }+q\left(\mathrm{s}\right)\phi \left(x\left(\tau \left(\mathrm{s}\right)\right)\right)=0,\mathrm{s}\ge {\mathrm{s}}_0>0,$$

(5)

where $\phi \left(u\right)={\left|u\right|}^{r-1}u,g^{\prime }\left(\mathrm{s}\right)\ge g_0>0$ and$\tau \circ g=g\circ \tau .$ The third- and first-order equation comparison principles provide the foundation for the obtained results. Below, we present some results obtained in previous studies to facilitate the reader’s understanding.

Theorem 1

([42]). Let $r\ge 1,\tau \in C^1\left(\left[{\mathrm{s}}_0,\infty \right)\right)$ and ${\tau }^{\prime }>0.$ Assume that

$${\int }_{{\mathrm{s}}_0}^{\infty }{\int }_v^{\infty }{\left(\frac{1}{a\left(g\left(u\right)\right)}{\int }_u^{\infty }Q\left(\theta \right)\mathrm{d}\theta \right)}^{1/r}\mathrm{d}u\mathrm{d}v=\infty ,$$

holds and $\tau \left(\mathrm{s}\right)\le g\left(\mathrm{s}\right)$. Moreover, assuming there is a function $\rho \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),\left(0,\infty \right)\right)$, for all ${\mathrm{s}}_1\ge {\mathrm{s}}_0$ large enough, there exists ${\mathrm{s}}_1\ge {\mathrm{s}}_0$ where

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{{\mathrm{s}}_2}^s\left(\frac{\rho \left(l\right)Q\left(l\right)}{2^{r+1}}-\frac{\left(1+\frac{{\eta }_0^r}{g_0}\right){\left({\left({\rho }^{\prime }\left(l\right)\right)}_{+}\right)}^{r+1}}{{\left(r+1\right)}^{r+1}{\left(\rho \left(l\right){\beta }_1\left(\tau \left(l\right),{\mathrm{s}}_1\right){\tau }^{\prime }\left(l\right)\right)}^r}\right)\mathrm{d}l=\infty ,$$

for $Q=$min$\left\{q\left(\mathrm{s}\right),q\left(g\left(\mathrm{s}\right)\right)\right\},{\left({\rho }^{\prime }\left(\mathrm{s}\right)\right)}_{+}:=max\left\{0,{\rho }^{\prime }\left(\mathrm{s}\right)\right\}$and ${\beta }_1\left(\mathrm{s},{\mathrm{s}}_1\right)={\int }_{{\mathrm{s}}_1}^{\mathrm{s}}1l{a}^{1/r}\left(\theta \right)\mathrm{d}\theta .$ Then, (1) is almost oscillatory.

Theorem 2

([44]). Let $\tau \left(\mathrm{s}\right)\le g\left(\mathrm{s}\right)\le 1.$ Assuming that $0<r\le 1,$

$${\int }_{{\mathrm{s}}_1}^{\mathrm{s}}{g}^{\prime }\left(v\right){\int }_v^{\infty }{\left(\frac{g^{\prime }\left(u\right)}{a\left(g\left(u\right)\right)}{\int }_u^{\infty }Q\left(t\right)\mathrm{d}t\right)}^{1/r}\mathrm{d}u\mathrm{d}v=\infty$$

and the first-order DDE

$$w^{\prime }\left(\mathrm{s}\right)+\frac{g_0}{g_o+{\eta }_0^r}{Q}_1\left(\mathrm{s}\right)w\left(g^{-1}\left(\tau \left(\mathrm{s}\right)\right)\right)=0$$

oscillates, then any positive solution of (5) meets ${lim}_{s\to \infty }$$x\left(\mathrm{s}\right)=0$, where $g^{-1}\left(\mathrm{s}\right)$ is an inverse function of $g\left(\mathrm{s}\right),$ and

$$Q_1\left(\mathrm{s}\right)=Q\left(\mathrm{s}\right)\left({\int }_{{\mathrm{s}}_1}^{\tau \left(\mathrm{s}\right)}\left(\mathcal{T}\left(l,t\right)-\mathcal{T}\left(l,t_1\right)\right)\mathrm{d}t\right).$$

Our goal in this study was to examine the asymptotic properties of a class of neutral third-order NDDEs. Based on the improved relationship between x and z that was derived in [45], we obtained new relationships between x and z. The new relationship is characterized by taking into account both cases $\eta \le 1$ and $\eta >1$; this was not common in previous third-order studies. We present Kamenev-type criteria that ensure that all solutions of the neutral DDE, (1), either converge to zero or are oscillatory. We begin by deducing some new relationships that help improve the approach. Then, we use the Philos function class to obtain the required conditions. The criteria we obtain improve and extend some results from previous studies. Finally, we employ the results in the special case of our studied equation.

2. Preliminaries

We begin with lemmas, notations that are required throughout this paper. For convenience, we use the symbol $\mathcal{P}$ to state the category of all eventually positive solutions to (1), the symbol ${\mathcal{P}}_{\downarrow }$ to denote the class of solutions $x\in \mathcal{P}$, whose corresponding function confirms $z^{\prime }\left(t\right)<0$, and the symbol ${\mathcal{P}}_{\uparrow }$ to denote the class of solutions $x\in \mathcal{P}$ whose corresponding function confirms $z^{\prime }\left(t\right)>0$.

Lemma 1

([41] (Lemma 1)). Assume that $x\in \mathcal{P}$. Then, z meets one of the following possible cases, eventually:

(i) 

$z>0,z^{\prime }>0$and $z^{″}>0;$

(ii) 

$z>0,z^{\prime }<0$and $z^{″}>0.$

Lemma 2

([41] (Lemma 2)). Suppose that $x\in {\mathcal{P}}_{\downarrow }$. If

$${\int }_{{\mathrm{s}}_0}^{\infty }{\int }_v^{\infty }{\left(\frac{1}{a\left(u\right)}{\int }_u^{\infty }q\left(\theta \right)\mathrm{d}\theta \right)}^{1/r}\mathrm{d}u\mathrm{d}v=\infty ,$$

(6)

then ${lim}_{\mathrm{s}\to \infty }x\left(\mathrm{s}\right)={lim}_{\mathrm{s}\to \infty }z\left(\mathrm{s}\right)=0.$

Lemma 3

([41] (Lemma 3)). Suppose that $u\in C^2\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right).$Assume that $u\left(\mathrm{s}\right)>0,u^{\prime }\left(\mathrm{s}\right)\ge 0$and$u^{″}\left(\mathrm{s}\right)\le 0,$on $\left[{\mathrm{s}}_0,\infty \right).$ Then, there exist a ${\mathrm{s}}_1\ge {\mathrm{s}}_0$ for each $k_1\in \left(0,1\right)$ such that

$$\frac{u\left(\tau \left(\mathrm{s}\right)\right)}{u\left(\mathrm{s}\right)}\ge k_1\frac{\tau \left(\mathrm{s}\right)}{\mathrm{s}},$$

where $\mathrm{s}\ge {\mathrm{s}}_1.$

Lemma 4

([46]). Suppose that $u\in C^{m+1}\left(\left[{\mathrm{s}}_0,\infty \right),\mathbb{R}\right)$, $u^{\left(j\right)}\left(\mathrm{s}\right)>0,$for $j=0,1,\dots ,m,$ and $u^{\left(m+1\right)}\left(\mathrm{s}\right)\le 0.$ Then, there exist a ${\mathrm{s}}_1\ge {\mathrm{s}}_0$, for each $k_2\in \left(0,1\right)$, such that

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

where$\mathrm{s}\ge {\mathrm{s}}_1.$

Notation 1.

For simplicity, let $G^{\left[0\right]}\left(\mathrm{s}\right):=\mathrm{s},G^{\left[j\right]}\left(\mathrm{s}\right)=G\left(G^{\left[j-1\right]}\left(\mathrm{s}\right)\right),G^{\left[-j\right]}\left(\mathrm{s}\right)=G^{-1}\left(G^{\left[-j+1\right]}\left(\mathrm{s}\right)\right),$ for $j=1,2,\dots$.

Lemma 5

([45]). Suppose that $x\in {\mathcal{P}}_{\uparrow }\cup {\mathcal{P}}_{\downarrow }$. Then,

$$x>\sum _{k=0}^m\left(\prod _{n=0}^{2k}\eta \left(g^{\left[n\right]}\right)\right)\left(\frac{z\left(g^{\left[2k\right]}\right)}{\eta \left(g^{\left[2k\right]}\right)}-z\left(g^{\left[2k+1\right]}\right)\right),$$

(7)

eventually, where $m>0,m\in \mathbb{Z}$.

Let ℜ be class of functions, the function $\mathcal{K}\in \Re ,$where $\mathcal{K}\in C\left(H,\mathbb{R}\right)$, $H=\left\{\left(\mathrm{s},\theta ,\ell \right):\right.$$\left.{\mathrm{s}}_0\le \ell \le \theta \le \mathrm{s}\le \infty \right\}$, if $\mathcal{K}$ satisfies the following hypotheses:

(1)

$\mathcal{K}\left(\mathrm{s},\mathrm{s},\ell \right)=0,\mathcal{K}\left(\mathrm{s},\ell ,\ell \right)=0,\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)\ne 0,$for $\ell <\theta <\mathrm{s};$

(2)

$\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)$ possesses the partial derivative $\partial \mathcal{K}/\partial \theta$ on H with the condition that $\partial \mathcal{K}/\partial \theta$ can be integrated locally in terms of $\theta$ in H and

$$\frac{\partial \mathcal{K}\left(\mathrm{s},\theta ,\ell \right)}{\partial \theta }=h\left(\mathrm{s},\theta ,\ell \right)\mathcal{K}\left(\mathrm{s},\theta ,\ell \right),$$

(8)

for some $h\in C\left(H,\mathbb{R}\right)$.

This class of functions is defined by Philos [47].

Notation 2.

During the main results, we need to define the following abbreviations:

$$\psi \left(\mathrm{s}\right)={\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\mathcal{T}\left({\mathrm{s}}_0,u\right)\mathrm{d}u,$$

$${\Theta }_1\left(\mathrm{s}\right)=\sum _{k=0}^m\left(\prod _{n=0}^{2k}\eta \left(g^{\left[n\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)\right)\left(\frac{1}{\eta \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}-1\right)\frac{\psi \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}{\psi \left(\tau \left(\mathrm{s}\right)\right)},$$

$${\Theta }_2\left(\mathrm{s}\right)=\sum _{k=1}^m\left(\prod _{j=1}^{2k-1}\frac{1}{\eta \left(g^{\left[-j\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}\right)\left(1-\frac{1}{\eta \left(g^{\left[-2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}\frac{\psi \left(g^{\left[-2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}{\psi \left(g^{\left[-2k+1\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}\right),$$

$$\Theta \left(\mathrm{s}\right)=\left\{\begin{array}{cc}1,\hfill & \mathit{for}\eta =0\hfill \\ {\Theta }_1\left(\mathrm{s}\right),\hfill & \mathit{for}0<\eta <1\hfill \\ {\Theta }_2\left(\mathrm{s}\right),\hfill & \mathit{for}\eta >\psi \left(g^{\left[-2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)/\psi \left(g^{\left[-2k+1\right]}\left(\tau \left(\mathrm{s}\right)\right)\right),\hfill \end{array}\right.$$

and

$$M_0=\sum _{j=0}^{r+1}\left(\genfrac{}{}{0pt}{}{r+1}{r-j+1}\right){\left(-1\right)}^{r-j+1}{\gamma }^{r-j+1}{\lambda }^j\frac{\Gamma \left(\gamma +j-r\right)\Gamma \left(\lambda -j+1\right)}{\Gamma \left(\gamma +\lambda -r+1\right)},\mathit{for}r\in {\mathbb{Z}}^{+},$$

where $\gamma ,$ $\lambda \in \left(r,\infty \right),$

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

and

$$k_0=\frac{1}{{\left(r+1\right)}^{\left(r+1\right)}}.$$

3. Main Results

We present new conditions that guarantee that each solution to DDE (1) oscillates or converges to zero.

Theorem 3.

Suppose that (6) holds and the function $\mathcal{K}\in \Re$. In the event that a function, $\rho \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),{\mathbb{R}}^{+}\right)$, is present and satisfies ${\rho }^{\prime }\left(\mathrm{s}\right)\ge 0$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)\rho \left(\theta \right)\left(k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}q\left(\theta \right){\Theta }^r\left(\theta \right)-k_{0}a\left(\theta \right){\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1}\right)\mathrm{d}\theta >0,$$

(9)

for any $k_1,k_2\in \left(0,1\right)$, then the solution $x\left(\mathrm{s}\right)$ oscillates or tends to zero.

Proof. 

Suppose that $x\in \mathcal{P}$. Suppose that there is an $\mathrm{s}\ge {\mathrm{s}}_1$ such that $x\left(\mathrm{s}\right)>0,x\left(g\left(\mathrm{s}\right)\right)>0,$and $x\left(\tau \left(\mathrm{s}\right)\right)>0$. Clearly, $z\left(\mathrm{s}\right)>0$, $\mathrm{s}\ge {\mathrm{s}}_1$. From Lemma 1, we can see that (i) or (ii) is satisfied.

Assume that (ii) is satisfied. Since (6) holds, following from Lemma 2 that ${lim}_{\mathrm{s}\to \infty }x\left(\mathrm{s}\right)=0$.

Now, assume that (i) is satisfied. We have

$$\begin{array}{ccc}\hfill z^{\prime }\left(\mathrm{s}\right)& \ge & {\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\frac{a^{1/r}\left(u\right)z^{″}\left(u\right)}{a^{1/r}\left(u\right)}\mathrm{d}u\ge a^{1/r}\left(\mathrm{s}\right)z^{″}\left(\mathrm{s}\right){\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\frac{1}{a^{1/r}\left(u\right)}\mathrm{d}u\hfill \\ & \ge & a^{1/r}\left(\mathrm{s}\right)z^{″}\left(\mathrm{s}\right)\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right),\mathrm{s}\ge {\mathrm{s}}_1\hfill \end{array}$$

therefore, we find

$${\left(\frac{z^{\prime }\left(\mathrm{s}\right)}{\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)}\right)}^{\prime }=\frac{\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)z^{″}\left(\mathrm{s}\right)-z^{\prime }\left(\mathrm{s}\right)a^{-1/r}\left(\mathrm{s}\right)}{{\mathcal{T}}^2\left({\mathrm{s}}_0,\mathrm{s}\right)}=\frac{a^{1/r}\left(\mathrm{s}\right)\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)z^{″}\left(\mathrm{s}\right)-z^{\prime }\left(\mathrm{s}\right)}{a^{1/r}\left(\mathrm{s}\right){\mathcal{T}}^2\left({\mathrm{s}}_0,\mathrm{s}\right)}\le 0,\mathrm{s}\ge {\mathrm{s}}_1.$$

(10)

Since

$$z\left(\mathrm{s}\right)\ge {\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\frac{\mathcal{T}\left({\mathrm{s}}_0,u\right)z^{\prime }\left(u\right)}{\mathcal{T}\left({\mathrm{s}}_0,u\right)}\mathrm{d}u,\mathit{for}\mathrm{s}\ge {\mathrm{s}}_1$$

by using (10), we obtain

$$z\left(\mathrm{s}\right)\ge \frac{z^{\prime }\left(\mathrm{s}\right)}{\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)}{\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\mathcal{T}\left({\mathrm{s}}_0,u\right)\mathrm{d}u\ge \frac{z^{\prime }\left(\mathrm{s}\right)}{\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)}\psi \left(\mathrm{s}\right)$$

and so

$${\left(\frac{z\left(\mathrm{s}\right)}{\psi \left(\mathrm{s}\right)}\right)}^{\prime }=\frac{\psi \left(\mathrm{s}\right)z^{\prime }\left(\mathrm{s}\right)-z\left(\mathrm{s}\right)\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)}{{\psi }^2\left(\mathrm{s}\right)}=\frac{{\mathcal{T}}^{-1}\left({\mathrm{s}}_0,\mathrm{s}\right)\psi \left(\mathrm{s}\right)z^{\prime }\left(\mathrm{s}\right)-z\left(\mathrm{s}\right)}{{\mathcal{T}}^{-1}\left({\mathrm{s}}_0,\mathrm{s}\right){\psi }^2\left(\mathrm{s}\right)}\le 0,\mathrm{s}\ge {\mathrm{s}}_1.$$

(11)

From (3), we have

$$x\left(\mathrm{s}\right)=z\left(\mathrm{s}\right)-\eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right).$$

Now, assume that $\eta <1$. Since $z\left(\mathrm{s}\right)$ satisfies (i), following Lemma 5, that (7) holds. Using $g^{\left[2k+1\right]}\left(\mathrm{s}\right)\le g^{\left[2k\right]}\left(\mathrm{s}\right)\le \mathrm{s},z^{\prime }\left(\mathrm{s}\right)>0$and (11), we obtain

$$z\left(g^{\left[2k+1\right]}\left(\mathrm{s}\right)\right)\le z\left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)\le z\left(\mathrm{s}\right),\mathrm{s}\ge {\mathrm{s}}_1$$

and

$$z\left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)\ge \frac{\psi \left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)z\left(\mathrm{s}\right)}{\psi \left(\mathrm{s}\right)},\mathit{for}k=0,1,\dots .$$

Thus, we see that (7) becomes

$$\begin{array}{ccc}\hfill x\left(\mathrm{s}\right)& >& \sum _{k=0}^m\left(\prod _{n=0}^{2k}\eta \left(g^{\left[n\right]}\left(\mathrm{s}\right)\right)\right)\left(\frac{z\left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)}{\eta \left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)}-z\left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)\right)\hfill \\ & >& \sum _{k=0}^m\left(\prod _{n=0}^{2k}\eta \left(g^{\left[n\right]}\left(\mathrm{s}\right)\right)\right)\left(\frac{1}{\eta \left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)}-1\right)z\left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)\hfill \\ & >& z\left(\mathrm{s}\right)\sum _{k=0}^m\left(\prod _{n=0}^{2k}\eta \left(g^{\left[n\right]}\left(\mathrm{s}\right)\right)\right)\left(\frac{1}{\eta \left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)}-1\right)\frac{\psi \left(g^{\left[2k\right]}\left(\mathrm{s}\right)\right)}{\psi \left(\mathrm{s}\right)},\mathrm{s}\ge {\mathrm{s}}_1.\hfill \end{array}$$

Using this inequality in (1), we obtain

$${\left(a\left(\mathrm{s}\right){\left({\left(x\left(\mathrm{s}\right)+\eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right)\right)}^{″}\right)}^r\right)}^{\prime }\le -q\left(\mathrm{s}\right)z^r\left(\tau \left(\mathrm{s}\right)\right){\Theta }^r\left(\mathrm{s}\right),\mathrm{s}\ge {\mathrm{s}}_1.$$

(12)

Now, assume that $\eta >1$. It follows from the (3) that

$$\begin{array}{ccc}\hfill x\left(\mathrm{s}\right)& =& \frac{1}{\eta \left(g^{-1}\left(\mathrm{s}\right)\right)}\left(z\left(g^{-1}\left(\mathrm{s}\right)\right)-x\left(g^{-1}\left(\mathrm{s}\right)\right)\right)\hfill \\ & =& \frac{z\left(g^{-1}\left(\mathrm{s}\right)\right)}{\eta \left(g^{-1}\left(\mathrm{s}\right)\right)}-\frac{1}{\eta \left(g^{-1}\left(\mathrm{s}\right)\right)\eta \left(g^{\left[-2\right]}\left(\mathrm{s}\right)\right)}\left(z\left(g^{\left[-2\right]}\left(\mathrm{s}\right)\right)-x\left(g^{\left[-2\right]}\left(\mathrm{s}\right)\right)\right)\hfill \\ & =& \frac{z\left(g^{-1}\left(\mathrm{s}\right)\right)}{\eta \left(g^{-1}\left(\mathrm{s}\right)\right)}\hfill \\ & & -\frac{1}{\eta \left(g^{-1}\left(\mathrm{s}\right)\right)\eta \left(g^{\left[-2\right]}\left(\mathrm{s}\right)\right)}\left(z\left(g^{\left[-2\right]}\left(\mathrm{s}\right)\right)-\frac{1}{\eta \left(g^{\left[-3\right]}\left(\mathrm{s}\right)\right)}\left(z\left(g^{\left[-3\right]}\left(\mathrm{s}\right)\right)-x\left(g^{\left[-3\right]}\left(\mathrm{s}\right)\right)\right)\right)\hfill \\ & =& \frac{z\left(g^{-1}\left(\mathrm{s}\right)\right)}{{\prod }_{j=1}^1\eta \left(g^{\left[-j\right]}\left(\mathrm{s}\right)\right)}-\frac{z\left(g^{\left[-2\right]}\left(\mathrm{s}\right)\right)}{{\prod }_{j=1}^2\eta \left(g^{\left[-j\right]}\left(\mathrm{s}\right)\right)}\hfill \\ & & +\frac{1}{{\prod }_{j=1}^3\eta \left(g^{\left[-j\right]}\left(\mathrm{s}\right)\right)}\left(z\left(g^{\left[-3\right]}\left(\mathrm{s}\right)\right)-x\left(g^{\left[-3\right]}\left(\mathrm{s}\right)\right)\right),\mathrm{s}\ge {\mathrm{s}}_1,\hfill \end{array}$$

and so on. Thus, we have

$$x\left(\mathrm{s}\right)>\sum _{k=1}^m\left(\prod _{j=1}^{2k-1}\frac{1}{\eta \left(g^{\left[-j\right]}\left(\mathrm{s}\right)\right)}\right)\left(z\left(g^{\left[-2k+1\right]}\left(\mathrm{s}\right)\right)-\frac{1}{\eta \left(g^{\left[-2k\right]}\left(\mathrm{s}\right)\right)}z\left(g^{\left[-2k\right]}\left(\mathrm{s}\right)\right)\right),\mathrm{s}\ge {\mathrm{s}}_1.$$

(13)

From the facts that $g^{\left[-2k\right]}\ge g^{\left[-2k+1\right]}\ge \mathrm{s},z^{\prime }>0$ and (11), we arrive at

$$z\left(g^{\left[-2k+1\right]}\left(\mathrm{s}\right)\right)\ge z\left(\mathrm{s}\right),\mathrm{s}\ge {\mathrm{s}}_1$$

(14)

and

$$z\left(g^{\left[-2k\right]}\left(\mathrm{s}\right)\right)\le \frac{\psi \left(g^{\left[-2k\right]}\left(\mathrm{s}\right)\right)z\left(g^{\left[-2k+1\right]}\left(\mathrm{s}\right)\right)}{\psi \left(g^{\left[-2k+1\right]}\left(\mathrm{s}\right)\right)},\mathrm{s}\ge {\mathrm{s}}_1.$$

(15)

Using (14) and (15) in (13), we obtain

$$x\left(\mathrm{s}\right)>z\left(\mathrm{s}\right)\sum _{k=1}^m\left(\prod _{j=1}^{2k-1}\frac{1}{\eta \left(g^{\left[-j\right]}\left(\mathrm{s}\right)\right)}\right)\left(1-\frac{1}{\eta \left(g^{\left[-2k\right]}\left(\mathrm{s}\right)\right)}\frac{\psi \left(g^{\left[-2k\right]}\left(\mathrm{s}\right)\right)}{\psi \left(g^{\left[-2k+1\right]}\left(\mathrm{s}\right)\right)}\right),\mathrm{s}\ge {\mathrm{s}}_1$$

and so

$$x\left(\tau \left(\mathrm{s}\right)\right)>z\left(\tau \left(\mathrm{s}\right)\right)\sum _{k=1}^m\left(\prod _{j=1}^{2k-1}\frac{1}{\eta \left(g^{\left[-j\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}\right)\left(1-\frac{1}{\eta \left(g^{\left[-2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}\frac{\psi \left(g^{\left[-2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}{\psi \left(g^{\left[-2k+1\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}\right),\mathrm{s}\ge {\mathrm{s}}_1.$$

From the above inequality and (1), we obtain (12), therefore,

$${\left(a\left(\mathrm{s}\right){\left({\left(x\left(\mathrm{s}\right)+\eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right)\right)}^{″}\right)}^r\right)}^{\prime }\le 0.$$

(16)

Using (16), $a^{\prime }\left(\mathrm{s}\right)\ge 0,$ and $z^{″}\left(\mathrm{s}\right)>0,$ we have $z^{‴}\left(\mathrm{s}\right)\le 0.$ Therefore, there exists an ${\mathrm{s}}_2\ge {\mathrm{s}}_1$ such that $z\left(\mathrm{s}\right)$ satisfies

$$z\left(\tau \left(\mathrm{s}\right)\right)>0,z^{\prime }\left(\mathrm{s}\right)>0,z^{″}\left(\mathrm{s}\right)>0,z^{‴}\left(\mathrm{s}\right)\le 0,\mathrm{s}\ge {\mathrm{s}}_2.$$

We define $\omega \left(\mathrm{s}\right)$ as follow:

$$\omega \left(\mathrm{s}\right)=\rho \left(\mathrm{s}\right)\frac{a\left(\mathrm{s}\right){\left(z^{″}\left(\mathrm{s}\right)\right)}^r}{{\left(z^{\prime }\left(\mathrm{s}\right)\right)}^r},\mathrm{s}\ge {\mathrm{s}}_2.$$

(17)

We see that $\omega \left(\mathrm{s}\right)>0$ and

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\mathrm{s}\right)& =& {\rho }^{\prime }\left(\mathrm{s}\right)\frac{a\left(\mathrm{s}\right){\left(z^{″}\left(\mathrm{s}\right)\right)}^r}{{\left(z^{\prime }\left(\mathrm{s}\right)\right)}^r}\hfill \\ & & +\frac{\rho \left(\mathrm{s}\right){\left(z^{\prime }\left(\mathrm{s}\right)\right)}^r{\left(a\left(\mathrm{s}\right){\left(z^{″}\left(\mathrm{s}\right)\right)}^r\right)}^{\prime }-r\rho \left(\mathrm{s}\right)a\left(\mathrm{s}\right){\left(z^{″}\left(\mathrm{s}\right)\right)}^r{\left(z^{\prime }\left(\mathrm{s}\right)\right)}^{r-1}{z}^{″}\left(\mathrm{s}\right)}{{\left(z^{\prime }\left(\mathrm{s}\right)\right)}^{2r}},\mathrm{s}\ge {\mathrm{s}}_2.\hfill \end{array}$$

By using (12) and (17), we have

$${\omega }^{\prime }\left(\mathrm{s}\right)\le {\rho }^{\prime }\left(\mathrm{s}\right)\frac{\omega \left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}-\rho \left(\mathrm{s}\right)\frac{q\left(\mathrm{s}\right)z^r\left(\tau \left(\mathrm{s}\right)\right){\Theta }^r\left(\mathrm{s}\right)}{{\left(z^{\prime }\left(\mathrm{s}\right)\right)}^r}-r\frac{{\omega }^{\left(r+1\right)/r}\left(\mathrm{s}\right)}{a^{1/r}\left(\mathrm{s}\right){\rho }^{1/r}\left(\mathrm{s}\right)},\mathrm{s}\ge {\mathrm{s}}_2.$$

(18)

By using Lemma 3 with $u\left(\mathrm{s}\right)=z^{\prime }\left(\mathrm{s}\right),$ there exists a ${\mathrm{s}}_3\ge {\mathrm{s}}_2$ such that

$$\frac{z^{\prime }\left(\tau \left(\mathrm{s}\right)\right)}{z^{\prime }\left(\mathrm{s}\right)}\ge k_1\frac{\tau \left(\mathrm{s}\right)}{\mathrm{s}},\mathrm{s}\ge {\mathrm{s}}_3\ge {\mathrm{s}}_2.$$

(19)

By using Lemma 4, we have

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

(20)

From (19) and (20), we obtain

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

(21)

Using (18) and (21), we obtain

$${\omega }^{\prime }\left(\mathrm{s}\right)\le {\rho }^{\prime }\left(\mathrm{s}\right)\frac{\omega \left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}-k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\mathrm{s}\right)}{{\left(2\mathrm{s}\right)}^r}\rho \left(\mathrm{s}\right)q\left(\mathrm{s}\right){\Theta }^r\left(\mathrm{s}\right)-r\frac{{\omega }^{\left(r+1\right)/r}\left(\mathrm{s}\right)}{a^{1/r}\left(\mathrm{s}\right){\rho }^{1/r}\left(\mathrm{s}\right)},\mathrm{s}\ge {\mathrm{s}}_3$$

and so

$$k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\mathrm{s}\right)}{{\left(2\mathrm{s}\right)}^r}\rho \left(\mathrm{s}\right)q\left(\mathrm{s}\right){\Theta }^r\left(\mathrm{s}\right)\le -{\omega }^{\prime }\left(\mathrm{s}\right)+{\rho }^{\prime }\left(\mathrm{s}\right)\frac{\omega \left(\mathrm{s}\right)}{\rho \left(\mathrm{s}\right)}-r\frac{{\omega }^{\left(r+1\right)/r}\left(\mathrm{s}\right)}{a^{1/r}\left(\mathrm{s}\right){\rho }^{1/r}\left(\mathrm{s}\right)},\mathrm{s}\ge {\mathrm{s}}_3.$$

(22)

Multiplying the above inequality by $\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)$ and integrating from $\ell \ge {\mathrm{s}}_3$ to $\mathrm{s}$, we obtain

$$\begin{array}{ccc}& & {\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}\rho \left(\theta \right)q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta \hfill \\ & \le & -{\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right){\omega }^{\prime }\left(\theta \right)\mathrm{d}\theta +{\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right){\rho }^{\prime }\left(\theta \right)\frac{\omega \left(\theta \right)}{\rho \left(\theta \right)}\mathrm{d}\theta \hfill \\ \hfill & & -r{\int }_{\ell }^{\mathrm{s}}\frac{\mathcal{K}\left(\mathrm{s},\theta ,\ell \right){\omega }^{\left(r+1\right)/r}\left(\theta \right)}{a^{1/r}\left(\theta \right){\rho }^{1/r}\left(\theta \right)}\mathrm{d}\theta ,\ell \ge {\mathrm{s}}_3.\hfill \end{array}$$

(23)

By using (8), for all $\mathrm{s}\ge \ell ,$we have

$$\begin{array}{ccc}& & {\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}\rho \left(\theta \right)q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta \hfill \\ & \le & {\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)\left(\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)\omega \left(\theta \right)-r\frac{{\omega }^{\left(r+1\right)/r}\left(\theta \right)}{a^{1/r}\left(\theta \right){\rho }^{1/r}\left(\theta \right)}\right)\mathrm{d}\theta ,\mathrm{s}\ge \ell .\hfill \end{array}$$

(24)

Set

$$F\left(\vartheta \right)=\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)\vartheta -r\frac{{\vartheta }^{\left(r+1\right)/r}}{a^{1/r}\left(\theta \right){\rho }^{1/r}\left(\theta \right)},\mathrm{s}\ge \ell .$$

A simple calculation implies when

$$\vartheta =k_0\rho \left(\theta \right)a\left(\theta \right){\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^r,\mathrm{s}\ge \ell$$

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

$$k_0\rho \left(\theta \right)a\left(\theta \right){\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1},\mathrm{s}\ge \ell$$

that is,

$$F\left(\vartheta \right)\le F_{max}=k_0\rho \left(\theta \right)a\left(\theta \right){\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1},\mathrm{s}\ge \ell .$$

(25)

Using (24) and (25), we have

$$\begin{array}{ccc}\hfill 0& \ge & {\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}\rho \left(\theta \right)q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta \hfill \\ & & -{\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)k_0\rho \left(\theta \right)a\left(\theta \right){\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1}\mathrm{d}\theta ,\mathrm{s}\ge \ell \hfill \end{array}$$

and so

$${\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)\rho \left(\theta \right)\left(k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}q\left(\theta \right){\Theta }^r\left(\theta \right)-k_{0}a\left(\theta \right){\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1}\right)\mathrm{d}\theta \le 0,\mathrm{s}\ge \ell .$$

Taking the super limit, we obtain

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\ell }^{\mathrm{s}}\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)\rho \left(\theta \right)\left(k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}q\left(\theta \right){\Theta }^r\left(\theta \right)-k_{0}a\left(\theta \right){\left(h\left(\mathrm{s},\theta ,\ell \right)+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1}\right)\mathrm{d}\theta \le 0,\mathrm{s}\ge \ell .$$

This contradicts (9) and the proof is complete. □

Theorem 4.

Assume that (6) holds and

$$\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)={\left(\mathrm{s}-\theta \right)}^{\sigma }{\left(\theta -\ell \right)}^{\vartheta },$$

where $\sigma ,\vartheta$ are constants greater than r. If there is a $\rho \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),{\mathbb{R}}^{+}\right)$ satisfying ${\rho }^{\prime }\left(\mathrm{s}\right)\ge 0$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\ell }^{\mathrm{s}}{\left(\mathrm{s}-\theta \right)}^{\sigma }{\left(\theta -\ell \right)}^{\vartheta }\rho \left(\theta \right)\Phi \left(\mathrm{s},l,\theta \right)\mathrm{d}\theta >0$$

(26)

for any $k_1,k_2\in \left(0,1\right),$then, the solution $x\left(\mathrm{s}\right)$ is oscillatory or converges to zero, where

$$\Phi \left(\mathrm{s},l,\theta \right):=k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}q\left(\theta \right){\Theta }^r\left(\theta \right)-k_{0}a\left(\theta \right){\left(\frac{\vartheta \mathrm{s}-\left(\sigma +\vartheta \right)\theta +\sigma \ell }{\left(\mathrm{s}-\theta \right)\left(\theta -\ell \right)}+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1}.$$

Proof. 

Suppose that $x\in \mathcal{P}$. Suppose that there is an $\mathrm{s}\ge {\mathrm{s}}_1$ such that $x\left(\mathrm{s}\right)>0,x\left(g\left(\mathrm{s}\right)\right)>0,$and $x\left(\tau \left(\mathrm{s}\right)\right)>0$. Clearly, $z\left(\mathrm{s}\right)>0$, $\mathrm{s}\ge {\mathrm{s}}_1$. Since

$$\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)={\left(\mathrm{s}-\theta \right)}^{\sigma }{\left(\theta -\ell \right)}^{\vartheta },\mathrm{s}\ge \ell ,$$

by using (8), we have

$$h\left(\mathrm{s},\theta ,\ell \right)=\frac{\vartheta \mathrm{s}-\left(\sigma +\vartheta \right)\theta +\sigma \ell }{\left(\mathrm{s}-\theta \right)\left(\theta -\ell \right)},\mathrm{s}\ge \ell .$$

Now, as in the proof of Theorem 3, we arrive at

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\ell }^{\mathrm{s}}{\left(\mathrm{s}-\theta \right)}^{\sigma }{\left(\theta -\ell \right)}^{\vartheta }\rho \left(\theta \right)\Phi \left(\mathrm{s},l,\theta \right)\mathrm{d}\theta \le 0,\mathrm{s}\ge \ell .$$

This contradicts (26) and the proof is complete. □

Theorem 5.

Assume that (6) holds and

$$\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)={\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda },$$

where $\gamma ,\lambda$ are constants greater than r. If there exists a function $\rho \in C^1\left(\left[{\mathrm{s}}_0,\infty \right),{\mathbb{R}}^{+}\right)$ satisfying ${\rho }^{\prime }\left(\mathrm{s}\right)\ge 0$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\ell }^{\mathrm{s}}{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\rho \left(\theta \right)\Psi \left(\mathrm{s},l,\theta \right)\mathrm{d}\theta >0$$

(27)

for any $k_1,k_2\in \left(0,1\right),$then, the solution $x\left(\mathrm{s}\right)$ is oscillatory or converges to zero, where

$$\begin{array}{ccc}\hfill \Psi \left(\mathrm{s},l,\theta \right)& :& =k_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}q\left(\theta \right){\Theta }^r\left(\theta \right)\hfill \\ & & -k_{0}a\left(\theta \right){\left(\frac{\lambda \mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\left(\gamma +\lambda \right)\mathcal{T}\left({\mathrm{s}}_0,\theta \right)+\gamma \mathcal{T}\left({\mathrm{s}}_0,\ell \right)}{a^{1/r}\left(\theta \right)\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}+\frac{{\rho }^{\prime }\left(\theta \right)}{\rho \left(\theta \right)}\right)}^{r+1}.\hfill \end{array}$$

Proof. 

Suppose that $x\in \mathcal{P}$. Suppose that there is an $\mathrm{s}\ge {\mathrm{s}}_1$ such that $x\left(\mathrm{s}\right)>0,x\left(g\left(\mathrm{s}\right)\right)>0,$and $x\left(\tau \left(\mathrm{s}\right)\right)>0$. Clearly, $z\left(\mathrm{s}\right)>0$, $\mathrm{s}\ge {\mathrm{s}}_1$. Since

$$\mathcal{K}\left(\mathrm{s},\theta ,\ell \right)={\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda },\mathrm{s}\ge \ell ,$$

by using (8), we have

$$h\left(\mathrm{s},\theta ,\ell \right)=\frac{\lambda \mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\left(\gamma +\lambda \right)\mathcal{T}\left({\mathrm{s}}_0,\theta \right)+\gamma \mathcal{T}\left({\mathrm{s}}_0,\ell \right)}{a^{1/r}\left(\theta \right)\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)},\mathrm{s}\ge \ell .$$

Now, as in the proof of Theorem 3, we arrive at

$$\underset{\mathrm{s}\to \infty }{limsup}{\int }_{\ell }^{\mathrm{s}}{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\rho \left(\theta \right)\Psi \left(\mathrm{s},l,\theta \right)d\theta \le 0,\mathrm{s}\ge \ell .$$

This contradicts (27) and the proof is complete. □

Corollary 1.

Suppose that (6) holds, r is an odd natural number and $\rho \left(\mathrm{s}\right)=1$. If there exist two constants $\gamma ,\lambda >r$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}\frac{{\int }_{\ell }^{\mathrm{s}}{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }{k}_1^r{k}_2^r\frac{{\tau }^{2r}\left(\theta \right)}{{\left(2\theta \right)}^r}q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta }{{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\gamma +\lambda -r}}>k_0{M}_0$$

(28)

for any $k_1,k_2\in \left(0,1\right)$, then, the solution $x\left(\mathrm{s}\right)$ is oscillatory or converges to zero.

Proof. 

As in Theorem 5 with $\rho \left(\mathrm{s}\right)=1$, we have to sufficiently prove that (28) leads to (27). From

$${\int }_0^1{y}^{\gamma -1}{\left(1-y\right)}^{\lambda -1}\mathrm{d}y=\frac{\Gamma \left(\gamma \right)\Gamma \left(\lambda \right)}{\Gamma \left(\gamma +\lambda \right)}.$$

Using $y=\varrho /\delta ,$ we obtain

$$\begin{array}{ccc}\hfill {\int }_0^{\delta }{\left(\delta -\varrho \right)}^{\gamma +j-r-1}{\varrho }^{\lambda -j}\mathrm{d}\varrho & =& {\int }_0^1{\delta }^{\gamma +\lambda -r}{\left(1-y\right)}^{\gamma +j-r-1}{y}^{\lambda -j}\mathrm{d}y\hfill \\ & =& {\delta }^{\gamma +\lambda -r}\frac{\Gamma \left(\gamma +j-r\right)\Gamma \left(\lambda -j+1\right)}{\Gamma \left(\gamma +\lambda -r+1\right)}.\hfill \end{array}$$

(29)

Let $\varrho =\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)$ and$\delta =\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right).$Then, by (9),

$$\begin{array}{ccc}& & {\int }_{\ell }^{\mathrm{s}}a\left(\theta \right){\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\mathsf{ϝ}\left(\mathrm{s},l,\theta \right)\mathrm{d}\theta \hfill \\ & =& {\int }_0^{\delta }{\left(\delta -\varrho \right)}^{\gamma -r-1}{\varrho }^{\lambda -r-1}{\left(\lambda \left(\delta -\varrho \right)-\gamma \varrho \right)}^{r+1}\mathrm{d}\varrho ,\hfill \end{array}$$

(30)

where

$$\mathsf{ϝ}\left(\mathrm{s},l,\theta \right):={\left(\frac{\lambda \mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\left(\gamma +\lambda \right)\mathcal{T}\left({\mathrm{s}}_0,\theta \right)+\gamma \mathcal{T}\left({\mathrm{s}}_0,\ell \right)}{a^{1/r}\left(\theta \right)\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}\right)}^{r+1},\mathrm{s}\ge \ell ,$$

and

$${\left(\lambda \left(\delta -\varrho \right)-\gamma \varrho \right)}^{r+1}=\sum _{j=0}^{r+1}{\left(-1\right)}^j\left(\genfrac{}{}{0pt}{}{r+1}{j}\right){\left(\lambda \left(\delta -\varrho \right)\right)}^j{\left(\gamma \varrho \right)}^{r+1-j}.$$

(31)

From (30) and (31), we have

$$\begin{array}{ccc}& & {\int }_{\ell }^{\mathrm{s}}a\left(\theta \right){\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\mathsf{ϝ}\left(\mathrm{s},l,\theta \right)\mathrm{d}\theta \hfill \\ & =& \sum _{j=0}^{r+1}\left(\genfrac{}{}{0pt}{}{r+1}{r-j+1}\right){\left(-1\right)}^{r-j+1}{\gamma }^{r-j+1}{\lambda }^j{\int }_0^{\delta }{\varrho }^{\lambda -j}{\left(\delta -\varrho \right)}^{\gamma +j-r-1}\mathrm{d}\varrho \hfill \\ & =& {\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\gamma +\lambda -r}{M}_0,\mathrm{s}\ge \ell .\hfill \end{array}$$

(32)

Hence, by (28) and (32), (27) holds. The proof is complete. □

Corollary 2.

Suppose that (6) holds, r is an odd natural number, and $\rho \left(\mathrm{s}\right)=1$. If there exist two constants $\gamma ,\lambda >r$ such that

$$\underset{\mathrm{s}\to \infty }{limsup}\frac{{\int }_{\ell }^{\mathrm{s}}{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\frac{{\tau }^{2r}\left(\theta \right)}{{\theta }^r}q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta }{{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\gamma +\lambda -r}}>2^r{k}_0{M}_0,$$

(33)

then, the solution $x\left(\mathrm{s}\right)$ is oscillatory or converges to zero.

Proof. 

We shall show (33) implies (28). Note that (33) implies

$${\left(\frac{k_1{k}_2}{2}\right)}^{r}q\left(\mathrm{s}\right){\Theta }^r\left(\theta \right){\left(\frac{{\tau }^2\left(\mathrm{s}\right)}{\mathrm{s}}\right)}^r={\left(\frac{k}{2}\right)}^{r}q\left(\mathrm{s}\right){\Theta }^r\left(\theta \right){\left(\frac{{\tau }^2\left(\mathrm{s}\right)}{\mathrm{s}}\right)}^r,$$

(34)

where $k=k_1{k}_2$. Conversely, (33) suggests, for $k\in \left(0,1\right)$,

$$\underset{\mathrm{s}\to \infty }{limsup}\frac{{\int }_{\ell }^{\mathrm{s}}{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\frac{{\tau }^{2r}\left(\theta \right)}{{\theta }^r}q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta }{{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\gamma +\lambda -r}}>\frac{1}{k^r}{2}^r{k}_0{M}_0,\mathrm{s}\ge \ell .$$

(35)

Combining (34) and (35), we obtain that (28) holds. Hence, by Corollary 1, we complete the proof. □

Example 1.

For the third-order NDDE

$${\left(x\left(\mathrm{s}\right)+\frac{1}{2}x\left(\frac{\mathrm{s}}{2}\right)\right)}^{‴}+\frac{\kappa }{{\mathrm{s}}^3}x\left(\frac{\mathrm{s}}{2}\right)=0,\mathrm{s}>1.$$

(36)

Note that $r=1,a\left(\mathrm{s}\right)=1,\eta \left(\mathrm{s}\right)=1/2<1,q\left(\mathrm{s}\right)=\kappa /{\mathrm{s}}^3,\kappa >0,$ $g\left(\mathrm{s}\right)=\mathrm{s}/2,$ and$\tau \left(\mathrm{s}\right)=\mathrm{s}/2.$ Condition (6) is satisfied, where

$${\int }_{{\mathrm{s}}_0}^{\infty }{\int }_v^{\infty }{\left(\frac{1}{a\left(u\right)}{\int }_u^{\infty }q\left(\theta \right)\mathrm{d}\theta \right)}^{1/r}\mathrm{d}u\mathrm{d}v={\int }_{{\mathrm{s}}_0}^{\infty }{\int }_v^{\infty }{\int }_u^{\infty }\frac{\kappa }{{\theta }^3}\mathrm{d}\theta \mathrm{d}u\mathrm{d}v=\infty .$$

Note that

$$\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)={\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\frac{\mathrm{d}\theta }{a^{1/r}\left(\theta \right)}={\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\mathrm{d}\theta =\left(\mathrm{s}-{\mathrm{s}}_0\right)=\left(\mathrm{s}-1\right).$$

We may choose $\gamma =4,\lambda =5$, then

$$\begin{array}{ccc}\hfill M_0& =& \sum _{j=0}^{r+1}\left(\genfrac{}{}{0pt}{}{r+1}{r-j+1}\right){\left(-1\right)}^{r-j+1}{\gamma }^{r-j+1}{\lambda }^j\frac{\Gamma \left(\gamma +j-r\right)\Gamma \left(\lambda -j+1\right)}{\Gamma \left(\gamma +\lambda -r+1\right)}\hfill \\ & =& \sum _{j=0}^{1+1}{C}_{1+1}^{1-j+1}{\left(-1\right)}^{1-j+1}{4}^{1-j+1}{5}^j\frac{\Gamma \left(4+j-1\right)\Gamma \left(5-j+1\right)}{\Gamma \left(4+5-1+1\right)}=4.1664\times {10}^{-2}\hfill \end{array}$$

and so

$$2^r{k}_0{M}_0=\left(2\right)\left(\frac{1}{4}\right)\left(4.1664\times {10}^{-2}\right)=2.0832\times {10}^{-2}.$$

Now,

$$\psi \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)=\frac{{\mathrm{s}}^2}{2^{4k+3}},$$

$$\begin{array}{ccc}\hfill {\Theta }_1\left(\mathrm{s}\right)& =& \sum _{k=0}^m\left(\prod _{n=0}^{2k}\eta \left(g^{\left[n\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)\right)\left(\frac{1}{\eta \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}-1\right)\frac{\psi \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}{\psi \left(\tau \left(\mathrm{s}\right)\right)}\hfill \\ & =& \sum _{k=0}^{20}{\left(\frac{1}{2}\right)}^{2k+1}\left(1\right)\frac{{\mathrm{s}}^2}{2^{4k+3}}\frac{2^3}{{\mathrm{s}}^2}=\sum _{k=0}^{20}{\left(\frac{1}{2}\right)}^{2k+1}\frac{1}{2^{4k}}\hfill \\ & \approx & 0.50794:={\mu }_0,\hfill \end{array}$$

Moreover, for $\mathrm{s}>\ell >1$, the left side of (33) is

$$\begin{array}{ccc}& & \underset{\mathrm{s}\to \infty }{limsup}\frac{{\int }_{\ell }^{\mathrm{s}}{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\frac{{\tau }^{2r}\left(\theta \right)}{{\theta }^r}q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta }{{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\gamma +\lambda -r}}\hfill \\ & =& \underset{\mathrm{s}\to \infty }{limsup}\frac{{\mu }_0\kappa }{4{\left(\mathrm{s}-\ell \right)}^8}{\int }_{\ell }^{\mathrm{s}}\frac{{\left(\mathrm{s}-\theta \right)}^4{\left(\theta -\ell \right)}^5}{{\theta }^2}\mathrm{d}\theta \hfill \\ & =& \frac{{\mu }_0}{1120}\kappa .\hfill \end{array}$$

Therefore, from Corollary 2, it confirms that every positive solution of (36) approaches zero and that $\kappa ⪆45.934$.

Example 2.

Consider the third-order NDDE

$$\left(\mathrm{s}{\left({\left(x\left(\mathrm{s}\right)+\frac{1}{3}x\left(\frac{\mathrm{s}}{2}\right)\right)}^{‴}\right)}^3\right)+\frac{\kappa }{{\mathrm{s}}^6}{x}^3\left(\frac{\mathrm{s}}{2}\right)=0,\mathrm{s}>1.$$

(37)

Note that $r=3,a\left(\mathrm{s}\right)=\mathrm{s},\eta \left(\mathrm{s}\right)=1/3<1,q\left(\mathrm{s}\right)=\kappa /{\mathrm{s}}^6,\kappa >0,$ $g\left(\mathrm{s}\right)=\mathrm{s}/2,$ and$\tau \left(\mathrm{s}\right)=\mathrm{s}/2.$Condition (6) is satisfied, where

$${\int }_{{\mathrm{s}}_0}^{\infty }{\int }_v^{\infty }{\left(\frac{1}{a\left(u\right)}{\int }_u^{\infty }q\left(\theta \right)\mathrm{d}\theta \right)}^{1/r}\mathrm{d}u\mathrm{d}v={\int }_{{\mathrm{s}}_0}^{\infty }{\int }_v^{\infty }{\left(\frac{1}{u}{\int }_u^{\infty }\frac{\kappa }{{\theta }^6}\mathrm{d}\theta \right)}^{1/3}\mathrm{d}u\mathrm{d}v=\infty .$$

Note that

$$\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)={\int }_{{\mathrm{s}}_0}^{\mathrm{s}}\frac{\mathrm{d}\theta }{a^{1/r}\left(\theta \right)}=\frac{3}{2}\left({\mathrm{s}}^{2/3}-1\right).$$

We may choose $\gamma =4,\lambda =5$, then

$$\begin{array}{ccc}\hfill M_0& =& \sum _{j=0}^{r+1}\left(\genfrac{}{}{0pt}{}{r+1}{r-j+1}\right){\left(-1\right)}^{r-j+1}{\gamma }^{r-j+1}{\lambda }^j\frac{\Gamma \left(\gamma +j-r\right)\Gamma \left(\lambda -j+1\right)}{\Gamma \left(\gamma +\lambda -r+1\right)}\hfill \\ & =& \sum _{j=0}^{3+1}{C}_{3+1}^{3-j+1}{\left(-1\right)}^{3-j+1}{4}^{3-j+1}{5}^j\frac{\Gamma \left(4+j-3\right)\Gamma \left(5-j+1\right)}{\Gamma \left(4+5-3+1\right)}=27.5\hfill \end{array}$$

and so

$$2^r{k}_0{M}_0=2^3\frac{1}{{\left(4\right)}^4}\left(27.5\right)=0.85938.$$

Now,

$$\psi \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)=\frac{9}{10}\frac{{\mathrm{s}}^{5/3}}{2^{\left(10k+5\right)/3}},$$

$$\begin{array}{ccc}\hfill {\Theta }_1\left(\mathrm{s}\right)& =& \sum _{k=0}^m\left(\prod _{n=0}^{2k}\eta \left(g^{\left[n\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)\right)\left(\frac{1}{\eta \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}-1\right)\frac{\psi \left(g^{\left[2k\right]}\left(\tau \left(\mathrm{s}\right)\right)\right)}{\psi \left(\tau \left(\mathrm{s}\right)\right)}\hfill \\ & =& \sum _{k=0}^{20}{\left(\frac{1}{3}\right)}^{2k+1}\left(2\right)\frac{{\mathrm{s}}^{5/3}}{2^{\left(10k+5\right)/3}}\frac{2^{5/3}}{{\mathrm{s}}^{5/3}}=\sum _{k=0}^{20}{\left(\frac{1}{3}\right)}^{2k+1}\left(2\right)\frac{1}{2^{10k/3}}\hfill \\ & \approx & 0.67410:={\mu }_0,\hfill \end{array}$$

and, for $\mathrm{s}>\ell >1$, the left side of (33) takes

$$\begin{array}{ccc}& & \underset{\mathrm{s}\to \infty }{limsup}\frac{{\int }_{\ell }^{\mathrm{s}}{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\theta \right)\right)}^{\gamma }{\left(\mathcal{T}\left({\mathrm{s}}_0,\theta \right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\lambda }\frac{{\tau }^{2r}\left(\theta \right)}{{\theta }^r}q\left(\theta \right){\Theta }^r\left(\theta \right)\mathrm{d}\theta }{{\left(\mathcal{T}\left({\mathrm{s}}_0,\mathrm{s}\right)-\mathcal{T}\left({\mathrm{s}}_0,\ell \right)\right)}^{\gamma +\lambda -r}}\hfill \\ & =& \underset{\mathrm{s}\to \infty }{limsup}{\left(\frac{3}{2}\right)}^9\frac{{\mu }_0^3\kappa }{2^6{\left({\mathrm{s}}^{2/3}-{\ell }^{2/3}\right)}^6}{\int }_{\ell }^{\mathrm{s}}\frac{{\left({\mathrm{s}}^{2/3}-{\theta }^{2/3}\right)}^4{\left({\theta }^{2/3}-{\ell }^{2/3}\right)}^5}{{\theta }^3}\mathrm{d}\theta \hfill \\ & =& {\left(\frac{3}{2}\right)}^9\frac{{\mu }_0^3\kappa }{2^6\left(20\right)}.\hfill \end{array}$$

Hence, by Corollary 2, it confirms that every nonoscillatory solution of (37) converges to zero provided that $\kappa ⪆93.412$.

Remark 1.

Consider the NDDE

$${\left(x\left(\mathrm{s}\right)+\frac{1}{2}x\left(\frac{\mathrm{s}}{4}\right)\right)}^{‴}+\frac{\kappa }{{\mathrm{s}}^3}x\left(\frac{\mathrm{s}}{2}\right)=0,\mathrm{s}>1.$$

(38)

We find that Theorem 1 in [42] and Theorem 2 in [44] cannot be applied to this equation because $\tau \left(\mathrm{s}\right)=\mathrm{s}/2>g\left(\mathrm{s}\right)=\mathrm{s}/4$. While using the results we obtained, we find that the solutions of (38) are oscillatory or tend to zero. Therefore, our results improve the results in [42,44].

Remark 2.

We note that additional conditions were mentioned in [42,44], including the composition condition $\left(\tau \circ g=g\circ \tau \right)$, which is a harsh condition on the delay functions, while we were able to dispense with these conditions in our results. We also note that the results we obtained are considered an expansion and extension of both [41,43], as we find that in [41], (1) was studied when $0\le \eta \left(\mathrm{s}\right)\le \eta <1$, and we find in [43] that Equation (1) was studied when $a=1$and $\eta \left(\mathrm{s}\right)\ge 1$, while in our study, Equation (1) was studied when $0\le \eta \left(\mathrm{s}\right)\le {\eta }_0<\infty$.

Remark 3.

From Example 1 in [41], we find that every nonoscillatory solution of (37) converges to zero provided that $\kappa >9^3/2$. However, by using our criterion (33), we find that every nonoscillatory solution of (37) converges to zero provided that $\kappa >93.412$. Hence, our findings enhance those presented in [41].

4. Conclusions

It is known that studying the solution behavior of odd-order differential equations is more difficult than studying even-order equations. This is due to several reasons, one of which is the ability to obtain relationships between the different derivatives of positive solutions, as well as the multiplicity of derivative possibilities for positive solutions. Based on the improved relationship between x and z that was derived in [45], we obtained new relationships between x and z. The new relationship takes into account the cases $\eta \le 1$ and $\eta >1$, and this was not usual in previous studies of neutral third-order differential equations. Using the appropriate Riccati substitution, we obtained the Riccati inequality and then applied the Philos approach to obtain new criteria for the asymptotic behavior of the studied equation. The new criteria ensure that all nonoscillatory solutions converge to zero. The results provided in this work improve and extend the well-known results in previous works; for instance, see [41,42,43,44]. It would also be of interest to use this approach to study the equation

$${\left(a\left(\mathrm{s}\right){\left({\left(x\left(\mathrm{s}\right)+\eta \left(\mathrm{s}\right)x\left(g\left(\mathrm{s}\right)\right)\right)}^{\left(n-1\right)}\right)}^r\right)}^{\prime }+q\left(\mathrm{s}\right)x^r\left(\tau \left(\mathrm{s}\right)\right)=0,$$

where $n\ge 3.$