Oscillation of Neutral Differential Equations with Damping Terms

Abstract

Our interest in this paper is to study and develop oscillation conditions for solutions of a class of neutral differential equations with damping terms. New oscillation criteria were obtained by using Riccati transforms. The criteria we obtained improved and completed some of the criteria in previous studies mentioned in the literature. Examples are provided to illustrate the applicability of our results.

1. Introduction

In this paper, we study the oscillatory behavior of the differential equations with damping terms

$${\left(r\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }+p\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }+q\left(s\right){\varkappa }^{\delta }\left(\sigma \left(s\right)\right)=0,s\ge s_0,$$

(1)

where

$$y\left(s\right)=\varkappa \left(s\right)+c\left(s\right)\varkappa \left(\tau \left(s\right)\right).$$

(2)

During this study, we will assume the following:

(H₁)

$\gamma$ and $\delta$ is a ratio of odd natural numbers;

(H₂)

$r\in C^1\left(I_0,{\mathbb{R}}^{+}\right),$$r^{\prime }\left(s\right)\ge 0,$$c\left(s\right)\in C\left(I_0,\mathbb{R}\right)$, $0<c\left(s\right)<1$ and $I_0:=\left[s_0,\infty \right);$

(H₃)

$p,q\in C\left(I_0,{\mathbb{R}}^{+}\right)$, $q\left(s\right)$ is not eventually zero on $\left[s^{*},\infty \right)$ for $s^{*}\ge s_0;$

(H₄)

$\tau \in C\left(I_0,\mathbb{R}\right)$, $\sigma \in C^1\left(I_0,\mathbb{R}\right)$, $\tau \left(s\right)\le s,$$\sigma \left(s\right)\le s,$${\sigma }^{\prime }\left(s\right)>0,$${lim}_{s\to \infty }\tau \left(s\right)=\infty$ and ${lim}_{s\to \infty }\sigma \left(s\right)=\infty$.

Let $S_{\varkappa }=min\left\{\tau \left(s\right),\sigma \left(s\right)\right\},$$s\ge s_0$. We say that a real-valued function $\varkappa \in C^3\left(I_{\varkappa }\right),$$s_{\varkappa }\ge s_0,$ is a solution of (1) if $r\left(y^{\prime \prime \prime }\right)\in C^1\left(I_{\varkappa },\mathbb{R}\right)$, $\varkappa$ satisfies (1) on $I_{\varkappa }$, and $sup\left\{\left|\varkappa \left(s\right)\right|:s_1\ge s\right\}>0$ for every $s\in I_{\varkappa }$. A solution of (1) is called oscillatory if it has arbitrarily large zeros on $I_{\varkappa }$; otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions oscillate.

There are many applications described by neutral differential equations. Such equations arise naturally in the modeling of physical and biological phenomena such as oscillations of neuromuscular systems, deformation of structures, or problems of elasticity (see [1,2] for more details).

We know that there is great interest in studying the qualitative behaviors of differential equations, such as the asymptotic behavior, the property of stability, the property of boundary, and the oscillatory properties (see [3,4,5,6,7,8,9,10,11,12,13,14,15]).

We will now present some of the previous results that were mentioned in the literature.

Grace and Akin [16] discuss the oscillations of nonlinear delay differential equations

$${\varkappa }^{\left(4\right)}\left(s\right)+{\varkappa }^{\left(2\right)}\left(s\right)+q\left(s\right)f\left(\varkappa \left(\sigma \left(s\right)\right)\right)=0,$$

where $f\left(\varkappa \right)/{\varkappa }^{\delta }\left(s\right)\ge k$ for $\varkappa \ne 0.$ They stipulated in their studies that the differential equation

$$z^{\left(2\right)}\left(s\right)+p\left(s\right)z\left(s\right)=0$$

is nonoscillatiory or oscillatory.

Elabbasy et al. [17] studied the fourth-order delay differential equation with the middle term

$${\left(r\left(s\right){\varkappa }^{\prime \prime \prime }\left(s\right)\right)}^{\prime }+p\left(s\right){\varkappa }^{\prime \prime \prime }\left(s\right)+q\left(s\right)\varkappa \left(\sigma \left(s\right)\right)=0,$$

(3)

where

$$\underset{s\to \infty }{lim}{\int }_{s_0}^s\frac{1}{r\left(\varrho \right)}exp\left(-{\int }_{s_0}^{\varrho }\frac{p\left(\varsigma \right)}{r\left(\varsigma \right)}\mathrm{d}\varsigma \right)\mathrm{d}\varrho =\infty .$$

They established new oscillation results by using generalized Riccati transformations and comparative principles.

In [18], Dassios and Bazighifan studied the oscillatory behaviour of nonlinear differential equations

$${\left(r\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }+q\left(s\right){\varkappa }^{\gamma }\left(\sigma \left(s\right)\right)=0,$$

(4)

where $\gamma =\delta ,$ and by using the Riccati technique, they proved that the solutions to (4) are oscillatory or converge to zero as $s\to \infty .$

Yang and Bai [19] investigated the oscillation behavior of solutions of the fourth-order p-Laplacian differential equations with the middle term

$${\left(r\left(s\right){\left|y^{\prime \prime \prime }\left(s\right)\right|}^{p_1-2}{y}^{\prime \prime \prime }\left(s\right)\right)}^{\prime }+p\left(s\right){\left|y^{\prime \prime \prime }\left(s\right)\right|}^{p_1-2}{y}^{\prime \prime \prime }\left(s\right)+q\left(s\right){\left|\varkappa \left(\sigma \left(s\right)\right)\right|}^{p_2-2}\varkappa \left(\sigma \left(s\right)\right)=0,$$

(5)

where $p_i>1,$$i=1,2$. Furthermore, they investigated (5) under the condition

$$\underset{s\to \infty }{lim}{\int }_{s_0}^s{\left(\frac{1}{r\left(\varrho \right)}exp\left(-{\int }_{s_0}^{\varrho }\frac{p\left(\varsigma \right)}{r\left(\varsigma \right)}\mathrm{d}\varsigma \right)\right)}^{1/\left(p_1-1\right)}\mathrm{d}\varrho =\infty ,$$

and by using the Riccati transformations and comparison method with first-order and second-order differential equations, they proved that the solutions to (5) are oscillatory.

We know that there are many works that deal with oscillating solutions of fourth-order differential equations with damping terms, but most of them—as far as we know—are concerned only with the canonical case. We also know that most of the results obtained in the noncanonical case guarantee that the fourth-order differential equations with damping terms are oscillatory or converge to zero. In light of this, this paper is a continuation of the above recent work on the noncanonical case in which we will introduce some new conditions that guarantee the oscillation of differential equations with damping terms that use Riccati transformations. We will also show that our criteria take into account the influence of the delay argument $\tau \left(s\right)$, which has been neglected in previous studies. The criteria we obtained improved and completed some of the criteria in previous studies. We will also mention some lemmas that will help us prove the main results of this paper. Finally, we present examples that show the applicability of our results.

Lemma 1.

([20]). Assume that $f\in C^n\left(I_0,{\mathbb{R}}^{+}\right),$$f^{\left(n\right)}$ is of one sign, eventually. Then, there exists a ${\varrho }_f\in I_0$ and $\kappa \in \left[0,n\right]$ is integer, with ${\left(-1\right)}^{n+\kappa }{f}^{\left(n\right)}\left(\varrho \right)\ge 0$, such that

$$\kappa >0give{f}^{\left(k\right)}\left(\varrho \right)>0,\mathrm{for}k=0,1,...,\kappa -1$$

and

$$\kappa \le n-1give{\left(-1\right)}^{\kappa +k}{f}^{\left(k\right)}\left(\varrho \right)>0\mathrm{for}k=\kappa ,\kappa +1,...,n-1,$$

for all $\varrho \in I_f$.

Lemma 2.

([21], Lemma 2.2.3). Let $\psi \in C^n(I_0,{\mathbb{R}}^{+})$ and ${\psi }^{\left(n\right)}$ be of fixed sign and not identically zero on a subray of $I_0$. Suppose that there exists a $s_1\in I_0$ such that ${\psi }^{\left(n-1\right)}{\psi }^{\left(n\right)}\le 0$ for $s\in I_1$. If ${lim}_{s\to \infty }\psi \left(s\right)\ne 0,$ then there exists a $s_{\lambda }\in I_1$ such that

$$\psi \ge \frac{\lambda }{\left(n-1\right)!}{s}^{n-1}\left|{\psi }^{\left(n-1\right)}\right|,$$

for every $\lambda \in (0,1)$ and $s\in I_{\lambda }.$

2. Main Results

Before presenting the main results of our paper, we will start by mentioning some notations. Let $R\left(s\right)=E\left(s\right)r\left(s\right)$, where

$$E\left(s\right)=exp\left({\int }_{s_0}^s\frac{p\left(\varrho \right)}{r\left(\varrho \right)}\mathrm{d}\varrho \right).$$

We define the functions

$${\varphi }_0\left(s\right):={\int }_s^{\infty }{R}^{-1/\gamma }\left(\varrho \right)\mathrm{d}\varrho ,s\ge s_0$$

(6)

and

$${\varphi }_{\kappa }\left(s\right):={\int }_s^{\infty }{\varphi }_{\kappa -1}\left(\varrho \right)\mathrm{d}\varrho ,\mathrm{for}\kappa =1,2.$$

Now, we define class $\mathsf{{\rm Y}}$ as the category of all eventually positive solutions of (1).

Lemma 3.

Assume that $\varkappa \in \mathsf{{\rm Y}}$. Then $y\left(s\right)>0$, ${\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le 0$, and one of the following cases hold, for $s\in \left[s_1,\infty \right),$$s_1\ge s_0$:

$\left(\mathbf{A}\right)$$y^{\prime }\left(s\right),y^{\prime \prime \prime }\left(s\right)\mathrm{are}\mathrm{positive}\mathrm{and}{y}^{\left(4\right)}\left(s\right)\mathrm{is}\mathrm{negative};$

$\left(\mathbf{B}\right)$$y^{\prime }\left(s\right),$$y^{\prime \prime }\left(s\right)$ are positive and $y^{\prime \prime \prime }\left(s\right)$ is negative; and

$\left(\mathbf{C}\right)$${\left(-1\right)}^n{y}^{\left(n\right)}\left(s\right)$ are positive for all $n=1,2,3$.

Proof. 

Assume that $\varkappa \in \mathsf{{\rm Y}}$, then, there exists $s_1\ge s_0$ such that $\varkappa \left(\sigma \left(s\right)\right)>0$ and $\varkappa \left(\tau \left(s\right)\right)>0$ for all $s\ge s_1$. Hence, we see that $y\left(s\right)>0$ for $s\ge s_1$. Multiplying (1) by $E\left(s\right)$, we get

$${\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }+E\left(s\right)q\left(s\right){\varkappa }^{\delta }\left(\sigma \left(s\right)\right)=0,s\ge s_0,$$

(7)

and so ${\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le 0.$ Now, by using Lemma 1 with $n=4$, we readily get the cases (A)–(C). □

Lemma 4.

Let $\varkappa \in \mathsf{{\rm Y}}$. Assume that ${\varphi }_0\left(s\right)<\infty$ and $y^{\prime \prime \prime }\left(s\right)<0,y^{\prime \prime }\left(s\right)>0$ and $y^{\prime }<0.$ Then

$${\left(R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }-E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }{y}^{\delta }\left(s\right)\ge 0.$$

(8)

Proof. 

Assume that $\varkappa \in \mathsf{{\rm Y}}$, and then there exists a $s_1\ge s_0$, such that $\varkappa \left(\sigma \left(s\right)\right)>0$ and $\varkappa \left(\tau \left(s\right)\right)>0$ for $s\ge s_1$. Considering the fact that $y^{\prime \prime \prime }\left(s\right)<0$, and from (7) we have

$${\left(R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }=E\left(s\right)q\left(s\right){\varkappa }^{\delta }\left(\sigma \left(s\right)\right)\ge 0,$$

(9)

from (9) we have that $R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }$ is increasing and thus

$${\left(\frac{R\left(s\right)}{R\left(\varrho \right)}\right)}^{1/\gamma }{y}^{\prime \prime \prime }\left(s\right)\ge y^{\prime \prime \prime }\left(\varrho \right),\varrho \ge s\ge s_1.$$

(10)

Integrating (10) from s to $\zeta$, we get

$$y^{\prime \prime }\left(\zeta \right)-y^{\prime \prime }\left(s\right)\le R^{1/\gamma }\left(s\right)y^{\prime \prime \prime }\left(s\right){\int }_s^{\zeta }{\left(\frac{1}{R\left(\varrho \right)}\right)}^{1/\gamma }\mathrm{d}\varrho ,$$

and so

$$-y^{\prime \prime }\left(s\right)\le R^{1/\gamma }\left(s\right)\left(y^{\prime \prime \prime }\left(s\right)\right){\varphi }_0\left(s\right).$$

(11)

From (11), we obtain

$${\left(\frac{y^{\prime \prime }\left(s\right)}{{\varphi }_0\left(s\right)}\right)}^{\prime }=\frac{y^{\prime \prime \prime }\left(s\right)}{{\varphi }_0\left(s\right)}+\frac{y^{\prime \prime }\left(s\right)R^{-1/\gamma }\left(s\right)}{{\varphi }_0^2\left(s\right)}\ge 0,$$

which leads to

$$-y^{\prime }\left(s\right)\ge {\int }_s^{\infty }\frac{y^{\prime \prime }\left(\varrho \right)}{{\varphi }_0\left(\varrho \right)}{\varphi }_0\left(\varrho \right)\mathrm{d}\varrho \ge \frac{y^{\prime \prime }\left(s\right)}{{\varphi }_0\left(s\right)}{\varphi }_1\left(s\right).$$

(12)

This implies

$${\left(\frac{y^{\prime }\left(s\right)}{{\varphi }_1\left(s\right)}\right)}^{\prime }=\frac{y^{\prime \prime }\left(s\right)}{{\varphi }_1\left(s\right)}+\frac{y^{\prime }\left(s\right){\varphi }_0\left(s\right)}{{\varphi }_1^2\left(s\right)}\le 0.$$

Now

$$-y\left(s\right)\le {\int }_s^{\infty }\frac{y^{\prime }\left(\varrho \right)}{{\varphi }_1\left(\varrho \right)}{\varphi }_1\left(\varrho \right)\mathrm{d}\varrho \le \frac{y^{\prime }\left(s\right)}{{\varphi }_1\left(s\right)}{\varphi }_2\left(s\right).$$

(13)

This implies

$${\left(\frac{y\left(s\right)}{{\varphi }_2\left(s\right)}\right)}^{\prime }=\frac{y^{\prime }\left(s\right)}{{\varphi }_2\left(s\right)}+\frac{y\left(s\right){\varphi }_1\left(s\right)}{{\varphi }_2^2\left(s\right)}\ge 0.$$

(14)

From (2) and (14), we have

$$\varkappa \left(s\right)=y\left(s\right)-c\left(s\right)\varkappa \left(\tau \left(s\right)\right)\ge y\left(s\right)-c\left(s\right)y\left(\tau \left(s\right)\right)\ge y\left(s\right)\left(1-c\left(s\right)\frac{{\varphi }_2\left(\tau \left(s\right)\right)}{{\varphi }_2\left(s\right)}\right).$$

(15)

From (9) and (15), we find (8) holds. This completes the proof. □

Lemma 5.

Let $\varkappa \in \mathsf{{\rm Y}}$. Assume that ${\varphi }_0\left(s\right)<\infty$ and $y^{\prime \prime \prime }\left(s\right)<0,y^{\prime \prime }\left(s\right)>0,$$y^{\prime }<0$ and

$$\upsilon \left(s\right):=\frac{R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\delta }},s\ge s_1.$$

(16)

Then

$\left(1\right)\upsilon \left(s\right){\varphi }_0^{\mu }\left(s\right)$ is bounded;

$\left(2\right){\upsilon }^{\prime }\left(s\right)\ge E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\delta m{\upsilon }^{\left(\mu +1\right)/\mu }\left(s\right)R^{-1/\gamma }\left(s\right),s\ge S_1,$

where $m>0$ and $\mu =max\left\{\gamma ,\delta \right\}$.

Proof. 

Assume that $\varkappa \in \mathsf{{\rm Y}}$, and then there exists a $s_1\ge s_0$, such that $\varkappa \left(\tau \left(s\right)\right)>0$ and $\varkappa \left(\sigma \left(s\right)\right)>0$ for $s\ge s_1$.

$\left(1\right)$ From Lemma 4, we have $R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }$ is increasing. By using (11) and (16), we obtain

$${\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma }\ge R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }{\varphi }_0^{\gamma }\left(s\right)=\upsilon \left(s\right){\left(y^{\prime \prime }\left(s\right)\right)}^{\delta }{\varphi }_0^{\gamma }\left(s\right).$$

It follows that

$${\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma -\delta }\ge \upsilon \left(s\right){\varphi }_0^{\gamma }\left(s\right),s\ge s_1.$$

(17)

If $\gamma >\delta ,$ from $y^{\prime \prime \prime }\left(s\right)<0$ and (17), we have that $\upsilon \left(s\right){\varphi }_0^{\gamma }\left(s\right)$ is bounded.

If $\delta \ge \gamma$, using (11), we have

$${\left(y^{\prime \prime }\left(s\right)\right)}^{\delta }\ge {\left(R^{1/\gamma }\left(s\right)\left(-y^{\prime \prime \prime }\left(s\right)\right)\right)}^{\delta -\gamma +\gamma }{\varphi }_0^{\delta }\left(s\right),$$

(18)

and by using (16), we obtain

$${\left(R^{1/\gamma }\left(s\right)\left(-y^{\prime \prime \prime }\left(s\right)\right)\right)}^{\gamma -\delta }\ge \upsilon \left(s\right){\varphi }_0^{\delta }\left(s\right).$$

Because ${\left(R^{1/\gamma }\left(s\right)\left(-y^{\prime \prime \prime }\left(s\right)\right)\right)}^{\gamma -\delta }$ is decreasing, then $\upsilon \left(s\right){\varphi }_0^{\delta }\left(s\right)$ is bounded. Thus, $\upsilon \left(s\right){\varphi }_0^{\mu }\left(s\right)$ is bounded, where $\mu =max\left\{\gamma ,\delta \right\}$.

$\left(2\right)$ From (16) and (8), we find

$$\begin{array}{ccc}\hfill {\upsilon }^{\prime }\left(s\right)& =& \frac{{\left(R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\delta }}+\frac{\delta R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma +1}}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\delta +1}}\hfill \\ & \ge & E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{y^{\delta }\left(s\right)}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\delta }}+\frac{\delta {\upsilon }^{\left(\gamma +1\right)/\gamma }\left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(y^{\prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\gamma },\hfill \end{array}$$

and from (12) and (13), we get

$${\upsilon }^{\prime }\left(s\right)\ge E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\frac{\delta {\upsilon }^{\left(\gamma +1\right)/\gamma }\left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(y^{\prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\gamma }.$$

(19)

If $\gamma >\delta$, from $y^{\prime \prime \prime }\left(s\right)<0$, then ${\left(y^{\prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\gamma }$ is increasing. Letting $m_1=$${\left(y^{\prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\gamma }$ (if $\delta =\gamma$, then $m_1=1$), the Inequality (19) becomes

$${\upsilon }^{\prime }\left(s\right)\ge E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\frac{\delta m_1{\upsilon }^{\left(\gamma +1\right)/\gamma }\left(s\right)}{R^{1/\gamma }\left(s\right)},s\ge S.$$

(20)

If $\delta \ge \gamma$, we get

$${\upsilon }^{\prime }\left(s\right)\ge E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\frac{\delta {\upsilon }^{\left(\delta +1\right)/\delta }\left(s\right)}{R^{1/\delta }\left(s\right)}{\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\delta }.$$

(21)

Because ${\left(R^{-1/\delta }\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\delta }\right)}^{\prime }>0$, then from (21) we have

$$\begin{array}{ccc}\hfill {\upsilon }^{\prime }\left(s\right)& \ge & E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\frac{\delta {\upsilon }^{\left(\delta +1\right)/\delta }\left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(R^{1/\gamma }\left(s\right)\left(-y^{\prime \prime \prime }\left(s\right)\right)\right)}^{\left(\delta -\gamma \right)/\delta }\hfill \\ & \ge & E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\frac{\delta m_2{\upsilon }^{\left(\delta +1\right)/\delta }\left(s\right)}{R^{1/\gamma }\left(s\right)},s\ge S_1\ge S,\hfill \end{array}$$

(22)

where $m_2={\left(R^{1/\gamma }\left(S_1\right)\left(-y^{\prime \prime \prime }\left(S_1\right)\right)\right)}^{\left(\delta -\gamma \right)/\delta }$(if $\gamma =\delta ,$ then $m_2=1$).

From (20) and (22), we obtain

$${\upsilon }^{\prime }\left(s\right)\ge E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\frac{\delta m{\upsilon }^{\left(\mu +1\right)/\mu }\left(s\right)}{R^{1/\gamma }\left(s\right)},s\ge S_1,$$

(23)

where $\mu =max\left\{\gamma ,\delta \right\},$ and

$$m=\left\{\begin{array}{c}1,\gamma =\delta \\ conss>0,\gamma \ne \delta .\end{array}\right.$$

This completes the proof. □

Lemma 6.

Let $\varkappa \in \mathsf{{\rm Y}}$. Assume that ${\varphi }_0\left(s\right)<\infty$, $y^{\left(4\right)}\left(s\right)<0,y^{\prime \prime \prime }\left(s\right)>0,$$y^{\prime }>0$ and

$$\omega \left(s\right):=\rho \left(s\right)\frac{R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }}{y^{\delta }\left(\sigma \left(s\right)\right)},s\ge s_1.$$

Then

$${\omega }^{\prime }\left(s\right)\le \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\frac{\delta \rho \left(s\right)R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }{y}^{\prime }\left(\sigma \left(s\right)\right){\sigma }^{\prime }\left(s\right)}{y^{\delta +1}\left(\sigma \left(s\right)\right)}.$$

(24)

Proof. 

Assume that $\varkappa \in \mathsf{{\rm Y}}$, and then there exists a $s_1\ge s_0$, such that $\varkappa \left(\tau \left(s\right)\right)>0$ and $\varkappa \left(\sigma \left(s\right)\right)>0$ for $s\ge s_1$.

Because $y^{\prime }>0$ for $s\ge s_1$. Recall that $\varkappa \left(s\right)=y\left(s\right)-c\left(s\right)\varkappa \left(\tau \left(s\right)\right)$. Hence,

$$\varkappa \left(s\right)\ge \left(1-c\left(s\right)\right)y\left(s\right).$$

(25)

By using (7) and (25) we get

$${\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le -E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{y}^{\delta }\left(\sigma \left(s\right)\right).$$

(26)

From the definition of the function $\omega \left(s\right)$, we have $\omega \left(s\right)>0$ and

$${\omega }^{\prime }\left(s\right)=\frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)+\rho \left(s\right)\left(\frac{{\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }}{y^{\delta }\left(\sigma \left(s\right)\right)}-\frac{\delta R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }{y}^{\prime }\left(\sigma \left(s\right)\right){\sigma }^{\prime }\left(s\right)}{y^{\delta +1}\left(\sigma \left(s\right)\right)}\right),$$

(27)

from (26) and (27), we find (24) holds. This completes the proof. □

Lemma 7.

Let $\varkappa \in \mathsf{{\rm Y}}$. Assume that ${\varphi }_0\left(s\right)<\infty$, $y^{\prime \prime \prime }\left(s\right)<0,y^{\prime \prime }\left(s\right)>0,$$y^{\prime }>0$ and

$$\varpi \left(s\right):=\vartheta \left(s\right)\left(\frac{R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma }}+\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right).$$

(28)

Then

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & \frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }{\left(y^{\prime \prime }\left(\sigma \left(s\right)\right)\right)}^{\delta -\gamma }\hfill \\ & & -\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right)}^{\left(\gamma +1\right)/\gamma }+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}.\hfill \end{array}$$

(29)

Proof. 

Assume that $\varkappa \in \mathsf{{\rm Y}}$, and then there exists a $s_1\ge s_0$, such that $\varkappa \left(\tau \left(s\right)\right)>0$ and $\varkappa \left(\sigma \left(s\right)\right)>0$ for $s\ge s_1$.

Because $R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }$ is nonincreasing, then

$$y^{\prime \prime }\left(\zeta \right)-y^{\prime \prime }\left(s\right)={\int }_s^{\zeta }\frac{1}{R^{1/\gamma }\left(\varrho \right)}{\left(R\left(\varrho \right){\left(y^{\prime \prime \prime }\left(\varrho \right)\right)}^{\gamma }\right)}^{1/\gamma }\mathrm{d}\varrho \le {\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{1/\gamma }{\int }_s^{\zeta }\frac{1}{R^{1/\gamma }\left(\varrho \right)}\mathrm{d}\varrho .$$

Letting $\zeta \to \infty$, we get

$$-y^{\prime \prime }\left(s\right)\le R^{1/\gamma }\left(s\right)\left(y^{\prime \prime \prime }\left(s\right)\right){\varphi }_0\left(s\right),$$

(30)

from (30), we have $\varpi >0,$ for $s\ge s_1.$ Therefore, we have

$${\varpi }^{\prime }\left(s\right)=\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)+\vartheta \left(s\right)\left(\frac{{\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma }}-\frac{\gamma R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma +1}}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma +1}}+\frac{\gamma R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}\right),$$

which, from (7), it follows that

$${\varpi }^{\prime }\left(s\right)=\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\frac{\vartheta \left(s\right)E\left(s\right)q\left(s\right){\varkappa }^{\delta }\left(\sigma \left(s\right)\right)}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma }}-\frac{\gamma \vartheta \left(s\right)R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma +1}}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma +1}}+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}.$$

By using (25) and (28), we have

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & -\frac{\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{y}^{\delta }\left(\sigma \left(s\right)\right)}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma }}-\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right)}^{\left(\gamma +1\right)/\gamma }\hfill \\ & & +\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}},\hfill \end{array}$$

(31)

which can be written as

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & \frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\frac{\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{y}^{\delta }\left(\sigma \left(s\right)\right)}{{\left(y^{\prime \prime }\left(\sigma \left(s\right)\right)\right)}^{\delta }}\frac{{\left(y^{\prime \prime }\left(\sigma \left(s\right)\right)\right)}^{\gamma }}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma }}{\left(y^{\prime \prime }\left(\sigma \left(s\right)\right)\right)}^{\delta -\gamma }\hfill \\ & & -\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right)}^{\left(\gamma +1\right)/\gamma }+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}.\hfill \end{array}$$

(32)

From Lemma 2, we get

$$y\left(s\right)\ge \frac{\lambda s^2}{2!}{y}^{\prime \prime }\left(s\right),$$

(33)

and hence (32), becomes

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & \frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }\frac{{\left(y^{\prime \prime }\left(\sigma \left(s\right)\right)\right)}^{\gamma }}{{\left(y^{\prime \prime }\left(s\right)\right)}^{\gamma }}{\left(y^{\prime \prime }\left(\sigma \left(s\right)\right)\right)}^{\delta -\gamma }\hfill \\ & & -\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right)}^{\left(\gamma +1\right)/\gamma }+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}.\hfill \end{array}$$

Thus, we find that (29) holds. This completes the proof. □

Theorem 1.

Assume that $\left(H_1\right)-\left(H_4\right)$ hold, $\gamma \ge 1,{\varphi }_0\left(s\right)<\infty$ and $c\left(s\right)<{\varphi }_2\left(s\right)/{\varphi }_2\left(\tau \left(s\right)\right).$ If there exists a positive nondecreasing function $\rho \in C^1\left(\left[s_0,\infty \right),\left(0,\infty \right)\right)$, such that

$$\underset{s\to \infty }{limsup}{\int }_S^s\left({\varphi }_0^{\mu }\left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(\varrho \right)\right)\right)}{{\varphi }_2\left(\sigma \left(\varrho \right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(\varrho \right)}{{\varphi }_0^{\delta }\left(\varrho \right)}-\frac{w}{{\varphi }_0\left(\varrho \right)R^{1/\gamma }\left(\varrho \right)}\right)\mathrm{d}\varrho =\infty ,$$

(34)

$$\underset{s\to \infty }{limsup}{\int }_S^s\left(\rho \left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }-\frac{{\left({\rho }^{\prime }\left(\varrho \right)\right)}^{\nu +1}R\left(\theta \left(\varrho \right)\right)}{{\left(\nu +1\right)}^{\nu +1}{\rho }^{\nu }\left(\varrho \right){\left({\sigma }^{\prime }\left(\varrho \right)K\right)}^{\nu }}{\left(\frac{2!}{\lambda {\sigma }^2\left(\varrho \right)}\right)}^{\nu }\right)\mathrm{d}\varrho =\infty$$

(35)

and

$$\underset{s\to \infty }{limsup}{\int }_{s_1}^s\left(\mathsf{\Theta }\left(\varrho \right)-\frac{R\left(\varrho \right)\vartheta \left(\varrho \right)}{{\left(\gamma +1\right)}^{\gamma +1}}{\left(\frac{{\vartheta }^{\prime }\left(\varrho \right)}{\vartheta \left(\varrho \right)}+\frac{(1+\gamma )}{R^{1/\gamma }\left(\varrho \right){\varphi }_0\left(\varrho \right)}\right)}^{\gamma +1}\right)\mathrm{d}\varrho =\infty$$

(36)

hold, where $K>0,\mu =max\left\{\gamma ,\delta \right\},\nu =min\left\{\gamma ,\delta \right\},$ and

$$\mathsf{\Omega }\left(s\right)=\left\{\begin{array}{c}{M}_2^{\delta -\gamma },\gamma >\delta ,\\ M_1^{\left(\delta -\gamma \right)/\gamma }{\varphi }_0^{\delta -\gamma }\left(s\right),\gamma \le \delta ,\end{array}\right.\theta \left(s\right)=\left\{\begin{array}{c}s,\gamma >\delta ,\\ \sigma \left(s\right),\gamma \le \delta ,\end{array}\right.$$

$$w=\left\{\begin{array}{c}{\left(\mu /\mu +1\right)}^{\mu +1}{\left(\mu /\delta m\right)}^{\mu },\gamma \ne \delta ,\\ {\left(\gamma /\gamma +1\right)}^{\gamma +1},\gamma =\delta \end{array}\right.m=\left\{\begin{array}{c}1,\gamma =\delta ,\\ conss>0,\gamma \ne \delta ,\end{array}\right.$$

and

$$\mathsf{\Theta }\left(s\right)=\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(s\right)-\left(\gamma -1\right)\frac{\vartheta \left(s\right)}{R^{1/\gamma }\left(s\right){\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}},$$

then (1) is oscillatory.

Proof. 

Assume that $\varkappa \in \mathsf{{\rm Y}}$, and then there exists a $s_1\ge s_0$, such that $\varkappa \left(\tau \left(s\right)\right)>0$ and $\varkappa \left(\sigma \left(s\right)\right)>0$ for $s\ge s_1$. From $y\left(s\right)\ge \varkappa \left(s\right)>0$ for $s\ge s_1$ and ( 7), we have

$${\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }=-E\left(s\right)q\left(s\right){\varkappa }^{\delta }\left(\sigma \left(s\right)\right)\le 0,$$

and hence $R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }$ is nonincreasing. Therefore, there exists a $s_2\ge s_1$, such that the cases $\left(\mathbf{A}\right)-\left(\mathbf{C}\right)$ hold, for all $s\ge s_2$.

Assume that case $\left(\mathbf{C}\right)$ holds. From Lemma 4, we get

$${\left(R\left(s\right){\left(-y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }-E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }{y}^{\delta }\left(s\right)\ge 0.$$

From (16), we have that $\upsilon \left(s\right)>0$ for all $s\ge s_2$. From Lemma 5, we have

$${\upsilon }^{\prime }\left(s\right)\ge E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}+\delta m{\upsilon }^{\left(\mu +1\right)/\mu }\left(s\right)R^{-1/\gamma }\left(s\right),\mathrm{for}\mathrm{all}s\ge s_2.$$

(37)

By taking the integral from $S\ge s_2$ to s after multiplying the Inequality (37) by ${\varphi }_0^{\mu }\left(s\right),$ we get

$$\begin{array}{ccc}& & {\int }_S^s{\varphi }_0^{\mu }\left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(\varrho \right)\right)\right)}{{\varphi }_2\left(\sigma \left(\varrho \right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(\varrho \right)}{{\varphi }_0^{\delta }\left(\varrho \right)}\mathrm{d}\varrho \hfill \\ & & \le {\int }_S^s{\varphi }_0^{\mu }\left(\varrho \right){\upsilon }^{\prime }\left(\varrho \right)\mathrm{d}\varrho -{\int }_S^s{\varphi }_0^{\mu }\left(\varrho \right)\delta m{\upsilon }^{\left(\mu +1\right)/\mu }\left(\varrho \right)R^{-1/\gamma }\left(\varrho \right)\mathrm{d}\varrho \hfill \\ & & \le {\int }_S^s{\varphi }_0^{\mu -1}\left(\varrho \right)R^{-1/\gamma }\left(\varrho \right)\left[\mu \upsilon \left(\varrho \right)-{\varphi }_0\left(\varrho \right)\delta m{\upsilon }^{\left(\mu +1\right)/\mu }\left(\varrho \right)\right]\mathrm{d}\varrho +{\varphi }_0^{\mu }\left(s\right)\upsilon \left(s\right).\hfill \end{array}$$

(38)

By using the inequality

$$D\upsilon -C{\upsilon }^{\left(\gamma +1\right)/\gamma }\le \frac{{\gamma }^{\gamma }}{{\left(\gamma +1\right)}^{\gamma +1}}\frac{D^{\gamma +1}}{C^{\gamma }},C>0,$$

(39)

we get

$$\begin{array}{ccc}& & {\int }_S^s{\varphi }_0^{\mu }\left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(\varrho \right)\right)\right)}{{\varphi }_2\left(\sigma \left(\varrho \right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(\varrho \right)}{{\varphi }_0^{\delta }\left(\varrho \right)}\mathrm{d}\varrho \hfill \\ & & \le {\int }_S^s{\varphi }_0^{\mu -1}\left(\varrho \right)R^{-1/\gamma }\left(\varrho \right)\left[\mu \upsilon \left(\varrho \right)-{\varphi }_0\left(\varrho \right)\delta m{\upsilon }^{\left(\mu +1\right)/\mu }\left(\varrho \right)\right]\mathrm{d}\varrho +{\varphi }_0^{\mu }\left(s\right)\upsilon \left(s\right)\hfill \\ & & \le {\int }_S^s\frac{{\mu }^{\mu }}{{\left(\mu +1\right)}^{\mu +1}}\frac{{\mu }^{\mu +1}}{{\left(\delta m\right)}^{\mu }}\frac{1}{{\varphi }_0\left(\varrho \right)R^{1/\gamma }\left(\varrho \right)}\mathrm{d}\varrho +{\varphi }_0^{\mu }\left(s\right)\upsilon \left(s\right)\hfill \\ & & \le {\int }_S^s\frac{w}{{\varphi }_0\left(\varrho \right)R^{1/\gamma }\left(\varrho \right)}\mathrm{d}\varrho +{\varphi }_0^{\mu }\left(s\right)\upsilon \left(s\right),\hfill \end{array}$$

and so

$${\int }_S^s\left({\varphi }_0^{\mu }\left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(\varrho \right)\right)\right)}{{\varphi }_2\left(\sigma \left(\varrho \right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(\varrho \right)}{{\varphi }_0^{\delta }\left(\varrho \right)}-\frac{w}{{\varphi }_0\left(\varrho \right)R^{1/\gamma }\left(\varrho \right)}\right)\mathrm{d}\varrho \le {\varphi }_0^{\mu }\left(s\right)\upsilon \left(s\right),$$

(40)

where

$$w=\left\{\begin{array}{c}{\left(\mu /\mu +1\right)}^{\mu +1}{\left(\mu /\delta m\right)}^{\mu },\gamma \ne \delta ,\\ {\left(\gamma /\gamma +1\right)}^{\gamma +1},\gamma =\delta \end{array}\right.$$

because ${\varphi }_0^{\mu }\left(s\right)\upsilon \left(s\right)$ is bounded. Letting $s\to \infty$ in (40), we obtain a contradiction with (34).

Assume that case $\left(\mathbf{A}\right)$ holds. From Lemma 6, we have that (24) holds.

If $\gamma <\delta$. Note that $R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }$ is a positive nonincreasing, and then

$$R^{1/\gamma }\left(s\right)y^{\prime \prime \prime }\left(s\right)\le R^{1/\gamma }\left(\sigma \left(s\right)\right)y^{\prime \prime \prime }\left(\sigma \left(s\right)\right).$$

(41)

By using Lemma 2 with $\psi =y^{\prime },$ we obtain

$$y^{\prime }\left(s\right)\ge \frac{\lambda s^2}{2!}{y}^{\prime \prime \prime }\left(s\right),$$

(42)

by using (41) and (42) in (24), we get

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(s\right)& \le & \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\frac{\delta \rho \left(s\right)R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }{y}^{\prime }\left(\sigma \left(s\right)\right){\sigma }^{\prime }\left(s\right)}{y^{\delta +1}\left(\sigma \left(s\right)\right)}\hfill \\ & \le & \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\delta {\sigma }^{\prime }\left(s\right)\frac{y^{\left(\delta /\gamma \right)-1}\left(\sigma \left(s\right)\right)}{{\rho }^{1/\gamma }\left(s\right)R^{1/\gamma }\left(s\right)}\frac{y^{\prime }\left(\sigma \left(s\right)\right)}{y^{\prime \prime \prime }\left(s\right)}{\omega }^{\left(\gamma +1\right)/\gamma }\left(s\right)\hfill \\ & \le & \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\delta {\sigma }^{\prime }\left(s\right)\frac{y^{\left(\delta /\gamma \right)-1}\left(\sigma \left(s\right)\right)}{{\rho }^{1/\gamma }\left(s\right)R^{1/\gamma }\left(\sigma \left(s\right)\right)}\frac{\lambda {\sigma }^2\left(s\right)}{2!}{\omega }^{\left(\gamma +1\right)/\gamma }\left(s\right).\hfill \end{array}$$

Because $y^{\prime }>0$, thus there exists a constant $K_1>0$ and $s_2\ge s_1$, such that

$$y^{\left(\delta /\gamma \right)-1}\left(\sigma \left(s\right)\right)\ge K_1,s\ge s_2.$$

Hence, we obtain

$${\omega }^{\prime }\left(s\right)\le \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\frac{\gamma {\sigma }^{\prime }\left(s\right)K_1\lambda {\sigma }^2\left(s\right)}{2!{\rho }^{1/\gamma }\left(s\right)R^{1/\gamma }\left(\sigma \left(s\right)\right)}{\omega }^{\left(\gamma +1\right)/\gamma }\left(s\right).$$

(43)

If $\gamma =\delta$, then $K_1=1$; thus, (43) still holds.

If $\gamma >\delta$, because $r^{\prime }\left(s\right)\ge 0$, we have $R^{\prime }\left(s\right)\ge 0$. Because ${\left(R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\right)}^{\prime }\le 0$, hence $y^{\left(4\right)}\le 0$, therefore ${\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\delta }$ is nondecreasing. Thus, there exist constant $K_2>0$, $s_3\ge s_2$, such that

$${\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\delta }\ge K_2,s\ge s_3.$$

(44)

By combining (24) and (44), we then obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(s\right)& \le & \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\delta {\sigma }^{\prime }\left(s\right)\frac{{\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\left(\delta -\gamma \right)/\delta }}{{\rho }^{1/\delta }\left(s\right)R^{1/\delta }\left(s\right)}\frac{\lambda {\sigma }^2\left(s\right)}{2!}{\omega }^{\left(\delta +1\right)/\delta }\left(s\right)\hfill \\ & \le & \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\frac{\delta {\sigma }^{\prime }\left(s\right)K_2\lambda {\sigma }^2\left(s\right)}{2!{\rho }^{1/\delta }\left(s\right)R^{1/\delta }\left(s\right)}{\omega }^{\left(\delta +1\right)/\delta }\left(s\right),\hfill \end{array}$$

which, together with (43), implies that

$${\omega }^{\prime }\left(s\right)\le \frac{{\rho }^{\prime }\left(s\right)}{\rho \left(s\right)}\omega \left(s\right)-\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }-\frac{\nu {\sigma }^{\prime }\left(s\right)K\lambda {\sigma }^2\left(s\right)}{2!{\rho }^{1/\nu }\left(s\right)R^{1/\nu }\left(\theta \left(s\right)\right)}{\omega }^{\left(\nu +1\right)/\nu }\left(s\right),$$

(45)

where $K=min\left\{K_1,K_2\right\},\nu =min\left\{\gamma ,\delta \right\}$ and

$$\theta \left(s\right)=\left\{\begin{array}{c}s,\gamma >\delta ,\\ \sigma \left(s\right),\gamma \le \delta .\end{array}\right.$$

By using (39) in (45), we have

$${\omega }^{\prime }\left(s\right)\le -\rho \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }+\frac{{\left({\rho }^{\prime }\left(s\right)\right)}^{\nu +1}R\left(\theta \left(s\right)\right)}{{\left(\nu +1\right)}^{\nu +1}{\rho }^{\nu }\left(s\right){\left({\sigma }^{\prime }\left(s\right)K\right)}^{\nu }}{\left(\frac{2!}{\lambda {\sigma }^2\left(s\right)}\right)}^{\nu }.$$

Integrating this inequality from $S\ge s_3$ to s, we get

$$\omega \left(s\right)\le \omega \left(S\right)-{\int }_S^s\left(\rho \left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }-\frac{{\left({\rho }^{\prime }\left(\varrho \right)\right)}^{\nu +1}R\left(\theta \left(\varrho \right)\right)}{{\left(\nu +1\right)}^{\nu +1}{\rho }^{\nu }\left(\varrho \right){\left({\sigma }^{\prime }\left(\varrho \right)K\right)}^{\nu }}{\left(\frac{2!}{\lambda {\sigma }^2\left(\varrho \right)}\right)}^{\nu }\right)\mathrm{d}\varrho .$$

(46)

Letting $s\to \infty$ in (46), we get a contradiction with (35).

Assume that case $\left(\mathbf{B}\right)$ holds. From Lemma 7, we see that (29) holds.

If $\gamma <\delta$. Note that $R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }$ is a nonincreasing, then

$$R\left(s\right){\left(y^{\prime \prime \prime }\left(s\right)\right)}^{\gamma }\le R\left(s_1\right){\left(y^{\prime \prime \prime }\left(s_1\right)\right)}^{\gamma }=-M_1<0,\mathrm{for}s\ge s_1,$$

which is

$$y^{\prime \prime \prime }\left(s\right)\le -\frac{M_1^{1/\gamma }}{R^{1/\gamma }\left(s\right)},$$

integrating from s to ∞, we obtain

$$0\le y^{\prime \prime }\left(s\right)-{\int }_s^{\infty }\frac{M_1^{1/\gamma }}{R^{1/\gamma }\left(\varrho \right)}\mathrm{d}\varrho .$$

By using (6), we get

$$y^{\prime \prime }\left(s\right)\ge M_1^{1/\gamma }{\varphi }_0\left(s\right).$$

Thus, we find

$${\left(y^{\prime \prime }\left(s\right)\right)}^{\delta -\gamma }\ge M_1^{\left(\delta -\gamma \right)/\gamma }{\varphi }_0^{\delta -\gamma }\left(s\right).$$

(47)

By using (47) in (24), we obtain

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & \frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }{M}_1^{\left(\delta -\gamma \right)/\gamma }{\varphi }_0^{\delta -\gamma }\left(s\right)\hfill \\ & & -\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right)}^{\left(\gamma +1\right)/\gamma }+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}.\hfill \end{array}$$

(48)

If $\gamma =\delta$, then $M_1^{\left(\delta -\gamma \right)/\gamma }{\varphi }_0^{\delta -\gamma }\left(s\right)=1$; thus, (48) still holds.

If $\gamma >\delta$, because $y^{\prime \prime }\left(s\right)$ is nonincreasing, there exists a $M_2>0$, such that

$$y^{\prime \prime }\le M_2$$

and so

$${\left(y^{\prime \prime }\left(s\right)\right)}^{\delta -\gamma }\ge M_2^{\delta -\gamma }.$$

(49)

By combining (29) and (49), we then have

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & \frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }{M}_2^{\delta -\gamma }\hfill \\ & & -\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right)}^{\left(\gamma +1\right)/\gamma }+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}},\hfill \end{array}$$

(50)

which, together with (48), implies that

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & \frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(s\right)\hfill \\ & & -\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}{\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}\right)}^{\left(\gamma +1\right)/\gamma }+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}},\hfill \end{array}$$

(51)

where

$$\mathsf{\Omega }\left(s\right)=\left\{\begin{array}{c}{M}_2^{\delta -\gamma },\gamma >\delta ,\\ M_1^{\left(\delta -\gamma \right)/\gamma }{\varphi }_0^{\delta -\gamma }\left(s\right),\gamma \le \delta .\end{array}\right.$$

By using the inequality

$$A^{(\gamma +1)/\gamma }-{(A-B)}^{(\gamma +1)/\gamma }\le \frac{B^{1/\gamma }}{\gamma }[(1+\gamma )A-B],AB\ge 0,$$

we find that

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & \frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}\varpi \left(s\right)-\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(s\right)+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}\hfill \\ & & -\frac{\gamma \vartheta \left(s\right)}{R^{1/\gamma }\left(s\right)}\left({\left(\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}\right)}^{(\gamma +1)/\gamma }-\frac{1}{\gamma {\varphi }_0\left(s\right)}[(1+\gamma )\frac{\varpi \left(s\right)}{\vartheta \left(s\right)}-\frac{1}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma }}]\right),\hfill \end{array}$$

which is

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & -\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(s\right)+\frac{\gamma \vartheta \left(s\right)R^{-1/\gamma }\left(s\right)}{{\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}\hfill \\ & & +\left(\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}+\frac{(1+\gamma )}{R^{1/\gamma }\left(s\right){\varphi }_0\left(s\right)}\right)\varpi \left(s\right)-\gamma \frac{{\varpi }^{(\gamma +1)/\gamma }\left(s\right)}{R^{1/\gamma }\left(s\right){\vartheta }^{1/\gamma }\left(s\right)}-\frac{\vartheta \left(s\right)}{R^{1/\gamma }\left(s\right){\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}.\hfill \end{array}$$

(52)

By using (39) and (52), we obtain

$$\begin{array}{ccc}\hfill {\varpi }^{\prime }\left(s\right)& \le & -\vartheta \left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(s\right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(s\right)+\left(\gamma -1\right)\frac{\vartheta \left(s\right)}{R^{1/\gamma }\left(s\right){\left({\varphi }_0\left(s\right)\right)}^{\gamma +1}}\hfill \\ & & +\frac{R\left(s\right)\vartheta \left(s\right)}{{\left(\gamma +1\right)}^{\gamma +1}}{\left(\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}+\frac{(1+\gamma )}{R^{1/\gamma }\left(s\right){\varphi }_0\left(s\right)}\right)}^{\gamma +1}.\hfill \end{array}$$

(53)

By integrating (53) from $s_1$ to s, we get

$$\begin{array}{ccc}\hfill \varpi \left(s_1\right)& \ge & {\int }_{s_1}^s\left(\vartheta \left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(\varrho \right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(\varrho \right)-\left(\gamma -1\right)\frac{\vartheta \left(\varrho \right)}{R^{1/\gamma }\left(\varrho \right){\left({\varphi }_0\left(\varrho \right)\right)}^{\gamma +1}}\right)\mathrm{d}\varrho \hfill \\ & & -{\int }_{s_1}^s\frac{R\left(\varrho \right)\vartheta \left(\varrho \right)}{{\left(\gamma +1\right)}^{\gamma +1}}{\left(\frac{{\vartheta }^{\prime }\left(\varrho \right)}{\vartheta \left(\varrho \right)}+\frac{(1+\gamma )}{R^{1/\gamma }\left(\varrho \right){\varphi }_0\left(\varrho \right)}\right)}^{\gamma +1}\mathrm{d}\varrho ,\hfill \end{array}$$

which is

$$\varpi \left(s_1\right)\ge {\int }_{s_1}^s\left(\mathsf{\Theta }\left(\varrho \right)-\frac{R\left(\varrho \right)\vartheta \left(\varrho \right)}{{\left(\gamma +1\right)}^{\gamma +1}}{\left(\frac{{\vartheta }^{\prime }\left(\varrho \right)}{\vartheta \left(\varrho \right)}+\frac{(1+\gamma )}{R^{1/\gamma }\left(\varrho \right){\varphi }_0\left(\varrho \right)}\right)}^{\gamma +1}\right)\mathrm{d}\varrho ,$$

which contradicts (36). This completes the proof. □

Theorem 2.

Theorem 1 still holds if conditions (34)–(36) are replaced by

$$\underset{s\to \infty }{liminf}{\varphi }_0^{\mu +1}\left(s\right)R^{1/\gamma }\left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}>w,$$

(54)

$$\underset{s\to \infty }{limsup}{\int }_S^{s}E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }\mathrm{d}\varrho =\infty$$

(55)

and

$$\underset{s\to \infty }{limsup}{\int }_{s_1}^s\left(E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(\varrho \right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(\varrho \right)-\frac{\gamma }{R^{1/\gamma }\left(\varrho \right){\varphi }_0^{\gamma +1}\left(\varrho \right)}\right)\mathrm{d}\varrho =\infty ,$$

(56)

respectively.

Proof. 

Suppose that (54) holds. Then there exists a sufficiently large $S\ge s_0$, for any $\epsilon >0$, such that

$${\varphi }_0^{\mu }\left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}>\frac{w-\epsilon }{{\varphi }_0\left(s\right)R^{1/\gamma }\left(s\right)},s\ge S.$$

(57)

By integrating (57) from S to s, we have

$$\begin{array}{ccc}& & {\int }_S^s\left({\varphi }_0^{\mu }\left(\varrho \right)E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(\varrho \right)\right)\right)}{{\varphi }_2\left(\sigma \left(\varrho \right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(\varrho \right)}{{\varphi }_0^{\delta }\left(\varrho \right)}-\frac{w}{{\varphi }_0\left(\varrho \right)R^{1/\gamma }\left(\varrho \right)}\right)\mathrm{d}\varrho \hfill \\ & & >-{\int }_S^s\frac{\epsilon }{{\varphi }_0\left(\varrho \right)R^{1/\gamma }\left(\varrho \right)}\mathrm{d}\varrho \hfill \\ & & =\epsilon \left(ln\frac{1}{{\varphi }_0\left(s\right)}-ln\frac{1}{{\varphi }_0\left(S\right)}\right).\hfill \end{array}$$

(58)

Letting $s\to \infty$ in (58), we have that (34) holds.

Moreover, condition (55) can be obtained directly by $\rho \left(s\right)=1$ into (35).

Now, condition (56) can be obtained directly by substituting $\vartheta \left(s\right)=1$ into (36).

This completes the proof. □

Now, we present some examples to illustrate the possibility of applying the results that we obtained. First, we present a special case, which is when $p\left(s\right)=0$.

Example 1.

Consider the differential equation

$${\left(s^4{\left(\varkappa \left(s\right)+c_0\varkappa \left({\lambda }_{1}s\right)\right)}^{\prime \prime \prime }\right)}^{\prime }+q_0\varkappa \left({\lambda }_{2}s\right)=0,$$

(59)

where $\gamma =\delta =\mu =1,r\left(s\right)=s^4,c\left(s\right)=c_0,p\left(s\right)=0,q\left(s\right)=q_0,\tau \left(s\right)={\lambda }_{1}s,c_0<{\lambda }_1,\sigma \left(s\right)={\lambda }_{2}s,$ and ${\lambda }_1,{\lambda }_2\in \left(0,1\right).$ It is easy to verify that ${\varphi }_0\left(s\right)=1/3s^3,{\varphi }_1\left(s\right)=1/6s^2$ and ${\varphi }_2\left(s\right)=1/6s.$ Moreover, we see that

$$\underset{s\to \infty }{limsup}{\int }_S^{s}E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }\mathrm{d}\varrho =q_0\left(1-c_0\right)\underset{s\to \infty }{limsup}{\int }_S^s\mathrm{d}\varrho =\infty ,$$

and then condition (55) holds.

Now, the condition (54) becomes

$$\begin{array}{ccc}& & \underset{s\to \infty }{liminf}{\varphi }_0^{\mu +1}\left(s\right)R^{1/\gamma }\left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}\hfill \\ & =& \underset{s\to \infty }{liminf}{\left(\frac{1}{3s^3}\right)}^2{s}^4{q}_0\left(1-c_0\frac{1}{{\lambda }_1}\right)\frac{s^2}{2}=\underset{s\to \infty }{liminf}\frac{q_0}{18}\left(1-c_0\frac{1}{{\lambda }_1}\right),\hfill \end{array}$$

so it is verified, if

$$q_0>\frac{18}{4\left(1-c_0\frac{1}{{\lambda }_1}\right)},$$

(60)

and condition (56) becomes

$$\begin{array}{ccc}& & \underset{s\to \infty }{limsup}{\int }_{s_1}^s\left(E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(\varrho \right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(\varrho \right)-\frac{\gamma }{R^{1/\gamma }\left(\varrho \right){\varphi }_0^{\gamma +1}\left(\varrho \right)}\right)\mathrm{d}\varrho \hfill \\ & =& \underset{s\to \infty }{limsup}{\int }_{s_1}^s\left(q_0\left(1-c_0\right)\frac{{\lambda }_2^2{\varrho }^2}{2!}-9{\varrho }^2\right)\mathrm{d}\varrho ,\hfill \end{array}$$

so it is verified, if

$$q_0>\frac{18}{\left(1-c_0\right){\lambda }_2^2}.$$

(61)

By using Theorem 2, we have that (59) is oscillatory if the conditions (60) and (61) verified.

Example 2.

Consider the differential equation

$${\left(\varkappa \left(s\right)+\frac{1}{2}\varkappa \left(\frac{2s}{3}\right)\right)}^{\left(4\right)}+\frac{4}{s}{\left(\varkappa \left(s\right)+\frac{1}{2}\varkappa \left(\frac{2s}{3}\right)\right)}^{\prime \prime \prime }+s^{-1}\varkappa \left(\frac{3s}{4}\right)=0,$$

(62)

where $r\left(s\right)=1,c\left(s\right)=1/2,p\left(s\right)=4/s,q\left(s\right)=s^{-1},\tau \left(s\right)=2s/3,\sigma \left(s\right)=3s/4.$ Let $s_0=1$, then we have $E\left(s\right)=s^4,R\left(s\right)=s^4,{\varphi }_0\left(s\right)=1/3s^3,{\varphi }_1\left(s\right)=1/6s^2$ and ${\varphi }_2\left(s\right)=1/6s.$

Now, the condition (54) becomes

$$\begin{array}{ccc}& & \underset{s\to \infty }{liminf}{\varphi }_0^{\mu +1}\left(s\right)R^{1/\gamma }\left(s\right)E\left(s\right)q\left(s\right){\left(1-c\left(\sigma \left(s\right)\right)\frac{{\varphi }_2\left(\tau \left(\sigma \left(s\right)\right)\right)}{{\varphi }_2\left(\sigma \left(s\right)\right)}\right)}^{\delta }\frac{{\varphi }_2^{\delta }\left(s\right)}{{\varphi }_0^{\delta }\left(s\right)}\hfill \\ & =& \underset{s\to \infty }{liminf}{\left(\frac{1}{3s^3}\right)}^{1+1}{s}^4{s}^4{s}^{-1}\left(1-\frac{1}{2}\frac{9s}{6s}\right)\frac{s^2}{2}=\infty .\hfill \end{array}$$

The condition (55) becomes

$$\underset{s\to \infty }{limsup}{\int }_S^{s}E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }\mathrm{d}\varrho =\underset{s\to \infty }{limsup}{\int }_S^s{\varrho }^4{\varrho }^{-1}\left(1-\frac{1}{2}\right)\mathrm{d}\varrho =\infty .$$

The condition (56) becomes

$$\begin{array}{ccc}& & \underset{s\to \infty }{limsup}{\int }_{s_1}^s\left(E\left(\varrho \right)q\left(\varrho \right){\left(1-c\left(\sigma \left(\varrho \right)\right)\right)}^{\delta }{\left(\frac{\lambda {\sigma }^2\left(\varrho \right)}{2!}\right)}^{\delta }\mathsf{\Omega }\left(\varrho \right)-\frac{\gamma }{R^{1/\gamma }\left(\varrho \right){\varphi }_0^{\gamma +1}\left(\varrho \right)}\right)\mathrm{d}\varrho \hfill \\ & =& \underset{s\to \infty }{limsup}{\int }_{s_1}^s\left({\varrho }^4{\varrho }^{-1}\left(1-\frac{1}{2}\right)\frac{{\left(\frac{3\varrho }{4}\right)}^2}{2}-\frac{1}{{\varrho }^4{\left(1/3{\varrho }^3\right)}^2}\right)\mathrm{d}\varrho =\infty .\hfill \end{array}$$

Hence (54)–(56) are satisfied, and by using Theorem 2, we see that (62) is oscillatory.

Remark 1.

Note that our criterion (60) essentially takes into account the influence of the delay argument $\tau \left(s\right)$, which has been neglected in [18]. Moreover, note that our criteria guarantee the oscillation of (59), whereas the criteria that were deduced in [18] guarantee that (59) is oscillatory or converges to zero. Therefore, our criteria are an improvement and complement the criteria in [18].

3. Conclusions

In this paper, very fourth-order differential equations with damping terms have been considered. There is not much work on fourth-order differential equations with damping terms in the noncanonical case. We established our results by using the Riccati transformations. We are sure that the results will give a direction of research to the researcher working in this field. Examples are also given for a better understanding of the results. The established criteria are new and can easily be extended for more classes of differential equations; it can also be used to plan future research papers. We mention, for example that

(1)

one can consider Equation (1) with

$$y\left(s\right)=\varkappa \left(s\right)+c\left(s\right){\varkappa }^{\alpha }\left(\tau \left(s\right)\right)$$

where $\alpha \ge 1$; and

(2)

it would be of interest to extend the results of this paper for higher order equations of type

$${\left(r\left(s\right){\left(y^{\left(n-1\right)}\left(s\right)\right)}^{\gamma }\right)}^{\prime }+p\left(s\right){\left(y^{\left(n-1\right)}\left(s\right)\right)}^{\gamma }+q\left(s\right){\varkappa }^{\delta }\left(\sigma \left(s\right)\right)=0,s\ge s_0,$$

where $n\ge 4$ is an even, natural number.