Oscillation of Third-Order Differential Equations with Advanced Arguments

Abstract

The main objective of this work was to study some oscillatory and asymptotic properties of a new class of advanced neutral differential equations. Using new relations to link the solution and its corresponding function, we introduced new oscillatory criteria that aim to enhance, simplify, and complement some of current results. We provide some examples to demonstrate the significance of our results.

1. Introduction

The nineteenth century witnessed remarkable progress in the field of studying differential equations. They became a central topic in mathematical analysis. The development of computers enabled faster solutions to previously intractable differential equations, leading to advances in various scientific and engineering disciplines. Today, they are indispensable in numerous branches of mathematics, engineering, physics, and other sciences. The study and analysis of differential equations is still an active field of research with the emergence of new applications and technologies. It has become an effective link between mathematics and various sciences. As they describe various phenomena, differential equations enable us to not only study and analyse these phenomena but also understand them in depth. See, for example, ref. [1,2,3,4,5,6] and the references therein.

Advanced differential equations are of great importance in most life sciences, such as physics and engineering. They serve as an effective language for accurately describing physical models and phenomena, including electromagnetism, classical mechanics, quantum mechanics, thermodynamics, and fluid dynamics. In addition, the resulting complex equations are often solved using techniques like the Fourier series and the Laplace transform. In engineering, advanced differential equations are used to study and model the behaviour of mechanical systems, signal processing, robotics, and control systems, which are important for analysing the response, stability, and control of these systems. They also prove effective for describing and modelling biological systems, such as neural networks, biochemical reactions, population dynamics, epidemiology, and physiological processes. Its importance also appears in studying the spread of diseases, analysing genetic networks, and modelling drug interactions. Recently, their significance has extended to simulation, computer graphics, artificial intelligence, and machine learning.

Oscillation theory, a vital branch of mathematics, studies oscillatory behaviour in diverse mathematical systems. This theory focuses on identifying the necessary conditions for the oscillation of differential equation solutions. It includes a large number of mathematical models and phenomena. The theory of oscillations covers a wide range of systems, including mechanical systems and electrical circuits. It focuses on some basic aspects and concepts related to oscillation theory, such as periodic solutions, stability criteria analysis, and forced oscillations. Bifurcation theory, closely related to oscillation theory, studies the qualitative changes that occur in the behaviour of a system as a variable parameter. It explores how having different parameter values causes the appearance of, disappearance of, or change in oscillatory behaviour.

Differential equations with delays of various orders have long been the focus of scientists and researchers. Consequently, numerous studies have emerged around them, for example, ref. [7,8,9,10,11].

The ability of advanced differential equations to describe and analyse complex phenomena in the real world makes their importance increase day by day. Advanced differential equations are used in many fields, such as thermodynamics, classical mechanics, and electromagnetism. They are also used to describe population growth, the spread of diseases, and the dynamics of ecosystems. They also show importance in medicine, where advanced differential equations help in understanding drug interactions within the body and disease dynamics. As for meteorology and earth sciences, they are used in weather forecasts, studying ocean movement, and analysing geological phenomena such as earthquakes. So, advanced differential equations have appeared as a response to the urgent need for formulating models that include a description of future time, which is taken into consideration in many of life’s problems. There has been a clear interest in advanced second-order differential equations, the simplest of which is

$${\varkappa }^{″}\left(\nu \right)+q\left(\nu \right)\varkappa \left(\omega \left(\nu \right)\right)=0.$$

(1)

Later, the results of (1) were subsequently extended to include noncanonical forms, see [8,12]. On the other hand, Baculikova and Džurina [13] extended Equation (1) to include more models, as shown in the following equation

$${\left(r_1\left(\nu \right){\left({\left(r_2\left(\nu \right){\varkappa }^{\prime }\left(\nu \right)\right)}^{\prime }\right)}^m\right)}^{\prime }+q\left(\nu \right)\varkappa \left(\omega \left(\nu \right)\right)=0.$$

Additionally, in [14,15], some oscillation results were presented by using the iterative construction method for equation

$${\left(r\left(\nu \right){\left({\varkappa }^{\prime }\left(\nu \right)\right)}^m\right)}^{″}+q\left(\nu \right)\varkappa \left(\omega \left(\nu \right)\right)=0.$$

(2)

Tongxing and Yuriy [16] established some conditions that guarantee the oscillation of all solutions to the equation

$${\left(r\left(\nu \right){\left({\left(\varkappa \left(\nu \right)+p_0\varkappa \left(\nu -{\delta }_0\right)\right)}^{″}\left(\nu \right)\right)}^m\right)}^{\prime }+q\left(\nu \right){\varkappa }^m\left(\omega \left(\nu \right)\right)=0,$$

where ${\delta }_0<0$ and $\omega \left(\nu \right)>\nu .$ Furthermore, various methods have been used to obtain oscillation criteria for solutions in special cases of the studied equation. We mention here [8,17,18,19] and the references cited therein.

In this work, we investigated the asymptotic properties of solutions to the third-order neutral differential equation with an advanced argument of the form:

$${\left(r\left(\nu \right){\left({\left(\varkappa \left(\nu \right)+p\left(\nu \right)\varkappa \left(\varsigma \left(\nu \right)\right)\right)}^{″}\right)}^m\right)}^{\prime }+{\int }_a^{b}q\left(\nu ,s\right)f\left(\varkappa \left(\omega \left(\nu ,s\right)\right)\right)\mathrm{d}s=0,\nu \ge {\nu }_0>0,$$

(3)

where m is a quotient of odd positive integers, and $0\le a<b$. Throughout this paper, the following conditions must be satisfied:

(C1)

$q\left(\nu ,s\right)\in C\left([{\nu }_0,\infty )\times \left(a,b\right),\left(0,\infty \right)\right),$ $\omega \left(\nu ,s\right)\in C^1\left([{\nu }_0,\infty )\times \left(a,b\right),\left(0,\infty \right)\right),$ $\varsigma ,$ $r,$ $p\in C\left([{\nu }_0,\infty ),\left(0,\infty \right)\right),$ $\omega \left(\nu ,s\right)\ge \nu ,$ $\varsigma \left(\nu \right)\ge \nu ,$ ${\varsigma }^{\prime }\left(\nu \right)\ge {\varsigma }_0>0,$ ${\omega }^{\prime }\left(\nu ,s\right)\ge {\omega }_0>0,$ $p\left(\nu \right)\le p_0<\infty ,$ ${lim}_{\nu \to \infty }\varsigma \left(\nu \right)=\infty ,$ ${lim}_{\nu \to \infty }\omega \left(\nu ,s\right)=\infty$ and $q\left(\nu ,s\right)\ge 0$ does not vanish identically, under the condition

$$\pi \left(\nu \right)={\int }_{{\nu }_0}^{\nu }\frac{1}{r^{1/m}\left(s\right)}\mathrm{d}s=\infty ;$$

(4)

(C2)

$f:[t_0,\infty )\times \mathbb{R}\to \mathbb{R}$ is a continuous function such that $\varkappa f\left(\varkappa \right)>0$, $f\left(\varkappa \right)/\kappa >{\varkappa }^m$ for all $\varkappa \ne 0,$ where $\kappa >0$ and $f^{\prime }\left(\varkappa \right)\le 0$.

Definition 1.

A solution of (3) means $\varkappa \in C^2\left([{\nu }_{\varkappa },\infty ),[0,\infty )\right)$, ${\nu }_{\varkappa }>{\nu }_0$, which satisfies the property $r{\left({\varkappa }^{″}\right)}^m\in C^1\left([{\nu }_{\varkappa },\infty ),[0,\infty )\right)$ and satisfies (3) on $[{\nu }_{\varkappa },\infty ).$ We consider those solutions of (3) existing on some half-line $[{\nu }_{\varkappa },\infty )$ and satisfying

$$sup\{\left|\varkappa \left(\nu \right)\right|:T\le \nu <\infty \}>0\mathit{for}\mathit{any}T\ge {\nu }_{\varkappa }.$$

Definition 2.

A solution ϰ of (3) is called oscillatory if it has arbitrarily large zeros on $[{\nu }_{\varkappa },\infty );$ otherwise, it is said to be nonoscillatory. Equation (3) is said to be oscillatory if all its solutions are oscillatory.

Remark 1.

We note that the solutions of (3) are either $S=\varkappa$ or $S=-\varkappa$. Therefore, when studying any nonoscillatory solution to this equation, we do not need to study them together; it is sufficient to study only the positive solutions.

To the best of our knowledge, there are no previous results related to the study of oscillatory behaviour of neutral differential equations with advanced arguments (3). The difficulty of finding a relationship that links the solution $\varkappa$ and the corresponding function

$$z\left(\nu \right)=\varkappa \left(\nu \right)+p\left(\nu \right)\varkappa \left(\varsigma \left(\nu \right)\right),$$

represents a challenge in establishing the necessary conditions to exclude nonoscillatory solutions and, consequently, a difficulty in finding criteria that ensure the oscillation of the solutions of Equation (3). Therefore, in this paper, we derive some new results related to the study of oscillatory and asymptotic behaviour of Equation (3), by utilizing important and new relations we obtained.

2. Preliminaries Results

In this section, we present two lemmas that are used to obtain the main results of this study. Throughout the paper, we use the following abbreviations:

$$L_{i}z\left(\nu \right)=\left\{\begin{array}{c}\frac{1}{{\omega }_0}r\left({\omega }^{-1}\left(\nu ,s\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\right)}^m\\ +\frac{p_0^{{}^m}}{{\omega }_0{\varsigma }_0}r\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right)\right)}^m\mathrm{if}i=1\\ r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m+\frac{p_0^{{}^m}}{{\varsigma }_0}r\left(\varsigma \left(\nu \right)\right){\left(z^{″}\left(\left(\varsigma \left(\nu \right)\right)\right)\right)}^m\mathrm{if}i=2\end{array}\right.$$

and

$${\tilde{q}}_i\left(\nu ,s\right)=\left\{\begin{array}{c}min\left\{\Omega \left({\omega }^{-1}\left(\nu ,s\right)\right),\Omega \left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right)\right\}\mathrm{if}i=1\\ min\left\{\Omega \left(\nu ,s\right),\Omega \left(\varsigma \left(\nu ,s\right)\right)\right\}\mathrm{if}i=2\end{array}\right.,$$

where $\Omega \left(\nu \right)={\int }_a^{b}q\left(\nu ,s\right)$.

Lemma 1

([20] (Lemmas 1 and 2)). Assume that $F_1$, $F_2\in [0,\infty ).$ If $m\ge 1,$ then

$$F_1^m+F_2^m\ge \frac{1}{2^{m-1}}{\left(F_1+F_2\right)}^m.$$

Also, if $0<m\le 1,$ then

$$F_1^m+F_2^m\ge {\left(F_1+F_2\right)}^m.$$

Lemma 2.

Let $\varkappa >0$ be a solution of (3). Then,

$$z\left(\nu \right)z^{″}\left(\nu \right)>0,$$

and $z\left(\nu \right)$ satisfies one of the following two cases:

$$z\left(\nu \right)z^{\prime }\left(\nu \right)<0,$$

(I₁)

or

$$z\left(\nu \right)z^{\prime }\left(\nu \right)>0,$$

(I₂)

eventually.

Proof. 

Assume that $\varkappa >0$ is a solution of (3) for $\nu \ge {\nu }_0$. From (3), we see that ${\left(r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m\right)}^{\prime }$ is eventually negative. That is, $r{\left(z^{″}\right)}^m$ is of fixed sign eventually. If $r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m$ is negative, then $z\left(\nu \right)$ is negative and decreasing. That is, we conclude that $r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m$ is positive and $z\left(\nu \right)$ satisfies either Case (I₁) or Case (I₂). The proof is complete. □

3. Main Results

In this section, we first present various criteria that guarantee the absence of solutions of type (I₂) in Theorems 1, 4, and 5. Furthermore, by combining these results with Theorem 2, we establish the oscillation criteria for Equation (3) in both Theorems 3 and 6.

Theorem 1.

Suppose that ${\left({\omega }^{-1}\left(\nu ,s\right)\right)}^{\prime }>0$. If

$$\underset{\nu \to \infty }{lim\; inf}\frac{1}{\varpi \left(\nu \right)}{\int }_{\nu }^{\infty }{\varpi }^{1+1/m}\left(s\right)\pi \left({\omega }^{-1}\left(\nu ,s\right)\right)\mathrm{d}s>{\left(m+1\right)}^{{}^{-\left(1+1/m\right)}},$$

(5)

where

$$\varpi \left(\nu \right)=\frac{\kappa }{\mu }{\int }_{\nu }^{\infty }{\tilde{q}}_1\left(\nu ,s\right)\mathrm{d}s,$$

(6)

then all nonoscillatory solutions of (3) satisfy only Case (I₁).

Proof. 

Let$\varkappa >0$ be a solution of (3). Using

$$F_1=\varkappa \left(\nu \right)\mathrm{and}{F}_2=p_0\varkappa \left(\varsigma \left(\nu \right)\right),$$

in Lemma 1, we obtain

$${\left(\varkappa \left(\nu \right)+p_0\varkappa \left(\varsigma \left(\nu \right)\right)\right)}^m\le \mu \left({\varkappa }^m\left(\nu \right)+p_0^m{\varkappa }^m\left(\varsigma \left(\nu \right)\right)\right),$$

and

$$\mu \left({\varkappa }^m\left(\nu \right)+p_0^m{\varkappa }^m\left(\varsigma \left(\nu \right)\right)\right)\ge z^m\left(\nu \right),$$

(7)

where

$$\mu :=\left\{\begin{array}{c}1\mathrm{when}0<m\le 1\\ 2^{1-m}\mathrm{when}m\ge 1\end{array}\right..$$

Using $\left(C_2\right)$ in (3), we obtain

$$\begin{array}{ccc}& & {\left(r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m\right)}^{\prime }+\kappa {\int }_a^{b}q\left(\nu ,s\right){\varkappa }^m\left(\omega \left(\nu ,s\right)\right)\mathrm{d}s\hfill \\ & \le & {\left(r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m\right)}^{\prime }+\kappa {\varkappa }^m\left(\nu \right){\int }_a^{b}q\left(\nu ,s\right)\mathrm{d}s\le 0.\hfill \end{array}$$

(8)

Also,

$$\frac{1}{{\omega }_0}{\left(r\left({\omega }^{-1}\left(\nu ,s\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\right)}^m\right)}^{\prime }+\kappa {\varkappa }^m\left(\nu \right)\Omega \left({\omega }^{-1}\left(\nu ,s\right)\right)\le 0.$$

It follows that

$$\frac{p_0^{{}^m}}{{\omega }_0{\varsigma }_0}{\left(r\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right)\right)}^m\right)}^{\prime }+\kappa p_0^{{}^m}{\varkappa }^m\left(\varsigma \left(\nu \right)\right)\Omega \left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right)\le 0.$$

(9)

Combining (8) and (9) and using (7), we have

$$\begin{array}{ccc}\hfill 0& \ge & \frac{1}{{\omega }_0}{\left(r\left({\omega }^{-1}\left(\nu ,s\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\right)}^m\right)}^{\prime }\hfill \\ & & +\frac{p_0^{{}^m}}{{\omega }_0{\varsigma }_0}{\left(r\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right)\right)}^m\right)}^{\prime }\hfill \\ & & +\kappa \left({\varkappa }^m\left(\nu \right)\Omega \left({\omega }^{-1}\left(\nu ,s\right)\right)+p_0^{{}^m}{\varkappa }^m\left(\varsigma \left(\nu \right)\right)\Omega \left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right)\right)\hfill \\ & \ge & {\left(\frac{1}{{\omega }_0}r\left({\omega }^{-1}\left(\nu ,s\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\right)}^m+\frac{p_0^{{}^m}}{{\omega }_0{\varsigma }_0}r\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\varsigma \left(\nu ,s\right)\right)\right)\right)}^m\right)}^{\prime }\hfill \\ & & +\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)z^m\left(\nu \right).\hfill \end{array}$$

Thus,

$${\left(L_{1}z\left(\nu \right)\right)}^{\prime }+\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)z^m\left(\nu \right)\le 0.$$

(10)

Now, let z satisfy Case (I₂). Define the positive function $\varphi$ as follows:

$$\varphi \left(\nu \right)=\frac{1}{z^m\left(\nu \right)}{L}_{1}z\left(\nu \right).$$

(11)

Since ${\left({\omega }^{-1}\left(\nu ,s\right)\right)}^{\prime }>0$ and ${\left(L_{1}z\left(\nu \right)\right)}^{\prime }>0,$ this implies

$$\varphi \left(\nu \right)<\frac{{\varsigma }_0+p_0^{{}^m}}{{\omega }_0{\varsigma }_0}\frac{1}{z^m\left(\nu \right)}r\left({\omega }^{-1}\left(\nu ,s\right)\right){\left(z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\right)}^m$$

or

$$\frac{{\omega }_0{\varsigma }_0}{{\varsigma }_0+p_0^{{}^m}}\varphi \left(\nu \right)<r\left({\omega }^{-1}\left(\nu ,s\right)\right)\frac{{\left(z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\right)}^m}{z^m\left(\nu \right)}.$$

(12)

Differentiating $\varphi \left(\nu \right)$, we have

$${\varphi }^{\prime }\left(\nu \right)=\frac{{\left(L_{1}z\left(\nu \right)\right)}^{\prime }}{z^m\left(\nu \right)}-m\frac{L_{1}z\left(\nu \right)}{z^m\left(\nu \right)}\frac{z^{\prime }\left(\nu \right)}{z\left(\nu \right)}.$$

Since $\omega \left(\nu \right)\ge \left(\nu ,s\right)$ and $z^{\prime }\left(\nu \right)>0,$ we see that

$${\varphi }^{\prime }\left(\nu \right)\le \frac{1}{z^m\left(\nu \right)}{\left(L_{1}z\left(\nu \right)\right)}^{\prime }-m\frac{1}{z^m\left(\nu \right)}{L}_{1}z\left(\nu \right)\frac{z^{\prime }\left({\omega }^{-1}\left(\nu ,s\right)\right)}{z\left(\nu \right)}$$

By using (10) and (11), we have

$${\varphi }^{\prime }\left(\nu \right)\le -\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)-m\frac{z^{\prime }\left({\omega }^{-1}\left(\nu ,s\right)\right)}{z\left(\nu \right)}\varphi \left(\nu \right).$$

(13)

From (10), we note that $r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m$ is nonincreasing, which implies

$$\begin{array}{ccc}\hfill z^{\prime }\left(\nu \right)& \ge & {\int }_{{\nu }_1}^{\nu }\frac{1}{r^{1/m}\left(s\right)}{\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^m\right)}^{1/m}\mathrm{d}s\hfill \\ & \ge & {\left(r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m\right)}^{1/m}{\int }_{{\nu }_1}^{\nu }\frac{1}{r^{1/m}\left(s\right)}\mathrm{d}s\hfill \\ & \ge & {\left(r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m\right)}^{1/m}\pi \left(\nu \right).\hfill \end{array}$$

(14)

That is,

$$z^{\prime }\left({\omega }^{-1}\left(\nu ,s\right)\right)\ge r^{1/m}\left({\omega }^{-1}\left(\nu ,s\right)\right)z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\pi \left({\omega }^{-1}\left(\nu ,s\right)\right).$$

(15)

From (13) and (15), one has

$${\varphi }^{\prime }\left(\nu \right)\le -\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)-m\frac{1}{z\left(\nu \right)}{r}^{1/m}\left({\omega }^{-1}\left(\nu ,s\right)\right)\left(z^{″}\left({\omega }^{-1}\left(\nu ,s\right)\right)\right)\pi \left({\omega }^{-1}\left(\nu ,s\right)\right)\varphi \left(\nu \right).$$

From (12), we find

$${\varphi }^{\prime }\left(\nu \right)\le -\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)-m{B}^{1/m}\pi \left({\omega }^{-1}\left(\nu ,s\right)\right){\varphi }^{1+1/m}\left(\nu \right).$$

where

$$B=\frac{{\omega }_0{\varsigma }_0}{{\varsigma }_0+p_0^{{}^m}}.$$

(16)

Integrating the previous inequality, we obtain

$$\varphi \left(\nu \right)\ge \varpi \left(\nu \right)+m{B}^{1/m}{\int }_{\nu }^{\infty }\pi \left({\omega }^{-1}\left(\nu ,s\right)\right){\varphi }^{1+1/m}\left(s\right)\mathrm{d}s,$$

(17)

thus,

$$\frac{\varphi \left(\nu \right)}{\varpi \left(\nu \right)}\ge 1+m{B}^{1/m}\frac{1}{\varpi \left(\nu \right)}{\int }_{\nu }^{\infty }{\left(\frac{\varphi \left(s\right)}{\varpi \left(s\right)}\right)}^{1+1/m}\pi \left({\omega }^{-1}\left(\nu ,s\right)\right){\varpi }^{1+1/m}\left(s\right)\mathrm{d}s.$$

By $\varphi \left(\nu \right)/\varpi \left(\nu \right)>1$, we see that

$$\underset{\nu \ge {\nu }_1}{inf}\varphi \left(\nu \right)/\varpi \left(\nu \right)=\lambda ,\mathrm{where}\lambda \in [1,\infty ),$$

where $\varpi \left(\nu \right)$ is defined as in (6). That is,

$$\frac{1}{\varpi \left(\nu \right)}\varphi \left(\nu \right)-1\ge m{B}^{1/m}\frac{{\lambda }^{1+1/m}}{\varpi \left(\nu \right)}{\int }_{\nu }^{\infty }\pi \left({\omega }^{-1}\left(\nu ,s\right)\right){\varpi }^{1+1/m}\left(s\right)\mathrm{d}s.$$

(18)

From (5), we have

$$\frac{1}{{\left(\theta +1\right)}^{{}^{1+1/\theta }}}<\eta <{\varpi }^{-1}\left(\nu \right){\int }_{\nu }^{\infty }\pi \left({\omega }^{-1}\left(\nu ,s\right)\right){\varpi }^{1+1/m}\left(\nu ,s\right)\mathrm{d}s,$$

(19)

where

$$\theta =m{\left({\omega }_0{\varsigma }_0\right)}^{1/m}/{\left({\varsigma }_0+p_0^{{}^m}\right)}^{1/m},$$

for $\eta >0.$ Using (18) in (19), we obtain

$$\frac{1}{\varpi \left(\nu \right)}\varphi \left(\nu \right)-1\ge \theta \left({\lambda }^{1+1/\theta }\right)\eta .$$

Also,

$$1+\frac{\theta \left({\lambda }^{1+1/m}\right)}{{\left(m+1\right)}^{{}^{1+1/m}}}<1+\theta \left({\lambda }^{1+1/m}\right)\eta \le \lambda$$

and

$$\frac{1}{\theta +1}\theta \left({\lambda }^{1+1/m}\right){\left(m+1\right)}^{{}^{-\left(1+1/m\right)}}+\frac{1}{\theta +1}<\frac{1}{\theta +1}\lambda .$$

This is in contradiction to the fact that the function

$$g\left(\widehat{\lambda }\right)=\frac{1}{\theta +1}+\frac{\theta }{\theta +1}{\widehat{\lambda }}^{1+1/m}-\widehat{\lambda },\widehat{\lambda },\mathrm{where}\widehat{\lambda }=\frac{1}{\theta +1}\lambda$$

is positive for all $\widehat{\lambda }>0.$ This completes the proof. □

Corollary 1.

Suppose that

$${\int }_{{\nu }_0}^{\infty }{\tilde{q}}_1\left(\nu ,s\right)\mathrm{d}s=\infty$$

(20)

or

$${\int }_{{\nu }_0}^{\infty }\varpi {\left(s\right)}^{1+1/m}\pi \left({\omega }^{-1}\left(\nu ,s\right)\right)\mathrm{d}s=\infty .$$

(21)

Then, all nonoscillatory solutions of (3) satisfy only Case (I₁).

Proof. 

As in the proof of Theorem 1, (17) contradicts (20). Now, from (17) and $\varphi \left(\nu \right)>\varpi \left(\nu \right),$ we see that

$$\varpi \left({\nu }_1\right)+{\int }_{{\nu }_1}^{\infty }m{\left(\frac{{\omega }_0{\varsigma }_0}{{\varsigma }_0+p_0^{{}^m}}\right)}^{1/m}{\varpi }^{1+1/m}\left(s\right)\pi \left({\omega }^{-1}\left(\nu ,s\right)\right)\mathrm{d}s-\varphi \left({\nu }_1\right)\le 0.$$

This completes the proof. □

Theorem 2.

Suppose that all nonoscillatory solutions of (3) satisfy only Case (I₁) and $\varsigma \circ \omega =\omega \circ \varsigma$. If

$${\int }_{{\nu }_0}^{\infty }{\int }_v^{\infty }\left[r^{-1/m}\left(u\right){\left({\int }_u^{\infty }{\tilde{q}}_2\left(\nu ,s\right)\mathrm{d}s\right)}^{1/m}\right]\mathrm{d}u\mathrm{d}v=\infty ,$$

(22)

then every nonoscillatory solution of (3) satisfies

$$\underset{\nu \to \infty }{lim}z\left(\nu \right)=0.$$

(23)

Proof. 

Let $\varkappa >0$ be a solution of (3). Using (7), we obtain

$$\varkappa {\left(\omega \left(\nu ,s\right)\right)}^m+p_0^m\varkappa {\left(\varsigma \left(\omega \left(\nu ,s\right)\right)\right)}^m\ge \frac{1}{\mu }{z}^m\left(\omega \left(\nu ,s\right)\right).$$

(24)

By ${\varsigma }^{\prime }\left(\nu \right)\ge {\varsigma }_0>0,$ we have

$$\frac{p_0^{{}^m}}{{\varsigma }_0}{\left(r\left(\varsigma \left(\nu \right)\right){\left(z^{″}\left(\varsigma \left(\nu \right)\right)\right)}^m\right)}^{\prime }+\kappa p_0^{{}^m}q\left(\varsigma \left(\nu ,s\right)\right){\varkappa }^m\left(\varsigma \left(\omega \left(\nu ,s\right)\right)\right)\le 0.$$

(25)

From (8) and (25), by using (24), we have

$${\left(r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m+\frac{p_0^{{}^m}}{{\varsigma }_0}r\left(\varsigma \left(\nu \right)\right){\left(z^{″}\left(\left(\varsigma \left(\nu \right)\right)\right)\right)}^m\right)}^{\prime }+\frac{\kappa }{\mu }{\tilde{q}}_2\left(\nu ,s\right)z^m\left(\omega \left(\nu ,s\right)\right)\le 0.$$

So, it is easy to see that

$${\left(L_{2}z\left(\nu \right)\right)}^{\prime }+\frac{\kappa }{\mu }{\tilde{q}}_2\left(\nu ,s\right)z^m\left(\omega \left(\nu ,s\right)\right)\le 0.$$

(26)

Now, since $\varsigma \left(\nu \right)>\nu$, we obtain

$$\begin{array}{ccc}\hfill L_{2}z\left(\nu \right)& =& r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m\hfill \\ & & +\frac{p_0^{{}^m}}{{\varsigma }_0}r\left(\varsigma \left(\nu \right)\right){\left(z^{″}\left(\left(\varsigma \left(\nu \right)\right)\right)\right)}^m\hfill \\ & \le & \frac{{\varsigma }_0+p_0^{{}^m}}{{\varsigma }_0}r\left(\nu \right){\left(z^{″}\left(\nu \right)\right)}^m.\hfill \end{array}$$

(27)

From Case (I₁), it follows that

$$lim{}_{{}_{\nu \to \infty }}z\left(\nu \right)=\varrho ,$$

where $\varrho \ge 0$. Let $\varrho >0$. Integrating (26) and using $\varrho <z\left(\omega \left(\nu ,s\right)\right)$, we obtain

$$L_{2}z\left(\nu \right)-\frac{\kappa }{\mu }{\varrho }^m{\int }_{\nu }^{\infty }{\tilde{q}}_2\left(\nu ,s\right)\mathrm{d}s\ge 0.$$

(28)

From (27), (28) becomes

$$z^{″}\left(\nu \right)-{\left(\frac{\kappa {\varsigma }_0}{\mu \left({\varsigma }_0+p_0^{{}^m}\right)}\right)}^{\frac{1}{m}}\frac{\varrho }{r^{\frac{1}{m}}\left(\nu \right)}{\left({\int }_{\nu }^{\infty }{\tilde{q}}_2\left(\nu ,s\right)\mathrm{d}s\right)}^{\frac{1}{m}}\ge 0.$$

(29)

Integrating (29), we see that

$$-z^{\prime }\left(\nu \right)-\varrho {\left(\frac{\kappa {\varsigma }_0}{\mu \left({\varsigma }_0+p_0^{{}^m}\right)}\right)}^{\frac{1}{m}}{\int }_{\nu }^{\infty }\frac{1}{r^{1/m}\left(u\right)}{\left({\int }_u^{\infty }{\tilde{q}}_2\left(\nu ,s\right)\mathrm{d}s\right)}^{1/m}\mathrm{d}u\ge 0.$$

Integrating again, we find

$$z\left({\nu }_1\right)-\varrho {\left(\frac{\kappa {\varsigma }_0}{\mu \left({\omega }_0+p_0^{{}^m}\right)}\right)}^{\frac{1}{m}}{\int }_{{\nu }_1}^{\infty }{\int }_v^{\infty }\frac{1}{r^{1/m}\left(u\right)}{\left({\int }_u^{\infty }{\tilde{q}}_2\left(\nu ,s\right)\mathrm{d}s\right)}^{1/m}\mathrm{d}u\mathrm{d}v\ge 0.$$

This contradiction with (22) leads us to (23). □

Theorem 3.

Assume that $\omega \left(\nu ,s\right)<\varsigma \left(\nu \right)$ and $\varsigma \circ \omega =\omega \circ \varsigma .$ If (5) or (20) or (21) and (22) hold, then every solution of (3) oscillates or satisfies (23).

Proof. 

By combining the results in Theorem 1 and Corollary 1 with the results in Theorem 2, it is easy to see that every solution of (3) oscillates or satisfies (23). □

Theorem 4.

Assume that there exists some ${\gamma }_n\left(\nu \right)$, where ${\left\{{\gamma }_n\left(\nu \right)\right\}}_{n=0}^{\infty }$ is a sequence such that

$${\int }_{{\nu }_0}^{\infty }{\tilde{q}}_1\left(\nu ,s\right)exp\left(mK{B}^{1/m}{\int }_{{\nu }_0}^{\nu }{\gamma }_n^{1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s\right)\mathrm{d}\nu =\infty ,\mathit{for}\mathit{some}K\in (0,1)$$

(30)

and

$${\gamma }_{n+1}\left(\nu \right)-{\gamma }_0\left(\nu \right)=m{B}^{1/m}{\int }_{\nu }^{\infty }\pi \left(s\right){\gamma }_n^{1+1/m}\left(s\right)\mathrm{d}s,n=0,1,...,$$

(31)

where $B={\omega }_0{\varsigma }_0/\left({\varsigma }_0+p_0^{{}^m}\right).$ Then, all nonoscillatory solutions of (3) satisfy only Case (I₁).

Proof. 

Suppose that $\varkappa$ is a solution of (3) and $z\left(\nu \right)>0$ satisfying Case (I₂). By using (17), it is easy to see that

$$\varphi \left(\nu \right)\ge {\gamma }_0\left(\nu \right).$$

Thus,

$$\begin{array}{ccc}\hfill {\gamma }_1\left(\nu \right)& =& {\gamma }_0\left(\nu \right)+mK{B}^{1/m}{\int }_{\nu }^{\infty }{\gamma }_0^{1+1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s\hfill \\ & \le & {\gamma }_0\left(\nu \right)+mK{B}^{1/m}{\int }_{\nu }^{\infty }{\varphi }^{1+1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s.\hfill \end{array}$$

That is

$${\gamma }_1\left(\nu \right)\le \varphi \left(\nu \right).$$

The sequence ${\left\{{\gamma }_n\left(\nu \right)\right\}}_{n=0}^{\infty }$ is nondecreasing (by induction) and

$$\varphi \left(\nu \right)\ge {\gamma }_n\left(\nu \right).$$

Hence,

$$\underset{n\to \infty }{lim}{\gamma }_n\left(\nu \right)=\gamma \left(\nu \right).$$

By using Lebesgue monotone convergence theorem, (31) becomes

$$\gamma \left(\nu \right)-{\gamma }_0\left(\nu \right)=mK{B}^{1/m}{\int }_{\nu }^{\infty }{\gamma }^{1+1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s,$$

and

$$\begin{array}{ccc}\hfill {\gamma }^{\prime }\left(\nu \right)+K\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)+mK{B}^{1/m}{\gamma }^{1+1/m}\left(\nu \right)\pi \left(\nu \right)& =& 0\hfill \\ \hfill {\gamma }^{\prime }\left(\nu \right)+K\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)+mK{B}^{1/m}\gamma \left(\nu \right){\gamma }_n^{1/m}\left(\nu \right)\pi \left(\nu \right)& \le & 0\mathrm{for}\nu \ge {\nu }_1.\hfill \end{array}$$

That is,

$$\begin{array}{ccc}\hfill 0& \ge & {\left(\gamma \left(\nu \right)exp\left(mK{B}^{1/m}{\int }_{{\nu }_1}^{\nu }{\gamma }_n^{1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s\right)\right)}^{\prime }\hfill \\ & & +K\frac{\kappa }{\mu }{\tilde{q}}_1\left(\nu ,s\right)exp\left(mK{B}^{1/m}{\int }_{{\nu }_1}^{\nu }{\gamma }_n^{1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s\right).\hfill \end{array}$$

(32)

Integrating (32), we have

$$\begin{array}{ccc}\hfill 0& \le & \gamma \left(\nu \right)exp\left(mK{B}^{1/m}{\int }_{{\nu }_1}^{\nu }{\gamma }_n^{1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s\right)\hfill \\ & \le & \gamma \left({\nu }_1\right)-K\frac{\kappa }{\mu }{\int }_{{\nu }_1}^{\nu }{\tilde{q}}_1\left(u,s\right)\left(exp\left(mK{B}^{1/m}{\int }_{{\nu }_1}^{\nu }{\gamma }_n^{1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s\right)\right)\mathrm{d}u.\hfill \end{array}$$

Therefore, this contradiction with (30) ends the proof. □

Theorem 5.

Suppose that there exists some ${\gamma }_n\left(\nu \right)$ such that

$$\underset{\nu \to \infty }{lim\; sup}{\left({\int }_{{\nu }_1}^{\nu }\widehat{\pi }\left(s\right)\mathrm{d}s\right)}^m{\gamma }_n\left(\nu \right)>1,$$

(33)

where $\widehat{\pi }\left(\nu \right)=\pi \left(\nu \right)-\pi \left({\nu }_1\right).$ Then, all nonoscillatory solutions of (3) satisfy only Case (I₁).

Proof. 

Let $\varkappa \left(\nu \right)>0$ be a solution of (3) and $z\left(\nu \right)>0$ satisfying (I₂). Integrating the inequality (14) from ${\nu }_1$ and $\nu$, we find that

$$z\left(\nu \right)-r^{\frac{1}{m}}\left(\nu \right)z^{″}\left(\nu \right){\int }_{{\nu }_1}^{\nu }{\int }_{{\nu }_1}^u\left(r^{-1/m}\left(s\right)\mathrm{d}s\right)\mathrm{d}u\ge 0.$$

(34)

From (12) and (34), we obtain

$$\frac{1}{\varphi \left(\nu \right)}=B\frac{1}{r\left(\nu \right)}{\left(\frac{z\left(\nu \right)}{z^{″}\left(\nu \right)}\right)}^m\ge B{\left({\int }_{{\nu }_1}^{\nu }\pi \left(s\right)-\pi \left({\nu }_1\right)\mathrm{d}s\right)}^m,$$

where B is defined as in (16). Therefore,

$${\left({\int }_{{\nu }_1}^{\nu }\widehat{\pi }\left(s\right)\mathrm{d}s\right)}^m{\gamma }_n\left(\nu \right)\le {\left({\int }_{{\nu }_1}^{\nu }\widehat{\pi }\left(s\right)\mathrm{d}s\right)}^m\varphi \left(\nu \right)\le 1.$$

This is a contradiction. □

Corollary 2.

If

$$\underset{\nu \to \infty }{lim\; sup}{\left({\int }_{\nu 1}^{\nu }\widehat{\pi }\left(s\right)\mathrm{d}s\right)}^m{\int }_{\nu }^{\infty }{\tilde{q}}_1\left(\nu ,s\right)\mathrm{d}s>1$$

(35)

or

$$\underset{\nu \to \infty }{lim\; sup}{\left({\int }_{{\nu }_1}^{\nu }\widehat{\pi }\left(s\right)\mathrm{d}s\right)}^m\left(\varpi \left(\nu \right)+m{B}^{1/m}{\int }_{\nu }^{\infty }{\varpi }^{1+1/m}\left(s\right)\pi \left(s\right)\mathrm{d}s\right)>1.$$

(36)

Then, all nonoscillatory solutions of (3) satisfy only Case (I₁).

Proof. 

In Theorem 5, set $n=0$ and $n=1$, respectively, we have (35) and (36), which is straightforward. □

Theorem 6.

Assume that there exists some ${\gamma }_n\left(\nu \right)$ such that (30) and (31) or (33) or (35) or (36) and (22). Then, every solution of (3) oscillates or converges to zero.

Proof. 

By combining the results in Theorems 4 and 5, and Corollary 2 with the results in Theorem 2, it is easy to see that every solution of (3) oscillates or satisfies (23). □

4. Applications

Example 1.

Consider the differential Equation (3) where

$$r\left(\nu \right)=\nu ,p\left(\nu \right)=p,\varsigma \left(\nu \right)=\lambda \nu ,\omega \left(\nu ,s\right)=\lambda \nu ,q\left(\nu ,s\right)=\beta {\nu }^{-6},m=3,a=0,b=1,$$

(E₁)

where $\beta \in (0,\infty ),\lambda \in [1,\infty ),\nu \ge 1.$ That is,

$${\tilde{q}}_1\left(\nu ,s\right)=\frac{\beta }{{\nu }^6}\mathrm{and}\varpi \left(\nu \right)=\frac{\kappa \beta }{5\mu {\nu }^5}.$$

From Theorem 1, we see that all nonoscillatory solutions of Equation (3) with (E₁) satisfy only Case (I₁) if

$$\beta \frac{\kappa }{\lambda }>5^4{\left(\frac{1}{6}\right)}^{1/3}.$$

Moreover, through Theorem 2, (22) holds. Hence, every nonoscillatory solution $z\left(\nu \right)$ of (3) with (E₁) converges to zero as $\nu \to \infty$.

Example 2.

Consider the differential Equation (3) where

$$r\left(\nu \right)={\nu }^2,\omega \left(\nu ,s\right)=\lambda \nu ,q\left(\nu ,s\right)=\beta {\nu }^{-5},m=3,a=0,b=1,$$

(E₂)

where $\beta \in (0,\infty ),\lambda \in [1,\infty ),\nu \ge 1.$ By Corollary 2, we see that all nonoscillatory solutions of (3) with (E₂) satisfy only Case (I₁)

$$\begin{array}{cc}\hfill when n=1 in \left(33\right)& if1-\frac{9^3}{4^4}{\lambda }^3\beta <\frac{9^4}{4^{16/3}}{\lambda }^4{\beta }^{4/3}\hfill \\ \hfill when n=0 in \left(33\right)& if\beta {\lambda }^3>\frac{4^4}{9^3}\hfill \end{array}$$

Also, from Theorem 2, (22) holds. Thus, every nonoscillatory solution $z\left(\nu \right)$ of (3) with (E₂) satisfies (23).

Example 3.

Consider the differential Equation (3) where

$$r\left(\nu \right)=\nu ,\omega \left(\nu ,s\right)={\nu }^2,q\left(\nu ,s\right)=\beta {\nu }^{-9},m=3,a=0,b=1,$$

(E₃)

where $\beta \in (0,\infty ),\nu \ge 1.$ From Theorem 1, we see that all nonoscillatory solutions of Equation (3) with (E₃) satisfy only Case (I₁) if

$$\beta >{\left(\frac{5}{4}\right)}^4\frac{8}{27}.$$

Remark 2.

Upon substituting $p=0$ in Example 1, it becomes evident that the criterion ensuring all nonoscillatory solutions of Equation (3) satisfy only Case (I₁) is predominantly reliant on the size of the advanced argument. Further examination reveals that $\omega \left(\nu \right)$ in Example 3 is greater than in Example 1, facilitating a reduction in the function $q\left(\nu ,s\right).$

Furthermore, wew apply our results to the following advanced third-order differential equations

$${\left(r\left(\nu \right){\left|\left(z^{″}\left(\nu \right)\right)\right|}^{m-1}{z}^{″}\left(\nu \right)\right)}^{\prime }+{\int }_a^{b}q\left(\nu ,s\right){\left|\left[\varkappa \left(\omega \left(\nu ,s\right)\right)\right]\right|}^{m-1}\varkappa \left(\omega \left(\nu ,s\right)\right)\mathrm{d}s=0,\mathrm{where}m>0.$$

(37)

Example 4.

Consider the differential Equation (37) where

$$r\left(\nu \right)={\nu }^a,\omega \left(\nu ,s\right)={\nu }^c,q\left(\nu ,s\right)=\beta {\nu }^{-\tau },a=0,b=1,\beta >0,c\in [1,\infty ),$$

(E₄)

where $\nu \ge 1,$ $m>a>0.$ When substituting the variables in (E₄), all nonoscillatory solutions of (37) satisfy only Case (I₁) if

$$1-m\ge \tau -mc$$

or

$$1+\frac{1}{m}-m\ge \frac{\tau }{m}-mc-\tau +\frac{a}{m+2}(\mathit{Corollary} 1)$$

and

$$1-m<\tau -mc(\mathit{Corollary} 2).$$

5. Conclusions

In this paper, we established some new results concerning the oscillation and convergence of Equation (3). We obtained new relationships that allowed us to overcome the difficulties encountered in the previous literature to attain conditions for the oscillation of the solutions; see [8,12,13]. Furthermore, these relationships can be utilised to derive various oscillation conditions using different techniques, such as the integrated average technique and comparison principles. On the other hand, using the iterative nature of the series ${\left\{{\gamma }_n\left(\nu \right)\right\}}_{n=0}^{\infty }$ enables us to test the oscillation of the solutions of Equation (3) several times, and this further improves the quality of the conditions, as is clear in Example 2. Providing a relationship between the solution and the corresponding function to eliminate the decreasing positive solutions is a valuable addition to oscillation theory and a subject of interest and research.