On the Monotonic and Asymptotic Properties of Positive Solutions to Third-Order Neutral Differential Equations and Their Effect on Oscillation Criteria

Abstract

The monotonic properties of positive solutions to functional differential equations of the third order are examined in this paper. It is generally known that by optimizing the relationships between a solution and its corresponding function, as well as its derivatives, one can improve the oscillation criterion for neutral differential equations. Based on this, we obtain new relationships and inequalities and test their effect on the oscillation parameters of the studied equation. To obtain the oscillation parameters, we used Riccati techniques and comparison with lower-order equations. Finally, the progress achieved in oscillation theory for third-order equations was measured by comparing our results with previous relevant results.

1. Introduction

Third-order differential equations are used in many models, such as the model for studying blood entry flows into a “stenosed artery”, an artery partially or totally occluded due to the thickening of the arterial wall [1]. In addition, in nuclear reactor kinetics [2], by constructing phase space solutions to third-order systems of equations and considering the solutions to the equations as explicit functions of the independent variable, which enables computer-aided phase space analysis, to perform a comprehensive and expeditious study of the system behavior for any combination of parameter values of interest. Although third-order differential equations have appeared in many models of life, they have received less attention from researchers of first- and second-order differential equations. This lethargy is due to the fact that this type of equation has a greater number of positive solution classifications than equations of the first and second order, which further complicates its study. In addition, its characteristic equation must contain a solution or solutions belonging to the set of real numbers [3].

On the other hand, functional differential equations (FDEs) are one of the classes of differential equations in which oscillatory behavior is common. The study of FDEs has attracted the attention of many researchers recently, in terms of the qualitative behavior of the solutions as well as the numerical solutions of these equations, see [4,5,6,7,8]. This type of equation deals with the after-effects of life phenomena, which means the presence of deviating arguments that express the previous and current times of a phenomenon, and it is known that these arguments increase the possibility of the existence of oscillatory solutions. The retarded functional differential equation, or the delay differential equation (DDE) is one of the basic subclasses of functional differential equations. This type is based on past and current values of the time derivatives, which leads to more accurate and effective future predictions. The deviating arguments, in this case, are called delays or time lags, see [9,10]. When the highest order derivative appears with and without delay, it creates another subclass of functional differential equations known as neutral delay differential equations (NDDEs). This subclass has a wide scope for modeling, as we find many models of chemistry, electricity, mechanics, and economics represented by NDDEs; see [11,12,13], where the study of the asymptotic and monotonic properties, together with the oscillatory behavior, of solutions to the third-order neutral delay differential equations was used to model many life phenomena.

The study of the oscillatory behavior of differential equations has received great and continuous attention from researchers. Philos [14] and Santra et al. [15,16] were interested in first-order differential equations. While, the works in [17,18,19,20,21,22] were concerned with even- and odd-order differential equations, with their various classifications. By reviewing the previous literature, we can note that it included three basic steps in the conclusion of oscillation criteria, regardless of the quality. These essential points can be summed up by first classifying all positive solutions to the studied equation and then developing a new or updated method to obtain improved relations and inequalities linking the solution, its derivatives, and its corresponding function; lastly, excluding the positive solutions using these improved relations and the chosen technique.

Therefore, this paper aimed to study and improve the monotonic properties of positive solutions and then use them to develop new oscillation criteria for the half-linear third-order neutral delay differential equation

$${\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }+q\left(s\right)x^{\alpha }\left(\sigma \left(s\right)\right)=0,$$

(1)

for all $s\ge s_0$, where $z\left(s\right)$ stands for the corresponding function

$$z\left(s\right)=x\left(s\right)+p\left(s\right)x\left(\tau \left(s\right)\right).$$

(2)

We assume the following assumptions hold:

(A₁)

$\alpha$ is a quotient of two odd positive integers;

(A₂)

$r\left(s\right)\in C^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right)$ and satisfies

$$R\left(a,b\right)={\int }_a^b\frac{1}{r^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu ,$$

with $R\left(s_0,\infty \right)=\infty$;

(A₃)

$q\left(s\right)\in C\left(\left[s_0,\infty \right),{\mathbb{R}}^{+}\right)$ and does not eventually vanish;

(A₄)

$p\left(s\right)\in C\left(\left[s_0,\infty \right),{\mathbb{R}}^{+}\right)$ and there exists a constant $p_0$ such that $p_0\ge p$;

(A₅)

$\tau \left(s\right),\sigma \left(s\right)\in C^1\left(\left[s_0,\infty \right),\mathbb{R}\right)$ symbolize the delayed functions, where $\tau \left(s\right)\le s$, $\sigma \left(s\right)\le s,$ and ${lim}_{s\to \infty }\tau \left(s\right)={lim}_{s\to \infty }\sigma \left(s\right)=\infty$.

Definition 1.

A solution to (1) is defined as a function $x\in C\left(\left[s_0,\infty \right),\mathbb{R}\right)$, which has the properties $z\in C^2\left(\left[s_0,\infty \right),\mathbb{R}\right)$ and $r{\left(z^{″}\right)}^{\alpha }\in C^1\left(\left[s_0,\infty \right),\mathbb{R}\right)$ and satisfies (1) on $\left[s_0,\infty \right)$.

Our interest is directed to the solutions of (1) that satisfy the condition

$$sup\left\{\left|x\left(s\right)\right|:s\ge s_{*}\right\}>0,$$

for all $s_{*}\ge s_0$.

Definition 2.

A nontrivial solution to (1) is said to be oscillatory if it has arbitrarily large zeros, and otherwise it is called non-oscillatory.

Definition 3.

Equation (1) is said to be oscillatory if all its solutions are oscillatory. Otherwise, it is called non-oscillatory.

Remark 1.

The term half-linear equation refers to the fact that the solution space of (1) has just one half of the properties that characterize linearity, namely homogeneity (but not additivity).

Finding solutions to differential equations is a rich research topic that has attracted great interest from researchers in the past decades, and this remains so to this day. This is because it was and still is one of the most significant tools used to describe and deduce the ways in which quantities change in systems, as well as to shed light on how and why these changes occur. However, the problem occurs when nonlinear differential equations are used to describe systems, because most of these equations are difficult to solve in a closed form. Therefore, researchers resort to using the qualitative theory of differential equations, where a topological description of the local and global properties of the solutions to these equations is developed, regardless of finding their exact form. Oscillation theory is one of these subfields of qualitative theory and is concerned with analyzing the oscillatory and non-oscillatory behavior of solutions to differential equations. For more information about oscillation theory, please see the monographs in [23,24] and the papers in [25,26].

However, the stage of classifying differential equation solutions is the first and most important step, which precedes the study of the asymptotic and monotonic properties of positive solutions, which in turn paves the way for determining the behavior of the oscillatory equation. The positive solutions to Equation (1) can be classified into four possible classes. However, under the condition $\left({\mathrm{A}}_2\right)$, these classifications are reduced to two, since the probabilities that the $z^{″}\left(s\right)$ derivative of solutions are negative are rejected. In light of this, we can conclude that the positive solutions to Equation (1) follow one of the following classes:

C₁:

$z>0,z^{\prime }>0,z^{″}>0,$ and $z^{‴}<0$;

C₂:

$z>0,z^{\prime }<0,z^{″}>0,$ and $z^{‴}<0$.

Studying the properties of solutions to third-order differential equations has many different applications. In addition to scientific applications, this study often contains many open and complex analytical problems and issues. Due to these relative difficulties, the previous works related to Equation (1) are few and appeared over long periods of time. The first study of the oscillation of third-order differential equations was in 1961 by Hanah [27]. She considered the linear case of (1) with $r\left(s\right)\equiv 1,p\left(s\right)\equiv 0,$ and $\alpha \equiv 1$, and established the following very famous sufficient criterion of oscillation:

$$\underset{s\to \infty }{\lim \inf }{s}^{3}q\left(s\right)>\frac{2}{3\sqrt{3}}.$$

(3)

Following that, authors were interested in studying the oscillatory behavior and the properties of solutions to (1), but in the case of $p\left(s\right)\equiv 0$; that is, in the case that the highest derivative appears with only a delay argument. For more details, one can see the works of Grace et al. [28] and Candan and Dahiya [29]. Li et al. [30] and Dzurina et al. [31], studied the half-linear type for $\alpha$ states for a quotient of two odd positive integers. Meanwhile, Qaraad et al. [32] considered the mixed type, in which the equation contains both delayed and advanced functions. On the other hand, Bohner et al. [33] obtained oscillation results for damped third-order differential equations. Chatzarakis et al. were interested in the Emden–Flower and quasi-linear type in [34] and [35], respectively, while Grace et al. [36] investigated oscillation criteria for the superlinear type.

In 2010, Li et al. [37] extended the results given in [27] by considering the neutral delay type, where $p\left(s\right)\in C\left(\left[s_0,\infty \right),{\mathbb{R}}^{+}\right)$, and relied on comparison theorem to confirm that the following differential equation

$${\left(x\left(s\right)-p\left(s\right)x\left(\tau \left(s\right)\right)\right)}^{‴}+q\left(s\right)x\left(\sigma \left(s\right)\right)=0,$$

is oscillatory or tends to zero if

$$\underset{s\to \infty }{lim}{\int }_{s_0}^s\left({\sigma }^2\left(\nu \right)q\left(\nu \right)-\frac{2}{3\sqrt{3}\nu }\right)\mathrm{d}\nu =\infty .$$

In the same year, Baculikova and Dzurina [38], used another methodology based on the Riccati technique, where for $𝓁\in \left(0,1\right)$ and $s_{𝓁}$ that are large enough, then every solution of the half-linear NDDE (1) oscillates or tends to zero if

$$\underset{s\to \infty }{liminf}\frac{s^{\alpha }}{r\left(s\right)}{\int }_s^{\infty }{𝓁}^{\alpha }{\left(1-p\left(\nu \right)\right)}^{\alpha }q\left(\nu \right){\left(\frac{\sigma \left(\nu \right)}{\nu }\right)}^{\alpha }{\left(\frac{\sigma \left(\nu \right)-{\nu }_{𝓁}}{2}\right)}^{\alpha }\mathrm{d}\nu >\frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}},$$

(4)

under the assumption that $r^{\prime }\left(s\right)\ge 0$ and

$${\int }_{s_0}^{\infty }{\int }_{\nu }^{\infty }{\left[\frac{1}{r\left(u\right)}{\int }_u^{\infty }q\left(\mu \right)\mathrm{d}\mu \right]}^{1/\alpha }\mathrm{d}u\mathrm{d}\nu =\infty$$

holds. Furthermore, they presented a simplified condition for (4), see Corollary 1 in [38], as follows:

$$\underset{s\to \infty }{liminf}\frac{s^{\alpha }}{r\left(s\right)}{\int }_s^{\infty }q\left(\nu \right)\frac{{\sigma }^{2\alpha }\left(\nu \right)}{{\nu }^{\alpha }}\mathrm{d}\nu >\frac{{\left(2\alpha \right)}^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}{\left(1-p_o\right)}^{\alpha }}.$$

(5)

In 2011, Li and Thandapani [39], used the same technique, but by obtaining improved properties of the solutions, they were able to set the condition

$$\underset{s\to \infty }{limsup}{\int }_{s_2}^s\left[\delta \left(\nu \right)\frac{q\left(\nu \right)}{2\alpha -1}-\frac{1+\frac{p_0^{\alpha }}{{\tau }_0}}{{\left(\alpha +1\right)}^{\alpha +1}}\left(\frac{{\left({\delta }^{\prime }{\left(\nu \right)}_{+}\right)}^{\alpha +1}}{{\left(\delta \left(\nu \right)R\left(\sigma \left(\nu \right),s_1\right){\sigma }^{\prime }\left(\nu \right)\right)}^{\alpha }}\right)\right]\mathrm{d}\nu =\infty ,s_2\ge s,$$

(6)

where $\alpha \ge 1,{\sigma }^{\prime }\left(s\right)>0,\sigma \left(s\right)\le \tau \left(s\right),$ $\delta \left(s\right)\in {\mathbf{C}}^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right),$ and

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

This criterion is considered simpler and more effective and does not assume the previous conditions, to ensure that every solution of (1) is either oscillatory or converges to zero (${\mathrm{C}}_1=⌀$). Later, Grace et al. [40], extended the previous condition under the same assumptions and $\alpha \ge 2$ to

$$\underset{s\to \infty }{limsup}{\int }_{s_2}^s\left[\delta \left(\nu \right)q\left(\nu \right){\left(p_{*}\left(\sigma \left(\nu \right)\right)\right)}^{\alpha }{\left(\frac{{\tau }^{-1}\left(\left(\sigma \left(\nu \right)\right)\right)}{\nu }\right)}^{2\alpha }-{\delta }_{+}^{\prime }\left(\nu \right){\left(\frac{8}{{\nu }^2}\right)}^{\alpha }\right]\mathrm{d}\nu =\infty ,$$

(7)

where

$$p_{*}\left(s\right)=\frac{1}{p\left({\tau }^{-1}\left(s\right)\right)}\left(1-\frac{1}{p\left({\tau }^{-1}\left({\tau }^{-1}\left(s\right)\right)\right)}·\frac{m\left({\tau }^{-1}\left({\tau }^{-1}\left(s\right)\right)\right)}{m\left({\tau }^{-1}\left(s\right)\right)}\right)$$

and $m\left(s\right)=s^2,m\left(s\right)=s^3,m\left(s\right)=e^s,$ or $m\left(s\right)=s^{\alpha }{e}^{ϵs}$.

On the other hand, Dzurina et al. [41] proved that case ${\mathrm{C}}_2$ does not happen if

$$\underset{s\to \infty }{liminf}{\int }_{{\tau }^{-1}\left(\varpi \left(s\right)\right)}^s{q}^{*}\left(\nu \right)\left({\int }_{\sigma \left(\nu \right)}^{\rho \left(\nu \right)}{\int }_u^{\rho \left(\nu \right)}\frac{1}{r\left(\mu \right)}\mathrm{d}\mu \mathrm{d}u\right)\mathrm{d}\nu >\frac{{\tau }_0+p_0}{{\tau }_0\mathrm{e}},$$

where $q^{*}\left(s\right)=min\left\{q\left(s\right),q\left(\tau \left(s\right)\right)\right\}$ and $\varpi \in \left(\left[s_0,\infty \right),\mathbb{R}\right)$ is a positive function that satisfies $\tau \left(s\right)>\varpi \left(s\right)>\sigma \left(s\right)$. Moaaz et al. [42,43] presented several interesting results on the oscillation of solutions to odd-order delay differential equations. Very recently, Moaaz et al. [44] developed new criteria for the nonexistence of class ${\mathrm{C}}_2$ of (1), under the condition

$$\underset{s\to \infty }{liminf}{\int }_{{\tau }^{-1}\left(\varpi \left(s\right)\right)}^s{q}^{*}\left(\nu \right){\Psi }_n^{\alpha }\left(\varpi \left(\nu \right),\sigma \left(\nu \right)\right)\mathrm{d}\nu >\frac{{\tau }_0+p_0^{\alpha }}{\delta {\tau }_0\mathrm{e}},$$

where $\varpi \left(s\right)\le \tau \left(\varpi \left(s\right)\right)$ and

$${\Psi }_{n+1}^{\alpha }\left(h,k\right)={\int }_h^k{\int }_u^k\frac{1}{r\left(\mu \right)}exp\left[\frac{\delta {\tau }_0}{{\tau }_0+p_0^{\alpha }}{\int }_{{\tau }^{-1}\left(\mu \right)}^k{q}^{*}\left(\nu \right){\Psi }_n^{\alpha }\left(\varpi \left(\nu \right),\sigma \left(\nu \right)\right)\mathrm{d}\nu \right]\mathrm{d}\mu \mathrm{d}u.$$

In this paper, we will derive some new monotonic properties and use them as in application for

• Improving the relationships between the solution and its derivatives;
• Improving the relationships between the solution and its corresponding function;
• Obtaining improved criteria that ensure that there are no positive solutions;
• Obtaining oscillation criteria that improve on the criteria mentioned in the previous literature.

Comparing these criteria mentioned in earlier works with our results revealed that our results improved them and provided a broader and greater scope of applicability.The following is a focused summary of what makes the results of this paper distinctive:

• The criteria require fewer assumptions about the coefficients and the auxiliary functions than their predecessors, which reduces the complexities when applying them;
• The half-linear property (exponent $\alpha$ of the first and second derivatives) allows for a larger area when determining where the same results can be applied to the linear ($\alpha =1$) and ordinary ($\tau \left(s\right)=\sigma \left(s\right)=1$) type;
• Our results consider two cases of the constant $p_0$; i.e., for $p_0>1$ and $p_0<1$.

The paper structure is divided into five basic sections. The first section is divided into two introductory parts. In the first part, we give introductions to the important points of the study, define the equation under our interest, and establish the major assumptions that have been applied to all of our results. It also contains a summary chronology of the most important previous works related to the studied equation, which we will use to compare our results later. In Section 2 and Section 3, we study the monotonicity properties of the positive solutions to (1) and improve these properties, in addition to giving criteria to ensure that there are no positive solutions for both class ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$. Section 4 relies on the previous criteria to derive theorems that ensure the oscillation of all solutions of (1). In the last section, a summary of the paper’s content is given, in terms of the basic theorems and results, as well as an explanation of the most important points that distinguish our results.

Remark 2.

All functional inequalities are assumed to hold for all sufficiently large s in following sections.

2. Improved Monotonic Properties

In this section, we study and improve the monotonic properties of positive solutions to (1), which we will rely on later to obtain our basic results. First, let us introduce some auxiliary lemmas to facilitate the study of the properties of later solutions.

Lemma 1

([45]). Assume that x is a positive real variable, and

$$F\left(x\right)=b_{1}x-b_2{x}^{\frac{\alpha +1}{\alpha }},$$

where $b_i$ are constants, $i=1, 2,$ $b_2>0,$ and α defined as in $\left({\mathrm{A}}_1\right)$. Then the following properties hold:

(P₁)

F has a maximum value at $x^{*}={\left(\frac{b_1\alpha }{b_2\left(\alpha +1\right)}\right)}^{\alpha };$

(P₂)

$F\left(x^{*}\right)={\displaystyle \underset{x\in \mathbb{R}}{max}}\left(F\right)=\frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}{b}_1^{\alpha +1}{b}_2^{-\alpha };$

(P₃)

$b_{1}x-b_2{x}^{\frac{\alpha +1}{\alpha }}\le \frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}{b}_1^{\alpha +1}{b}_2^{-\alpha }.$

The following lemma considers an improvement of the known relationship between variables x and z

$$x\left(s\right)\ge \left(1-p\left(s\right)\right)z\left(s\right),$$

(8)

introduced by Moaaz et al. [46]. Based on this lemma, we can obtain results that are superior to those obtained by employing (8).

Lemma 2.

Assume that x is a positive solution of (1). Then

$$x\left(s\right)>\sum _{𝓁=0}^k\left(\prod _{m=0}^{2𝓁}p\left({\tau }^{\left[m\right]}\left(s\right)\right)\right)\left[\frac{z\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}{p\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}-z\left({\tau }^{\left[2𝓁+1\right]}\left(s\right)\right)\right],$$

(9)

eventually holds for any nonnegative integer k.

2.1. Properties for Solutions to Class $C_1$

This subsection concerns studying the monotonic properties of the positive solutions of (1) that belong to Class ${\mathrm{C}}_1$. To simplify the basic lemmas and main results, let us define the following notations for sufficiently large s, and j stands for any nonnegative integer, then ${\tau }^{\left[0\right]}\left(s\right):=s$,

$${\tau }^{\left[j\right]}\left(s\right)=\tau \left({\tau }^{\left[j-1\right]}\left(s\right)\right),$$

(10)

and

$${\tau }^{\left[-j\right]}\left(s\right)={\tau }^{-1}\left({\tau }^{\left[-j+1\right]}\left(s\right)\right).$$

(11)

Moreover,

$$\eta \left(s\right):={\int }_{s_0}^{s}R\left(s_0,\nu \right)\mathrm{d}\nu ,$$

(12)

and

$${\Phi }_k\left(s\right):=\left\{\begin{array}{cc}1\hfill & \mathrm{for}{p}_0=0;\hfill \\ \sum _{𝓁=0}^k\left({\displaystyle \prod _{m=0}^{2𝓁}}p\left({\tau }^{\left[m\right]}\left(s\right)\right)\right)\left[\frac{1}{p\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}-1\right]\frac{\eta \left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}{\eta \left(s\right)}\hfill & \mathrm{for}{p}_0<1;\hfill \\ {\sum }_{𝓁=1}^k\left({\displaystyle \prod _{m=1}^{2𝓁-1}}\frac{1}{p\left({\tau }^{\left[-m\right]}\left(s\right)\right)}\right)\left[1-\frac{1}{p\left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right)}\frac{\eta \left({\tau }^{-2𝓁}\left(s\right)\right)}{\eta \left({\tau }^{\left[-2𝓁+1\right]}\left(s\right)\right)}\right]\hfill & \mathrm{for}{p}_0>\frac{\eta \left(s\right)}{\eta \left(\tau \right)},\hfill \end{array}\right.$$

(13)

for any nonnegative integers $𝓁,m,$ and k. Additionally, let’s also define some notations that we will use for the improved lemmas in this section. So, let

$${\lambda }_n\left(s\right):=R\left(s_0,s\right)+{\int }_{s_0}^{s}R\left(s_0,\nu \right){\rho }_n\left(\nu \right)\eta \left(\sigma \left(\nu \right)\right)\mathrm{d}\nu$$

(14)

where

$${\rho }_n\left(s\right)=\frac{1}{\alpha }{\eta }^{\alpha -1}\left(\sigma \left(s\right)\right){\overline{\Phi }}_{k,n}^{\alpha }\left(\sigma \left(s\right)\right)q\left(s\right),$$

(15)

and ${\overline{\Phi }}_{k,n}\left(s\right)$ is an improved coefficient, to be specified later. In addition, let

$${\tilde{\lambda }}_n\left(s\right)={\int }_{s_0}^s{\lambda }_n\left(\nu \right)\mathrm{d}\nu ,$$

(16)

$${\tilde{R}}_n\left(s\right):=exp\left[{\int }_{s_0}^s\frac{\mathrm{d}\nu }{{\lambda }_n\left(\nu \right)r^{1/\alpha }\left(\nu \right)}\right],$$

(17)

and

$${\tilde{\eta }}_{n+1}\left(s\right):={\int }_{s_0}^s{\tilde{R}}_n\left(\nu \right)\mathrm{d}\nu$$

(18)

for any nonnegative integer n.

Lemma 3.

Assume that $x\in {\mathrm{C}}_1,$ then the following properties hold, for sufficiently large s,

(P₄)

${\left(z^{\prime }\left(s\right)/R\left(s_0,s\right)\right)}^{\prime }\le 0$;

(P₅)

${\left(z\left(s\right)/\eta \left(s\right)\right)}^{\prime }\le 0.$

Proof. 

Assume that $x\in {\mathrm{C}}_1$, then from the nature of $r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }$, which is an eventually positive non-increasing function, we obtain

$$z^{\prime }\left(s\right)\ge {\int }_{s_0}^s\frac{r^{1/\alpha }\left(\nu \right)z^{″}\left(\nu \right)}{r^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu \ge r^{1/\alpha }\left(s\right)z^{″}\left(s\right){\int }_{s_0}^s\frac{1}{r^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu =r^{1/\alpha }\left(s\right)z^{″}\left(s\right)R\left(s_0,s\right).$$

(19)

But

$$\begin{array}{ccc}\hfill {\left(\frac{z^{\prime }\left(s\right)}{R\left(s_0,s\right)}\right)}^{\prime }& =& \frac{R\left(s_0,s\right)z^{″}\left(s\right)-z^{\prime }\left(s\right)r^{-1/\alpha }\left(s\right)}{R^2\left(s_0,s\right)}\hfill \\ & =& \frac{r^{-1/\alpha }\left(s\right)}{R^2\left(s_0,s\right)}\left[r^{1/\alpha }\left(s\right)z^{″}\left(s\right)R\left(s_0,s\right)-z^{\prime }\left(s\right)\right]\le 0.\hfill \end{array}$$

So, $\left({\mathrm{P}}_4\right)$-part holds.

Similarly, we prove $\left({\mathrm{P}}_5\right)$-part but through the increasing monotonicity of $z^{\prime }\left(s\right)$, where

$$z\left(s\right)\ge {\int }_{s_0}^s\frac{z^{\prime }\left(\nu \right)}{R\left(s_0,\nu \right)}R\left(s_0,\nu \right)\mathrm{d}\nu \ge \frac{z^{\prime }\left(s\right)}{R\left(s_0,s\right)}{\int }_{s_0}^{s}R\left(s_0,\nu \right)\mathrm{d}\nu =\frac{z^{\prime }\left(s\right)}{R\left(s_0,s\right)}\eta \left(s\right),$$

which shows that

$$\begin{array}{ccc}\hfill {\left(\frac{z\left(s\right)}{\eta \left(s\right)}\right)}^{\prime }& =& \frac{R\left(s_0,s\right)}{{\eta }^2\left(s\right)}\left[\frac{z^{\prime }\left(s\right)}{R\left(s_0,s\right)}\eta \left(s\right)-z\left(s\right)\right]\hfill \\ & \le & 0.\hfill \end{array}$$

and this completes the proof. □

Lemma 4.

Assume that $x\in {\mathrm{C}}_1$. Then, for sufficiently large s,

(P₆)

$x\left(s\right)>{\Phi }_k\left(s\right)z\left(s\right)$;

(P₇)

${\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }+q\left(s\right){\left({\Phi }_k\left(\sigma \left(s\right)\right)z\left(\sigma \left(s\right)\right)\right)}^{\alpha }\le 0.$

Proof. 

Assume that $x\in {\mathrm{C}}_1$. We can see from the definition of ${\Phi }_k$ in (13) that its value depends on the value of $p_0$, where $p_0$ has three possible cases.

• In the first case, where $p_0=0$, the proof is obvious, so we omit it.

• Case two: for $p_0<1$, it is obvious from (10) and $\left({\mathrm{A}}_5\right)$ that

  $$s\ge {\tau }^{\left[2𝓁\right]}\left(s\right)\ge {\tau }^{\left[2𝓁+1\right]}\left(s\right).$$

  Since $z\left(s\right)$ is a positive increasing function, then

  $$z\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)\ge z\left({\tau }^{\left[2𝓁+1\right]}\left(s\right)\right),$$

  but from $\left({\mathrm{P}}_5\right)$-part of Lemma 3 we have that

  $$z\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)\ge \frac{\eta \left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}{\eta \left(s\right)}z\left(s\right),$$

  for 𝓁 any nonnegative integer. Substituting (9) into Lemma 2, yields

  $$\begin{array}{ccc}\hfill x\left(s\right)& >& \sum _{𝓁=0}^k\left(\prod _{m=0}^{2𝓁}p\left({\tau }^{\left[m\right]}\left(s\right)\right)\right)\left[\frac{1}{p\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}-1\right]z\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)\hfill \\ & >& \sum _{𝓁=0}^k\left(\prod _{m=0}^{2𝓁}p\left({\tau }^{\left[m\right]}\left(s\right)\right)\right)\left[\frac{1}{p\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}-1\right]\frac{\eta \left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}{\eta \left(s\right)}z\left(s\right),\hfill \end{array}$$

  which in turn with (1), verifies $\left({\mathrm{P}}_7\right)$.

Now, for $p_0>1$ case. It is obvious from the definition of the corresponding function in (2) that

$$\begin{array}{ccc}\hfill z\left({\tau }^{-1}\left(s\right)\right)& =& x\left({\tau }^{-1}\left(s\right)\right)+p\left({\tau }^{-1}\left(s\right)\right)x\left(s\right)\hfill \\ & =& \left[\frac{z\left({\tau }^{\left[-2\right]}\left(s\right)\right)-x\left({\tau }^{\left[-2\right]}\left(s\right)\right)}{p\left({\tau }^{\left[-2\right]}\left(s\right)\right)}\right]+p\left({\tau }^{-1}\left(s\right)\right)x\left(s\right)\hfill \\ & =& \prod _{m=2}^2\frac{z\left({\tau }^{\left[-2\right]}\left(s\right)\right)}{p\left({\tau }^{\left[-m\right]}\left(s\right)\right)}-\prod _{m=2}^3\frac{\left[z\left({\tau }^{\left[-3\right]}\left(s\right)\right)-x\left({\tau }^{\left[-3\right]}\left(s\right)\right)\right]}{p\left({\tau }^{\left[-m\right]}\left(s\right)\right)}+p\left({\tau }^{-1}\left(s\right)\right)x\left(s\right).\hfill \end{array}$$

Substituting (9) into Lemma 2, we obtain

$$x\left(s\right)>\sum _{𝓁=1}^k\left(\prod _{m=1}^{2𝓁-1}\frac{1}{p\left({\tau }^{\left[-m\right]}\left(s\right)\right)}\right)\left[z\left({\tau }^{\left[-2𝓁+1\right]}\left(s\right)\right)-\frac{1}{p\left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right)}z\left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right)\right].$$

(20)

Now, again from (11) and $\left({\mathrm{A}}_5\right)$, we have

$${\tau }^{\left[-2𝓁\right]}\left(s\right)\ge {\tau }^{\left[-2𝓁+1\right]}\left(s\right)\ge s,$$

using the above inequality in the $\left({\mathrm{P}}_5\right)$-part of Lemma 3 and the monotonicity of $z\left(s\right)$ yields

$$\frac{\eta \left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right)}{\eta \left({\tau }^{\left[-2𝓁+1\right]}\left(s\right)\right)}z\left({\tau }^{\left[-2𝓁+1\right]}\left(s\right)\right)\ge z\left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right),$$

and

$$z\left({\tau }^{\left[-2𝓁+1\right]}\left(s\right)\right)\ge z\left(s\right).$$

As a result, inequality (20) turns into

$$x\left(s\right)>z\left(s\right)\sum _{𝓁=1}^k\left(\prod _{m=1}^{2𝓁-1}\frac{1}{p\left({\tau }^{\left[-m\right]}\left(s\right)\right)}\right)\left[1-\frac{1}{p\left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right)}\frac{\eta \left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right)}{\eta \left({\tau }^{\left[-2𝓁+1\right]}\left(s\right)\right)}\right],$$

which, when combined with (1), yields $\left({\mathrm{P}}_7\right)$. And this completes the proof. □

Remark 3.

By choosing $k=0$, then $\left({\mathrm{P}}_6\right)$ reduces to obtain the well known classical relation (8) for $p_0<1$.

Lemma 5.

Assume that $x\in {\mathrm{C}}_1$ and $\alpha \ge 1$. Then,

$${\left(r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\right)}^{\prime }+\rho \left(s\right)z\left(\sigma \left(s\right)\right)\le 0,$$

(21)

where

$$\rho \left(s\right)=\frac{1}{\alpha }{\eta }^{\alpha -1}\left(\sigma \left(s\right)\right){\Phi }_k^{\alpha }\left(\sigma \left(s\right)\right)q\left(s\right),$$

holds eventually.

Proof. 

Assume that $x\in {\mathrm{C}}_1$. Since

$$\begin{array}{ccc}\hfill {\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }& =& {\left({\left(r^{1/\alpha }\left(s\right)\left(z^{″}\left(s\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & =& \alpha {\left(r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\right)}^{\alpha -1}{\left(r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\right)}^{\prime }.\hfill \end{array}$$

(22)

From the $\left({\mathrm{P}}_4\right)$ and $\left({\mathrm{P}}_5\right)$ parts of Lemma 3, we have

$$r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\le \frac{z^{\prime }\left(s\right)}{R\left(s_0,s\right)}\le \frac{z\left(s\right)}{\eta \left(s\right)}$$

and so

$$r^{1/\alpha }\left(\sigma \left(s\right)\right)z^{″}\left(\sigma \left(s\right)\right)\le \frac{z\left(\sigma \left(s\right)\right)}{\eta \left(\sigma \left(s\right)\right)}.$$

The monotonicity of $r^{1/\alpha }\left(s\right)z^{″}\left(s\right)$ implies

$$r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\le r^{1/\alpha }\left(\sigma \left(s\right)\right)z^{″}\left(\sigma \left(s\right)\right),$$

then

$$r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\le r^{1/\alpha }\left(\sigma \left(s\right)\right)z^{″}\left(\sigma \left(s\right)\right)\le \frac{z\left(\sigma \left(s\right)\right)}{\eta \left(\sigma \left(s\right)\right)}.$$

By taking power $\alpha -1$ for both sided, we obtain

$${\left(r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\right)}^{\alpha -1}\le {\left(\frac{1}{\eta \left(\sigma \left(s\right)\right)}\right)}^{\alpha -1}{z}^{\alpha -1}\left(\sigma \left(s\right)\right).$$

Substituting from the last inequality into (22)

$${\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }\ge \alpha {\left(\frac{1}{\eta \left(\sigma \left(s\right)\right)}\right)}^{\alpha -1}{z}^{\alpha -1}\left(\sigma \left(s\right)\right){\left(r^{1/\alpha }\left(s\right)\left(z^{″}\left(s\right)\right)\right)}^{\prime },$$

and once again substituting from the last inequality into (1), we obtain

$$\begin{array}{ccc}\hfill -q\left(s\right)x^{\alpha }\left(\sigma \left(s\right)\right)& =& {\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & \ge & \alpha {\left(\frac{1}{\eta \left(\sigma \left(s\right)\right)}\right)}^{\alpha -1}{z}^{\alpha -1}\left(\sigma \left(s\right)\right){\left(r^{1/\alpha }\left(s\right)\left(z^{″}\left(s\right)\right)\right)}^{\prime }.\hfill \end{array}$$

But from $\left({\mathrm{A}}_4\right)$ and $\left({\mathrm{P}}_6\right)$, we obtain

$$-x^{\alpha }\left(\sigma \left(s\right)\right)\le -{\Phi }_k^{\alpha }\left(\sigma \left(s\right)\right)z^{\alpha }\left(\sigma \left(s\right)\right),$$

and the monotonicity of $z\left(s\right)$ implies that

$$\begin{array}{ccc}\hfill -{\Phi }_k^{\alpha }\left(\sigma \left(s\right)\right)q\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)& \ge & -q\left(s\right)x^{\alpha }\left(\sigma \left(s\right)\right)\hfill \\ & \ge & \alpha {\left(\frac{1}{\eta \left(\sigma \left(s\right)\right)}\right)}^{\alpha -1}{z}^{\alpha -1}\left(\sigma \left(s\right)\right){\left(r^{1/\alpha }\left(s\right)\left(z^{″}\left(s\right)\right)\right)}^{\prime }.\hfill \end{array}$$

i.e.,

$${\left(r^{1/\alpha }\left(s\right)\left(z^{″}\left(s\right)\right)\right)}^{\prime }+\frac{1}{\alpha }{\Phi }_k^{\alpha }\left(\sigma \left(s\right)\right)q\left(s\right){\left(\eta \left(\sigma \left(s\right)\right)\right)}^{\alpha -1}z\left(\sigma \left(s\right)\right)\le 0.$$

which gives (21). And this completes the proof. □

Remark 4.

The functions $\rho \left(s\right)$, ${\rho }_0\left(s\right)$ defined in Lemma 5 and (15) are equivalent, i.e., $\rho \left(s\right)={\rho }_0\left(s\right)$.

In the following lemma, we use the definition of functional sequences given in (14), (17) and (18) to obtain improved monotonic properties of class ${\mathrm{C}}_1$ solutions.

Lemma 6.

Assume that $x\in {\mathrm{C}}_1$ and $\alpha \ge 1$. Then, the following improved properties hold for a sufficiently large s and n any positive integer:

(P₈)

${\left(z^{\prime }\left(s\right)/{\tilde{R}}_{n-1}\left(s\right)\right)}^{\prime }\le 0$;

(P₉)

${\left(z\left(s\right)/{\tilde{\eta }}_n\left(s\right)\right)}^{\prime }\le 0$;

(P₁₀)

${\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }+q\left(s\right){\left({\overline{\Phi }}_{k,n}\left(\sigma \left(s\right)\right)z\left(\sigma \left(s\right)\right)\right)}^{\alpha }\le 0$,

where

$${\overline{\Phi }}_{k,n}\left(s\right):=\left\{\begin{array}{cc}1\hfill & \mathit{for}{p}_0=0;\hfill \\ {\sum }_{𝓁=0}^k\left({\displaystyle \prod _{m=0}^{2𝓁}}p\left({\tau }^{\left[m\right]}\left(s\right)\right)\right)\left[\frac{1}{p\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}-1\right]\frac{{\tilde{\eta }}_n\left({\tau }^{\left[2𝓁\right]}\left(s\right)\right)}{{\tilde{\eta }}_n\left(s\right)}\hfill & \mathit{for}{p}_0<1;\hfill \\ {\sum }_{𝓁=1}^k\left({\displaystyle \prod _{m=1}^{2𝓁-1}}\frac{1}{p\left({\tau }^{\left[-m\right]}\left(s\right)\right)}\right)\left[1-\frac{1}{p\left({\tau }^{\left[-2𝓁\right]}\left(s\right)\right)}\frac{{\tilde{\eta }}_n\left({\tau }^{-2𝓁}\left(s\right)\right)}{{\tilde{\eta }}_n\left({\tau }^{\left[-2𝓁+1\right]}\left(s\right)\right)}\right]\hfill & \mathit{for}{p}_0>\frac{\eta \left(s\right)}{\eta \left(\tau \right)}\hfill \end{array}\right.$$

(23)

and ${\overline{\Phi }}_{k,0}\left(s\right)={\Phi }_k\left(s\right).$

Proof. 

Assume that $x\in {\mathrm{C}}_1$. Define the function $G\left(s\right)$, where

$$G\left(s\right)=-z^{\prime }\left(s\right)+R\left(s_0,s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right).$$

Then, it is obvious that

$$\begin{array}{ccc}\hfill G^{\prime }\left(s\right)& =& -z^{″}\left(s\right)+R\left(s_0,s\right)r^{1/\alpha }\left(s\right)z^{‴}\left(s\right)+R\left(s_0,s\right){\left(r^{1/\alpha }\left(s\right)\right)}^{\prime }{z}^{″}\left(s\right)+r^{-1/\alpha }\left(s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\hfill \\ & =& R\left(s_0,s\right)r^{1/\alpha }\left(s\right)z^{‴}\left(s\right)+R\left(s_0,s\right){\left(r^{1/\alpha }\left(s\right)\right)}^{\prime }{z}^{″}\left(s\right)\hfill \\ & =& R\left(s_0,s\right){\left(r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\right)}^{\prime }.\hfill \end{array}$$

From (21), we obtain

$${\left(r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\right)}^{\prime }\le -\rho \left(s\right)z\left(\sigma \left(s\right)\right),$$

and so

$$G^{\prime }\left(s\right)\le -R\left(s_0,s\right)\rho \left(s\right)z\left(\sigma \left(s\right)\right).$$

Integrating the last inequality from $s_0$ to s, then

$$z^{\prime }\left(s\right)\ge R\left(s_0,s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right)+{\int }_{s_0}^{s}R\left(s_0,\nu \right)\rho \left(\nu \right)z\left(\sigma \left(\nu \right)\right)\mathrm{d}\nu .$$

(24)

Again, by integrating (19) from $s_0$ to s and using (12), we obtain

$$\begin{array}{ccc}\hfill z\left(s\right)& \ge & {\int }_{s_0}^s{r}^{1/\alpha }\left(\nu \right)z^{″}\left(\nu \right)R\left(s_0,\nu \right)\mathrm{d}\nu \hfill \\ & \ge & r^{1/\alpha }\left(s\right)z^{″}\left(s\right){\int }_{s_0}^{s}R\left(s_0,\nu \right)\mathrm{d}\nu \hfill \\ & =& r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\eta \left(s\right).\hfill \end{array}$$

But the non-increasing monotonicity of $r^{1/\alpha }\left(s\right)z^{″}\left(s\right)$ implies

$$z\left(\sigma \left(s\right)\right)\ge r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\eta \left(\sigma \left(s\right)\right).$$

(25)

Substituting from (25) into (24), we obtain

$$\begin{array}{ccc}\hfill z^{\prime }\left(s\right)& \ge & R\left(s_0,s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right)+{\int }_{s_0}^{s}R\left(s_0,\nu \right)\rho \left(\nu \right)r^{1/\alpha }\left(\nu \right)z^{″}\left(\nu \right)\eta \left(\sigma \left(\nu \right)\right)\mathrm{d}\nu \hfill \\ & \ge & R\left(s_0,s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right)+r^{1/\alpha }\left(s\right)z^{″}\left(s\right){\int }_{s_0}^{s}R\left(s_0,\nu \right)\rho \left(\nu \right)\eta \left(\sigma \left(\nu \right)\right)\mathrm{d}\nu \hfill \\ & =& r^{1/\alpha }\left(s\right)z^{″}\left(s\right)\left[R\left(s_0,s\right)+{\int }_{s_0}^{s}R\left(s_0,\nu \right)\rho \left(\nu \right)\eta \left(\sigma \left(\nu \right)\right)\mathrm{d}\nu \right]\hfill \\ & =& {\lambda }_0\left(s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right).\hfill \end{array}$$

(26)

Now, by multiplying the last inequality by ${\tilde{R}}_0^{-1}\left(s\right)$, then

$$\frac{{\tilde{R}}_0^{-1}\left(s\right)}{{\lambda }_0\left(s\right)r^{1/\alpha }\left(s\right)}{z}^{\prime }\left(s\right)\ge {\tilde{R}}_0^{-1}\left(s\right)z^{″}\left(s\right).$$

(27)

From (17), it is clear that

$${\tilde{R}}_0^{\prime }\left(s\right)=\frac{{\tilde{R}}_0\left(s\right)}{{\lambda }_0\left(s\right)r^{1/\alpha }\left(s\right)}.$$

So,

$$\begin{array}{ccc}\hfill {\left(\frac{z^{\prime }\left(s\right)}{{\tilde{R}}_0\left(s\right)}\right)}^{\prime }& =& \frac{{\tilde{R}}_0\left(s\right)z^{″}\left(s\right)-\frac{{\tilde{R}}_0\left(s\right)}{{\lambda }_0\left(s\right)r^{1/\alpha }\left(s\right)}{z}^{\prime }\left(s\right)}{{\tilde{R}}_0^2\left(s\right)}\hfill \\ & =& \frac{1}{{\lambda }_0\left(s\right)r^{1/\alpha }\left(s\right){\tilde{R}}_0\left(s\right)}\left[{\lambda }_0\left(s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right)-z^{\prime }\left(s\right)\right].\hfill \end{array}$$

which, in view of (27), implies $\left({\mathrm{P}}_8\right)$; i.e.,

$$\frac{z^{\prime }\left(s\right)}{{\tilde{R}}_0\left(s\right)}\mathrm{is}\mathrm{decreasing}.$$

The monotonicity of $z^{\prime }\left(s\right)$ gives

$$z\left(s\right)\ge {\int }_{s_0}^s\frac{z^{\prime }\left(\nu \right)}{{\tilde{R}}_0\left(s\right)}{\tilde{R}}_0\left(\nu \right)\mathrm{d}\nu \ge {\tilde{\eta }}_1\left(s\right)\frac{z^{\prime }\left(\nu \right)}{{\tilde{R}}_0\left(s\right)},$$

therefore,

$$\frac{z\left(s\right)}{{\tilde{\eta }}_1\left(s\right)}\mathrm{is}\mathrm{also}\mathrm{decreasing}.$$

Now, the $\left({\mathrm{P}}_{10}\right)$ part is clearly proven using the last monotonicity and (23) into (9), which becomes

$$x\left(s\right)>{\overline{\Phi }}_{k,1}\left(\sigma \left(s\right)\right)z\left(s\right),$$

and as a result (1) implies $\left({\mathrm{P}}_{10}\right)$ for $n=1$, i.e.,

$${\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }+q\left(s\right){\left({\overline{\Phi }}_{k,1}\left(\sigma \left(s\right)\right)z\left(\sigma \left(s\right)\right)\right)}^{\alpha }\le 0.$$

For $n=2$, we obtain $\left({\mathrm{P}}_{10}\right)$ by replacing $\left({\mathrm{P}}_7\right)$ with the last inequality and concluding the proof using the same technique as before. For $n=3,4,\dots$, we can similarly follow the same technique and complete the proof. □

2.2. Properties for Solutions of Class $C_2$

This subsection concerns the study of the monotonic properties of the positive solutions to (1) that belong to Class ${\mathrm{C}}_2$. First, let us define auxiliary notations such as

$$\breve{{\eta }_0}\left(h,k\right):={\int }_h^{k}R\left(\nu ,k\right)\mathrm{d}\nu ,$$

and

$$q^{*}\left(s\right)=min\left\{q\left(s\right),q\left(\tau \left(s\right)\right)\right\}.$$

Additionally, this section’s proofs need to add another assumption to the basic ones $\left({\mathrm{A}}_1\right)$–$\left({\mathrm{A}}_5\right)$ in the introduction section, in which

(A₆)

$\tau \left(\sigma \left(s\right)\right)=\sigma \left(\tau \left(s\right)\right)$ and ${\tau }^{\prime }\left(s\right)\ge {\tau }_0>0.$

Lemma 7.

Assume that $x\in {\mathrm{C}}_2$ and there exists ϖ a positive function $\varpi \in \left(\left[s_0,\infty \right),\mathbb{R}\right)$ such that

$$\tau \left(s\right)>\varpi \left(s\right)>\sigma \left(s\right).$$

Then, for ${\tau }^{-1}\left(h\right)\le k,$

$$z\left(h\right)\ge {\breve{\eta }}_n\left(h,k\right)r^{1/\alpha }\left(k\right)z^{″}\left(k\right),n=0,1,\dots ,$$

(28)

where

$${\breve{\eta }}_{n+1}\left(h,k\right):={\int }_h^k{\int }_u^k\frac{1}{r^{1/\alpha }\left(\mu \right)}exp\frac{1}{\alpha }\left[\frac{{\tau }_0}{{\tau }_0+p_0}{\int }_{{\tau }^{-1}\left(\mu \right)}^k{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu \right]\mathrm{d}\mu \mathrm{d}u.$$

(29)

Proof. 

Assume that $x\in {\mathrm{C}}_2$. From the non-increasing monotonicity of $r^{1/\alpha }\left(s\right)z^{″}\left(s\right)$, then

$$\begin{array}{ccc}\hfill -z^{\prime }\left(h\right)& \ge & {\int }_h^k\frac{r^{1/\alpha }\left(\nu \right)z^{″}\left(\nu \right)}{r^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu \ge r^{1/\alpha }\left(k\right)z^{″}\left(k\right){\int }_h^k\frac{1}{r^{1/\alpha }\left(\nu \right)}\mathrm{d}\nu \hfill \\ & =& r^{1/\alpha }\left(k\right)z^{″}\left(k\right)R\left(h,k\right),\hfill \end{array}$$

where $h\le k$. Integrating the last inequality again from h to k, we obtain

$$\begin{array}{ccc}\hfill z\left(h\right)& \ge & r^{1/\alpha }\left(k\right)z^{″}\left(k\right){\int }_h^{k}R\left(\nu ,k\right)\mathrm{d}\nu \hfill \\ & =& r^{1/\alpha }\left(k\right)z^{″}\left(k\right)\breve{{\eta }_0}\left(h,k\right),\hfill \end{array}$$

also for all $h\le k$. Next, we employ the mathematical induction to prove the rest of the proof by assuming for every $n\in {\mathbb{N}}_0$ and sufficiently large s that

$$z\left(h\right)\ge {\breve{\eta }}_n\left(h,k\right)r^{1/\alpha }\left(k\right)z^{″}\left(k\right).$$

(30)

In the following, we prove that (28) is valid for $n+1$. From (1) it is clear that

$$q\left(\tau \left(s\right)\right)x^{\alpha }\left(\sigma \left(\tau \left(s\right)\right)\right)=-\frac{{\left(r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{\tau }^{\prime }\left(s\right)},$$

but $\left({\mathrm{A}}_6\right)$ implies that

$$\begin{array}{ccc}\hfill p_{0}q\left(\tau \left(s\right)\right)x^{\alpha }\left(\tau \left(\sigma \left(s\right)\right)\right)& =& p_{0}q\left(\tau \left(s\right)\right)x^{\alpha }\left(\sigma \left(\tau \left(s\right)\right)\right)\hfill \\ & =& -\frac{p_0}{{\tau }^{\prime }\left(s\right)}{\left(r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & \le & -\frac{p_0}{{\tau }_0}{\left(r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }\right)}^{\prime }.\hfill \end{array}$$

By adding the above inequality to (1), we obtain

$$\begin{array}{ccc}\hfill q^{*}\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)& \le & q\left(s\right)x^{\alpha }\left(\sigma \left(s\right)\right)+p_{0}q\left(\tau \left(s\right)\right)x^{\alpha }\left(\tau \left(\sigma \left(s\right)\right)\right)\hfill \\ & \le & -{\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }-\frac{p_0}{{\tau }_0}{\left(r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & =& -{\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }+\frac{p_0}{{\tau }_0}r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }\right)}^{\prime }.\hfill \end{array}$$

(31)

Putting $h\left(s\right)=\sigma \left(s\right)$ and $k\left(s\right)=\varpi \left(s\right)$ in (30), yields

$$z^{\alpha }\left(\sigma \left(s\right)\right)\ge {\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)r\left(\varpi \left(s\right)\right){\left(z^{″}\left(\varpi \left(s\right)\right)\right)}^{\alpha }.$$

Substituting into (31), then

$$q^{*}\left(s\right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)r\left(\varpi \left(s\right)\right){\left(z^{″}\left(\varpi \left(s\right)\right)\right)}^{\alpha }\le -{\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }+\frac{p_0}{{\tau }_0}r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }\right)}^{\prime }.$$

(32)

Now, let us define the auxiliary function

$$M\left(s\right):=r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }+\frac{p_0}{{\tau }_0}r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }.$$

Using ${\mathrm{C}}_1$ or ${\mathrm{C}}_2$ and $\left({\mathrm{A}}_5\right)$, we obtain

$$\left(\frac{{\tau }_0+p_0}{{\tau }_0}\right)r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\le M\left(s\right)\le \left(\frac{{\tau }_0+p_0}{{\tau }_0}\right)r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha },$$

(33)

and so

$$\left(\frac{{\tau }_0}{{\tau }_0+p_0}\right)M\left({\tau }^{-1}\left(s\right)\right)\le r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }.$$

Substituting into (32), we arrive at

$$\begin{array}{ccc}\hfill M^{\prime }\left(s\right)& \le & -q^{*}\left(s\right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)r\left(\varpi \left(s\right)\right){\left(z^{″}\left(\varpi \left(s\right)\right)\right)}^{\alpha }\hfill \\ & \le & -\frac{{\tau }_0}{{\tau }_0+p_0}{q}^{*}\left(s\right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)M\left({\tau }^{-1}\left(\varpi \left(s\right)\right)\right),\hfill \end{array}$$

(34)

which indicates that $M^{\prime }\left(s\right)\le 0$. So, we conclude that $M\left(s\right)$ is a non-increasing function. As a result, (34) becomes

$$M^{\prime }\left(s\right)\le -\frac{{\tau }_0}{{\tau }_0+p_0}{q}^{*}\left(s\right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)M\left(s\right).$$

Integrating the last inequality again from h to k, we obtain

$$ln\left(\frac{M\left(h\right)}{M\left(k\right)}\right)\ge \frac{{\tau }_0}{{\tau }_0+p_0}{\int }_h^k{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu ,$$

or

$$M\left(h\right)\ge M\left(k\right)exp\left[\frac{{\tau }_0}{{\tau }_0+p_0}{\int }_h^k{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu \right].$$

Using (33), we obtain

$$M\left({\tau }^{-1}\left(h\right)\right)\ge \left(\frac{{\tau }_0+p_0}{{\tau }_0}\right)r\left(k\right){\left(z^{″}\left(k\right)\right)}^{\alpha }exp\left[\frac{{\tau }_0}{{\tau }_0+p_0}{\int }_{{\tau }^{-1}\left(h\right)}^k{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu \right],$$

i.e.,

$$z^{″}\left(h\right)\ge \frac{1}{r^{1/\alpha }\left(h\right)}{r}^{1/\alpha }\left(k\right)z^{″}\left(k\right)exp\frac{1}{\alpha }\left[\frac{{\tau }_0}{{\tau }_0+p_0}{\int }_{{\tau }^{-1}\left(h\right)}^k{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu \right].$$

Integrating the last inequality from h to k, we have

$$-z^{\prime }\left(h\right)\ge r^{1/\alpha }\left(k\right)z^{″}\left(k\right){\int }_h^k\frac{1}{r^{1/\alpha }\left(\mu \right)}exp\frac{1}{\alpha }\left[\frac{{\tau }_0}{{\tau }_0+p_0}{\int }_{{\tau }^{-1}\left(\mu \right)}^k{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu \right]\mathrm{d}\mu ,$$

once more, from h to k

$$\begin{array}{ccc}\hfill z\left(h\right)& \ge & r^{1/\alpha }\left(k\right)z^{″}\left(k\right){\int }_h^k{\int }_u^k\frac{1}{r^{1/\alpha }\left(\mu \right)}exp\frac{1}{\alpha }\left[\frac{{\tau }_0}{{\tau }_0+p_0}{\int }_{{\tau }^{-1}\left(\mu \right)}^k{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu \right]\mathrm{d}\mu \mathrm{d}u\hfill \\ & =& r^{1/\alpha }\left(k\right)z^{″}\left(k\right){\breve{\eta }}_{n+1}\left(h,k\right),\hfill \end{array}$$

for every $n+1$, $n\in {\mathbb{N}}_0$. And this completes the proof. □

3. Nonexistence of Positive Solution Theorems

In this section, we will use the comparison method, the Riccati technique, and the improved monotonic properties that we obtained in the previous section as an application to exclude the existence of any positive solutions to (1).

3.1. Nonexistence of Solutions in Class $C_1$

Theorem 1.

Assume that there exists a differentiable function $\delta \left(s\right)\in {\mathbf{C}}^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right)$ satisfies that

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left[\delta \left(\nu \right)·q\left(\nu \right)·{\Phi }_k^{\alpha }\left(\sigma \left(\nu \right)\right)·{\left(\frac{\eta \left(\sigma \left(\nu \right)\right)}{\eta \left(\nu \right)}\right)}^{\alpha }-\frac{{\left({\delta }^{\prime }\left(\nu \right)\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}·{\left(\delta \left(\nu \right)R\left(s_0,\nu \right)\right)}^{\alpha }}\right]\mathrm{d}\nu =\infty .$$

(35)

Then, the class ${\mathrm{C}}_1$ is empty.

Proof. 

Contrarily, assume that $x\in {\mathrm{C}}_1$. Let us define the positive function

$$\omega :=\delta ·\frac{r{\left(z^{″}\right)}^{\alpha }}{z^{\alpha }}.$$

Differentiating the last equation implies

$$\begin{array}{ccc}\hfill {\omega }^{\prime }& =& {\delta }^{\prime }·r{\left(z^{″}\right)}^{\alpha }·z^{-\alpha }+\delta ·{\left(r{\left(z^{″}\right)}^{\alpha }\right)}^{\prime }·z^{-\alpha }-\alpha \delta ·r{\left(z^{″}\right)}^{\alpha }·z^{-\alpha -1}{z}^{\prime }\hfill \\ & =& \frac{{\delta }^{\prime }}{\delta }\omega +\delta ·{\left(r{\left(z^{″}\right)}^{\alpha }\right)}^{\prime }·z^{-\alpha }-\alpha \delta ·\frac{r{\left(z^{″}\right)}^{\alpha }}{z^{\alpha +1}}{z}^{\prime }.\hfill \end{array}$$

(36)

Substituting from $\left({\mathrm{P}}_7\right)$ into (36), we obtain

$${\omega }^{\prime }\le \frac{{\delta }^{\prime }}{\delta }\omega -\frac{\delta ·q·{\left[{\Phi }_k\left(\sigma \right)z\left(\sigma \right)\right]}^{\alpha }}{z^{\alpha }}-\alpha \delta ·\frac{r{\left(z^{″}\right)}^{\alpha }}{z^{\alpha +1}}{z}^{\prime },$$

and from (19)

$$\begin{array}{ccc}\hfill {\omega }^{\prime }& \le & \frac{{\delta }^{\prime }}{\delta }\omega -\frac{\delta ·q·{\left[{\Phi }_k\left(\sigma \right)z\left(\sigma \right)\right]}^{\alpha }}{z^{\alpha }}-\alpha \delta ·R·{\left(\frac{r^{1/\alpha }{z}^{″}}{z}\right)}^{\alpha +1}\hfill \\ & =& \frac{{\delta }^{\prime }}{\delta }\omega -\delta ·q·{\Phi }_k^{\alpha }\left(\sigma \right)·{\left(\frac{z\left(\sigma \right)}{z}\right)}^{\alpha }-\alpha \delta ·R·{\left(\frac{r^{1/\alpha }{z}^{″}}{z}\right)}^{\alpha +1}\hfill \\ & =& \frac{{\delta }^{\prime }}{\delta }\omega -\delta ·q·{\Phi }_k^{\alpha }\left(\sigma \right)·{\left(\frac{z\left(\sigma \right)}{z}\right)}^{\alpha }-\alpha {\delta }^{-1/\alpha }·R·{\left(\omega \right)}^{1+1/\alpha }.\hfill \end{array}$$

By using the monotonicity of $z\left(s\right)/\eta \left(s\right)$ given in $\left({\mathrm{P}}_5\right)$-part of Lemma 3, we have

$${\omega }^{\prime }\le \frac{{\delta }^{\prime }}{\delta }\omega -\delta ·q·{\Phi }_k^{\alpha }\left(\sigma \right)·{\left(\frac{\eta \left(\sigma \right)}{\eta }\right)}^{\alpha }-\alpha {\delta }^{-1/\alpha }·R·{\left(\omega \right)}^{1+1/\alpha }.$$

(37)

Now, for $\frac{{\delta }^{\prime }}{\delta }\omega -\alpha {\delta }^{-1/\alpha }·R·{\left(\omega \right)}^{1+1/\alpha }$, by using $\left({\mathrm{P}}_3\right)$ in Lemma 1 with $b_1=\frac{{\delta }^{\prime }}{\delta },b_2=\alpha {\delta }^{-1/\alpha }·R,$ and $x=\omega ,$ then

$$\begin{array}{ccc}\hfill \frac{{\delta }^{\prime }}{\delta }\omega -\alpha ·{\delta }^{-1/\alpha }·R·{\omega }^{\frac{\alpha +1}{\alpha }}& \le & \frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}{\left(\frac{{\delta }^{\prime }}{\delta }\right)}^{\alpha +1}{\left(\alpha ·{\delta }^{-1/\alpha }·R\right)}^{-\alpha }\hfill \\ & \le & \frac{{\left({\delta }^{\prime }\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}·{\left(\delta R\right)}^{\alpha }}.\hfill \end{array}$$

Substituting into (37), we have

$${\omega }^{\prime }\le -\delta ·q·{\Phi }_k^{\alpha }\left(\sigma \right)·{\left(\frac{\eta \left(\sigma \right)}{\eta }\right)}^{\alpha }+\frac{{\left({\delta }^{\prime }\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}·{\left(\delta R\right)}^{\alpha }}.$$

Integrating the last inequality from $s_0$ into s, yields

$$\omega \left(s_0\right)\ge {\int }_{s_0}^s\delta \left(\nu \right)·q\left(\nu \right)·{\Phi }_k^{\alpha }\left(\sigma \left(\nu \right)\right)·{\left(\frac{\eta \left(\sigma \left(\nu \right)\right)}{\eta \left(\nu \right)}\right)}^{\alpha }-\frac{{\left({\delta }^{\prime }\left(\nu \right)\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}·{\left(\delta \left(\nu \right)R\left(s_0,\nu \right)\right)}^{\alpha }}\mathrm{d}\nu .$$

A contradicts (35). And this completes the proof. □

Example 1.

Consider the following half-linear NDDE:

$${\left({\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }+\frac{b_0}{s^{2\alpha +1}}{y}^{\alpha }\left(\mu s\right)=0,s>0,$$

(38)

where the corresponding function $z\left(s\right)$ is defined as

$$z\left(s\right)=y\left(s\right)+a_{0}y\left(\gamma s\right).$$

Moreover, we assume that α is a quotient of two odd positive integers, $a_0\in \left[0,\infty \right),b_0\in \left(0,\infty \right),$ and $\gamma ,\mu \in \left(0,1\right)$. Since $\tau \left(s\right)=\gamma s,\sigma \left(s\right)=\mu s,$ and $R\left(0,s\right)=s,$ then assumptions $\left({\mathrm{A}}_1\right)$–$\left({\mathrm{A}}_5\right)$ are easily satisfied. With some calculations, we obtain from (10)–(13) that

$${\tau }^{\left[i\right]}\left(s\right):={\gamma }^{i}s,{\tau }^{\left[-i\right]}\left(s\right):={\gamma }^{-i}s,\eta \left(s\right):=\frac{s^2}{2},$$

and

$${\Phi }_k\left(s\right):=\left\{\begin{array}{cc}1\hfill & \mathit{for}{a}_0=0;\hfill \\ \left(1-a_0\right)\sum _{i=0}^k{a}_0^{2i}·{\gamma }^{4i}\hfill & \mathit{for}{a}_0<1;\hfill \\ \sum _{i=1}^k{a}_0^{-2i}·\left[a_0-{\gamma }^{-2}\right]\hfill & \mathit{for}{a}_0>\frac{1}{{\gamma }^2}.\hfill \end{array}\right.$$

By taking $\delta \left(s\right)=s^{2\alpha }$, it follows from Theorem 1 that (38) does not possess a increasing positive solution (${\mathrm{C}}_1=⌀$) if

$$b_0>{\left(\frac{2\alpha }{\alpha +1}\right)}^{\alpha +1}·\frac{1}{R_0^{\alpha }·{\mu }^{2\alpha }},$$

(39)

for $R_0={\Phi }_k\left(s\right)$.

Theorem 2.

Assume that $\alpha \ge 1$ and there exists a differentiable function $\delta \left(s\right)\in {\mathbf{C}}^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right)$ satisfies that

$$\underset{s\to \infty }{limsup}{\int }_{s_0}^s\left[\delta \left(\nu \right)·q\left(\nu \right)·{\overline{\Phi }}_{k,n}^{\alpha }\left(\sigma \left(\nu \right)\right)·{\left(\frac{{\tilde{\eta }}_n\left(\sigma \left(\nu \right)\right)}{{\tilde{\eta }}_n\left(\nu \right)}\right)}^{\alpha }-\frac{{\left({\delta }^{\prime }\left(\nu \right)\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}·{\left(\delta \left(\nu \right)R\left(s_0,\nu \right)\right)}^{\alpha }}\right]\mathrm{d}\nu =\infty ,$$

(40)

for any nonnegative integer $n,k$. Then, the class ${\mathrm{C}}_1$ is empty.

Proof. 

Contrarily, assume that $x\in {\mathrm{C}}_1$. By using (23) and replacing $\left({\mathrm{P}}_4\right)$ and $\left({\mathrm{P}}_5\right)$ with $\left({\mathrm{P}}_8\right)$ and $\left({\mathrm{P}}_9\right)$, the proof of this theorem becomes similar to the proof of Theorem 1, so we omit it. □

Remark 5.

Criterion (40) given in the previous theorem is considered an improvement on Criterion (35) in Theorem 1; i.e., it gives better results when applied.

Example 2.

As in the last example, consider the half-linear NDDE (38), where $\alpha \ge 1$. To apply Theorem 2 in (38), we need to first calculate the iterative functions given in (14)–(18); so, let us define the following auxiliary sequences $\left\{R_i\right\}$ and $\left\{T_i\right\}$ for $i=0,1,\dots$ and $T_0=1,$ as

$$R_i:=\left\{\begin{array}{cc}1\hfill & \mathit{for}{a}_0=0;\hfill \\ \left(1-a_0\right)\sum _{i=0}^k{a}_0^{2i}·{\gamma }^{2\left(T_i+1\right)i}\hfill & \mathit{for}{a}_0<1;\hfill \\ \left(a_0-{\gamma }^{-1-T_i}\right)\sum _{i=1}^k{a}_0^{-2i}\hfill & \mathit{for}{a}_0>\frac{1}{{\gamma }^2}\hfill \end{array}\right.$$

(41)

and

$$T_{i+1}:=\frac{1}{1+\frac{{\mu }^{2\alpha }·R_i^{\alpha }·b_0}{\alpha ·2^{\alpha }}}.$$

(42)

Using the previous notations and mathematical induction yields

$$R_i={\Phi }_{k,i}\left(s\right).$$

Hence,

$${\rho }_i\left(s\right)=\frac{{\mu }^{2\left(\alpha -1\right)}·R_i^{\alpha }·b_0}{\alpha ·2^{\alpha -1}}·\frac{1}{s^3},$$

$${\lambda }_i\left(s\right)=\left(1+\frac{{\mu }^{2\alpha }·R_i^{\alpha }·b_0}{\alpha ·2^{\alpha }}\right)s=\frac{s}{T_{i+1}},$$

$${\tilde{R}}_i\left(s\right)=s^{1/\left(1+\frac{{\mu }^{2\alpha }·R_i^{\alpha }·b_0}{\alpha ·2^{\alpha }}\right)}=s^{T_{i+1}},$$

and

$${\tilde{\eta }}_i\left(s\right)=\frac{s^{T_i+1}}{T_i+1}.$$

Now, Theorem 2 implies that (38) does not possess any increasing positive solutions (${\mathrm{C}}_1=⌀$), if

$$b_0·R_i^{\alpha }·{\left({\mu }^{T_i+1}\right)}^{\alpha }>{\left(\frac{2\alpha }{\alpha +1}\right)}^{\alpha +1}.$$

(43)

Theorem 3.

Assume that $\alpha \ge 1$ and

$$\underset{s\to \infty }{liminf}{\int }_{\sigma \left(s\right)}^{s}q\left(\nu \right){\overline{\Phi }}_{k,n}^{\alpha }\left(\sigma \left(\nu \right)\right){\tilde{\lambda }}_n^{\alpha }\left(\sigma \left(\nu \right)\right)\mathrm{d}\nu >\frac{1}{\mathrm{e}},$$

(44)

for any nonnegative integer $n,k$ and ${\tilde{\lambda }}_n\left(s\right)$ are defined as in (16). Then, the class ${\mathrm{C}}_1$ is empty.

Proof. 

Contrarily, assume that $x\in {\mathrm{C}}_1$. As in (26), we obtain

$$z^{\prime }\left(s\right)\ge {\lambda }_n\left(s\right)r^{1/\alpha }\left(s\right)z^{″}\left(s\right).$$

Integrating the above inequality from $s_0$ to s, then

$$\begin{array}{ccc}\hfill z\left(s\right)& \ge & r^{1/\alpha }\left(s\right)z^{″}\left(s\right){\int }_{s_0}^s{\lambda }_n\left(\nu \right)\mathrm{d}\nu \hfill \\ & =& r^{1/\alpha }\left(s\right)z^{″}\left(s\right){\tilde{\lambda }}_n\left(s\right).\hfill \end{array}$$

(45)

Combining (45) and the $\left({\mathrm{P}}_{10}\right)$ part of lemma 6, we arrive at

$${\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(s\right)r\left(\sigma \left(s\right)\right){\left({\overline{\Phi }}_{k,n}\left(\sigma \left(s\right)\right)z^{″}\left(\sigma \left(s\right)\right){\tilde{\lambda }}_n\left(\sigma \left(s\right)\right)\right)}^{\alpha }.$$

(46)

Now, let us define the positive function

$$U\left(s\right):=r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }.$$

Therefore, (46) becomes

$$U^{\prime }\left(s\right)+q\left(s\right){\overline{\Phi }}_{k,n}^{\alpha }\left(\sigma \left(s\right)\right){\tilde{\lambda }}_n^{\alpha }\left(\sigma \left(s\right)\right)U\left(\sigma \left(s\right)\right)\le 0.$$

(47)

But Theorem 2.1.1 in [47] indicates that under (44) every solution of (47) oscillates. A contradiction, and this completes the proof. □

Example 3.

Recall the NDDE (38) for $\alpha \ge 1$. Exactly as in Example 2, where we used auxiliary sequences (41), (42), and mathematical induction to obtain that $R_i:={\Phi }_{k,i}\left(s\right)$ for $i=0,1,\dots$. Consequently,

$${\tilde{\lambda }}_i\left(s\right):=\frac{s^2}{2T_{i+1}}.$$

Substituting this into (44) in Theorem 3 ensures that (38) does not possess any increasing positive solution (${\mathrm{C}}_1=⌀$) if

$$\frac{b_0·R_i^{\alpha }·{\mu }^{2\alpha }}{2^{\alpha }{T}_{i+1}^{\alpha }}ln\frac{1}{\mu }>\frac{1}{\mathrm{e}}.$$

(48)

Theorem 4.

Assume that $\alpha \ge 1$ and

$$\underset{s\to \infty }{liminf}\frac{s^{\alpha }}{r\left(s\right)}{\int }_s^{\infty }q\left(\nu \right)\frac{{\sigma }^{2\alpha }\left(\nu \right)}{{\nu }^{\alpha }}\mathrm{d}𝓁>\frac{{\left(2\alpha \right)}^{\alpha }}{{\left(1+\alpha \right)}^{\alpha +1}{\overline{\Phi }}_{k,n}^{\alpha }\left(s\right)},$$

(49)

for any nonnegative integer $n,k$. Then, the class ${\mathrm{C}}_1$ is empty.

Proof. 

Contrarily, assume that $x\in {\mathrm{C}}_1$. From Corollary 1 in [38] and using $\left({\mathrm{P}}_{10}\right)$ instead of the inequality

$${\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(s\right){\left(1-p\left(\sigma \left(s\right)\right)\right)}^{\alpha }{z}^{\alpha }\left(\sigma \left(s\right)\right).$$

The proof becomes similar to the proof of Theorem 1 in [38], so we can omit it. □

Example 4.

Again, consider the half-linear NDDE (38) where $\alpha \ge 1$. As in the previous examples, Theorem 4 implies that (38) does not possess any increasing positive solutions (${\mathrm{C}}_1=⌀$) if

$$b_0>\frac{2^{\alpha }{\alpha }^{\alpha +1}}{{\left(1+\alpha \right)}^{\alpha +1}{\mu }^{2\alpha }{R}_i^{\alpha }}.$$

(50)

Remark 6.

Applying criteria (5)–(7) given in the works of Baculikova and Dzurina [38], Li and Thandapani [39], and Grace et al. [40] to (38), we obtain the following criteria:

$$b_0>{\left(\frac{\alpha }{\alpha +1}\right)}^{\alpha +1}·{\left(\frac{2}{\left(1-a_0\right){\mu }^2}\right)}^{\alpha },$$

(51)

$$b_0>\left(2\alpha -1\right)·{\left(\frac{2\alpha }{\alpha +1}\right)}^{\alpha +1}·\frac{1+\frac{a_0^{\alpha }}{\gamma }}{{\mu }^{2\alpha }},$$

(52)

and

$$b_0>\frac{\alpha ·2^{3\alpha +1}}{{\left(a_0-{\gamma }^{-2}\right)}^{\alpha }}{\left(\frac{a_0·\gamma }{\mu }\right)}^{2\alpha },$$

(53)

respectively. In the following table, we compare the effectiveness and novelty of our results for Theorems 3 and 4 with the criteria (51)–(53) in the previous works. In

Table 1. Comparison of oscillation criteria using the lower bounds of the value of the coefficient $b_0$.

	$\mathit{\alpha }$	$\mathit{\gamma }$	$\mathit{\mu }$	${\mathit{a}}_0$	(48)	(50)	(51)	(52)	(53)
$\left(\mathrm{a}\right)$	1	$0.1$	$0.5$	$0.4$	$5.1143$	$3.3332$	$3.3333$	20	Fail
$\left(\mathrm{b}\right)$	3	$0.1$	$0.5$	$0.4$	$766.3601$	$749.9242$	$750.0043$	$2656.8000$	Fail
$\left(\mathrm{c}\right)$	1	$0.6$	$0.3$	5	$52.6300$	$53.5450$	Fail	$103.7037$	720
$\left(\mathrm{d}\right)$	1	$0.9$	$0.7$	$2.5$	$10.3430$	$4.1661$	Fail	$7.7097$	$130.6322$

, we determined the lower bounds of the coefficient $b_0$ for different values of $a_0,\alpha ,\gamma ,$ and $\mu$, as follows:

We can notice from the previous table that

• Criterion (50) produced by applying Theorem 3 provides the best results for Cases a, b, and d;
• Criterion (48) produced by applying Theorem 4 provides the best results for Case c;
• Our results improved on the previous results in the literature, which demonstrates the importance of improving the relationships between a solution and its corresponding function and derivatives.

Figure 1 illustrates this comparison on a larger scale.

3.2. Nonexistance of Solutions in Class $C_2$

Theorem 5.

Assume that there exists a positive function $\varpi \in \left(\left[s_0,\infty \right),\mathbb{R}\right)$, such that

$$\tau \left(s\right)>\varpi \left(s\right)>\sigma \left(s\right),$$

and $\varpi \left(s\right)\ge {\tau }^{-1}\left(\sigma \left(s\right)\right)$. If

$$\underset{s\to \infty }{liminf}{\int }_{{\tau }^{-1}\left(\varpi \left(s\right)\right)}^s{q}^{*}\left(\nu \right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(\nu \right),\varpi \left(\nu \right)\right)\mathrm{d}\nu >\frac{{\tau }_0+p_0}{\mathrm{e}{\tau }_0},$$

(54)

Then, the class ${\mathrm{C}}_2$ is empty.

Proof. 

Assume that $x\in {\mathrm{C}}_2$. From (32) in the proof of Lemma 7, we can define the positive auxiliary function

$$F\left(s\right):=r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }+\frac{p_0}{{\tau }_0}r\left(\tau \left(s\right)\right){\left(z^{″}\left(\tau \left(s\right)\right)\right)}^{\alpha }.$$

But from ${\mathrm{C}}_1$ or ${\mathrm{C}}_2$ and (32), we have

$$\begin{array}{ccc}\hfill F^{\prime }\left(s\right)& \le & -q^{*}\left(s\right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)r\left(\varpi \left(s\right)\right){\left(z^{″}\left(\varpi \left(s\right)\right)\right)}^{\alpha }\hfill \\ & \le & \frac{-{\tau }_0}{{\tau }_0+p_0}{q}^{*}\left(s\right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)F\left({\tau }^{-1}\left(\varpi \left(s\right)\right)\right),\hfill \end{array}$$

then

$$F^{\prime }\left(s\right)+\frac{{\tau }_0}{{\tau }_0+p_0}{q}^{*}\left(s\right){\breve{\eta }}_n^{\alpha }\left(\sigma \left(s\right),\varpi \left(s\right)\right)F\left({\tau }^{-1}\left(\varpi \left(s\right)\right)\right)\le 0.$$

(55)

This means that F is a positive solution to (55). Now, using (54) and Theorem 2.1.1 in [47], we arrive at a contradiction with (55). And this completes the proof. □

Example 5.

Again, recall the half-linear NDDE (38). For $0<\mu <\frac{{\gamma }^2}{2-\gamma }$ and $\varpi \left(s\right)=\varrho s=\frac{\gamma +\mu }{2}s$, then it becomes clear that assumptions $\left({\mathrm{A}}_6\right)$ and

$$\gamma s>\frac{\gamma +\mu }{2}s>\mu s$$

are easily satisfied. In order to apply this example to Theorem 5, we first need to calculate the iterative function (29), which in turn requires defining some auxiliary sequences, just as we did previously in Example 2. So, let us define the sequence $\left\{{\tilde{R}}_i\right\}$ and $\left\{{\tilde{T}}_i\right\}$ for $i=0,1,\dots$, as:

$${\tilde{R}}_{i+1}:=\frac{{\gamma }^{{\tilde{T}}_i}{\varrho }^{{\tilde{T}}_i}}{{\tilde{T}}_i-1}\left[\mu {\varrho }^{1-{\tilde{T}}_i}+\frac{{\mu }^{2-{\tilde{T}}_i}}{{\tilde{T}}_i-2}-\frac{{\tilde{T}}_i-1}{{\tilde{T}}_i-2}{\varrho }^{2-{\tilde{T}}_i}\right],$$

$${\tilde{T}}_i:=\frac{\gamma b_0}{\gamma +a_0}{\tilde{R}}_i^{\alpha },$$

and

$${\tilde{R}}_0:={\breve{\eta }}_0\left(\mu ,\varrho \right)=\frac{{\left(\varrho -\mu \right)}^2}{2}.$$

Then, by using mathematical induction, we obtain

$${\breve{\eta }}_i\left(h,k\right):=\frac{k^{{\tilde{T}}_i}{\gamma }^{{\tilde{T}}_i}}{\left({\tilde{T}}_i-1\right)\left({\tilde{T}}_i-2\right)}\left[\left({\tilde{T}}_i-2\right)h{k}^{1-{\tilde{T}}_i}+h^{2-{\tilde{T}}_i}-\left({\tilde{T}}_i-1\right)k^{2-{\tilde{T}}_i}\right].$$

Substituting into (29) in Theorem 5 implies that (38) does not possess any decreasing positive solutions (${\mathrm{C}}_2=⌀$), if

$$b_0>\frac{\gamma +a_0}{\mathrm{e}\gamma }·\frac{1}{{\tilde{R}}_i^{\alpha }ln\frac{\gamma }{\varrho }}.$$

(56)

4. Oscillation Theorems

This section concerns giving oscillation theorems for (1) by combining criteria that ensure that the class ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are both empty.

Theorem 6.

Assume that there exists a function $\delta \left(s\right)\in {\mathbf{C}}^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right)$ and $\varpi \in \left(\left[s_0,\infty \right),\mathbb{R}\right)$ such that $\tau \left(s\right)>\varpi \left(s\right)>\sigma \left(s\right)$, $\varpi \left(s\right)\ge {\tau }^{-1}\left(\sigma \left(s\right)\right)$. If (35) and (54) hold, then every solution to (1) is oscillatory.

Theorem 7.

Assume that for $\alpha \ge 1$ there exists a function $\delta \left(s\right)\in {\mathbf{C}}^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right),$ and $\varpi \in \left(\left[s_0,\infty \right),\mathbb{R}\right)$ such that $\tau \left(s\right)>\varpi \left(s\right)>\sigma \left(s\right)$, $\varpi \left(s\right)\ge {\tau }^{-1}\left(\sigma \left(s\right)\right)$. If (40) and (54) hold, for any nonnegative integer $n,k$, then every solution to (1) is oscillatory.

Theorem 8.

Assume that for $\alpha \ge 1$ there exists a function $\delta \left(s\right)\in {\mathbf{C}}^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right),$ and $\varpi \in \left(\left[s_0,\infty \right),\mathbb{R}\right)$ such that $\tau \left(s\right)>\varpi \left(s\right)>\sigma \left(s\right)$, $\varpi \left(s\right)\ge {\tau }^{-1}\left(\sigma \left(s\right)\right)$. If (44) and (54) hold, for any nonnegative integer $n,k$, then every solution to (1) is oscillatory.

Theorem 9.

Assume that for $\alpha \ge 1$ there exists a function $\delta \left(s\right)\in {\mathbf{C}}^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right),$ and $\varpi \in \left(\left[s_0,\infty \right),\mathbb{R}\right)$ such that $\tau \left(s\right)>\varpi \left(s\right)>\sigma \left(s\right)$, $\varpi \left(s\right)\ge {\tau }^{-1}\left(\sigma \left(s\right)\right)$ If (49) and (54) hold, for any nonnegative integer $n,k$, then every solution to (1) is oscillatory.

Example 6.

Recall the half-linear NDDE (38). Exactly as we applied in Examples 1 and 5, we can obtain that (35) in Theorem 1 reduces to (39) to ensure that there are no positive solutions in Class ${\mathrm{C}}_1$, and (54) in Theorem 5 reduces to (56) to ensure that there are no positive solutions in Class ${\mathrm{C}}_2$. By combining these two criteria (39) and (56), we determine that all solutions of (1) are oscillatory if

$$b_0>max\left\{{\left(\frac{2\alpha }{\alpha +1}\right)}^{\alpha +1}·\frac{1}{R_0^{\alpha }·{\mu }^{2\alpha }},\frac{\gamma +a_0}{\mathrm{e}\gamma }·\frac{1}{{\tilde{R}}_i^{\alpha }ln\frac{\gamma }{\varrho }}\right\}.$$

5. Conclusions

In this paper, we deduced and improved some monotonic properties of positive solutions to (1) and their corresponding functions for classes ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$. After that, these relationships were used to set simple criteria with only one condition, to ensure that there are no positive solutions for either class, and then used them to ensure that all solutions to (1) oscillate. The results and criteria obtained in the previous sections were distinguished by several important points that confirm their originality and novelty. Our results were applied in Examples 1–6 and compared with previous works in Remarks 5 and 6. Through these comparisons, we noted that our results were an improvement on the oscillation criteria in many previous works. This requires fewer restrictions on coefficients and covers a larger area when applied. There were nine fundamental theorems provided, and their applicability and effectiveness were verified by testing the conditions they contained using more than one example.