New Criteria for Oscillation of Advanced Noncanonical Nonlinear Dynamic Equations

Abstract

In this study, novel criteria are derived to ensure the oscillation of solutions in nonlinear advanced noncanonical dynamic equations. The obtained results are reminiscent of the criteria proposed by Hille and Ohriska for canonical dynamic equations. Additionally, this paper addresses a previously unresolved issue found in numerous existing works in the literature on advanced dynamic equations. This study provides a range of illustrative examples to showcase the precision, practicality, and adaptability of the obtained findings.

1. Introduction

The theory of dynamic equations on time scales was proposed by Stefan Hilger [1] in order to establish a unified framework for analyzing both continuous and discrete systems. Different time scales can be utilized in a wide range of applications. The theory of dynamic equations encompasses classical theories pertaining to both differential equations and difference equations, while also encompassing cases that lie within the spectrum between these classical scenarios. The q-difference equations, which possess notable implications in the realm of quantum theory (refer to [2]), can be examined in the context of various time scales. These time scales include $\mathbb{T}=q^{{\mathbb{N}}_0}:=\{q^{\lambda }:\lambda \in {\mathbb{N}}_0$ for $q>1\}$, as well as $\mathbb{T}=h\mathbb{N}$, $\mathbb{T}{=\mathbb{N}}^2,$ and $\mathbb{T}={\mathbb{T}}_n,$ where ${\mathbb{T}}_n$ represents the set of harmonic numbers. Please refer to [3,4,5] for further information regarding the calculus of time scales.

The phenomenon of oscillation has attracted considerable interest from researchers in various applied disciplines, primarily because of its roots in mechanical vibrations and its wide-ranging utilization in the fields of science and engineering. To account for the influence of temporal contexts on solutions, oscillation models sometimes include delays or advanced terms. The topic of oscillation in delay equations has been extensively researched, as evidenced by [6,7,8,9,10]. The current body of research on advanced oscillation is relatively limited, with only a few studies, such as [11,12,13,14], specifically addressing this topic.

Oscillation phenomena are commonly observed in a wide range of real-world applications, where various models are employed to study and understand these phenomena. In the field of mathematical biology, certain models have been developed to incorporate oscillation and/or delay effects by incorporating cross-diffusion terms. To delve deeper into this subject, it is recommended to consult the scholarly articles [15,16]. The present study focuses on the investigation of differential equations, as they play a crucial role in understanding and analyzing a wide range of real-world phenomena. Specifically, this research explores the application of differential equations in the fields of non-Newtonian fluid theory and the turbulent flow of a polytrophic gas in a porous medium. These areas of study have significant practical implications and require a thorough understanding of the mathematical principles underlying them. For further elaboration, individuals who are interested can consult the papers [17,18,19,20,21,22].

This research paper aims to study the oscillatory behavior of a specific class of advanced nonlinear second-order noncanonical dynamic equations of the form

$${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }+q\left(\tau \right)z^{\gamma }\left(\phi \left(\tau \right)\right)=0$$

(1)

on an unbounded above arbitrary time scale $\mathbb{T}$, where $\tau \in {[{\tau }_0,\infty )}_{\mathbb{T}}$, ${\tau }_0\ge 0$, ${\tau }_0\in \mathbb{T}$, $\gamma$ is a quotient of odd positive integers, r and q are positive rd-continuous functions on $\mathbb{T}$, and $\phi :\mathbb{T}\to \mathbb{T}$ is an rd-continuous nondecreasing function satisfying $\phi \left(\tau \right)\ge \tau$ on ${[{\tau }_0,\infty )}_{\mathbb{T}}$ and ${lim}_{\tau \to \infty }\phi \left(\tau \right)=\infty$.

By a solution to Equation (1) we mean a nontrivial real-valued function $z\in {\mathrm{C}}_{\mathrm{rd}}^1{[T_z,\infty )}_{\mathbb{T}}$, $T_z\in {[{\tau }_0,\infty )}_{\mathbb{T}}$ such that $r{z}^{\Delta }\in {\mathrm{C}}_{\mathrm{rd}}^1{[T_z,\infty )}_{\mathbb{T}}$ and z satisfies (1) on ${[T_z,\infty )}_{\mathbb{T}}$, where ${\mathrm{C}}_{\mathrm{rd}}$ is the set of rd-continuous functions. In accordance with the findings of Trench [23], Equation (1) is considered to be in noncanonical form when

$${\int }_{{\tau }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}<\infty .$$

(2)

Conversely, Equation (1) is deemed to be in canonical form when

$${\int }_{{\tau }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}=\infty .$$

(3)

A solution z for (1) is considered oscillatory if it does not become eventually positive or eventually negative. Otherwise, we refer to it as nonoscillatory. We will exclude from consideration solutions that vanish in the vicinity of infinity.

It is worth mentioning that in a certain case where $\mathbb{T}=\mathbb{R}$, then

$$\mu \left(\tau \right)=0,{\eta }^{\Delta }\left(\tau \right)={\eta }^{\prime }\left(\tau \right),{\int }_a^b\eta \left(\tau \right)\Delta \tau ={\int }_a^b\eta \left(\tau \right)d\tau ,$$

and Equation (1) becomes the differential equation

$${\left[r\left(\tau \right)z^{\prime }\left(\tau \right)\right]}^{\prime }+q\left(\tau \right)z^{\gamma }\left(\phi \left(\tau \right)\right)=0.$$

(4)

The oscillatory characteristics of particular cases of Equation (4) are examined by Fite [24], who showed that if

$${\int }_{{\tau }_0}^{\infty }q\left(\omega \right)\mathrm{d}\omega =\infty ,$$

(5)

then every solution of the differential equation

$$z^{″}\left(\tau \right)+q\left(\tau \right)z\left(\tau \right)=0,$$

(6)

oscillates. Hille [25] improved condition (5) and proved that if

$$\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\tau {\int }_{\tau }^{\infty }q\left(\omega \right)\mathrm{d}\omega >\frac{1}{4},$$

(7)

then every solution of Equation (6) oscillates. Ohriska [26] obtained that if

$$\underset{\tau \to \infty }{\lim \sup }\tau {\int }_{\tau }^{\infty }q\left(\omega \right)\mathrm{d}\omega >1,$$

(8)

then every solution of Equation (6) oscillates. We will demonstrate that our findings not only expand upon existing oscillation conclusions for differential equations but that we can also apply these findings to other cases where the oscillatory behavior of solutions to these equations on different time scales is unknown. Notice that, if $\mathbb{T}=\mathbb{Z}$, then

$$\mu \left(\tau \right)=1,{\eta }^{\Delta }\left(\tau \right)=\Delta \eta \left(\tau \right),{\int }_a^b\eta \left(\tau \right)\Delta \tau =\sum _{\tau =a}^{b-1}\eta \left(\tau \right),$$

and (1) becomes the difference equation

$$\Delta \left[r\left(\tau \right)\Delta z\left(\tau \right)\right]+q\left(\tau \right)z^{\gamma }\left(\phi \left(\tau \right)\right)=0.$$

(9)

If $\mathbb{T}=h\mathbb{Z}$, $h>0;$ thus,

$$\mu \left(\tau \right)=h,{\eta }^{\Delta }\left(\tau \right)={\Delta }_h\eta \left(\tau \right)={\displaystyle \frac{\eta (\tau +h)-\eta \left(\tau \right)}{h}},$$

$${\int }_a^b\eta \left(\tau \right)\Delta \tau =\sum _{k=0}^{\frac{b-a-h}{h}}\eta (a+kh)h,$$

and (1) gets the difference equation

$${\Delta }_h\left[r\left(\tau \right){\Delta }_{h}z\left(\tau \right)\right]+q\left(\tau \right)z^{\gamma }\left(\phi \left(\tau \right)\right)=0.$$

(10)

If

$$\mathbb{T}=q^{{\mathbb{N}}_0}=\{\tau :\tau =q^k,k\in {\mathbb{N}}_0,q>1\},$$

then

$$\mu \left(\tau \right)=(q-1)\tau ,{\eta }^{\Delta }\left(\tau \right)={\Delta }_q\eta \left(\tau \right)=\frac{y\left(q\tau \right)-y\left(\tau \right)}{(q-1)\tau },$$

$${\int }_{{\tau }_0}^{\infty }\eta \left(\tau \right)\Delta \tau =\sum _{k=n_0}^{\infty }\eta \left(q^k\right)\mu \left(q^k\right),$$

where ${\tau }_0=q^{n_0}$, and (1) becomes the q-difference equation

$${\Delta }_q\left[r\left(\tau \right){\Delta }_{q}z\left(\tau \right)\right]+q\left(\tau \right)z^{\gamma }\left(\phi \left(\tau \right)\right)=0.$$

(11)

If

$$\mathbb{T}={\mathbb{N}}_0^2:=\{n^2:n\in {\mathbb{N}}_0\},$$

then

$$\mu \left(\tau \right)=1+2\sqrt{\tau },{\Delta }_q\eta \left(\tau \right)={\displaystyle \frac{\eta \left({(\sqrt{\tau }+1)}^2\right)-\eta \left(\tau \right)}{1+2\sqrt{\tau }}},$$

and (1) converts to the difference equation

$${\Delta }_N\left[r\left(\tau \right){\Delta }_{N}z\left(\tau \right)\right]+q\left(\tau \right)z^{\gamma }\left(\phi \left(\tau \right)\right)=0.$$

(12)

Regarding dynamic equations, Erbe et al. [27] extended the Hille oscillation criterion to include the dynamic equation

$$z^{\Delta \Delta }\left(\tau \right)+q\left(\tau \right)z\left(\tau \right)=0,$$

(13)

where

$${\int }_{{\tau }_0}^{\infty }\omega q\left(\omega \right)\Delta \omega =\infty .$$

(14)

They showed that if

$$\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\tau {\int }_{\sigma \left(\tau \right)}^{\infty }\left(\frac{\omega }{\sigma \left(\omega \right)}\right)q\left(\omega \right)\Delta \omega >\frac{1}{4l},$$

where $l:={\lim \inf }_{\tau \to \infty }\frac{\tau }{\sigma \left(\tau \right)}>0,$ then every solution of Equation (13) oscillates. Karpuz [28] considered the canonical form of the linear dynamic equation

$${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }+q\left(\tau \right)z\left(\sigma \left(\tau \right)\right)=0,$$

(15)

and determined that if

$$\underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}\frac{\mu \left(\tau \right)}{r\left(\tau \right)}<\infty ,{\int }_{{\tau }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}=\infty ,$$

and

$$\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\int }_{{\tau }_0}^{\tau }{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}{\int }_{\tau }^{\infty }q\left(\omega \right)\Delta \omega >\frac{1}{4}$$

then every solution of Equation (15) oscillates. For the canonical form of the advanced dynamic equation

$${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }+q\left(\tau \right)z\left(\phi \left(\tau \right)\right)=0,$$

(16)

where $\phi \left(\tau \right)\ge \tau$ and (3) holds, Hassan et al. [29] found that if

$$\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\int }_{{\tau }_0}^{\tau }{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}{\int }_{\sigma \left(\tau \right)}^{\infty }q\left(\omega \right)\Delta \omega >\frac{1}{4l},$$

(17)

where

$$l:=\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\displaystyle \frac{{\int }_{{\tau }_0}^{\tau }{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}}{{\int }_{{\tau }_0}^{\sigma \left(\tau \right)}{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}}}>0,$$

then every solution of Equation (16) oscillates. Hassan et al. [30] improved criterion (17) for the dynamic Equation (16) and proved that if

$$\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\int }_{{\tau }_0}^{\tau }{\displaystyle \frac{\Delta \omega }{r\left(\omega \right)}}{\int }_{\tau }^{\infty }q\left(\omega \right)\Delta \omega >\frac{1}{4},$$

then every solution of Equation (16) oscillates. For further Hille-type criteria, see the papers [31,32,33,34].

It is important to point out that all of the prior findings concentrate on the canonical form, which means that condition (3) is satisfied.

Regarding the noncanonical form (that is, (2) holds), Hassan et al. [35] found some interesting oscillation criteria—namely, Hille-type and Ohriska-type criteria—for the delay linear dynamic equation

$${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }+q\left(\tau \right)z\left(\phi \left(\tau \right)\right)=0,$$

(18)

where $\phi \left(\tau \right)\le \tau$ and (2) holds, which are as follows.

Theorem 1

(see [35]). Every solution of Equation (18) oscillates if any of the following conditions are satisfied:

$$\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\left\{\left({\int }_{\tau }^{\infty }\frac{\Delta \omega }{r\left(\omega \right)}\right)\left({\int }_T^{\tau }q\left(\omega \right)\Delta \omega \right)\right\}>\frac{1}{4};$$

(19)

$$\underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}\left\{\left({\int }_{\tau }^{\infty }\frac{\Delta \omega }{r\left(\omega \right)}\right)\left({\int }_T^{\tau }q\left(\omega \right)\Delta \omega \right)\right\}>1,$$

(20)

for sufficiently large $T\in {[{\tau }_0,\infty )}_{\mathbb{T}}$.

Therefore, in this paper, we will extend the results of [35] to include the advanced situation of nonlinear dynamic Equation (1) in the noncanonical case. These results solved an open problem represented in many of Hassan’s papers; e.g., [17,30,31]. The derivation of advanced dynamic equations has been influenced by numerous practical domains in which the rates of change are influenced by both present and future conditions. To effectively incorporate the impact of prospective future factors into the decision-making process, it is imperative to introduce a sophisticated variable into the equation. Population dynamics, economic issues, and mechanical control engineering are examples of fields that are influenced by future factors, leading to dynamic growth. Please refer to source [36,37,38,39,40] for more information.

2. Criteria for Oscillation (1) when $\gamma \le 1$

In this section, we will demonstrate the existence of new oscillatory criteria corresponding to the Hille and Ohriska types in the noncanonical case when $\gamma \le 1$.

Theorem 2.

Suppose that (2) holds. If, for sufficiently large $T\in {[{\tau }_0,\infty )}_{\mathbb{T}},$

$$\mathbb{A}:=\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_T^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega >\frac{1}{4},$$

(21)

where

$$\zeta \left(\tau \right):={\int }_{\tau }^{\infty }\frac{\Delta \omega }{r\left(\omega \right)},$$

then every solution of Equation (1) oscillates.

Proof. 

Let z be a nonoscillatory solution z of Equation (1) on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. Assume, without loss of generality, $z\left(\tau \right)>0$ on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. In view of (2) and (21), we have $\tau \in {[{\tau }_0,\infty )}_{\mathbb{T}},$

$${\int }_{\tau }^{\infty }{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega =\infty .$$

(22)

From (1), we have ${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }<0$ for $\tau \in {[{\tau }_0,\infty )}_{\mathbb{T}}$. This yields that $z^{\Delta }\left(\tau \right)$ will eventually be of one sign. There are two possibilities:

(a)

$z^{\Delta }\left(\tau \right)>0$ eventually;

(b)

$z^{\Delta }\left(\tau \right)<0$ eventually.

If (a) holds, then there is ${\tau }_1\in {[{\tau }_0,\infty )}_{\mathbb{T}}$ such that $z^{\Delta }\left(\tau \right)>0$ and $\zeta \left(\tau \right)\le 1$ on ${[{\tau }_1,\infty )}_{\mathbb{T}}$. Replacing $\tau$ with $\omega$ in Equation (1), integrating from ${\tau }_1$ to $v\in {[{\tau }_1,\infty )}_{\mathbb{T}}$, we see

$$\begin{array}{ccc}\hfill r\left({\tau }_1\right)z^{\Delta }\left({\tau }_1\right)& >& -r\left(v\right)z^{\Delta }\left(v\right)+r\left({\tau }_1\right)z^{\Delta }\left({\tau }_1\right)\hfill \\ & =& {\int }_{{\tau }_1}^{v}q\left(\omega \right)z^{\gamma }\left(\phi \left(\omega \right)\right)\Delta \omega \hfill \\ & \ge & z^{\gamma }\left(\phi \left({\tau }_1\right)\right){\int }_{{\tau }_1}^{v}q\left(\omega \right)\Delta \omega .\hfill \end{array}$$

Taking limits as $v\to \infty$ gives

$$r\left({\tau }_1\right)z^{\Delta }\left({\tau }_1\right)>z^{\gamma }\left(\phi \left({\tau }_1\right)\right){\int }_{{\tau }_1}^{\infty }q\left(\omega \right)\Delta \omega \ge z^{\gamma }\left(\phi \left({\tau }_1\right)\right){\int }_{{\tau }_1}^{\infty }{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega ,$$

which, in view of (22) contradicts the fact that $r{z}^{\Delta }$ is decreasing.

If (b) holds, then there is ${\tau }_1\in {[{\tau }_0,\infty )}_{\mathbb{T}}$ such that $z^{\Delta }\left(\tau \right)<0$ on ${[{\tau }_1,\infty )}_{\mathbb{T}}$. Define

$$u\left(\tau \right):=-\frac{z\left(\tau \right)}{r\left(\tau \right)z^{\Delta }\left(\tau \right)}.$$

(23)

Then,

$$\begin{array}{ccc}\hfill u^{\Delta }\left(\tau \right)& =& -\frac{1}{r\left(\tau \right)}-{\left(\frac{1}{r\left(\tau \right)z^{\Delta }\left(\tau \right)}\right)}^{\Delta }{z}^{\sigma }\left(\tau \right)\hfill \\ & =& -\frac{1}{r\left(\tau \right)}+\frac{{\left(r\left(\tau \right)z^{\Delta }\left(\tau \right)\right)}^{\Delta }}{r\left(\tau \right)z^{\Delta }\left(\tau \right){\left(r\left(\tau \right)z^{\Delta }\left(\tau \right)\right)}^{\sigma }}{z}^{\sigma }\left(\tau \right).\hfill \end{array}$$

Thanks to (1) and (23), we have

$$\begin{array}{cc}\hfill u^{\Delta }\left(\tau \right)& =-\frac{1}{r\left(\tau \right)}-q\left(\tau \right)\frac{z^{\gamma }\left(\phi \left(\tau \right)\right)}{r\left(\tau \right)z^{\Delta }\left(\tau \right)}{\left(\frac{z\left(\tau \right)}{r\left(\tau \right)z^{\Delta }\left(\tau \right)}\right)}^{\sigma }\hfill \\ \hfill & =-\frac{1}{r\left(\tau \right)}-q\left(\tau \right)\frac{z^{\gamma }\left(\phi \left(\tau \right)\right)}{z\left(\tau \right)}\frac{z\left(\tau \right)}{r\left(\tau \right)z^{\Delta }\left(\tau \right)}{\left(\frac{z\left(\tau \right)}{r\left(\tau \right)z^{\Delta }\left(\tau \right)}\right)}^{\sigma }\hfill \\ \hfill & =-\frac{1}{r\left(\tau \right)}-\frac{z^{\gamma }\left(\phi \left(\tau \right)\right)}{z\left(\tau \right)}q\left(\tau \right)u\left(\tau \right)u^{\sigma }\left(\tau \right),\hfill \end{array}$$

(24)

which gives that $u^{\Delta }<0$. Using ${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }<0,$ we obtain

$$-z\left(\tau \right)<{\int }_{\tau }^{\infty }\frac{r\left(\omega \right)z^{\Delta }\left(\omega \right)}{r\left(\omega \right)}\Delta \omega \le r\left(\tau \right)z^{\Delta }\left(\tau \right){\int }_{\tau }^{\infty }\frac{\Delta \omega }{r\left(\omega \right)}=r\left(\tau \right)z^{\Delta }\left(\tau \right)\zeta \left(\tau \right),$$

which implies that

$$\begin{array}{ccc}\hfill {\left(\frac{z\left(\tau \right)}{\zeta \left(\tau \right)}\right)}^{\Delta }& =& \frac{\zeta \left(\tau \right)z^{\Delta }\left(\tau \right)-{\zeta }^{\Delta }\left(\tau \right)z\left(\tau \right)}{\zeta \left(\tau \right){\zeta }^{\sigma }\left(\tau \right)}\hfill \\ & =& \frac{r\left(\tau \right)\zeta \left(\tau \right)z^{\Delta }\left(\tau \right)+z\left(\tau \right)}{r\left(\tau \right)\zeta \left(\tau \right){\zeta }^{\sigma }\left(\tau \right)}>0.\hfill \end{array}$$

(25)

Therefore,

$$\frac{z\left(\phi \left(\tau \right)\right)}{z\left(\tau \right)}\ge \frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}.$$

(26)

From $z^{\Delta }\left(\tau \right)<0$ and for an arbitrary $k\in \left(0,1\right)$ there is ${\tau }_2\in {[{\tau }_1,\infty )}_{\mathbb{T}}$ such that

$$z^{\gamma -1}\left(\tau \right)\ge k\mathrm{for}\tau \in {[{\tau }_2,\infty )}_{\mathbb{T}}.$$

Hence,

$$\frac{z^{\gamma }\left(\phi \left(\tau \right)\right)}{z\left(\tau \right)}={\left(\frac{z\left(\phi \left(\tau \right)\right)}{z\left(\tau \right)}\right)}^{\gamma }{z}^{\gamma -1}\left(\tau \right)\ge k{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }.$$

(27)

Substituting (27) into (24), we have

$$u^{\Delta }\left(\tau \right)\le -\frac{1}{r\left(\tau \right)}-k{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }q\left(\tau \right)u\left(\tau \right)u^{\sigma }\left(\tau \right).$$

(28)

Let

$$B\left(\tau \right):={\int }_{{\tau }_1}^{\tau }{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega .$$

By integrating (1) and using the fact that ${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }<0,$$z^{\Delta }\left(\tau \right)<0,$ and ${\left({\displaystyle \frac{z\left(\tau \right)}{\zeta \left(\tau \right)}}\right)}^{\Delta }>0,$ we find that

$$\begin{array}{ccc}\hfill r\left(\tau \right)z^{\Delta }\left(\tau \right)& \le & r\left(\tau \right)z^{\Delta }\left(\tau \right)-r\left({\tau }_1\right)z^{\Delta }\left({\tau }_1\right)\hfill \\ & =& -{\int }_{{\tau }_1}^{\tau }q\left(\omega \right)z^{\gamma }\left(\phi \left(\omega \right)\right)\Delta \omega \hfill \\ & \le & -{\int }_{{\tau }_1}^{\tau }{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)z^{\gamma }\left(\omega \right)\Delta \omega \hfill \\ & \le & -z^{\gamma }\left(\tau \right){\int }_{{\tau }_1}^{\tau }{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \hfill \\ & \le & -kz\left(\tau \right)B\left(\tau \right),\hfill \end{array}$$

which implies that

$$0\le \mathbb{U}:=\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}B\left(\tau \right)u\left(\tau \right)\le \frac{1}{k}.$$

Therefore, for any $\epsilon \in \left(0,1\right)$, there is ${\tau }_3\in {[{\tau }_2,\infty )}_{\mathbb{T}}$ such that, for $\tau \in {[{\tau }_3,\infty )}_{\mathbb{T}}$,

$$B^{\sigma }\left(\tau \right){\zeta }^{\sigma }\left(\tau \right)\ge \epsilon \mathbb{A}\mathrm{and}B\left(\tau \right)u\left(\tau \right)\ge \epsilon \mathbb{U}.$$

(29)

Then, (28) can be written as

$$\begin{array}{ccc}\hfill u^{\Delta }\left(\tau \right)& \le & -\frac{1}{r\left(\tau \right)}-\frac{k}{B\left(\tau \right)B^{\sigma }\left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }q\left(\tau \right)B\left(\tau \right)u\left(\tau \right)B^{\sigma }\left(\tau \right)u^{\sigma }\left(\tau \right)\hfill \\ & \le & -\frac{1}{r\left(\tau \right)}-\frac{k{\epsilon }^2{\mathbb{U}}^2}{B\left(\tau \right)B^{\sigma }\left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }q\left(\tau \right).\hfill \end{array}$$

(30)

By integrating (30) from $\sigma \left(\tau \right)$ to v, we get

$$u\left(v\right)-u^{\sigma }\left(\tau \right)\le -{\int }_{\sigma \left(\tau \right)}^v\frac{\Delta \omega }{r\left(\omega \right)}-k{\epsilon }^2{\mathbb{U}}^2{\int }_{\sigma \left(\tau \right)}^v\frac{1}{B\left(\omega \right)B^{\sigma }\left(\omega \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega .$$

By virtue of $u>0,$ and $u^{\Delta }<0,$ we see

$$\begin{array}{ccc}\hfill -u^{\sigma }\left(\tau \right)& \le & -{\int }_{\sigma \left(\tau \right)}^v\frac{\Delta \omega }{r\left(\omega \right)}-k{\epsilon }^2{\mathbb{U}}^2{\int }_{\sigma \left(\tau \right)}^v\frac{1}{B\left(\omega \right)B^{\sigma }\left(\omega \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \hfill \\ & =& -{\int }_{\sigma \left(\tau \right)}^v\frac{\Delta \omega }{r\left(\omega \right)}-k{\epsilon }^2{\mathbb{U}}^2{\int }_{\sigma \left(\tau \right)}^v{\left(\frac{-1}{B\left(\omega \right)}\right)}^{\Delta }\Delta \omega \hfill \\ & =& -{\int }_{\sigma \left(\tau \right)}^v\frac{\Delta \omega }{r\left(\omega \right)}-k{\epsilon }^2{\mathbb{U}}^2\left(\frac{1}{B^{\sigma }\left(\tau \right)}-\frac{1}{B\left(v\right)}\right).\hfill \end{array}$$

Due to (22) and letting $v\to \infty$, we obtain

$${\zeta }^{\sigma }\left(\tau \right)\le u^{\sigma }\left(\tau \right)-\frac{k{\epsilon }^2{\mathbb{U}}^2}{B^{\sigma }\left(\tau \right)}.$$

(31)

By multiplying both sides of (31) by $B^{\sigma }\left(\tau \right)$, we see

$$B^{\sigma }\left(\tau \right){\zeta }^{\sigma }\left(\tau \right)\le B^{\sigma }\left(\tau \right)u^{\sigma }\left(\tau \right)-k{\epsilon }^2{\mathbb{U}}^2.$$

(32)

From (29) and (32), we obtain

$$\epsilon \mathbb{A}\le B^{\sigma }\left(\tau \right)u^{\sigma }\left(\tau \right)-k{\epsilon }^2{\mathbb{U}}^2.$$

(33)

We obtain, by utilizing the liminf of the inequality (33) as $\tau \to \infty$,

$$\epsilon \mathbb{A}\le \mathbb{U}-k{\epsilon }^2{\mathbb{U}}^2.$$

Since $\epsilon ,k>0$ are arbitrary, we obtain

$$\mathbb{A}\le \mathbb{U}-{\mathbb{U}}^2\le \frac{1}{4},$$

which is in opposition to (21). □

Example 1.

Consider the second-order nonlinear advanced dynamic equation

$${\left[{\tau }^2{z}^{\Delta }\left(\tau \right)\right]}^{\Delta }+\sqrt{\zeta \left(\tau \right)\phi \left(\tau \right)z\left(\phi \left(\tau \right)\right)}=0for\tau \in {[{\tau }_0,\infty )}_{\mathbb{T}},$$

(34)

It is easy to see that (2) holds due to

$${\int }_{{\tau }_0}^{\infty }\frac{\Delta \omega }{{\omega }^2}<\infty ,$$

over such time scales as ${[{\tau }_0,\infty )}_{\mathbb{T}}$, when ${\int }_{{\tau }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{{\omega }^p}}<\infty$ with $p>1$. We have

$$\begin{array}{cc}\hfill & \underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_T^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \hfill \\ \hfill & =\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\int }_{\sigma \left(\tau \right)}^{\infty }\frac{\Delta \omega }{{\omega }^2}{\int }_T^{\sigma \left(\tau \right)}\sqrt{\phi \left(\omega \right)\zeta \left(\phi \left(\omega \right)\right)}\Delta \omega \hfill \\ \hfill & \ge \underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\int }_{\sigma \left(\tau \right)}^{\infty }{\left(\frac{-1}{\omega }\right)}^{\Delta }\Delta \omega {\int }_T^{\sigma \left(\tau \right)}\Delta \omega =1.\hfill \end{array}$$

Using Theorem 2, every solution of Equation (34) oscillates over such time scales when ${\int }_{{\tau }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{{\omega }^p}}<\infty$ with $p>1$.

Theorem 3.

Suppose that (2) holds. If, for sufficiently large $T\in {[{\tau }_0,\infty )}_{\mathbb{T}},$

$$\underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_T^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega >1,$$

(35)

then every solution of Equation (1) oscillates.

Proof. 

Let z be a nonoscillatory solution z of Equation (1) on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. Assume, without loss of generality, $z\left(\tau \right)>0$ on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. By virtue of the fact that $r\left(\tau \right)z^{\Delta }\left(\tau \right)$ is strictly decreasing on ${[{\tau }_0,\infty )}_{\mathbb{T}},$ we get that $z^{\Delta }\left(\tau \right)$ will eventually be of one sign. There are two possibilities:

(a)

$z^{\Delta }\left(\tau \right)>0$ eventually;

(b)

$z^{\Delta }\left(\tau \right)<0$ eventually.

If (a) holds, the proof is the same as that of case (a) in Theorem 2; hence, it is omitted.

If (b) holds, then there is a ${\tau }_1\in [{\tau }_0,\infty )$ such that $z^{\Delta }\left(\tau \right)<0$ on $[{\tau }_1,\infty )$. Using the fact that $z^{\Delta }\left(\tau \right)<0$ and ${\left({\displaystyle \frac{z\left(\tau \right)}{\zeta \left(\tau \right)}}\right)}^{\Delta }>0,$ we have, for an arbitrary $k\in \left(0,1\right),$

$$z^{\gamma }\left(\phi \left(\tau \right)\right)\ge {\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }{z}^{\gamma }\left(\tau \right)\ge k{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }z\left(\tau \right),$$

for $\tau \in {[{\tau }_2,\infty )}_{\mathbb{T}}\subseteq {[{\tau }_1,\infty )}_{\mathbb{T}}$. Hence, (1) becomes, for $\tau \in {[{\tau }_2,\infty )}_{\mathbb{T}},$

$${\left[r\left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }+k{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }q\left(\tau \right)z\left(\tau \right)\le 0.$$

(36)

By integrating (36) from ${\tau }_2$ to $\tau$, we obtain

$$\begin{array}{ccc}\hfill {\left(r\left(\tau \right)z^{\Delta }\left(\tau \right)\right)}^{\sigma }& \le & {\left(r\left(\tau \right)z^{\Delta }\left(\tau \right)\right)}^{\sigma }-r\left({\tau }_2\right)z^{\Delta }\left({\tau }_2\right)\hfill \\ & \le & -k{\int }_{{\tau }_2}^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)z\left(\omega \right)\Delta \omega \hfill \\ & \le & -kz\left(\sigma \left(\tau \right)\right){\int }_{{\tau }_2}^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega ,\hfill \end{array}$$

which implies

$$r\left(\omega \right)z^{\Delta }\left(\omega \right)\le {\left(r\left(\tau \right)z^{\Delta }\left(\tau \right)\right)}^{\sigma }\le -kz\left(\sigma \left(\tau \right)\right){\int }_{{\tau }_2}^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega ,$$

(37)

for $\omega \in [\sigma \left(\tau \right),\infty )$ and $\tau \in [{\tau }_2,\infty )$. We have, for $v\in [\sigma \left(\tau \right),\infty ),$

$$-z\left(\sigma \left(\tau \right)\right)\le -z\left(\sigma \left(\tau \right)\right)+z\left(v\right)={\int }_{\sigma \left(\tau \right)}^v\frac{r\left(\omega \right)z^{\Delta }\left(\omega \right)}{r\left(\omega \right)}\Delta \omega .$$

(38)

Substituting (37) into (38), we get

$$-z\left(\sigma \left(\tau \right)\right)\le -kz\left(\sigma \left(\tau \right)\right)\left({\int }_{{\tau }_2}^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \right)\left({\int }_{\sigma \left(\tau \right)}^v\frac{\Delta \omega }{r\left(\omega \right)}\right),$$

so

$$k\left({\int }_{{\tau }_2}^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \right)\left({\int }_{\sigma \left(\tau \right)}^v\frac{\Delta \omega }{r\left(\omega \right)}\right)\le 1.$$

Letting $v\to \infty$, we obtain

$$k\zeta \left(\sigma \left(\tau \right)\right){\int }_{{\tau }_2}^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \le 1.$$

Since k is arbitrary, we have

$$\underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_{{\tau }_2}^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \le 1,$$

which is in opposition to (35). □

Example 2.

Consider the second-order advanced dynamic equation

$${\left[\tau \sigma \left(\tau \right)z^{\Delta }\left(\tau \right)\right]}^{\Delta }+\beta {\left(\frac{\phi \left(\tau \right)}{\tau }\right)}^{\gamma }{z}^{\gamma }\left(\phi \left(\tau \right)\right)=0\mathit{for}\tau \in {[{\tau }_0,\infty )}_{\mathbb{T}},$$

(39)

where $\gamma \le 1$ and $\beta >0$ are constants. It is easy to see that (2) holds since

$${\int }_{{\tau }_0}^{\infty }\frac{\Delta \omega }{\omega \sigma \left(\omega \right)}={\int }_{{\tau }_0}^{\infty }{\left(\frac{-1}{\omega }\right)}^{\Delta }\Delta \omega <\infty .$$

We have

$$\begin{array}{cc}\hfill & \underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_T^{\sigma \left(\tau \right)}{\left(\frac{\zeta \left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}\right)}^{\gamma }q\left(\omega \right)\Delta \omega \hfill \\ \hfill & =\beta \underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}{\int }_{\sigma \left(\tau \right)}^{\infty }\frac{\Delta \omega }{\omega \sigma \left(\omega \right)}{\int }_T^{\sigma \left(\tau \right)}{\left(\frac{{\int }_{\phi \left(\omega \right)}^{\infty }\frac{\Delta \tau }{\tau \sigma \left(\tau \right)}}{{\int }_{\omega }^{\infty }\frac{\Delta \tau }{\tau \sigma \left(\tau \right)}}\right)}^{\gamma }\frac{\phi \left(\omega \right)}{\omega }\Delta \omega \hfill \\ \hfill & =\beta \underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}{\int }_{\sigma \left(\tau \right)}^{\infty }{\left(\frac{-1}{\omega }\right)}^{\Delta }\Delta \omega {\int }_T^{\sigma \left(\tau \right)}{\left(\frac{{\int }_{\phi \left(\omega \right)}^{\infty }{\left(\frac{-1}{\tau }\right)}^{\Delta }\Delta \tau }{{\int }_{\omega }^{\infty }{\left(\frac{-1}{\tau }\right)}^{\Delta }\Delta \tau }\right)}^{\gamma }{\left(\frac{\phi \left(\omega \right)}{\omega }\right)}^{\gamma }\Delta \omega =\beta .\hfill \end{array}$$

Therefore, every solution of Equation (39) oscillates if $\beta >1$, according to an application of Theorem 3.

3. Criteria for Oscillation (1) when $\gamma \ge 1$

In the following, we shall apply the oscillation criteria that were established in the preceding section to the case of $\gamma \ge 1$.

Theorem 4.

Suppose that (2) holds. If, for sufficiently large $T\in {[{\tau }_0,\infty )}_{\mathbb{T}},$

$$\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_T^{\sigma \left(\tau \right)}\frac{{\zeta }^{\gamma }\left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}q\left(\omega \right)\Delta \omega >\frac{1}{4},$$

then every solution of Equation (1) oscillates.

Proof. 

Let z be a nonoscillatory solution z of Equation (1) on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. Assume, without loss of generality, $z\left(\tau \right)>0$ on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. By virtue of the fact that $r\left(\tau \right)z^{\Delta }\left(\tau \right)$ is strictly decreasing on ${[{\tau }_0,\infty )}_{\mathbb{T}},$ it follows that $z^{\Delta }\left(\tau \right)$ will eventually be of one sign. There are two possibilities:

(a)

$z^{\Delta }\left(\tau \right)>0$ eventually;

(b)

$z^{\Delta }\left(\tau \right)<0$ eventually.

If (a) holds, the proof is the same as that of case (a) in Theorem 2; hence, it is omitted.

If (b) holds, then there is ${\tau }_1\in [{\tau }_0,\infty )$ such that $z^{\Delta }\left(\tau \right)<0$ on $[{\tau }_1,\infty )$. As demonstrated in the proof of Theorem 2, we have

$$u^{\Delta }\left(\tau \right)=-\frac{1}{r\left(\tau \right)}-\frac{z^{\gamma }\left(\phi \left(\tau \right)\right)}{z\left(\tau \right)}q\left(\tau \right)u\left(\tau \right)u^{\sigma }\left(\tau \right),$$

(40)

where u is defined by (23). By using the fact that ${\left({\displaystyle \frac{z\left(\tau \right)}{\zeta \left(\tau \right)}}\right)}^{\Delta }>0$, and for an arbitrary $k\in \left(0,1\right),$ there is ${\tau }_2\in {[{\tau }_1,\infty )}_{\mathbb{T}}$ such that for $\tau \in {[{\tau }_2,\infty )}_{\mathbb{T}},$

$$\frac{z^{\gamma }\left(\phi \left(\tau \right)\right)}{z\left(\tau \right)}\ge {\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }{z}^{\gamma -1}\left(\tau \right)\ge k{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }{\zeta }^{\gamma -1}\left(\tau \right)=k\frac{{\zeta }^{\gamma }\left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}.$$

Therefore, (40) becomes, for $\tau \in {[{\tau }_2,\infty )}_{\mathbb{T}},$

$$u^{\Delta }\left(\tau \right)\le -\frac{1}{r\left(\tau \right)}-k\frac{{\zeta }^{\gamma }\left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}q\left(\tau \right)u\left(\tau \right)u^{\sigma }\left(\tau \right).$$

The rest of the evidence is the same as it is in the proof of Theorem 2; hence, it is omitted. □

Example 3.

Consider the second-order nonlinear dynamic equation

$${\left[{\tau }^3{z}^{\Delta }\left(\tau \right)\right]}^{\Delta }+\beta \zeta \left(\tau \right){\sigma }^5\left(\tau \right)z^2\left(\sigma \left(\tau \right)\right)=0\mathit{for}\tau \in {[{\tau }_0,\infty )}_{\mathbb{T}},$$

(41)

where $\beta >0$ is a constant. It is obvious that (2) holds since

$${\int }_{{\tau }_0}^{\infty }\frac{\Delta \omega }{{\omega }^3}<\infty$$

over such time scales as ${[{\tau }_0,\infty )}_{\mathbb{T}}$ in which ${\int }_{{\tau }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{{\omega }^p}}<\infty$ with $p>1$. Note that

$$\begin{array}{cc}\hfill & \underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_T^{\sigma \left(\tau \right)}\frac{{\zeta }^{\gamma }\left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}q\left(\omega \right)\Delta \omega \hfill \\ \hfill & =\beta \underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\int }_{\sigma \left(\tau \right)}^{\infty }\frac{\Delta \omega }{{\omega }^3}{\int }_T^{\sigma \left(\tau \right)}{\left({\zeta }^{\sigma }\left(\omega \right)\right)}^2{\sigma }^5\left(\omega \right)\Delta \omega \hfill \\ \hfill & \ge \frac{\beta }{8}\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}{\int }_{\sigma \left(\tau \right)}^{\infty }{\left(\frac{-1}{{\omega }^2}\right)}^{\Delta }\Delta \omega {\int }_T^{\sigma \left(\tau \right)}\sigma \left(\omega \right)\Delta \omega \hfill \\ \hfill & \ge \frac{\beta }{16}\underset{\tau \to \infty }{\lim \inf \hspace{0.17em}}\frac{1}{{\sigma }^2\left(\tau \right)}{\int }_T^{\sigma \left(\tau \right)}{\left({\omega }^2\right)}^{\Delta }\Delta \omega =\frac{\beta }{16}.\hfill \end{array}$$

We deduce that, according to Theorem 4, every solution of Equation (41) oscillates if $\beta >4$ over such time scales as ${[{\tau }_0,\infty )}_{\mathbb{T}}$, in which ${\int }_{{\tau }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{{\omega }^p}}<\infty$ with $p>1$.

Theorem 5.

Suppose that (2) holds. If, for sufficiently large $T\in {[{\tau }_0,\infty )}_{\mathbb{T}},$

$$\underset{\tau \to \infty }{\lim \sup \hspace{0.17em}}\zeta \left(\sigma \left(\tau \right)\right){\int }_T^{\sigma \left(\tau \right)}\frac{{\zeta }^{\gamma }\left(\phi \left(\omega \right)\right)}{\zeta \left(\omega \right)}q\left(\omega \right)\Delta \omega >1,$$

(42)

then every solution of Equation (1) oscillates.

Proof. 

Let z be a nonoscillatory solution z of Equation (1) on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. Assume, without loss of generality, $z\left(\tau \right)>0$ on ${[{\tau }_0,\infty )}_{\mathbb{T}}$. Since $r\left(\tau \right)z^{\Delta }\left(\tau \right)$ is strictly decreasing on ${[{\tau }_0,\infty )}_{\mathbb{T}}$, then $z^{\Delta }\left(\tau \right)$ will eventually be of one sign. There are two possibilities:

(a)

$z^{\Delta }\left(\tau \right)>0$ eventually;

(b)

$z^{\Delta }\left(\tau \right)<0$ eventually.

If (a) holds, the proof is the same as that of case (a) in Theorem 2; hence, it is omitted.

If (b) holds, then in accordance with ${\left({\displaystyle \frac{z\left(\tau \right)}{\zeta \left(\tau \right)}}\right)}^{\Delta }>0,$ we will have

$$z^{\gamma }\left(\phi \left(\tau \right)\right)\ge {\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }{z}^{\gamma }\left(\tau \right)\ge k{\left(\frac{\zeta \left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}\right)}^{\gamma }{\zeta }^{\gamma -1}\left(\tau \right)z\left(\tau \right)=k\frac{{\zeta }^{\gamma }\left(\phi \left(\tau \right)\right)}{\zeta \left(\tau \right)}z\left(\tau \right),$$

eventually, for an arbitrary $k\in \left(0,1\right).$ The rest of the evidence is the same as it is in the proof of Theorem 3; hence, it is omitted. □

4. Discussion and Conclusions

(1)

The findings presented in this research are applicable to all time scales without any restriction conditions, such as: $\mathbb{T}=\mathbb{R},$$\mathbb{T}=\mathbb{Z},$$\mathbb{T}=h\mathbb{Z}$ with $h>0$, $\mathbb{T}=q^{{\mathbb{N}}_0}$ with $q>1$, etc. (see [5]).

(2)

In this work, our results differ from previous findings in the literature as we do not assume the fulfillment of a certain condition ((3), the canonical case) and therefore solve an open problem mentioned in many papers (see [30]).

(3)

It would be interesting to develop conditions for half-linear noncanonical dynamic equations of the form

$${\left(r\left(\tau \right){\left|z^{\Delta }\left(\tau \right)\right|}^{\gamma -1}{z}^{\Delta }\left(\tau \right)\right)}^{\Delta }+q\left(\tau \right){\left|z\left(\phi \left(\tau \right)\right)\right|}^{\gamma -1}z\left(\phi \left(\tau \right)\right)=0,$$

where $\gamma >0$ is a constant.