Criteria of Oscillation for Second-Order Mixed Nonlinearities in Dynamic Equations

Abstract

This paper investigates second-order functional dynamic equations with mixed nonlinearities on an arbitrary unbounded above-time scale, T. We will use a unified time scale approach and the well-known Riccati technique to derive oscillation criteria of the Nehari-type for second-order dynamic equations. The findings demonstrate a significant improvement in the literature on dynamic equations. The symmetry is beneficial and influential in defining the right style of study for the qualitative behavior of solutions to dynamic equations. We include an example to demonstrate the significance of our results.

1. Introduction

This study deals with nonlinear dynamic equations, which are present in a variety of real-world issues, such as the turbulent flow of a polytrophic gas in a porous media, non-Newtonian fluid theory, and the study of $p$—Laplace equations. Consequently, we are intrigued by the oscillatory solutions to the second-order functional dynamic equation that involves mixed nonlinearities

$${\left[a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\Delta }+p\left(\xi \right){\phi }_{\beta }\left(y\left(k_0\left(\xi \right)\right)\right)+p_1\left(\xi \right){\phi }_{\alpha }\left(y\left(k_1\left(\xi \right)\right)\right)+p_2\left(\xi \right){\phi }_{\gamma }\left(y\left(k_2\left(\xi \right)\right)\right)=0$$

(1)

where $\xi \in {[{\xi }_0,\infty )}_{\mathbb{T}}:=[{\xi }_0,\infty )\cap \mathbb{T}$, ${\xi }_0\ge 0$, ${\xi }_0\in \mathbb{T}$, $\alpha >\beta >\gamma >0,$${\phi }_{\mu }\left(\upsilon \right):={\left|\upsilon \right|}^{\mu -1}\upsilon$, $\mu >0$, $a\in {\mathrm{C}}_{\mathrm{rd}}\left(\mathbb{T},\left(0,\infty \right)\right)$, such that $a^{\Delta }\ge 0$ and

$${\int }_{{\xi }_0}^{\infty }{a}^{-{\displaystyle \frac{1}{\beta }}}\left(\omega \right)\Delta \omega =\infty ,$$

(2)

p, $p_1$, $p_2\in {\mathrm{C}}_{\mathrm{rd}}\left(\mathbb{T},\left[0,\infty \right)\right)$ with $p\not\equiv 0$, and $k_i\in {\mathrm{C}}_{\mathrm{rd}}\left(\mathbb{T},\mathbb{T}\right)$, $i=0,1,2,$ satisfying ${lim}_{\xi \to \infty }{k}_i\left(\xi \right)=\infty$.

We note that $\mathbb{T}$ is an arbitrary unbounded above-time scale and ${\mathrm{C}}_{\mathrm{rd}}$ is the space of right-dense continuous functions; we will refrain from examining solutions that vanish in the vicinity of infinity. If a solution y of (1) is neither eventually positive nor negative, it is regarded as oscillatory; otherwise, it is defined as nonoscillatory. The symmetry of dynamic equations in terms of non-oscillatory solutions determines how to examine their oscillatory behavior. By a solution of Equation (1), we mean a nontrivial real-valued function $y\in {\mathrm{C}}_{\mathrm{rd}}^1{[{\xi }_y,\infty )}_{\mathbb{T}}$ for some ${\xi }_y\ge {\xi }_0$ with ${\xi }_0\in \mathbb{T}$, such that $y^{\Delta },a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)\in {\mathrm{C}}_{\mathrm{rd}}^1{[{\xi }_y,\infty )}_{\mathbb{T}}$ and $y\left(\xi \right)$ satisfies Equation (1) on ${[{\xi }_y,\infty )}_{\mathbb{T}}$.

A time scale $\mathbb{T}$ is any closed real set. The operator $\sigma :\mathbb{T}\to \mathbb{T}$ is called the forward-jump operator and is defined by

$$\sigma \left(\xi \right)=inf\{\upsilon \in \mathbb{T}:\upsilon >\xi \},$$

and $y^{\Delta }:\mathbb{T}\to \mathbb{R}$ is called the derivative of y on $\mathbb{T}$ and is given by

$$y^{\Delta }\left(\xi \right)=\underset{\omega \to \xi }{lim}{\displaystyle \frac{y\left(\sigma \left(\xi \right)\right)-y\left(\omega \right)}{\sigma \left(\xi \right)-\omega }}.$$

The theory of dynamic equations on time scales was proposed by Hilger [1] in order to unify discrete and continuous analysis. Numerous applications can make use of different time scales. The classical theories of differential equations, difference equations, and other cases that lie between these classical cases make up the theory of dynamic equations. Consider the $q$—difference equations where $\mathbb{T}=q^{{\mathbb{N}}_0}:=\{q^{\lambda }:$$\lambda \in {\mathbb{N}}_0$ for $q>1\}$. These equations have important applications in quantum theory (see [2]). Other time scales that can be examined are $\mathbb{T}=h\mathbb{N}$, ${\mathbb{T}=\mathbb{N}}^2,$ and $\mathbb{T}={\mathbb{T}}_n,$ where ${\mathbb{T}}_n$ is the set of the harmonic numbers. For further information on the calculus of time scales, see [3,4,5].

Advanced dynamic equations have been developed using numerous practical domains in which the rates of change are impacted by both current and future factors. To account for the impact of hypothetical future factors on the decision-making process, a complex term must be introduced into the equation. Disciplines such as population dynamics, economic concerns, and mechanical control engineering are frequently distinguished by the impact of future factors on the growth of the dynamic component. Please see sources [6,7] for more information.

Oscillation has sparked substantial interest among applied researchers due to its beginnings in mechanical vibrations and its use in science and engineering. Oscillation models may include delays or advanced terms to represent the dependence of solutions on temporal contexts. The works of [8,9,10,11,12,13,14] demonstrate extensive research on oscillation in delay equations. There is a scarcity of studies focusing on advanced oscillation, as evidenced by the restricted number of works such as [15,16,17].

Several models in real-world applications incorporate oscillation phenomena. For example, in mathematical biology, certain models use cross-diffusion terms to integrate oscillation and/or delay effects. The publications [18,19] provide additional information on this topic. This work focuses on differential equations because they are useful in addressing a variety of real-world phenomena, including non-Newtonian fluid theory and the turbulent flow of a polytrophic gas in porous media. For more information, see papers [20,21,22,23,24].

The following section shows the oscillation results for differential, which are related to the oscillation results for (1) on time scales. It also provides an outline of the significant contributions that this publication made. We will demonstrate how our findings not only unify some differential and difference equation oscillation results, but can also be used to predict oscillatory behavior in other cases. If $\mathbb{T}=\mathbb{R},$ then

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

and (1) reduces to the differential equation

$${\left[a\left(\xi \right){\phi }_{\beta }\left(y^{\prime }\left(\xi \right)\right)\right]}^{\prime }+p\left(\xi \right){\phi }_{\beta }\left(y\left(k_0\left(\xi \right)\right)\right)+p_1\left(\xi \right){\phi }_{\alpha }\left(y\left(k_1\left(\xi \right)\right)\right)+p_2\left(\xi \right){\phi }_{\gamma }\left(y\left(k_2\left(\xi \right)\right)\right)=0.$$

(3)

In the following, Nehari [25] examines the oscillation characteristics of particular instances of Equation (3). The oscillatory solutions to the linear differential equation

$$y^{″}\left(\xi \right)+p\left(\xi \right)y\left(\xi \right)=0,$$

(4)

was inspected in Nehari [25] and inferred that, if

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_{{\xi }_0}^{\xi }{\omega }^{2}p\left(\omega \right)\mathrm{d}\omega >{\displaystyle \frac{1}{4}},$$

(5)

then every solution of (4) is oscillatory. We observe that (5) is an exact inequality that cannot be made weaker. In fact, let $p\left(\xi \right)={\displaystyle \frac{1}{4{\xi }^2}}$ for $\xi \ge 1.$ Thus, we have

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_{{\xi }_0}^{\xi }{\omega }^{2}p\left(\omega \right)\mathrm{d}\omega ={\displaystyle \frac{1}{4}},$$

and the Euler–Cauchy differential equation

$$y^{″}\left(\xi \right)+{\displaystyle \frac{1}{4{\xi }^2}}y\left(\xi \right)=0\mathrm{for}\xi \ge 1,$$

(6)

has a non-oscillatory solution $y\left(\xi \right)=\sqrt{\xi }$. In other words, for all solutions of (6), the lower bound for oscillation is the constant ${\displaystyle \frac{1}{4}}.$

We will investigate the fact that our findings not only extend certain established oscillation results for differential equations but also allow us to apply these findings to other cases where the oscillatory behavior of solutions to these equations on a variety of time scales was unknown. Note that, if $\mathbb{T}=\mathbb{Z}$, then

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

and (1) obtains the following difference equation:

$$\Delta \left[a\left(\xi \right){\phi }_{\beta }\left(\Delta y\left(\xi \right)\right)\right]\left(\xi \right)+p\left(\xi \right){\phi }_{\beta }\left(y\left(k_0\left(\xi \right)\right)\right)+p_1\left(\xi \right){\phi }_{\alpha }\left(y\left(k_1\left(\xi \right)\right)\right)+p_2\left(\xi \right){\phi }_{\gamma }\left(y\left(k_2\left(\xi \right)\right)\right)=0.$$

(7)

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

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

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

and (1) reduces to the difference equation

$${\Delta }_h\left[a\left(\xi \right){\phi }_{\beta }\left({\Delta }_{h}y\left(\xi \right)\right)\right]+p\left(\xi \right){\phi }_{\beta }\left(y\left(k_0\left(\xi \right)\right)\right)+p_1\left(\xi \right){\phi }_{\alpha }\left(y\left(k_1\left(\xi \right)\right)\right)+p_2\left(\xi \right){\phi }_{\gamma }\left(y\left(k_2\left(\xi \right)\right)\right)=0.$$

(8)

If

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

then

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

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

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

$${\Delta }_q\left[a\left(\xi \right){\phi }_{\beta }\left({\Delta }_{q}y\left(\xi \right)\right)\right]+p\left(\xi \right){\phi }_{\beta }\left(y\left(k_0\left(\xi \right)\right)\right)+p_1\left(\xi \right){\phi }_{\alpha }\left(y\left(k_1\left(\xi \right)\right)\right)+p_2\left(\xi \right){\phi }_{\gamma }\left(y\left(k_2\left(\xi \right)\right)\right)=0.$$

(9)

Regarding the Nehari-type oscillation criteria for dynamic equations, Erbe et al. [26] considered the super-linear half-linear delay dynamic equation

$${\left({\left(y^{\Delta }\left(\xi \right)\right)}^{\beta }\right)}^{\Delta }+p\left(\xi \right)y^{\beta }\left(k\left(\xi \right)\right)=0,$$

(10)

where $\beta \ge 1$ be a quotient of odd positive integers and $k\left(\xi \right)\le \xi$ for $\xi \in \mathbb{T}$ and proved that (10) is oscillatory, if

$${\int }_{{\xi }_0}^{\infty }{k}^{\beta }\left(\omega \right)p\left(\omega \right)\Delta \omega =\infty$$

(11)

and

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_{{\xi }_0}^{\xi }{\omega }^{\beta +1}{\left({\displaystyle \frac{k\left(\omega \right)}{\sigma \left(\omega \right)}}\right)}^{\beta }p\left(\omega \right)\Delta \omega +\underset{\xi \to \infty }{lim\; inf}{\xi }^{\beta }{\int }_{\sigma \left(\xi \right)}^{\infty }{\left({\displaystyle \frac{k\left(\omega \right)}{\sigma \left(\omega \right)}}\right)}^{\beta }p\left(\omega \right)\Delta \omega >{\displaystyle \frac{1}{{\ell }^{\beta (\beta +1)}}},$$

(12)

where $\ell :={lim\; inf}_{\xi \to \infty }{\displaystyle \frac{\xi }{\sigma \left(\xi \right)}}>0$. Erbe et al. [27] deduced Nehari-type oscillation criterion for the sub-linear half-linear delay dynamic equation

$${\left(a\left(\xi \right){\left(y^{\Delta }\left(\xi \right)\right)}^{\beta }\right)}^{\Delta }+p\left(\xi \right)y^{\beta }\left(k\left(\xi \right)\right)=0,$$

(13)

where $0<\beta \le 1$ be a quotient of odd positive integers, $a^{\Delta }\ge 0,$ and $k\left(\xi \right)\le \xi$ for $\xi \in \mathbb{T}$ and shown that (13) is oscillatory, if (11) holds,

$${\int }_{{\xi }_0}^{\infty }{a}^{-{\displaystyle \frac{1}{\beta }}}\left(\omega \right)\Delta \omega =\infty ,$$

(14)

and

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_{\xi }^{\xi }{\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}{\left({\displaystyle \frac{k\left(\omega \right)}{\sigma \left(\omega \right)}}\right)}^{\beta }p\left(\omega \right)\Delta \omega +\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{{\xi }^{\beta }}{a\left(\xi \right)}}{\int }_{\sigma \left(\xi \right)}^{\infty }{\left({\displaystyle \frac{k\left(\omega \right)}{\sigma \left(\omega \right)}}\right)}^{\beta }p\left(\omega \right)\Delta \omega >{\displaystyle \frac{1}{{\ell }^{\beta (\beta +1)}}}.$$

(15)

Recently, Hassan et al. [28] established some Nehari-type oscillation criteria for (13) which improved the results of [26,27], one of which is presented below.

Theorem 1. 

Every solution of Equation (13) is oscillatory if one of the following criteria is satisfied:

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}{\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}\varkappa \left(\omega \right)p\left(\omega \right)\Delta \omega >{\displaystyle \frac{1}{{\ell }^{\beta \left(\beta +1\right)}}}\left(1-{\displaystyle \frac{{\ell }^{\beta }}{\beta +1}}\right),$$

(16)

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_T^{\sigma \left(\xi \right)}{\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}\varkappa \left(\omega \right)p\left(\omega \right)\Delta \omega >{\displaystyle \frac{1}{{\ell }^{\beta \left(\beta +1\right)+1}}}\left(1-{\displaystyle \frac{{\ell }^{\beta +1}}{\beta +1}}\right),$$

(17)

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\xi }{\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}\varkappa \left(\omega \right)p\left(\omega \right)\Delta \omega >{\displaystyle \frac{1}{{\ell }^{\beta \left(\beta +1\right)}}}\left(1-{\displaystyle \frac{{\ell }^{\beta +1}}{\beta +1}}\right),$$

(18)

where $\varkappa \left(\omega \right):={\left[{\displaystyle \frac{k\left(\omega \right)}{\omega }}\right]}^{\beta }.$

The reader is directed to related results [29,30,31,32,33] and the cited sources within.

It should be mentioned that Nehari [25] provided valuable contributions that greatly inspired the research in this work. This goal of this paper is to extend the Nehari-type oscillation conditions to the generalized dynamic Equation (1) using the uniform time scale approach and the well-known Riccati technique. Furthermore, we point out that in contrast to [26,27], our oscillation theories do not require the restriction condition (11), and our results extend and enhance the oscillation conditions in [28] to the Equation (13) and can be applied for all $\alpha >\beta >\gamma >0.$

2. Main Results

The main findings will be illustrated in this section, and then examples will be provided to show the significance of the findings. Throughout the paper, we will assume that

$$\ell :=\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{\xi }{\sigma \left(\xi \right)}}\mathrm{and}k\left(\xi \right):=min\left\{\xi ,k_0\left(\xi \right),k_1\left(\xi \right),k_2\left(\xi \right)\right\}.$$

Theorem 2. 

If $\ell >0$ and for sufficiently large $T\in {[{\xi }_0,\infty )}_{\mathbb{T}},$

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega >{\displaystyle \frac{\beta }{{\ell }^{\beta (\beta +1)}\left(\beta +1\right)}},$$

(19)

where

$$Q\left(\xi \right):=(\alpha -\gamma ){\left({\displaystyle \frac{p_1\left(\xi \right)}{\beta -\gamma }}\right)}^{(\beta -\gamma )/(\alpha -\gamma )}{\left({\displaystyle \frac{p_2\left(\xi \right)}{\alpha -\beta }}\right)}^{\left(\alpha -\beta \right)/\left(\alpha -\gamma \right)},$$

(20)

then every solution of Equation (1) is oscillatory.

Proof. 

Let y be a non-oscillatory solution of Equation (1) on ${[{\xi }_0,\infty )}_{\mathbb{T}}$. We assume, without loss of generality, $y\left(\xi \right)>0$ and $y\left(k_i\left(\xi \right)\right)>0,$ $i=0,1,2,$ for $\xi \in {[{\xi }_0,\infty )}_{\mathbb{T}}$. By virtue of [20] ([Lemmas 2.1 and 2.2]), there exists ${\xi }_1\in {({\xi }_0,\infty )}_{\mathbb{T}}$ satisfying

$$y^{\Delta }\left(\xi \right)>0,{\left({\displaystyle \frac{y\left(\xi \right)}{\xi -{\xi }_0}}\right)}^{\Delta }<0,\mathrm{and}{\left[a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\Delta }<0\mathrm{for}\xi \in {[{\xi }_1,\infty )}_{\mathbb{T}}.$$

(21)

Define

$$x\left(\xi \right):={\displaystyle \frac{a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)}{y^{\beta }\left(\xi \right)}}.$$

(22)

Hence,

$$\begin{array}{ccc}\hfill x^{\Delta }\left(\xi \right)& =& {\displaystyle \frac{1}{y^{\beta }\left(\xi \right)}}{\left[a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\Delta }\hfill \\ & & +{\left({\displaystyle \frac{1}{y^{\beta }\left(\xi \right)}}\right)}^{\Delta }{\left[a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\sigma }\hfill \\ & =& {\displaystyle \frac{{\left[a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\Delta }}{y^{\beta }\left(\xi \right)}}-{\displaystyle \frac{{\left(y^{\beta }\left(\xi \right)\right)}^{\Delta }}{y^{\beta }\left(\xi \right)y^{\beta }\left(\sigma \left(\xi \right)\right)}}{\left[a\left(\xi \right){\phi }_{\beta }\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\sigma }.\hfill \end{array}$$

(23)

From (1) and (22), we have

$$\begin{array}{ccc}\hfill x^{\Delta }\left(\xi \right)& =& -p\left(\xi \right){\displaystyle \frac{y^{\beta }\left(k_1\left(\xi \right)\right)}{y^{\beta }\left(\xi \right)}}-p_1\left(\xi \right){\displaystyle \frac{y^{\alpha }\left(k_1\left(\xi \right)\right)}{y^{\beta }\left(\xi \right)}}-p_2\left(\xi \right){\displaystyle \frac{y^{\gamma }\left(k_2\left(\xi \right)\right)}{y^{\beta }\left(\xi \right)}}\hfill \\ & & -{\displaystyle \frac{{\left(y^{\beta }\left(\xi \right)\right)}^{\Delta }}{y^{\beta }\left(\xi \right)}}x\left(\sigma \left(\xi \right)\right)\hfill \\ & \le & -p\left(\xi \right){\displaystyle \frac{y^{\beta }\left(k\left(\xi \right)\right)}{y^{\beta }\left(\xi \right)}}-p_1\left(\xi \right){\displaystyle \frac{y^{\alpha }\left(k\left(\xi \right)\right)}{y^{\beta }\left(\xi \right)}}-p_2\left(\xi \right){\displaystyle \frac{y^{\gamma }\left(k\left(\xi \right)\right)}{y^{\beta }\left(\xi \right)}}\hfill \\ & & -{\displaystyle \frac{{\left(y^{\beta }\left(\xi \right)\right)}^{\Delta }}{y^{\beta }\left(\xi \right)}}x\left(\sigma \left(\xi \right)\right).\hfill \end{array}$$

Let $\xi \in {[{\xi }_0,\infty )}_{\mathbb{T}}$ be fixed. Since ${\left({\displaystyle \frac{y\left(\xi \right)}{\xi -{\xi }_0}}\right)}^{\Delta }<0$ on ${({\xi }_0,\infty )}_{\mathbb{T}}$, we obtain

$${\displaystyle \frac{y\left(k\right(\xi \left)\right)}{y\left(\xi \right)}}\ge {\displaystyle \frac{k\left(\xi \right)-{\xi }_0}{\xi }}\mathrm{for}\xi \ge k\left(\xi \right)\ge {\xi }_0.$$

Then there is ${\xi }_{\lambda }\in {[{\xi }_0,\infty )}_{\mathbb{T}}$, for each $0<\lambda <1$, such that

$$y\left(k\left(\xi \right)\right)\ge \lambda {\displaystyle \frac{k\left(\xi \right)}{\xi }}y\left(\xi \right)\mathrm{for}\xi \ge {\xi }_{\lambda }.$$

(24)

Therefore for $\xi \in {[{\xi }_{\lambda },\infty )}_{\mathbb{T}},$

$$\begin{array}{ccc}\hfill x^{\Delta }\left(\xi \right)& \le & -{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }p\left(\xi \right)-{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\alpha }{p}_1\left(\xi \right)y^{\alpha -\beta }\left(\xi \right)\hfill \\ & & -{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\gamma }{p}_2\left(\xi \right)y^{\gamma -\beta }\left(\xi \right)-{\displaystyle \frac{{\left(y^{\beta }\left(\xi \right)\right)}^{\Delta }}{y^{\beta }\left(\xi \right)}}x\left(\sigma \left(\xi \right)\right).\hfill \end{array}$$

(25)

Define

$$g\left(\xi ,y\right):={\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\alpha }{p}_1\left(\xi \right)y^{\alpha -\beta }+{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\gamma }{p}_2\left(\xi \right)y^{\gamma -\beta }.$$

We obtain that g obtains its minimum with respect to y at

$$y={\displaystyle \frac{\xi }{\lambda k\left(\xi \right)}}{\left({\displaystyle \frac{p_2\left(\xi \right)}{p_1\left(\xi \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -\gamma }}}{\left({\displaystyle \frac{\beta -\gamma }{\alpha -\beta }}\right)}^{{\displaystyle \frac{1}{\alpha -\gamma }}}$$

and

$$g_{min}={\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }Q\left(\xi \right),$$

where

$$Q\left(\xi \right)=(\alpha -\gamma ){\left({\displaystyle \frac{p_1\left(\xi \right)}{\beta -\gamma }}\right)}^{(\beta -\gamma )/(\alpha -\gamma )}{\left({\displaystyle \frac{p_2\left(\xi \right)}{\alpha -\beta }}\right)}^{\left(\alpha -\beta \right)/\left(\alpha -\gamma \right)}.$$

Therefore,

$${\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\alpha }{p}_1\left(\xi \right)y^{\alpha -\beta }\left(\xi \right)-{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\gamma }{p}_2\left(\xi \right)y^{\gamma -\beta }\left(\xi \right)\le -{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }Q\left(\xi \right).$$

Substituting the last inequality into (25) obtainig

$$x^{\Delta }\left(\xi \right)\le -{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]-{\displaystyle \frac{{\left(y^{\beta }\left(\xi \right)\right)}^{\Delta }}{y^{\beta }\left(\xi \right)}}x\left(\sigma \left(\xi \right)\right).$$

Use the Pötzsche chain rule to obtain

$$\begin{array}{ccc}\hfill {\left(y^{\beta }\left(\xi \right)\right)}^{\Delta }& =& \beta {\int }_0^1{\left[\left(1-h\right)y\left(\xi \right)+hy\left(\sigma \left(\xi \right)\right)\right]}^{\beta -1}\mathrm{d}h{y}^{\Delta }\left(\xi \right)\hfill \\ & \ge & \left\{\begin{array}{ccc}\beta y^{\beta -1}\left(\sigma \left(\xi \right)\right)y^{\Delta }\left(\xi \right),\hfill & & 0<\beta \le 1,\hfill \\ & & \\ \beta y^{\beta -1}\left(\xi \right)y^{\Delta }\left(\xi \right),\hfill & & \beta \ge 1.\hfill \end{array}\right.\hfill \end{array}$$

If $0<\beta \le 1,$ then

$$x^{\Delta }\left(\xi \right)<-{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]-\beta {\displaystyle \frac{y^{\Delta }\left(\xi \right)}{y\left(\sigma \left(\xi \right)\right)}}{\left({\displaystyle \frac{y\left(\sigma \left(\xi \right)\right)}{y\left(\xi \right)}}\right)}^{\beta }x\left(\sigma \left(\xi \right)\right);$$

and if $\beta \ge 1$, then

$$x^{\Delta }\left(\xi \right)\le -{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]-\beta {\displaystyle \frac{y^{\Delta }\left(\xi \right)}{y\left(\sigma \left(\xi \right)\right)}}{\displaystyle \frac{y\left(\sigma \left(\xi \right)\right)}{y\left(\xi \right)}}x\left(\sigma \left(\xi \right)\right).$$

From (21), we obtain for $\beta >0,$

$$\begin{array}{ccc}\hfill x^{\Delta }\left(\xi \right)& \le & -{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]-\beta {\displaystyle \frac{y^{\Delta }\left(\xi \right)}{y\left(\sigma \left(\xi \right)\right)}}x\left(\sigma \left(\xi \right)\right)\hfill \\ & =& -{\left({\displaystyle \frac{\lambda k\left(\xi \right)}{\xi }}\right)}^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]-\beta a^{-{\displaystyle \frac{1}{\beta }}}\left(\xi \right)x^{1+1/\beta }\left(\sigma \left(\xi \right)\right),\hfill \end{array}$$

(26)

for $\xi \in {[{\xi }_{\lambda },\infty )}_{\mathbb{T}}$. Multiplying (26) by ${\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}$, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}{x}^{\Delta }\left(\xi \right)& \le & -{\lambda }^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]{\displaystyle \frac{\xi k^{\beta }\left(\xi \right)}{a\left(\xi \right)}}-\beta {\left({\displaystyle \frac{{\xi }^{\beta }}{a\left(\xi \right)}}x\left(\sigma \left(\xi \right)\right)\right)}^{1+1/\beta }\hfill \\ & \le & -{\lambda }^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]{\displaystyle \frac{\xi k^{\beta }\left(\xi \right)}{a\left(\xi \right)}}-\beta {\left({\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }\right)}^{1+1/\beta }.\hfill \end{array}$$

For any $\epsilon >0$, there exists $T\in {[{\xi }_{\lambda },\infty )}_{\mathbb{T}}$, such that

$${\displaystyle \frac{\xi }{\sigma \left(\xi \right)}}\ge \ell -\epsilon \mathrm{and}{a}_{*}-\epsilon \le {\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }\le R+\epsilon \mathrm{for}\xi \in {[T,\infty )}_{\mathbb{T}},$$

(27)

where

$$a_{*}:=\underset{\xi \to \infty }{lim\; inf}{\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }\mathrm{and}R:=\underset{\xi \to \infty }{lim\; sup}{\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }.$$

(28)

Using (27) gives for $\xi \in {[T,\infty )}_{\mathbb{T}},$

$${\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}{x}^{\Delta }\left(\xi \right)\le -{\lambda }^{\beta }\left[p\left(\xi \right)+Q\left(\xi \right)\right]{\displaystyle \frac{\xi k^{\beta }\left(\xi \right)}{a\left(\xi \right)}}-\beta {\left(a_{*}-\epsilon \right)}^{1+1/\beta }.$$

(29)

Integrating (29) from T to $\sigma \left(\xi \right)$, we obtain

$$\begin{array}{ccc}\hfill {\int }_T^{\sigma \left(\xi \right)}{\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}{x}^{\Delta }\left(\omega \right)\Delta \omega & \le & -{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & -\beta {\left(a_{*}-\epsilon \right)}^{1+1/\beta }\left(\sigma \left(\xi \right)-T\right).\hfill \end{array}$$

Using integrating by parts, we obtain

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }-{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)& \le & {\int }_T^{\sigma \left(\xi \right)}{\left({\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}\right)}^{\Delta }{x}^{\sigma }\left(\omega \right)\Delta \omega \hfill \\ & & -{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & -\beta {\left(a_{*}-\epsilon \right)}^{1+1/\beta }\left(\sigma \left(\xi \right)-T\right).\hfill \end{array}$$

Now,

$${\left({\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}\right)}^{\Delta }={\displaystyle \frac{{\left({\omega }^{\beta +1}\right)}^{\Delta }}{a^{\sigma }\left(\omega \right)}}-{\displaystyle \frac{{\omega }^{\beta +1}{a}^{\Delta }\left(\omega \right)}{a\left(\omega \right)a^{\sigma }\left(\omega \right)}}\le (\beta +1){\left({\displaystyle \frac{{\omega }^{\beta }}{a\left(\omega \right)}}\right)}^{\sigma }.$$

(30)

Therefore,

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }-{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)& \le & (\beta +1){\int }_T^{\sigma \left(\xi \right)}{\left({\displaystyle \frac{{\omega }^{\beta }}{a\left(\omega \right)}}x\left(\omega \right)\right)}^{\sigma }\Delta \omega \hfill \\ & & -{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & -\beta {\left(a_{*}-\epsilon \right)}^{1+1/\beta }\left(\sigma \left(\xi \right)-T\right)\hfill \\ & \le & (\beta +1)\left(R+\epsilon \right)\left(\sigma \left(\xi \right)-T\right)\hfill \\ & & -{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega .\hfill \end{array}$$

(31)

Dividing (31) by $\sigma \left(\xi \right),$ we obtain

$$\begin{array}{ccc}\hfill {\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }& \le & {\left({\displaystyle \frac{{\xi }^{\beta }}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }\le {\displaystyle \frac{{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)}{\sigma \left(\xi \right)}}+\left(R+\epsilon \right)\left(\beta +1\right)\left[1-{\displaystyle \frac{T}{\sigma \left(\xi \right)}}\right]\hfill \\ & & -{\displaystyle \frac{{\lambda }^{\beta }}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega .\hfill \end{array}$$

Take the $limsup$ as $\xi \to \infty$ and by (28), we obtain

$$R\le \left(\beta +1\right)\left(R+\epsilon \right)-\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{{\lambda }^{\beta }}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega .$$

Due to $0<\lambda <1$ and $\epsilon >0$ are arbitrary, we have

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \le \beta R.$$

(32)

Multiplying (26) by ${\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}$ and integrate from T to $\sigma \left(\xi \right)$, obtaining

$$\begin{array}{ccc}\hfill {\int }_T^{\sigma \left(\xi \right)}{\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}{x}^{\Delta }\left(\omega \right)\Delta \omega & \le & -{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & -\beta {\int }_T^{\sigma \left(\xi \right)}{\left[{\displaystyle \frac{{\omega }^{\beta }{x}^{\sigma }\left(\omega \right)}{a\left(\omega \right)}}\right]}^{1+1/\beta }\Delta \omega .\hfill \end{array}$$

(33)

By using (27), we obtain

$$\begin{array}{ccc}\hfill {\int }_T^{\sigma \left(\xi \right)}{\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}{x}^{\Delta }\left(\omega \right)\Delta \omega & \le & -{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & -\beta {\left(\ell -\epsilon \right)}^{\beta +1}{\int }_T^{\sigma \left(\xi \right)}{\left[{\left({\displaystyle \frac{{\omega }^{\beta }}{a\left(\omega \right)}}x\left(\omega \right)\right)}^{\sigma }\right]}^{1+1/\beta }\Delta \omega .\hfill \end{array}$$

By integration by parts, we see

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }& \le & {\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)-{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & +{\int }_T^{\sigma \left(\xi \right)}{\left({\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}\right)}^{\Delta }{x}^{\sigma }\left(\omega \right)\Delta \omega \hfill \\ & & -\beta {\left(\ell -\epsilon \right)}^{\beta +1}{\int }_T^{\sigma \left(\xi \right)}{\left[{\left({\displaystyle \frac{{\omega }^{\beta }}{a\left(\omega \right)}}x\left(\omega \right)\right)}^{\sigma }\right]}^{1+1/\beta }\Delta \omega .\hfill \end{array}$$

(34)

Since

$${\left({\displaystyle \frac{{\omega }^{\beta +1}}{a\left(\omega \right)}}\right)}^{\Delta }={\displaystyle \frac{{\left({\omega }^{\beta +1}\right)}^{\Delta }}{a^{\sigma }\left(\omega \right)}}-{\displaystyle \frac{{\omega }^{\beta +1}{a}^{\Delta }\left(\omega \right)}{a\left(\omega \right)a^{\sigma }\left(\omega \right)}}\le (\beta +1){\displaystyle \frac{{\sigma }^{\beta }\left(\omega \right)}{a\left(\omega \right)}}.$$

Hence,

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }& \le & {\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)-{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & +{\int }_T^{\sigma \left(\xi \right)}\left[\left(\beta +1\right){\left({\displaystyle \frac{{\omega }^{\beta }}{a\left(\omega \right)}}x\left(\omega \right)\right)}^{\sigma }\right.\hfill \\ & & \left.-\beta {\left(\ell -\epsilon \right)}^{\beta +1}{\left[{\left({\displaystyle \frac{{\omega }^{\beta }}{a\left(\omega \right)}}x\left(\omega \right)\right)}^{\sigma }\right]}^{1+1/\beta }\right]\Delta \omega .\hfill \end{array}$$

Using the inequality

$$Au-B{u}^{1+1/\beta }\le {\displaystyle \frac{{\beta }^{\beta }}{{(\beta +1)}^{\beta +1}}}{\displaystyle \frac{A^{\beta +1}}{B^{\beta }}},$$

(35)

with $A=\beta +1,$ $B=\beta {\left(\ell -\epsilon \right)}^{\beta +1}$ and $u={\left({\displaystyle \frac{{\omega }^{\beta }}{a\left(\omega \right)}}x\left(\omega \right)\right)}^{\sigma }$, we obtain

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }& \le & {\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)-{\lambda }^{\beta }{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & +\left(\sigma \left(\xi \right)-T\right){\displaystyle \frac{1}{{\left(\ell -\epsilon \right)}^{\beta (\beta +1)}}}.\hfill \end{array}$$

(36)

Dividing (36) by $\sigma \left(\xi \right),$ we see

$$\begin{array}{ccc}\hfill {\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }& \le & {\left({\displaystyle \frac{{\xi }^{\beta }}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }\hfill \\ & \le & {\displaystyle \frac{{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)}{\sigma \left(\xi \right)}}-{\displaystyle \frac{{\lambda }^{\beta }}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & +{\displaystyle \frac{1}{{\left(\ell -\epsilon \right)}^{\beta (\beta +1)}}}\left(1-{\displaystyle \frac{T}{\sigma \left(\xi \right)}}\right).\hfill \end{array}$$

Take the lim sup as $\xi \to \infty$ we obtain

$$R\le -\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{{\lambda }^{\beta }}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega +{\displaystyle \frac{1}{{\left(\ell -\epsilon \right)}^{\beta (\beta +1)}}}.$$

Thanks to $\epsilon >0$ and $0<\lambda <1$ are arbitrary, we deduce

$$R\le -\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega +{\displaystyle \frac{1}{{\ell }^{\beta (\beta +1)}}}.$$

(37)

Substituting (37) into (32), it follows that

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \le {\displaystyle \frac{\beta }{{\ell }^{\beta (\beta +1)}\left(\beta +1\right)}},$$

which contadicts (19). □

Theorem 3. 

If $\ell >0$ and for sufficiently large $T\in {[{\xi }_0,\infty )}_{\mathbb{T}}$,

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega >{\displaystyle \frac{1}{{\ell }^{\beta \left(\beta +1\right)+1}}}\left(1-{\displaystyle \frac{\ell }{\beta +1}}\right),$$

(38)

where is defined by (20), then all solutions of (1) are oscillatory.

Proof. 

Let y be a non-oscillatory solution of Equation (1) on ${[{\xi }_0,\infty )}_{\mathbb{T}}$. We assume, without loss of generality, $y\left(\xi \right)>0$ and $y\left(k_i\left(\xi \right)\right)>0,$ $i=0,1,2,$ for $\xi \in {[{\xi }_0,\infty )}_{\mathbb{T}}$. As shown in the proof of Theorem 2, (36) and (31) hold for $\xi \in {[T,\infty )}_{\mathbb{T}},$ for sufficiently large $T\in {[{\xi }_0,\infty )}_{\mathbb{T}}$. Dividing (31) by $\xi ,$ we obtain

$$\begin{array}{ccc}\hfill {\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }& \le & {\displaystyle \frac{1}{\xi }}{\left({\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }\le {\displaystyle \frac{{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)}{\xi }}+(\beta +1)\left(R+\epsilon \right)\left[{\displaystyle \frac{\sigma \left(\xi \right)}{\xi }}-{\displaystyle \frac{T}{\xi }}\right]\hfill \\ & & -{\displaystyle \frac{{\lambda }^{\beta }}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & \le & {\displaystyle \frac{{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)}{\xi }}+(\beta +1)\left(R+\epsilon \right)\left[{\displaystyle \frac{1}{\ell -\epsilon }}-{\displaystyle \frac{T}{\xi }}\right]\hfill \\ & & -{\displaystyle \frac{{\lambda }^{\beta }}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega .\hfill \end{array}$$

Take the $limsup$ as $\xi \to \infty$ and by (28), we obtain

$$R\le \left(\beta +1\right){\displaystyle \frac{R+\epsilon }{\ell -\epsilon }}-\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{{\lambda }^{\beta }}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega .$$

By dint of $\epsilon >0$ and $0<\lambda <1$ are arbitrary, we achieve

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \le {\displaystyle \frac{R}{\ell }}\left(\beta +1-\ell \right).$$

(39)

Dividing (36) by $\xi$, we obtain

$$\begin{array}{ccc}\hfill {\xi }^{\beta }{\left({\displaystyle \frac{x\left(\xi \right)}{a\left(\xi \right)}}\right)}^{\sigma }& \le & {\displaystyle \frac{1}{\xi }}{\left({\displaystyle \frac{{\xi }^{\beta +1}}{a\left(\xi \right)}}x\left(\xi \right)\right)}^{\sigma }\hfill \\ & \le & {\displaystyle \frac{{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)}{\xi }}-{\displaystyle \frac{{\lambda }^{\beta }}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & +{\displaystyle \frac{1}{{\left(\ell -\epsilon \right)}^{\beta (\beta +1)}}}\left({\displaystyle \frac{\sigma \left(\xi \right)}{\xi }}-{\displaystyle \frac{T}{\xi }}\right)\hfill \\ & \le & {\displaystyle \frac{{\displaystyle \frac{T^{\beta +1}}{a\left(T\right)}}x\left(T\right)}{\xi }}-{\displaystyle \frac{{\lambda }^{\beta }}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & & +{\displaystyle \frac{1}{{\left(\ell -\epsilon \right)}^{\beta (\beta +1)}}}\left({\displaystyle \frac{1}{\ell -\epsilon }}-{\displaystyle \frac{T}{\xi }}\right).\hfill \end{array}$$

Take the lim sup as $\xi \to \infty$ to obtain

$$R\le -\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{{\lambda }^{\beta }}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega +{\displaystyle \frac{1}{{\left(\ell -\epsilon \right)}^{\beta (\beta +1)+1}}}.$$

Hence,

$$R\le -\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega +{\displaystyle \frac{1}{{\ell }^{\beta (\beta +1)+1}}}.$$

(40)

Substituting (40) into (39), we obtain

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\xi }}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\le {\displaystyle \frac{1}{{\ell }^{\beta \left(\beta +1\right)+1}}}\left(1-{\displaystyle \frac{\ell }{\beta +1}}\right),$$

which contadicts (38). □

Example 1. 

Consider a second-order mixed nonlinearities dynamic equation

$${\left[{\xi }^2{\phi }_3\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\Delta }+{\displaystyle \frac{\delta \xi }{2k^3\left(\xi \right)}}{\phi }_3\left(y\left(k_0\left(\xi \right)\right)\right)+{\displaystyle \frac{\delta {\xi }^2}{8k^2\left(\xi \right)}}{\phi }_4\left(y\left(k_1\left(\xi \right)\right)\right)+{\displaystyle \frac{\delta }{2k^4\left(\xi \right)}}{\phi }_2\left(y\left(k_2\left(\xi \right)\right)\right)=0$$

(41)

where $\delta >0$ is a constant and $\ell ={lim\; inf}_{\xi \to \infty }{\displaystyle \frac{\xi }{\sigma \left(\xi \right)}}>0$. It is easy to show that (2) holds since

$${\int }_{{\xi }_0}^{\infty }{a}^{-{\displaystyle \frac{1}{\beta }}}\left(\omega \right)\Delta \omega ={\int }_{{\xi }_0}^{\infty }{\displaystyle \frac{\Delta \omega }{{\omega }^{2/3}}}=\infty .$$

Now

$$\begin{array}{ccc}& & \underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[p\left(\omega \right)+Q\left(\omega \right)\right]{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}\Delta \omega \hfill \\ & =& \underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}\left[{\displaystyle \frac{\delta }{2}}{\displaystyle \frac{\omega }{k^3\left(\omega \right)}}+2\sqrt{{\displaystyle \frac{\delta {\omega }^2}{8k^2\left(\omega \right)}}}\sqrt{{\displaystyle \frac{\delta }{2k^4\left(\omega \right)}}}\right]{\displaystyle \frac{k^3\left(\omega \right)}{\omega }}\Delta \omega \hfill \\ & =& \delta \underset{\xi \to \infty }{lim\; inf}\left(1-{\displaystyle \frac{T}{\sigma \left(\xi \right)}}\right)=\delta .\hfill \end{array}$$

Usage of Theorems 2 and 3 mean that Equation (41) is oscillatory if either $\delta >{\displaystyle \frac{3}{4{\ell }^{12}}}$ or $\delta >{\displaystyle \frac{1}{{\ell }^{13}}}\left(1-{\displaystyle \frac{\ell }{4}}\right).$

3. Results and Discussion

(I)

The results of this paper extend the Nehari-type oscillation criteria to the generalized mixed nonlinearities dynamic Equation (1) in all cases, $k_i\left(\xi \right)\le \xi$ and $k_i\left(\xi \right)\ge \xi$ and on any arbitrary time scale, for all $\alpha >\beta >\gamma >0.$ Also, in contrast to the previous literature, the results we have obtained in this work do not presume the fulfillment of condition (11).

(II)

The results can be applicable to all time scales, including $\mathbb{T}=\mathbb{R},$$\mathbb{T}=\mathbb{Z},$$\mathbb{T}=h\mathbb{Z}$ with $h>0$, $\mathbb{T}=q^{{\mathbb{N}}_0}$ with $q>1$, and so forth (see [5]).

(III)

If $p_1\left(\xi \right)=p_2\left(\xi \right)=0$, then criterion (19) reduces to

$$\underset{\xi \to \infty }{lim\; inf}{\displaystyle \frac{1}{\sigma \left(\xi \right)}}{\int }_T^{\sigma \left(\xi \right)}{\displaystyle \frac{\omega k^{\beta }\left(\omega \right)}{a\left(\omega \right)}}p\left(\omega \right)\Delta \omega >{\displaystyle \frac{\beta }{{\ell }^{\beta (\beta +1)}\left(\beta +1\right)}}$$

By virtue of

$${\displaystyle \frac{\beta }{{\ell }^{\beta (\beta +1)}\left(\beta +1\right)}}\le {\displaystyle \frac{1}{{\ell }^{\beta \left(\beta +1\right)}}}\left({\displaystyle \frac{\beta +1-{\ell }^{\beta }}{\beta +1}}\right),$$

criterion (19) improves (16).

(IV)

It would be intriguing to apply the sharp Nehari-type criterion that the solutions of the second-order Euler differential equation $y^{″}\left(t\right)+{\displaystyle \frac{\gamma }{t^2}}y\left(t\right)=0$ are oscillatory when $\gamma >{\displaystyle \frac{1}{4}}$ to a second-order dynamic equation, as demonstrated in [25].