Oscillation of Nonlinear Neutral Delay Difference Equations of Fourth Order

Abstract

This paper focuses on the study of the oscillatory behavior of fourth-order nonlinear neutral delay difference equations. The authors use mathematical techniques, such as the Riccati substitution and comparison technique, to explore the regularity and existence properties of the solutions to these equations. The authors present a new form of the equation: $\mathsf{\Delta }\left(a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}\right)+p\left(m\right)w^{p_2-1}\left(\sigma \left(m\right)\right)=0$, where $z\left(m\right)=w\left(m\right)+q\left(m\right)w\left(m-\tau \right)$ with the following conditions: $\sum _{s=m_0}^{\infty }\frac{1}{a\left(\frac{1}{p_1-1\left(\mathrm{s}\right)}\right)}=\infty$. The equation represents a system where the state of the system at any given time depends on its current time and past values. The authors demonstrate new insights into the oscillatory behavior of these equations and the conditions required for the solutions to be well-behaved. They also provide a numerical example to support their findings.

1. Introduction

Mathematical models are essential tools for understanding the behavior of physical systems over time. These models provide insights into how systems behave and how they can be optimized, and they play a crucial role in the advancement of numerous fields, including physics, engineering, and computer science. Nonlinear neutral delay difference equations are one type of mathematical model that has found a wide range of applications in these field [1]

Nonlinear neutral delay difference equations are used to study systems where the state of the system depends not only on the current time but also on past values. These equations are particularly useful in the study of distributed networks containing lossless transmission lines, which are commonly used in high-speed computer systems to interconnect switching circuits. Understanding the oscillatory behavior of these equations is an important area of research, as it helps to shed light on the behavior of these complex systems and to identify ways to optimize the [2]

An equation of the fourth order is a mathematical expression that involves four variables and the relationships between them. In this study, we considered a particular form of the fourth-order equation that involves a delay term, which means that the state of the system at any given time depends on its past values. The equation we studied also has a nonlinear component, meaning that the relationship between the variables is not a simple linear one [3,4].

We studied the oscillatory behavior of the nonlinear neutral delay differential equation of the fourth order, which means we looked at how the solution to the equation changes over time. Our study is based on two mathematical techniques, the Riccati substitution and the comparison technique, which allow us to obtain new results about the oscillatory behavior of the equation [5].

Several fields of science, including biology, architecture, chemistry, and medicine, use delay differential equations. There has been a study on both oscillatory and non-oscillatory solutions to such solutions because nonlinear differential equations are crucial to many fields [6]. Several different physical, biochemical, and biological processes are mathematically modeled using fourth-order differential equations. Applications of this kind of equation include issues with modulus, architectural deformation, and soil settling. Also, problems regarding the existence of oscillatory and non-oscillatory solutions are often raised in the mechanical and engineering domains, and the answers depend on the existence of the previously described equation [7].

Mathematical expressions are used in a wide variety of real-world situations, as is widely known. Advanced differential equations in particular are used in dynamical systems, network mathematics, optimization, and the mathematical modeling of engineering processes including those in materials, energy, and electrical power systems. The investigation of the qualitative oscillation behavior of differential equations has received significant attention in recent years [8]. Fourth-order differential equations are found in a wide variety of domains and challenges, including engineering, physics, chemical processes, and biological modeling. Applications that deal with issues of elasticity, physical deformation, and surface settlement demonstrate the significance of these sorts of equations. Neutral differential equation studies have focused on the oscillatory characteristics of solutions [9]. Many environmental processes may be modeled with the help of the study of oscillatory solutions to differential equations. Several researchers have stressed the significance of ocean eddies and chaos in controlling how the ocean reacts to weather-related events. A physical mechanism in the water causes a nonlinear or chaotic oscillation [10].

We studied the oscillatory properties of solutions of the following fourth-order neutral difference equation:

$$\mathsf{\Delta }\left(a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}\right)+p\left(m\right)w^{p_2-1}\left(\sigma \left(m\right)\right)=0$$

(1)

where

$$z\left(m\right)=w\left(m\right)+q\left(m\right)w\left(m-\tau \right)$$

(2)

subject to the following conditions:

• $\left(S_1\right)p_1-1,p_2-1$are ratios and $p_1, p_{2 }$are positive constants $>1$;
• $\left(S_2\right)\left\{a\left(m\right)\right\}, \left\{p\left(m\right)\right\}$, more real sequences with $\mathsf{\Delta }a\left(m\right) \ge 0, a\left(m\right) \ge 0, p\left(m\right)>0$;
• $\left(S_3\right)\sigma \left(m\right)$is a sequence of positive integers;
• $\left(S_4\right)\tau$ is a positive integer.
• $\left(S_5\right){\displaystyle {\displaystyle \sum }_{s=m_0}^{\infty }}\frac{1}{a^{\left(\frac{1}{{\mathrm{p}}_1-1\left(\mathrm{s}\right)}\right)}}=\infty .$

Neutral difference equations are mathematical models that describe the behavior of various systems over time. These equations have found a wide range of applications in both technology and natural science, making them valuable tools for understanding complex processes [11,12]. One of the key applications of neutral difference equations is in the study of distributed networks containing lossless transmission lines. These transmission lines are commonly used in high-speed computer systems to interconnect switching circuits, and the study of their behavior is critical to ensuring optimal performance. The study of the oscillatory behavior of neutral difference equations in this context is particularly important as it helps us to understand how these systems behave and how they can be improved [13]. The study of neutral difference equations has the potential to shed light on a wide range of technological and scientific problems. For example, these equations can be used to study the behavior of complex systems such as power grids, transportation networks, and communication systems. The results of this research can lead to a deeper understanding of these systems and provide insights into how they can be improved and optimized [14].

The oscillation of a neutral differential equation of the fourth order is being studied. New oscillation criteria that guarantee that all solutions to the examined equation are oscillatory are achieved by applying the Riccati substitution and comparison approach [15]. Fundamental necessary conditions are provided for the oscillation of neutral differential equations of the fourth order. Our paper’s main goal is to further enhance and supplement certain well-known findings that have lately been published in the field [16].

In recent years, the mathematical study of neutral difference equations has become increasingly important due to their wide range of applications in technology and natural science. One particular area of interest is the study of the oscillatory behavior of these equations in the context of distributed networks containing lossless transmission lines. This type of research is crucial in understanding the behavior of these systems and how they can be optimized for improved performance.

One of the key goals of this line of research is to explore the regularity and existence properties of the solutions to oscillation difference equations. This type of study aims to understand how the solutions to these equations change over time and what conditions are necessary for them to exist and be well-behaved. This type of research requires a deep understanding of the behavior of these equations and the use of advanced mathematical techniques to analyze them.

By exploring the regularity and existence properties of solutions to oscillation difference equations, researchers have been able to gain new insights into the behavior of these equations and how they can be used to optimize the design of distributed networks containing lossless transmission lines. These studies have helped to further our understanding of the behavior of neutral difference equations and the conditions that are necessary for their solutions to be well-behaved. This has led to the development of new and improved methods for analyzing these equations, which can be used to improve the performance of these systems in a variety of applications.

One of the key challenges in studying the oscillatory behavior of neutral difference equations is that these equations are often nonlinear and contain a delay term. This means that the state of the system at any given time depends not only on the current time but also on past values. To overcome this challenge, researchers have developed advanced mathematical techniques, such as the Riccati substitution and comparison technique, which allow them to obtain new results about the oscillatory behavior of these equations.

The use of mathematical techniques such as the Riccati substitution and comparison technique have significantly advanced the study of neutral difference equations. These techniques have allowed researchers to uncover new oscillatory behavior of these equations and to gain a deeper understanding of the conditions necessary for the solutions to be well-behaved. The discovery that the behavior of neutral difference equations can be highly sensitive to the choice of initial conditions highlights the importance of accurately determining these conditions for the prediction of system behavior. Additionally, the discovery that the solutions to these equations can exhibit complex, nonlinear behavior even for relatively simple systems underscores the need for sophisticated mathematical methods to accurately describe the behavior of these systems over time.

One important application of the study of neutral difference equations is in the design and optimization of distributed networks containing lossless transmission lines. The oscillatory behavior of these equations provides valuable insights into the performance of these systems, allowing researchers to identify areas where improvements can be made. For example, by understanding the conditions necessary for solutions to be well-behaved, researchers can design systems that are less prone to unwanted oscillations, leading to improved performance. The use of mathematical techniques, such as the Riccati substitution and comparison technique, has allowed researchers to better understand the behavior of these systems and to develop new design and optimization strategies that take into account the complexities of neutral difference equations.

In [17], the paper studied oscillation conditions for $n$th order difference equation with the damping term is

$$\mathsf{\Delta }\left(a\left(m\right)\zeta \left({\mathsf{\Delta }}^{n-1}w\left(m\right)\right)\right)+r\left(m\right)\zeta \left({\mathsf{\Delta }}^{n-1}w\left(m\right)\right)+p\left(m\right)\zeta \left(w\left(m-\alpha \right)\right)=0,$$

where $\zeta ={\left|s\right|}^{p-2}.sandn$ is even. The authors used Riccati transformation substitution together with the integral averaging technique.

In [18], Tripathy established new oscillation criteria for all solution oscillatory for the class of fourth-order neutral functional differential equations of the form

$${\mathsf{\Delta }}^2\left(r\left(n\right){\mathsf{\Delta }}^2\left(y\left(n\right)+p\left(n\right)y\left(n-m\right)\right)\right)+q\left(n\right)G\left(y\left(n-k\right)\right)=0$$

is studied under the assumption

$${\displaystyle \sum }_{n=0}^{\infty }\frac{n}{r\left(n\right)}<\infty .$$

In this article, we provide oscillatory properties of solutions of Equation (1). By using the comparison method and Riccati transformation, we get new oscillatory results for Equation (1). Our results complement some known results. Further, an example is provided.

2. Preliminaries

The following lemmas will be useful for establishing oscillation criteria:

Lemma 1 ([19]). 

Let $E>0 and F$ be constants. Then,

$$Fz\left(m\right)-E{z}^{\frac{\beta +1}{\beta }}\left(m\right) \le \frac{{\beta }^{\beta }}{{\left(\beta +1\right)}^{\beta +1}}\frac{F^{\beta +1}}{E^{\beta }}$$

Lemma 2 ([20]). 

Let ${\mathsf{\Delta }}^{n}z\left(m\right)$ be a fixed sign and ${\mathsf{\Delta }}^{n-1}z\left(m\right){\mathsf{\Delta }}^{n}z\left(m\right) \le 0$ for all $m\ge m_1.$ If $\underset{m\to \infty }{lim}z\left(m\right) \ne 0,$ then for every $\lambda \in \left(0,1\right),$ there exists $m_{\lambda }\ge m_1$ such that

$$z\left(m\right) \ge \frac{\lambda }{\left(n-1\right)!}{m}^{n-1}\left|{\mathsf{\Delta }}^{n-1}z\left(m\right)\right|form\ge m_{\lambda }$$

Lemma 3 ([1]). 

Let $z^n\left(m\right)>0, n=0,1,\dots p, and z^{\left(p+1\right)}\left(m\right)<0$.  Then,

$$\frac{z\left(m\right)}{\raisebox{1ex}{$m^p$}\!\left/ \!\raisebox{-1ex}{$p!$}\right.} \ge \frac{\mathsf{\Delta }z\left(m\right)}{\raisebox{1ex}{$m^{p-1}$}\!\left/ \!\raisebox{-1ex}{$\left(p-1\right)!$}\right.}$$

Lemma 4 ([21]). 

Let $A \ge 0, B \ge 0 and \alpha \ge 1$. Then ${\left(A+B\right)}^{\alpha } \le 2^{\alpha -1}\left(A^{\alpha }+B^{\alpha }\right)$.

Lemma 5 ([21]). 

Let $A \ge 0, B \ge 0 and 0 \le \alpha \le 1$. Then ${\left(A+B\right)}^{\alpha } \le A^{\alpha }+B^{\alpha }$.

Lemma 6 ([22]). 

Assume $w\left(m\right)$ is an ultimately positive solution to Equation (1). Then, there exist two possible cases, $m\ge m_1,$ for large enough where $m_1\ge m_0$ is sufficiently large.

$$\left(\mathrm{C}11\right) z\left(m\right)>0, \mathsf{\Delta }z\left(m\right)>0, {\mathsf{\Delta }}^{2}z\left(m\right)>0, {\mathsf{\Delta }}^{3}z\left(m\right)>0 and {\mathsf{\Delta }}^{4}z\left(m\right)<0$$

$$\left(\mathrm{C}22\right) z\left(m\right)>0, \mathsf{\Delta }z\left(m\right)>0, {\mathsf{\Delta }}^{2}z\left(m\right)<0, {\mathsf{\Delta }}^{3}z\left(m\right)>0 and {\mathsf{\Delta }}^{4}z\left(m\right)<0$$

3. Oscillation Criteria

We consider the following notations:

$$\left(N_1\right)q_1\left(m\right)=\frac{1}{q\left(m+\tau \right)}\left[1-\frac{{\left(m+2\tau \right)}^3}{{\left(m+\tau \right)}^{3}q\left(m+2\tau \right)}\right]$$

$$\left(N_2\right)q_2\left(m\right)=\frac{1}{q\left(m+\tau \right)}\left[1-\frac{\left(m+2\tau \right)}{\left(m+\tau \right)q\left(m+2\tau \right)}\right]$$

$$\left(N_3\right)D_1\left(m\right)={\left(\frac{\lambda {\left[\eta \left(m+\tau \right)\right]}^3}{6}\right)}^{p_2-1}p\left(m\right)q_1^{p_2-1}\left(\eta \left(m\right)\right)a^{\frac{-\left(p_2-1\right)}{\left(p_1-1\right)}}\left(\eta \left(m+\tau \right)\right)$$

$$\left(N_4\right)D_2\left(m\right)={\displaystyle \sum }_{t=m}^{\infty }{\left[\frac{1}{a\left(t\right)}{\displaystyle \sum }_{s=t}^{\infty }p\left(s\right){\left(\frac{\varphi \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\frac{1}{p_1-1}}$$

In this section, we establish some new oscillation criteria for Equation (1).

Lemma 7. 

Let $w\left(m\right)$ be an ultimately positive solution to Equation (1). Then,

$$w\left(m\right) \ge \frac{1}{q\left(m+\tau \right)}\left[z\left(m+\tau \right)-\frac{z\left(m+2\tau \right)}{q\left(m+2\tau \right)}\right]$$

(3)

Proof. 

Let $w\left(m\right)$ be an ultimately positive solution to Equation (1). The definition of $z\left(m\right)$ we know that

$$z\left(m\right)=w\left(m\right)+q\left(m\right)w\left(m-\tau \right)$$

$$q\left(m\right)w\left(m-\tau \right)=z\left(m\right)-w\left(m\right).$$

We substitute, $m=m+\tau ,$we see that

$$q\left(m+\tau \right)w\left(m\right)=z\left(m+\tau \right)-w\left(m+\tau \right)$$

$$w\left(m\right)=\frac{1}{q\left(m+\tau \right)}\left[z\left(m+\tau \right)-w\left(m+\tau \right)\right].$$

Repeating the same process, we obtain

$$w\left(m\right) = \frac{1}{q\left(m+\tau \right)}[z\left(m+\tau \right)-(\frac{z\left(m+2\tau \right)-w\left(m+2\tau \right)}{q\left(m+2\tau \right)})]$$

$$w\left(m\right) \ge \frac{1}{q\left(m+\tau \right)}\left[z\left(m+\tau \right)-\frac{z\left(m+2\tau \right)}{q\left(m+2\tau \right)}\right].$$

Thus, Equation (3) holds. □

Theorem 1. 

Let $\sigma \left(m\right) \le m-\tau .$ Assume that there exist positive real sequences $\theta , {\theta }_1 and {\lambda }_1 \in \left(0,1\right)$ and for every constant $N_1, N_2>0$ such that

$${\displaystyle \sum }_{s=m_0}^{\infty }\left[B_1\left(s\right)-\frac{2^{p_1-1}}{p_1{}^{p_1}}\frac{a\left(\sigma \left(s+\tau \right)\right){\left(\mathsf{\Delta }\theta \left(s\right)\right)}^{p_1}}{{\left[{\lambda }_1\theta \left(s+1\right)\mathsf{\Delta }\left(\sigma \left(s+\tau \right)\right)\mathsf{\Delta }\left(\sigma \left(s\right)\right){\left(\sigma \left(s+\tau \right)\right)}^2\right]}^{p_1-1}}\right]=\infty$$

(4)

$${\displaystyle \sum }_{s=m_0}^{\infty }\left[B_2\left(s\right)-\frac{{\left({\theta }_1^{\prime }\left(s\right)\right)}^2}{4{\theta }_1\left(s\right)}\right]=\infty$$

(5)

where $B_1\left(m\right)=N_1^{p_2-p_1}\theta \left(m\right)p\left(m\right)q_1^{p_2-1}\left(\sigma \left(m\right)\right)$ and $B_2\left(m\right)=q_2^{\frac{p_2-1}{p_1-1}}{\theta }_1(m+1)N_2^{\frac{p_2-p_1}{p_1-1}} \sum _{t=m}^{\infty }{\left[\frac{1}{a\left(t\right)}\sum _{s=t}^{\infty }p\left(s\right){\left(\frac{\sigma \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p_1-1$}\right.}$ then Equation (1) is oscillatory.

Proof. 

Let $w\left(m\right)$ be a non-oscillatory solution of Equation (1) on $[m_0,\infty ).$ □

Without the loss of generality, we assume $w\left(m\right)$ is an ultimately positive solution.

It follows from Lemma (5) that there exist two possible cases, $\left(C_{11}\right) and \left(C_{22}\right)$.

Let $\left(C_{11}\right)$ hold, and from Lemma (3), we obtain

$$\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m\right)} \le \frac{3}{m}$$

Summing from $\left(m+\tau \right) to \left(m-1\right),$we obtain

$$\frac{z\left(m+\tau \right)}{z\left(m\right)} \ge \frac{{\left(m+\tau \right)}^3}{m^3}$$

(6)

$$m^{3}z\left(m+\tau \right) \ge z\left(m\right){\left(m+\tau \right)}^3$$

It follows that

$${\left(m+\tau \right)}^{3}z(m+2\tau ) \le z\left(m+\tau \right){\left(m+2\tau \right)}^3$$

(7)

Substituting Equation (7) in (3), we obtain

$$w\left(m\right) \ge \frac{1}{q\left(m+\tau \right)}\left[z\left(m+\tau \right)-\frac{z\left(m+2\tau \right)}{q\left(m+2\tau \right)}\right]$$

$$w\left(m\right) \ge \frac{z\left(m+\tau \right)}{q\left(m+\tau \right)}\left[1-\frac{{\left(m+2\tau \right)}^3}{q\left(m+2\tau \right){\left(m+\tau \right)}^3}\right]$$

Using the notations $\left(N_{11}\right),$ we obtain

$$w\left(m\right) \ge q_1\left(m\right)z(m+\tau )$$

(8)

Substituting Equation (8) in (1), we obtain

$$\mathsf{\Delta }\left(a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}\right)+p\left(m\right)q_1^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-1}\left(\sigma \left(m+\tau \right)\right) \le 0$$

(9)

Define

$${\chi }_1\left(m\right)=\theta \left(m\right)\frac{a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}}{z^{p_1-1}\left(\sigma \left(m+\tau \right)\right)}$$

(10)

$$\mathsf{\Delta }{\chi }_1\left(m\right)=\mathsf{\Delta } \left[\theta \left(m\right)\frac{a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}}{z^{p_1-1}\left(\sigma \left(m+\tau \right)\right)}\right]$$

Using Equations (9) and (10), we obtain

$$\begin{gathered} \mathsf{\Delta }{\chi }_1\left(m\right) \le \frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)}{\chi }_1\left(m+1\right)-\frac{\theta \left(m\right)p\left(m\right)}{z^{p_1-1}\left(\sigma \left(m+\tau \right)\right)}{q}_1^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-1}\left(\sigma \left(m+\tau \right)\right) \\ +\theta \left(m\right)a\left(m+1\right){\left({\mathsf{\Delta }}^{3}z\left(m+1\right)\right)}^{p_1-1}\mathsf{\Delta }\left(\frac{1}{z^{p_1-1}\left(\sigma \left(m+\tau \right)\right)}\right) \end{gathered}$$

$$\begin{gathered} \mathsf{\Delta }{\chi }_1\left(m\right)\le \frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)}{\chi }_1\left(m+1\right)-\theta \left(m\right)p\left(m\right)q_1^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-p_1}\left(\sigma \left(m+\tau \right)\right) \\ -\theta \left(m\right)a\left(m+1\right){\left({\mathsf{\Delta }}^{3}z\left(m+1\right)\right)}^{p_1-1}\frac{\mathsf{\Delta }{z}^{p_1-1}\left(\sigma \left(m+\tau \right)\right)}{z^{p_1-1}\left(\sigma \left(m+1+\tau \right)\right)z^{p_1-1}\left(\sigma \left(m+\tau \right)\right)} \\ \le \frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)}{\chi }_1\left(m+1\right)-\theta \left(m\right)p\left(m\right)q_1^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-p_1}\left(\sigma \left(m+\tau \right)\right) \\ -\frac{\theta \left(m\right)a\left(m+1\right){\left({\mathsf{\Delta }}^{3}z\left(m+1\right)\right)}^{p_1-1}\left(p_1-1\right) \mathsf{\Delta }\left(\sigma \left(m\right)\right)\mathsf{\Delta }\left(\sigma \left(m+\tau \right)\right) \mathsf{\Delta }z\left(\sigma \left(m+\tau \right)\right)}{z^{p_1-1}\left(\sigma \left(m+1+\tau \right)\right)z\left(\sigma \left(m+\tau \right)\right)} \end{gathered}$$

(11)

Recalling that $a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}$ is decreasing, we obtain

$$a\left(\sigma \left(m+\tau \right)\right){\left({\mathsf{\Delta }}^{3}z\left(\sigma \left(m+\tau \right)\right)\right)}^{p_1-1} \ge a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}$$

This is reached at

$${\left({\mathsf{\Delta }}^{3}z\left(\sigma \left(m+\tau \right)\right)\right)}^{p_1-1} \ge \frac{a\left(m\right)}{a\left(\sigma \left(m+\tau \right)\right)}{\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}$$

(12)

It follows that from Lemma (2), we obtain

$$\mathsf{\Delta }z\left(\sigma \left(m+\tau \right)\right) \ge {\lambda }_1\frac{{\left(\sigma \left(m+\tau \right)\right)}^2}{2}{\mathsf{\Delta }}^{3}z\left(\sigma \left(m+\tau \right)\right)$$

(13)

For all ${\lambda }_1 \in (0,1)$. Thus, by Equations (11)–(13) we obtain

$$\begin{gathered} \mathsf{\Delta }{\chi }_1\left(m\right) \le \frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)}{\chi }_1\left(m+1\right)-\theta \left(m\right)p\left(m\right)q_1^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-p_1}\left(\sigma \left(m+\tau \right)\right) \\ -\frac{\left(p_1-1\right)\theta \left(m\right)a\left(m+1\right) \mathsf{\Delta }\left(\sigma \left(m\right)\right) \mathsf{\Delta }\left(\sigma \left(m+\tau \right)\right)}{z^{p_1-1}\left(\sigma \left(m+1+\tau \right)\right)z\left(\sigma \left(m+\tau \right)\right)}{\left({\mathsf{\Delta }}^{3}z\left(m+1\right)\right)}^{p_1-1}{\lambda }_1\frac{{\left(\sigma \left(m+\tau \right)\right)}^2}{2} {\mathsf{\Delta }}^{3}z\left(\sigma \left(m+\tau \right)\right) \end{gathered}$$

If $m+\tau \le m+\tau +1 then\sigma \left(m\right) \le \sigma \left(m+1\right)$,  we say that $z\left(\sigma \left(m+1\right)\right) \ge z\left(\sigma \left(m\right)\right)$

$$\begin{gathered} \mathsf{\Delta }{\chi }_1\left(m\right) \le \frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)}{\chi }_1\left(m+1\right)-\theta \left(m\right)p\left(m\right)q_1^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-p_1}\left(\sigma \left(m+\tau \right)\right) \\ -\left(p_1-1\right)\theta \left(m+1\right)\frac{{\lambda }_1}{2}{\left(\frac{a\left(m\right)}{a\left(\sigma \left(m+\tau \right)\right)}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(p_1-1\right)$}\right.}a\left(m+1\right){\left({\mathsf{\Delta }}^{3}z\left(m+1\right)\right)}^{p_1} \\ \frac{\mathsf{\Delta }\left(\sigma \left(m+\tau \right)\right) \mathsf{\Delta }\left(\sigma \left(m\right)\right){\left(\sigma \left(m+\tau \right)\right)}^2}{z^{p_1}\left(\sigma \left(m+1+\tau \right)\right)} \\ \mathsf{\Delta }{\chi }_1\left(m\right) \le \frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)}{\chi }_1\left(m+1\right)-B_1\left(m\right)-\left(p_1-1\right)\frac{{\lambda }_1}{2}{\left(\frac{a\left(m\right)}{a\left(\sigma \left(m+\tau \right)\right)}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(p_1-1\right)$}\right.} \end{gathered}$$

$$\frac{\mathsf{\Delta }\left(\sigma \left(m+\tau \right)\right) \mathsf{\Delta }\left(\sigma \left(m\right)\right){\left(\sigma \left(m+\tau \right)\right)}^2}{{\left[\theta \left(m+1\right)a\left(m+1\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(p_1-1\right)$}\right.}}{\chi }_1^{\raisebox{1ex}{$p_1$}\!\left/ \!\raisebox{-1ex}{$\left(p_1-1\right)$}\right.}\left(m+1\right)$$

(14)

Because $\mathsf{\Delta }z\left(m\right)>0,$there exist $m_2 \ge m_1$ and a constant $N > 0$ such that $z\left(m\right)>N,$for all $m \ge m_2.$

Using Lemma (1), we obtain

$$\begin{gathered} F=\frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)},E=\left(p_1-1\right)\frac{{\lambda }_1}{2}{\left(\frac{a\left(m\right)}{a\left(\sigma \left(m+\tau \right)\right)}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(p_1-1\right)$}\right.}\frac{\mathsf{\Delta }\left(\sigma \left(m+\tau \right)\right) \mathsf{\Delta }\left(\sigma \left(m\right)\right){\left(\sigma \left(m+\tau \right)\right)}^2}{{\left[\theta \left(m+1\right)a\left(m+1\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(p_1-1\right)$}\right.}} \\ and z\left(m\right)={\chi }_1\left(m\right) \\ Fz\left(m\right)-E{z}^{\frac{\beta +1}{\beta }}\left(m\right) \le \frac{{\beta }^{\beta }}{{\left(\beta +1\right)}^{\left(\beta +1\right)}}\frac{F^{\left(\beta +1\right)}}{E^{\beta }} \end{gathered}$$

From Equation (14), we have

$$\begin{array}{l}\mathsf{\Delta }{\chi }_1\left(m\right)\\ \le -B_1\left(m\right)\\ +\frac{{\left(p_1-1\right)}^{\left(p_1-1\right)}}{p_1{}^{p_1}}{\left(\frac{\mathsf{\Delta }\theta \left(m\right)}{\theta \left(m+1\right)}\right)}^{p_1}\frac{{\left(\frac{2}{{\lambda }_1}\right)}^{\left(p_1-1\right)}\frac{1}{{\left(p_1-1\right)}^{\left(p_1-1\right)}}\left[\frac{a\left(\sigma \left(m+\tau \right)\right)}{a\left(m\right)}\right]\left[\theta \left(m+1\right)a\left(m+1\right)\right]}{{\left[\mathsf{\Delta }\left(\sigma \left(m+\tau \right)\right) \mathsf{\Delta }\left(\sigma \left(m\right)\right){\left(\sigma \left(m+\tau \right)\right)}^2\right]}^{p_1-1}}\end{array}$$

$$\mathsf{\Delta }{\chi }_1\left(m\right) \le -B_1\left(m\right)+\frac{2^{\left(p_1-1\right)}}{p_1{}^{p_1}}\frac{a\left(\sigma \left(m+\tau \right)\right){\left(\mathsf{\Delta }\theta \left(m\right)\right)}^{p_1}}{{\left[{\lambda }_1\theta \left(m+1\right)\mathsf{\Delta }\left(\sigma \left(m+\tau \right)\right) \mathsf{\Delta }\left(\sigma \left(m\right)\right){\left(\sigma \left(m+\tau \right)\right)}^2\right]}^{p_1-1}}$$

Summing from $m_1$ to $m-1,$ we obtain

$$\begin{gathered} {\chi }_1\left(m\right)-{\chi }_1\left(m_1\right) \\ \le {\displaystyle \sum }_{s=m_1}^{m-1}\left(-B_1\left(s\right)+\frac{2^{\left(p_1-1\right)}}{p_1{}^{p_1}}\frac{a\left(\sigma \left(s+\tau \right)\right){\left(\mathsf{\Delta }\theta \left(s\right)\right)}^{p_1}}{{\left[{\lambda }_1\theta \left(s+1\right)\mathsf{\Delta }\left(\sigma \left(s+\tau \right)\right) \mathsf{\Delta }\left(\sigma \left(s\right)\right){\left(\sigma \left(s+\tau \right)\right)}^2\right]}^{p_1-1}}\right) \end{gathered}$$

Because $m \ge m_1,$ we obtain

$${\displaystyle \sum }_{s=m_1}^{m-1}\left(B_1\left(s\right)-\frac{2^{\left(p_1-1\right)}}{p_1{}^{p_1}}\frac{a\left(\sigma \left(s+\tau \right)\right){\left(\mathsf{\Delta }\theta \left(s\right)\right)}^{p_1}}{{\left[{\lambda }_1\theta \left(s+1\right)\mathsf{\Delta }\left(\sigma \left(s+\tau \right)\right) \mathsf{\Delta }\left(\sigma \left(s\right)\right){\left(\sigma \left(s+\tau \right)\right)}^2\right]}^{p_1-1}}\right) \le {\chi }_1\left(m_1\right)$$

which is a contradiction to Equation (4).

Let $\left(C_{22}\right)$ hold.

By using Lemma (3), we have

$$\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m\right)} \le \frac{1}{m}$$

Summing from $\left(m+\tau \right)tom-1,$ we obtain

$$\frac{z\left(\sigma \left(m+\tau \right)\right)}{z\left(m\right)} \ge \frac{\sigma \left(m+\tau \right)}{m}$$

$$\sigma \left(m+\tau \right)z\left(m\right) \le mz\left(\sigma \left(m+\tau \right)\right)$$

(15)

It follows that

$$\left(m+\tau \right)z\left(m+2\tau \right) \le \left(m+2\tau \right)z\left(m+\tau \right)$$

(16)

Substituting Equation (16) in (3), we obtain

$$w\left(m\right) \ge \frac{1}{q\left(m+\tau \right)}\left[z\left(m+\tau \right)-\frac{z\left(m+2\tau \right)}{q\left(m+2\tau \right)}\right]$$

$$w\left(m\right) \ge \frac{1}{q\left(m+\tau \right)}\left[z\left(m+\tau \right)-\frac{\left(m+2\tau \right)z\left(m+\tau \right)}{\left(m+\tau \right)q\left(m+2\tau \right)}\right]$$

$$w\left(m\right) \ge \frac{z\left(m+\tau \right)}{q\left(m+\tau \right)}\left[1-\frac{\left(m+2\tau \right)}{\left(m+\tau \right)q\left(m+2\tau \right)}\right]$$

Using the notations $\left(N_{22}\right)$ we obtain

$$w\left(m\right) \ge q_2\left(m\right)z\left(m+\tau \right)$$

Equation (1) gives

$$\begin{gathered} \mathsf{\Delta }\left[a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}\right]=-p\left(m\right)w^{p_2-1}\left(\sigma \left(m\right)\right) \\ \le -p\left(m\right)q_2^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-1}\left(\sigma \left(m+\tau \right)\right) \end{gathered}$$

(17)

Summing from $mtot-1,$we obtain

$$a\left(t\right){\left({\mathsf{\Delta }}^{3}z\left(t\right)\right)}^{p_1-1}-a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1} \le -{\displaystyle \sum }_{s=m}^{t-1}\left[p\left(s\right)q_2^{p_2-1}\left(\sigma \left(s\right)\right)z^{p_2-1}\left(\sigma \left(s+\tau \right)\right)\right]$$

(18)

Let $t \to \infty$in Equation (18) and using Equation (15), we obtain

$$a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1} \ge q_2^{p_2-1}\left(\sigma \left(m\right)\right)z^{p_2-1}\left(m\right){\displaystyle \sum }_{s=m}^{t-1}\left[p\left(s\right){\left(\frac{\sigma \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]$$

$${\mathsf{\Delta }}^{3}z\left(m\right) \ge q_2^{\frac{p_2-1}{p_1-1}}\left(\sigma \left(m\right)\right)z^{\frac{p_2-1}{p_1-1}}\left(m\right){\left[\frac{1}{a\left(m\right)}{\displaystyle \sum }_{s=m}^{t-1}p\left(s\right){\left(\frac{\sigma \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p_1-1$}\right.}$$

Summing again from $mto \infty ,$we obtain

$$-{\mathsf{\Delta }}^{2}z\left(m\right) \ge q_2^{\frac{p_2-1}{p_1-1}}\left(\sigma \left(m\right)\right)z^{\frac{p_2-1}{p_1-1}}\left(m\right){\displaystyle \sum }_{t=m}^{\infty }{\left[\frac{1}{a\left(t\right)}{\displaystyle \sum }_{s=t}^{\infty }p\left(s\right){\left(\frac{\sigma \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p_1-1$}\right.}$$

$${\mathsf{\Delta }}^{2}z\left(m\right) \le -q_2^{\frac{p_2-1}{p_1-1}}\left(\sigma \left(m\right)\right)z^{\frac{p_2-1}{p_1-1}}\left(m\right){\displaystyle \sum }_{t=m}^{\infty }{\left[\frac{1}{a\left(t\right)}{\displaystyle \sum }_{s=t}^{\infty }p\left(s\right){\left(\frac{\sigma \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p_1-1$}\right.}$$

(19)

Now, we define

$${\chi }_2\left(m\right)={\theta }_1\left(m\right)\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m\right)}$$

Then, ${\chi }_2\left(m\right)>0 form \ge m_1.$

$$\begin{gathered} \mathsf{\Delta }{\chi }_2\left(m\right)=\mathsf{\Delta } \left({\theta }_1\left(m\right)\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m\right)}\right) \\ =\mathsf{\Delta }{\theta }_1\left(m\right)\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m\right)}+{\theta }_1\left(m+1\right) \mathsf{\Delta }\left(\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m\right)}\right) \\ =\frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}{\chi }_2\left(m\right)+{\theta }_1\left(m+1\right)\frac{{\mathsf{\Delta }}^{2}z\left(m\right)}{z\left(m\right)}-{\theta }_1\left(m+1\right)\mathsf{\Delta }z\left(m\right)\left[\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m+1\right)z\left(m\right)}\right] \end{gathered}$$

$$\mathsf{\Delta }{\chi }_2\left(m\right)\le \frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}{\chi }_2\left(m\right)+{\theta }_1\left(m+1\right)\frac{{\mathsf{\Delta }}^{2}z\left(m\right)}{z\left(m\right)}-{\theta }_1\left(m\right){\left[\frac{\mathsf{\Delta }z\left(m\right)}{z\left(m\right)}\right]}^2$$

$$\mathsf{\Delta }{\chi }_2\left(m\right)\le \frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}{\chi }_2\left(m\right)-\frac{1}{{\theta }_1\left(m\right)}{\chi }_2^2\left(m\right)+{\theta }_1\left(m+1\right)\frac{{\mathsf{\Delta }}^{2}z\left(m\right)}{z\left(m\right)}$$

Using Equation (19), we obtain

$$\begin{array}{l}\mathsf{\Delta }{\chi }_2\left(m\right)\\ \le \frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}{\chi }_2\left(m\right)-\frac{1}{{\theta }_1\left(m\right)}{\chi }_2^2\left(m\right)\\ -\frac{{\theta }_1\left(m+1\right)}{z\left(m\right)}{q}_2^{\frac{p_2-1}{p_1-1}}\left(\sigma \left(m\right)\right)z^{\frac{p_2-1}{p_1-1}}\left(m\right){\displaystyle \sum _{t=m}^{\infty }}{\left[\frac{1}{a\left(t\right)}{\displaystyle \sum _{s=t}^{\infty }}p\left(s\right){\left(\frac{\sigma \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p_1-1$}\right.}\\ \le \frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}{\chi }_2\left(m\right)-\frac{1}{{\theta }_1\left(m\right)}{\chi }_2^2\left(m\right)\\ -{\theta }_1\left(m+1\right)q_2^{\frac{p_2-1}{p_1-1}}\left(\sigma \left(m\right)\right)z^{\frac{p_2-p_1}{p_1-1}}\left(m\right){\displaystyle \sum _{t=m}^{\infty }}{\left[\frac{1}{a\left(t\right)}{\displaystyle \sum _{s=t}^{\infty }}p\left(s\right){\left(\frac{\sigma \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p_1-1$}\right.}\end{array}$$

$$\mathsf{\Delta }{\chi }_2\left(m\right) \le \frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}{\chi }_2\left(m\right)-\frac{1}{{\theta }_1\left(m\right)}{\chi }_2^2\left(m\right)-B_2\left(m\right)$$

Using Lemma (1), we obtain

$$F=\frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}, E=\frac{1}{{\theta }_1\left(m\right)}and z={\chi }_2$$

$$\mathsf{\Delta }{\chi }_2\left(m\right) \le -B_2\left(m\right)+\frac{1^1}{2^2}{\left(\frac{\mathsf{\Delta }{\theta }_1\left(m\right)}{{\theta }_1\left(m\right)}\right)}^2{\theta }_1\left(m\right)$$

$$B_2\left(m\right)-\frac{{\left[\mathsf{\Delta }{\theta }_1\left(m\right)\right]}^2}{4{\theta }_1\left(m\right)} \le -\mathsf{\Delta }{\chi }_2\left(m\right)$$

Summing from $m_{1}tom-1$ and since $m \ge m_1,$ we obtain

$${\displaystyle \sum }_{s=m_1}^{m-1}{B}_2\left(m\right)-\frac{{\left[\mathsf{\Delta }{\theta }_1\left(m\right)\right]}^2}{4{\theta }_1\left(m\right)} \le {\chi }_2\left(m_1\right)$$

which is a contradiction to Equation (5).

Hence, it is proven.

Theorem 2. 

$$Let\frac{{\left(m+2\tau \right)}^3}{{\left(m+\tau \right)}^{3}q\left(m+2\tau \right)} \le 1$$

(20)

Assume that there exist positive sequences $\left\{\eta \left(m\right)\right\}, \left\{\varphi \left(m\right)\right\} \in q\left(m\right)$ satisfying

$$\eta \left(m\right) \le \sigma \left(m\right), \eta \left(m\right)<m-\tau ,$$

$$\varphi \left(m\right) \le \sigma \left(m\right),\varphi \left(m\right)<m-\tau , \mathsf{\Delta }\varphi \left(m\right)\ge 0 \mathrm{and}$$

$$\underset{m\to \infty }{\lim }\eta \left(m\right)=\underset{m\to \infty }{\lim }\varphi \left(m\right)=\infty$$

(21)

If there exists ${\epsilon }_1,\lambda ϵ \left(0,1\right)$ such that the difference equations

$$\mathsf{\Delta }{A}_1^{\prime }\left(m\right)+D_1\left(m\right)A_1^{\frac{p_2-1}{p_1-1}}\left(\eta \left(m+\tau \right)\right)=0$$

(22)

$$and{A}_2^{\prime }\left(m\right)+q_2^{\frac{p_2-1}{p_1-1}}{\epsilon }_1{\left[\varphi \left(m+\tau \right)\right]}^{\frac{p_2-1}{p_1-1}}{D}_2\left(m\right)A_2^{\frac{p_2-1}{p_1-1}}\left(\varphi \left(m+\tau \right)\right)=0$$

(23)

are oscillatory, then Equation (1) is oscillatory.

Proof. 

The above is in the proof of Theorem (1). □

Let $\left(C_{11}\right)$ hold and from Equation (9).

Because $\eta \left(m\right) \le \sigma \left(m\right)and \mathsf{\Delta }z\left(m\right)>0,$  we obtain

$$\mathsf{\Delta }\left[a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}\right] \le -p\left(m\right)q_1^{p_2-1}\left(\eta \left(m\right)\right)z^{p_2-1}\left(\eta \left(m+\tau \right)\right)$$

(24)

Using Lemma (2), we obtain

$$z\left(m\right) \ge \frac{\lambda }{6}{m}^3{\mathsf{\Delta }}^{3}z\left(m\right)$$

(25)

For some $\lambda ϵ\left(0,1\right).$

Substituting Equation (25) in (24), we obtain

$$\mathsf{\Delta }\left[a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}\right]+p\left(m\right)q_1^{p_2-1}\left(\eta \left(m\right)\right){\left(\frac{\lambda }{6}\right)}^{p_2-1}{\left[{\left(\eta \left(m+\tau \right)\right)}^3\right]}^{p_2-1}{\left[{\mathsf{\Delta }}^{3}z\left(\eta \left(m+\tau \right)\right)\right]}^{p_2-1}\le 0$$

We choose

$$A_1\left(m\right)=a\left(m\right){\left({\mathsf{\Delta }}^{3}z\left(m\right)\right)}^{p_1-1}$$

So, we find out that $A_1\left(m\right)$ is a positive solution to inequality.

$$\mathsf{\Delta }{A}_1\left(m\right)+D_1\left(m\right)a^{\frac{p_2-1}{p_1-1}}\left(\eta \left(m+\tau \right)\right){\left[{\mathsf{\Delta }}^{3}z\left(\eta \left(m+\tau \right)\right)\right]}^{p_2-1} \le 0$$

$$\mathsf{\Delta }{A}_1\left(m\right)+D_1\left(m\right)A_1^{\frac{p_2-1}{p_1-1}}\left(\eta \left(m+\tau \right)\right) \le 0$$

Using (see [19], Lemma 2.7), we see that Equation (22) has a positive solution, which is a contradiction.

Let $\left(C_{12}\right)$ hold.

From Theorem (1), we prove that Equation (19) holds.

Because $\varphi \left(m\right) \le \sigma \left(m\right)and \mathsf{\Delta }z\left(m\right)>0,$ we obtain

$${\mathsf{\Delta }}^{2}z\left(m\right) \le -q_2^{\frac{p_2-1}{p_1-1}}{z}^{\frac{p_2-1}{p_1-1}}\left(\varphi \left(m+\tau \right)\right){\displaystyle \sum }_{t=m}^{\infty }{\left[\frac{1}{a\left(t\right)}{\displaystyle \sum }_{s=t}^{\infty }p\left(s\right){\left(\frac{\varphi \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\frac{1}{p_1-1}}$$

(26)

Using Lemma (2), we obtain

$$z\left(m\right) \ge \lambda m \mathsf{\Delta }z\left(m\right)$$

(27)

From Equations (16) and (27), we obtain

$$z\left(\varphi \left(m+\tau \right)\right) \ge \lambda \left(\varphi \left(m+\tau \right)\right)m \mathsf{\Delta }z\left(\varphi \left(m+\tau \right)\right)$$

From Equation (27),

$${\mathsf{\Delta }}^{2}z\left(m\right) \le -q_2^{\frac{p_2-1}{p_1-1}}\lambda {\left[\mathsf{\Delta }z\left(\varphi \left(m+\tau \right)\right)\right]}^{\frac{p_2-1}{p_1-1}}{\left[\varphi \left(m+\tau \right)\right]}^{\frac{p_2-1}{p_1-1}}$$

$${\displaystyle \sum }_{t=m}^{\infty }{\left[\frac{1}{a\left(t\right)}{\displaystyle \sum }_{s=t}^{\infty }p\left(s\right){\left(\frac{\varphi \left(s+\tau \right)}{s}\right)}^{p_2-1}\right]}^{\frac{1}{p_1-1}}$$

$${\mathsf{\Delta }}^{2}z\left(m\right) \le -q_2^{\frac{p_2-1}{p_1-1}}\lambda {\left[\mathsf{\Delta }z\left(\varphi \left(m+\tau \right)\right)\right]}^{\frac{p_2-1}{p_1-1}}{\left[\varphi \left(m+\tau \right)\right]}^{\frac{p_2-1}{p_1-1}}{D}_2\left(m\right)$$

We choose $A_2\left(m\right)=\mathsf{\Delta }z\left(m\right)$ to find $A_2\left(m\right)$ is a positive solution of

$$\mathsf{\Delta }{\mathrm{A}}_2\left(m\right)+q_2^{\frac{p_2-1}{p_1-1}}\lambda {\left[{\mathrm{A}}_2\left(\varphi \left(m+\tau \right)\right)\right]}^{\frac{p_2-1}{p_1-1}}{\left[\varphi \left(m+\tau \right)\right]}^{\frac{p_2-1}{p_1-1}}{D}_2\left(m\right) \le 0$$

(28)

Using (see [19], Lemma 2.7), we see that Equation (23) has a positive solution, which is a contradiction.

Hence, it is proven.

Example 1. 

Consider the equation

$${\mathsf{\Delta }}^4\left[w\left(m\right)+16w\left(m-1\right)\right]+240w\left(m-2\right)=0.$$

(29)

We take $a\left(m\right)=1, \tau =1, p_1=p_2=2, \sigma \left(m\right)=m-2, q\left(m\right)=16 andp\left(m\right)=240$.

Thus, we obtain

$$q_1\left(m\right)=\frac{1}{q\left(m+\tau \right)}\left[1-\frac{{\left(m+2\tau \right)}^3}{{\left(m+\tau \right)}^{3}q\left(m+2\tau \right)}\right]=\frac{1}{16}\left(1-\frac{1}{16}\right)=\frac{15}{256},$$

$$q_2\left(m\right)=\frac{1}{q\left(m+\tau \right)}\left[1-\frac{\left(m+2\tau \right)}{\left(m+\tau \right)q\left(m+2\tau \right)}\right]=\frac{1}{16}\left(1-\frac{1}{16}\right)=\frac{15}{256} ,$$

 substitute values are $\theta \left(m\right)={\theta }_1\left(m+1\right)=16m,$ we obtain

$$B_1\left(m\right)=B_2\left(m\right)=225m$$

Applying conditions (4) and (5), we obtain

$$\begin{gathered} {\displaystyle \sum }_{s=m_0}^{\infty }\left[B_1\left(s\right)-\frac{2^{p_1-1}}{p_1{}^{p_1}}\frac{a\left(\sigma \left(s+\tau \right)\right){\left(\mathsf{\Delta }\theta \left(s\right)\right)}^{p_1}}{{\left[{\lambda }_1\theta \left(s+1\right)\mathsf{\Delta }\left(\sigma \left(s+\tau \right)\right)\mathsf{\Delta }\left(\sigma \left(s\right)\right){\left(\sigma \left(s+\tau \right)\right)}^2\right]}^{p_1-1}}\right] \\ ={\displaystyle \sum }_{s=m_0}^{\infty }\left[225s-\frac{8}{{\lambda }_{1}s{\left(s-1\right)}^2}\right] \\ =\infty \\ {\displaystyle \sum }_{s=m_0}^{\infty }\left[B_2\left(s\right)-\frac{{\left({\theta }_1^{\prime }\left(s\right)\right)}^2}{4{\theta }_1\left(s\right)}\right]={\displaystyle \sum }_{s=m_0}^{\infty }\left[225s-\frac{4}{s}\right]=\infty \end{gathered}$$

Hence, it is easy to verify that all conditions of Theorem (1) are satisfied. Hence, Equation (29) is oscillatory.

This study explores the advantages of the lemma technique as well as the comparison method. These new elements of complexity are being added to certain previously recognized examples of neutrally differentiated outcomes.

4. Conclusions

In conclusion, this study provides new insights into the oscillation behavior of fourth-order nonlinear neutral delay difference equations through the application of the comparison and Riccati techniques. Our results demonstrate a promising advancement in understanding these complex systems and pave the way for further research in this area, specifically through a comparison with second-order difference equations. Overall, this work adds to the growing body of knowledge in the field and has the potential to contribute to the optimization of distributed networks containing lossless transmission lines. Consider using the derivation technique as well as the comparison method in your approach. On top of some instances of neutrally differentiated outcomes that have been recognized in the past, these newly introduced layers of complexity are being added.