Third-Order Neutral Differential Equations with Damping and Distributed Delay: New Asymptotic Properties of Solutions

Abstract

In this paper, we are interested in studying the oscillation of differential equations with a damping term and distributed delay. We establish new criteria that guarantee the oscillation of the third-order differential equation in terms of oscillation of the second-order linear differential equation without a damping term. By using the Riccati transformation technique and the principle of comparison, we obtain new results on the oscillation for the studied equation. The results show significant improvement and extend the previous works. Symmetry contributes to determining the correct methods for solving neutral differential equations. Some examples are provided to show the significance of our results.

1. Introduction

In recent years, differential equations and their symmetric properties have received much attention, since they have a wide application related to the various phenomena of life. For instance, differential equations appeared in the modeling of population growth, as they were associated with various life sciences such as biology, neural networks and chemical reactions (see [1,2,3] and, for nonlinear dynamic systems [4,5,6]).

In particular, in order to develop qualitative theory and practical reasons, researchers began to study the oscillation of the first-order and second-order equations (see [7,8,9,10]). For example, oscillation properties appear in mathematical biology problems when their formulation includes cross-propagation terms [11,12].

Tiryaki et al. [13] studied the oscillatory and nonoscillatory behavior of the following ordinary differential equation:

$$\nu \left(\iota \right){\theta }^{″}\left(\iota \right)+b\left(\iota \right)\theta \left(\iota \right)=0,$$

(1)

where $b\in C\left(\left[{\iota }_0,\infty \right),\left(0,\infty \right)\right)$ and ${\int }_{{\iota }_1}^{\iota }{\nu }^{-1}\left(\xi \right)\mathrm{d}\xi =\infty .$ Meanwhile, Bohner et al. [14] investigated the oscillation criteria for the second-order linear delay differential equation

$${\theta }^{″}\left(\iota \right)-\left(\iota \right)\theta \left(\varphi \left(\iota \right)\right)=0,$$

(2)

where $\in C\left(\left[{\iota }_0,\infty \right),\left[0,\infty \right)\right),$$\varphi \in C^1\left(\left[{\iota }_0,\infty \right),\left[0,\infty \right)\right),$$\varphi \left(\iota \right)<\iota$ and ${\varphi }^{\prime }\left(\iota \right)\ge 0$ for $\iota \in [{\iota }_0,\infty )$ and ${lim}_{\iota \to \infty }$ $\varphi \left(\iota \right)=\infty$, and they concluded that Equation (2) is oscillatory if

$$\underset{\iota \to \infty }{lim\; sup}{\int }_{\varphi \left(\iota \right)}^{\iota }\left(\xi \right)\mathrm{d}\xi >1$$

(3)

or

$$\underset{\iota \to \infty }{lim\; sup}{\int }_{\varphi \left(\iota \right)}^{\iota }\left({\int }_u^{\iota }\left(\xi \right)\mathrm{d}\xi \right)\mathrm{d}u>1,$$

(4)

where

$$\tilde{V}\left(\iota ,{\iota }_1\right)={\int }_{{\iota }_1}^{\iota }\frac{1}{\nu \left(\xi \right)}\mathrm{d}\xi .$$

(5)

Within a not so long time, many research activities have emerged which are concerned with studying the oscillatory and asymptotic properties of third-order neutral differential equations, where some results can be followed up on in [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. For the equations on time scales, see [30,31,32] and the references therein. However, there is a very limited number of results that deal with the oscillation and asymptotic behavior of third-order neutral differential equations that contain damping terms in the previous literature.

Graef [33] discussed the oscillatory behavior of the differential equation

$${\left({\nu }_1\left(\iota \right){\left({\nu }_2\left(\iota \right){\left(\psi \left(\iota \right)+\rho y\left(\iota -\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right)}^{″}+\gamma \left(\iota \right)ϝ\left(\Omega \left(\varrho \left(\iota -\varrho \right)\right)\right)=0,$$

where $0\le \rho <1$ and ${\int }_{{\iota }_0}^{\infty }{\nu }_1^{-1}\left(\iota \right)\mathrm{d}\iota ={\int }_{{\iota }_0}^{\infty }{\nu }_2^{-1}\left(\iota \right)\mathrm{d}\iota =\infty .$ Later, Chatzarakis [7] improved and extended the results.

Zhang et al. [34] established some oscillation criteria (Philos-type) equations:

$${\left(\nu \left(\iota \right){\left(\psi \left(\iota \right)+{\int }_c^d\rho \left(\iota ,\zeta \right)\psi \left(\varsigma \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta \right)}^{″}\right)}^{\prime }+{\int }_c^d\gamma \left(\iota ,\zeta \right)ϝ\left(\Omega \left(\varrho \left(\iota ,\zeta \right)\right)\right)\mathrm{d}\zeta =0,$$

(6)

where ${\int }_{{\iota }_0}^{\infty }{\nu }^{-1}\left(\iota \right)\mathrm{d}\iota =\infty$ and

$$0\le \rho \equiv {\int }_c^d\gamma \left(\iota ,\zeta \right)\mathrm{d}\zeta \le \rho <1.$$

The results for Equation (6) were completed in [35,36,37] by using both the Riccati transformation technique and integral averaging technique.

The third-order damped neutral differential equation

$${\left(\nu \left(\iota \right){\left({\psi }^{\prime }\left(\iota \right)\right)}^{\alpha }\right)}^{″}+\eta \left(\iota \right){\left({\psi }^{\prime }\left(\iota \right)\right)}^{\alpha }+\gamma \left(\iota \right)ϝ\left(\Omega \left(\varrho \left(\iota \right)\right)\right)=0$$

(7)

and its special cases were studied by [7,13,14,38,39,40] with both conditions of ${\int }_{{\iota }_0}^{\infty }{\nu }^{-1}\left(\iota \right)\mathrm{d}\iota <\infty$ and ${\int }_{{\iota }_0}^{\infty }{\nu }^{-1}\left(\iota \right)\mathrm{d}\iota =\infty .$

In this paper, we focus exclusively on the oscillation behavior of a third-order neutral differential equation with a damping term and distributed delay of the form

$${\left(\nu \left(\iota \right){\left({\psi }^{\prime }\left(\iota \right)\right)}^{\alpha }\right)}^{″}+\eta \left(\iota \right){\left({\psi }^{\prime }\left(\iota \right)\right)}^{\alpha }+{\int }_c^d\gamma \left(\iota ,\zeta \right)ϝ\left(\Omega \left(\varrho \left(\iota ,\zeta \right)\right)\right)\mathrm{d}\zeta =0,$$

(8)

where $\psi \left(\iota \right)=\Omega \left(\iota \right)+\rho \left(\iota \right)\Omega \left(\varsigma \left(\iota \right)\right),$ $\iota \ge {\iota }_0>0$ and $\alpha$ is a quotient of odd positive integers. Throughout this paper, we assume some hypotheses as follows:

(s₁)

$\nu ,$ $\eta \in C\left(\left[{\iota }_0,\infty \right),\left(0,\infty \right)\right)$:

$${\int }_{\iota }^{\infty }{\left(\frac{1}{\nu \left(\iota \right)}exp\left(-{\int }_{{\iota }_0}^{\iota }\frac{\rho \left(\xi \right)}{\nu \left(\xi \right)}\right)\mathrm{d}\xi \right)}^{1/\alpha }\mathrm{d}\iota =\infty ;$$

(s₂)

$\rho \in C\left(\left[{\iota }_0,\infty \right),\left[0,\infty \right)\right),$$\gamma \in C\left(\left[{\iota }_0,\infty \right)\times \left[c,d\right],\left[0,\infty \right)\right)$ and $\rho \left(\iota \right)\le {\rho }_0<\infty$;

(s₃)

$\varrho ,\varsigma \in C^1\left(\left[{\iota }_0,\infty \right),\mathbb{R}\right),$$\varrho \left(\iota ,\zeta \right)<\iota ,$$\varsigma \left(\iota \right)<\iota ,$${\varsigma }^{\prime }\ge {\varsigma }_0\ge 0,$${\varrho }^{\prime }\left(\iota \right)\ge {\varrho }_0\ge 0,$$\varrho \circ \varsigma =\varsigma \circ \varrho$, $\varrho \left(\iota ,\zeta \right)\to \infty$ as $\left(\iota ,\zeta \right)\to \infty$ and $\varsigma \left(\iota \right)\to \infty$ as $\iota \to \infty$;

(s₄)

$ϝ\in C\left(\mathbb{R},\mathbb{R}\right)$ such that

$$ϝ\left(\Omega \right)\ge \beta x^{\alpha }\mathrm{for}\Omega \ne 0.$$

For the sake of brevity, it would be better to define the following:

$${\acute{L}}_1\psi :=\nu {\left({\psi }^{\prime }\right)}^{\alpha },{\acute{L}}_2\psi :={\left(\nu {\left({\psi }^{\prime }\right)}^{\alpha }\right)}^{\prime },{\acute{L}}_3\psi :={\left({\left(\nu {\left({\psi }^{\prime }\right)}^{\alpha }\right)}^{\prime }\right)}^{\prime }.$$

Under a solution for Equation (8) with mean that $\Omega \in C\left([{\iota }_a,\infty ),\mathbb{R}\right)$ for some ${\iota }_a\ge {\iota }_0$ and has the property

$$\psi ,\nu {\left({\psi }^{\prime }\right)}^{\alpha },{\left(\nu {\left({\psi }^{\prime }\right)}^{\alpha }\right)}^{\prime }\in C^1\left([{\iota }_a,\infty ),[0,\infty )\right)$$

which satisfies (8) on $[T_a,\infty ).$ We consider those solutions to Equation (8) which exist on some half-line $[{\iota }_n,\infty )$ and satisfy the condition $sup\{\left|\Omega \left(\iota \right)\right|:{\iota }_m\le \iota <\infty \}>0$ for ${\iota }_m\in [{\iota }_n,\infty )$. A solution $\Omega$ is called oscillatory if it has arbitrarily large zeros; if not, then it is called nonoscillatory. In addition, if all of its solutions are oscillatory, then Equation (8) is called oscillatory.

It is known that the oscillation criteria of the second-order differential Equation (1) (without delay) have been extensively studied by researchers. On the basis of the ideas exploited in [13], we offer some new criteria which ensure that any solution to Equation (8) oscillates when Equation (1) is nonoscillatory:

Lemma 1

([9]). Assume that $m_1,m_2\in [0,\infty )$ and $j>0$. Then, we have

$$\frac{{\left(m_1+m_2\right)}^j}{\left(m_1^j+m_2^j\right)}\le \mu ,$$

where

$$\mu :=\left\{\begin{array}{cc}{2}^{j-1}\hfill & ifj>1\\ 1\hfill & ifj\le 1.\end{array}\right.$$

Remark 1.

Without loss of generality, in this paper, since the negative solutions are similar to the positive solutions, we will just discuss the positive solutions.

2. Preliminaries

In this section, we present some results that will be useful in establishing the oscillation criteria of Equation (8). First, we define the following function:

$$\tilde{V}\left(\iota ,{\iota }_2\right)={\int }_{{\iota }_2}^{\iota }{\left(\frac{\xi -{\iota }_2}{\nu \left(\xi \right)}\right)}^{1/\alpha }\mathrm{d}\xi .$$

Lemma 2.

Suppose that Equation (1) is nonoscillatory. If Ω is a nonoscillatory solution to Equation (8) on $[{\iota }_2,\infty )$, ${\iota }_2\ge {\iota }_0$, then $\exists {\iota }_3\in [{\iota }_2,\infty )$ such that

$$\psi \left(\iota \right){\acute{L}}_1\psi \left(\iota \right)>0$$

(9)

or

$$\psi \left(\iota \right){\acute{L}}_1\psi \left(\iota \right)<0,\mathit{for}\iota \ge {\iota }_3.$$

(10)

Proof. 

Let $\Omega >0$ be a solution to Equation (8) on $[{\iota }_2,\infty )$. Then, $\psi \left(\iota \right)>0,$$\psi \left(\varsigma \left(\iota \right)\right)>0$ and $\psi \left(\varrho \left(\iota ,\zeta \right)\right)>0$ for $\iota \ge {\iota }_2\ge {\iota }_1$. From Equation (8) and assumption s${}_4$, we obtain the inequality

$${\acute{L}}_3\psi \left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_1\psi \left(\iota \right)\le -\beta {\int }_c^d\gamma \left(\iota ,\zeta \right){\Omega }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta$$

(11)

and

$$w^{″}\left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}w\left(\iota \right)\ge \beta {\int }_c^d\gamma \left(\iota ,\zeta \right){\Omega }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta ,\iota \ge {\iota }_2,$$

where $\Theta =-{\acute{L}}_1\psi$. Let $\varkappa \left(\iota \right)>0$ be a solution to Equation (1) for $\iota \ge {\iota }_2\ge {\iota }_1$. Let $\omega$ be oscillatory. We can see that at c and $d({\iota }_2<c<d)$, the function $\omega$ has consecutive zeros such that ${\Theta }^{\prime }\left(c\right)\ge 0,{\Theta }^{\prime }\left(d\right)\le 0$ and $\Theta \left(\iota \right)\ge 0$ for $\iota \in (c,d)$. This implies that

$$\begin{array}{ccc}\hfill 0& <& {\int }_c^d\left({\Theta }^{″}\left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\Theta \left(\iota \right)\right)\varkappa \left(\iota \right)\mathrm{d}\iota \hfill \\ & =& {\int }_c^d\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\Theta \left(\iota \right)\varkappa \left(\iota \right)\mathrm{d}\iota +{\left.{\psi }^{\prime }\varkappa \right|}_c^d-{\int }_c^d{\Theta }^{\prime }\left(\iota \right){\varkappa }^{\prime }\left(\iota \right)\mathrm{d}\iota \hfill \\ & =& {\int }_c^d{\varkappa }^{″}\left(\iota \right)\Theta \left(\iota \right)\mathrm{d}\iota +{\int }_c^d\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\Theta \left(\iota \right)\varkappa \left(\iota \right)\mathrm{d}\iota +{\left.{\Theta }^{\prime }\varkappa \right|}_c^d-{\left.w{u}^{\prime }\right|}_c^d\hfill \\ & =& {\left.{\Theta }^{\prime }\varkappa \right|}_c^d+{\int }_c^d\left({\varkappa }^{″}\left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\varkappa \left(\iota \right)\right)\Theta \left(\iota \right)\mathrm{d}\iota \hfill \\ & =& {\left.{\Theta }^{\prime }\varkappa \right|}_c^d\le 0.\hfill \end{array}$$

This completes the proof.  □

Remark 2.

A class $N_1$ means a set which includes all solutions Ω to Equation (8) where their corresponding ψ satisfies the property in Equation (9), and a class $N_2$ is a set which includes all solutions Ω to Equation (8) where their corresponding ψ satisfies the property in Equation (10).

Lemma 3.

Suppose that Ω is a nonoscillatory solution to Equation (8) and ψ satisfies Equation (9). Then, we have

$${\acute{L}}_1\psi \left(\iota \right)\ge \left(\iota -{\iota }_2\right){\acute{L}}_2\psi \left(\iota \right)$$

(12)

and

$$\psi \left(\iota \right)\ge \tilde{V}\left(\iota ,{\iota }_2\right){\acute{L}}_2^{1/\alpha }\psi \left(\iota \right),$$

(13)

for all $\iota \ge {\iota }_2.$ Moreover, if ψ satisfies Equation (10), then we have

$$\psi \left(u\right)\ge \tilde{V}\left(v,u\right){\left(-{\acute{L}}_1\psi \left(v\right)\right)}^{1/\alpha },\mathit{for}v\ge u\ge \iota .$$

(14)

Proof. 

Assume that $\Omega >0$ is a solution to Equation (8). Then, $\psi \left(\iota \right)>0,$$\psi \left(\varsigma \left(\iota \right)\right)>0$ and $\psi \left(\varrho \left(\iota ,\zeta \right)\right)>0$ for $\iota \ge {\iota }_2\ge {\iota }_1$. Let $\psi$ satisfy Equation (9). From Equation (11), we see that ${\acute{L}}_2\psi \left(\iota \right)$ is non-increasing on $[{\iota }_2,\infty )$, and thus

$$\begin{array}{ccc}\hfill {\acute{L}}_1\psi \left(\iota \right)& =& {\int }_{{\iota }_2}^{\iota }{\left({\acute{L}}_1\psi \left(\xi \right)\right)}^{\prime }\mathrm{d}\xi +{\acute{L}}_1\psi \left({\iota }_2\right)\ge {\int }_{{\iota }_2}^{\iota }{\left({\acute{L}}_1\psi \left(\xi \right)\right)}^{\prime }\mathrm{d}\xi \hfill \\ & =& {\acute{L}}_2\psi \left(\iota \right){\int }_{{\iota }_2}^{\iota }\frac{1}{\nu \left(\xi \right)}\mathrm{d}\xi \hfill \\ & =& {\acute{L}}_2\psi \left(\iota \right)\left(\iota -{\iota }_2\right).\hfill \end{array}$$

That is, we have

$${\psi }^{\prime }\left(\iota \right)\ge \frac{{\left(\iota -{\iota }_2\right)}^{1/\alpha }}{{\nu }^{1/\alpha }\left(\iota \right)}{\acute{L}}_2^{{}^{1/\alpha }}\psi \left(\iota \right).$$

(15)

Additionally, integrating Equation (15) from ${\iota }_2$ to $\iota$ implies that

$$\psi \left(\iota \right)\ge {\int }_{{\iota }_2}^{\iota }\frac{{\left(\iota -{\iota }_2\right)}^{1/\alpha }}{{\nu }^{1/\alpha }\left(\xi \right)}{\acute{L}}_2^{1/\alpha }\psi \left(\xi \right)\mathrm{d}\xi \ge {\acute{L}}_2^{1/\alpha }\psi \left(\iota \right)\tilde{V}\left(\iota ,{\iota }_2\right).$$

Now, let $\psi$ satisfy Equation (10). Since ${\acute{L}}_2\psi \left(\iota \right)\ge 0$, for $\iota \le u\le v,$ we obtain

$$\begin{array}{ccc}\hfill \psi \left(u\right)& =& \psi \left(v\right)-{\int }_u^v\frac{1}{{\nu }^{\frac{1}{\alpha }}\left(\xi \right)}{\left({\acute{L}}_1\psi \left(\xi \right)\right)}^{1/\alpha }\mathrm{d}\xi \hfill \\ & \ge & -{\left({\acute{L}}_1\psi \left(v\right)\right)}^{1/\alpha }{\int }_u^v\frac{1}{{\nu }_1^{\frac{1}{\alpha }}\left(\xi \right)}\mathrm{d}\xi \hfill \\ & =& {\left(-{\acute{L}}_1\psi \left(v\right)\right)}^{1/\alpha }\tilde{V}\left(v,u\right),\hfill \end{array}$$

In other words, we have

$$\psi \left(u\right)\ge {\left(-{\acute{L}}_1\psi \left(v\right)\right)}^{1/\alpha }\tilde{V}\left(v,u\right),$$

for $\iota \ge {\iota }_2.$ This completes the proof.  □

Lemma 4.

Let Ω be a positive solution to Equation (8). If ψ satisfies Equation (9), and $0\le \rho \left(\iota \right)<1$, then we obtain

$${\acute{L}}_3\psi \left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_1\psi \left(\iota \right)+\beta {\int }_c^d\widehat{\gamma }\left(\iota ,\zeta \right){\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta \le 0.$$

(16)

Furthermore, if ψ satisfies Equation (10), then

$${\acute{L}}_3\psi \left(\iota \right)+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}{\acute{L}}_3\psi \left(\varsigma \left(\iota \right)\right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_1\psi \left(\iota \right)+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}\frac{\eta \left(\varsigma \left(\iota \right)\right)}{\nu \left(\varsigma \left(\iota \right)\right)}{\acute{L}}_1\psi \left(\varsigma \left(\iota \right)\right)+\beta {\int }_c^d\tilde{\gamma }\left(\iota ,\zeta \right){\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta \le 0,$$

(17)

where

$$\breve{O}\left(\iota ,\zeta \right)={\int }_c^d\widehat{\gamma }\left(\iota ,\zeta \right)\mathrm{d}\zeta ={\int }_c^d\gamma \left(\iota ,\zeta \right){\left(1-\rho \left(\varrho \left(\iota ,\zeta \right)\right)\right)}^{\alpha }\mathrm{d}\zeta$$

and

$$\tilde{\gamma }\left(\iota ,\zeta \right)=min\{\gamma \left(\iota ,\zeta \right),\gamma \left(\varsigma \left(\iota \right),\zeta \right)\}.$$

Proof. 

Assume that $\Omega >0$ is a solution to Equation (8). Then, $\psi \left(\iota \right)>0,$$\psi \left(\varsigma \left(\iota \right)\right)>0$ and $\psi \left(\varrho \left(\iota ,\zeta \right)\right)>0$ for $\iota \ge {\iota }_2\ge {\iota }_1$. First, let $\psi$ satisfy Equation (9). The corresponding $\psi \left(\iota \right)$ satisfies

$$\begin{array}{ccc}\hfill \Omega \left(\iota \right)& =& \psi \left(\iota \right)-\rho \left(\iota \right)\Omega \left(\varsigma \left(\iota \right)\right)\ge \psi \left(\iota \right)-\rho \left(\iota \right)\psi \left(\varsigma \left(\iota \right)\right)\hfill \\ & \ge & \psi \left(\iota \right)\left(1-\rho \left(\iota \right)\right),\hfill \end{array}$$

Because of this, it follows that

$${\Omega }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\ge {\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right){\left(1-\rho \left(\varrho \left(\iota ,\zeta \right)\right)\right)}^{\alpha }.$$

(18)

In Equation (16), we find

$${\acute{L}}_3\psi \left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_1\psi \left(\iota \right)+\beta {\int }_c^d\widehat{\gamma }\left(\iota ,\zeta \right){\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta \le 0.$$

Now, assume that $\psi$ satisfies Equation (10). By Lemma 1, we conclude that

$${\psi }^{\alpha }\left(\iota \right)\le \mu \left({\Omega }^{\alpha }\left(\iota \right)+{\rho }_0^{\alpha }{\Omega }^{\alpha }\left(\varsigma \left(\iota \right)\right)\right)$$

(19)

and

$${\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\le \mu \left({\Omega }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)+{\rho }_0^{\alpha }{\Omega }^{\alpha }\left(\varrho \left(\varsigma \left(\iota ,\zeta \right)\right)\right)\right).$$

(20)

From Equation (11), we have

$$\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}{\acute{L}}_3\psi \left(\varsigma \left(\iota \right)\right)+{\rho }_0^{\alpha }\frac{\eta \left(\varsigma \left(\iota \right)\right)}{{\varsigma }_0^{{}^{\frac{1}{\alpha }}}\nu \left(\varsigma \left(\iota \right)\right)}{\acute{L}}_1\psi \left(\varsigma \left(\iota \right)\right)\le -{\rho }_0^{\alpha }\beta {\int }_c^d\gamma \left(\varsigma \left(\iota \right),\zeta \right){\Omega }^{\alpha }\left(\varrho \left(\varsigma \left(\iota \right),\zeta \right)\right)\mathrm{d}\zeta .$$

(21)

By combining Equations (11) and (21), we see that

$$\begin{array}{ccc}& & {\acute{L}}_3\psi \left(\iota \right)+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}{\acute{L}}_3\psi \left(\varsigma \left(\iota \right)\right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_1\psi \left(\iota \right)+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}\frac{\eta \left(\varsigma \left(\iota \right)\right)}{\nu \left(\varsigma \left(\iota \right)\right)}{\acute{L}}_1\psi \left(\varsigma \left(\iota \right)\right)\hfill \\ & & +\beta {\int }_c^d\tilde{\gamma }\left(\iota ,\zeta \right){\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta \le 0.\hfill \end{array}$$

The proof of the lemma is completed.  □

3. Oscillation Results

In this section, we present some new criteria that ensure the oscillation of all solutions to Equation (8).

Theorem 1.

Assume that Equation (1) is nonoscillatory. Let us say that $\exists \tilde{L}\left(\iota \right)\in C^1\left(\left[{\iota }_0,\infty \right),\left(0,\infty \right)\right)$ for all sufficiently large ${\iota }_2\ge {\iota }_0$ such that

$$\underset{\iota \to \infty }{limsup}{\int }_{{\iota }_3}^{\iota }\left[\frac{\beta }{\mu }\tilde{L}\left(\xi \right)\breve{O}\left(\iota ,\xi \right)-\frac{{\left({\tilde{L}}^{\prime }\left(\xi \right)\nu \left(\xi \right)-\tilde{L}\left(\xi \right)\eta \left(\xi \right)\right)}^2}{4\alpha \delta \left(\xi \right){\nu }^2\left(\xi \right){\varrho }^{\prime }\left(\iota \right){\tilde{V}}^{\alpha -1}\left(\varrho \left(\xi ,\zeta \right),{\iota }_2\right)}\right]\mathrm{d}\xi =\infty ,\mathit{for}{\iota }_2\ge {\iota }_3.$$

(22)

Then, $N_1=\varnothing$ or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Proof. 

Let $\Omega >0$ be a solution to Equation (8). Then, $\psi \left(\iota \right)>0,$$\psi \left(\varsigma \left(\iota \right)\right)>0$ and $\psi \left(\varrho \left(\iota ,\zeta \right)\right)>0$ for $\iota \ge {\iota }_2$. Let Equation (9) be satisfied. Since ${\acute{L}}_3\psi \left(\iota \right)\le 0$, it is easy to see that, for any ${\iota }_3\ge {\iota }_2,$ we obtain

$$\begin{array}{ccc}\hfill {\acute{L}}_1\psi \left(\iota \right)& =& {\int }_{{\iota }_3}^{\iota }{\acute{L}}_2\psi \left(\xi \right)\mathrm{d}\xi +{\acute{L}}_1\psi \left({\iota }_2\right)\hfill \\ & =& {\acute{L}}_1\psi \left({\iota }_3\right)+{\acute{L}}_2\psi \left({\iota }_3\right)\left(\iota -{\iota }_3\right).\hfill \end{array}$$

That is to say, ${\acute{L}}_2\psi \left({\iota }_3\right)>0$; otherwise, ${lim}_{\iota \to \infty }{\acute{L}}_1\psi \left(\iota \right)=-\infty ,$ a contradiction to ${\acute{L}}_1\psi \left(\iota \right)>0$. Now, since ${\acute{L}}_2\psi \left(\iota \right)$ is positive on [${\iota }_2,\infty$), we define

$$\Psi =\tilde{L}\frac{{\acute{L}}_2\psi }{{\psi }^{\alpha }\left(\varrho \right)}>0$$

(23)

on [${\iota }_2,\infty$). Since $\varrho \left(\iota ,\zeta \right)$ is non-increasing with respect to $\xi ,$ we note that ${\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\le {\psi }^{\alpha }\left(\varrho \left(\iota ,c\right)\right)$ and

$$\beta {\int }_c^d\widehat{\gamma }\left(\iota ,\zeta \right){\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta \ge k{y}^{\alpha }\left(\varrho \left(\iota ,c\right)\right)\breve{O}\left(\iota ,\zeta \right)\mathrm{for}\zeta \in \left(c,d\right).$$

By substituting in Equation (16), we have

$${\acute{L}}_3\psi \left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_1\psi \left(\iota \right)+k{y}^{\alpha }\left(\varrho \left(\iota ,c\right)\right)\breve{O}\left(\iota ,\zeta \right)\le 0.$$

(24)

Moreover, we have

$$k{y}^{\alpha }\left(\varrho \left(\iota ,c\right)\right){\int }_c^d\widehat{\gamma }\left(\iota ,\zeta \right)\mathrm{d}\zeta \frac{\Psi \left(\iota \right)}{{\acute{L}}_2\psi }\ge k\delta \left(\iota \right)\breve{O}\left(\iota ,\zeta \right).$$

(25)

Using Equation (15), we obtain

$${\psi }^{\prime }\left(\varrho \left(\iota ,\zeta \right)\right)\ge \frac{1}{{\nu }^{1/\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}{\acute{L}}_2^{1/\alpha }\psi \left(\varrho \left(\iota ,\zeta \right)\right)\ge \frac{1}{{\nu }^{1/\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}{\acute{L}}_2^{1/\alpha }\psi \left(\iota \right).$$

Thus, we have

$$\begin{array}{ccc}\hfill \frac{{\psi }^{\prime }\left(\varrho \left(\iota ,\zeta \right)\right)}{\psi \left(\varrho \left(\iota ,\zeta \right)\right)}& \ge & \frac{1}{{\nu }^{1/\alpha }\left(\varrho \left(\iota ,\zeta \right)\right){\tilde{L}}^{1/\alpha }\left(\iota \right)}{\left(\frac{\tilde{L}\left(\iota \right){\acute{L}}_2\psi \left(\iota \right)}{{\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}\right)}^{1/\alpha }\hfill \\ & =& \frac{1}{{\nu }^{1/\alpha }\left(\varrho \left(\iota ,\zeta \right)\right){\tilde{L}}^{1/\alpha }\left(\iota \right)}{\Psi }^{1/\alpha }\left(\iota \right).\hfill \end{array}$$

(26)

From Equation (13), we obtain

$$\Psi \left(\iota \right)=\tilde{L}\left(\iota \right)\frac{{\acute{L}}_2\psi \left(\iota \right)}{{\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}\le \tilde{L}\left(\iota \right)\frac{{\acute{L}}_2\psi \left(\varrho \left(\iota ,\zeta \right)\right)}{{\psi }^{\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}\le \frac{\tilde{L}\left(\iota \right)}{{\tilde{V}}^{\alpha }\left(\varrho \left(\iota ,\zeta \right),{\iota }_2\right)}.$$

Hence, we find that

$$\Psi {\left(\iota \right)}^{\left(1-\alpha \right)/\alpha }\le \frac{{\left(\tilde{L}\left(\iota \right)\right)}^{\left(1-\alpha \right)/\alpha }}{{\tilde{V}}^{1-\alpha }\left(\varrho \left(\iota ,\zeta \right),{\iota }_2\right)}.$$

(27)

By differentiating Equation (23) and then using Equations (12), (24) and (25), we obtain

$$\begin{array}{ccc}\hfill {\Psi }^{\prime }& =& {\tilde{L}}^{\prime }\left(\iota \right)\frac{{\acute{L}}_2\psi }{{\psi }^{\alpha }\left(\varrho \right)}+\tilde{L}\left(\iota \right)\frac{{\acute{L}}_3\psi }{{\psi }^{\alpha }\left(\varrho \right)}-\frac{\alpha \delta y^{\alpha -1}\left(\varrho \right){\psi }^{\prime }\left(\varrho \right){\varrho }^{\prime }{\acute{L}}_2\psi }{{\psi }^{2\alpha }\left(\varrho \right)}\hfill \\ \hfill {\Psi }^{\prime }\left(\iota \right)& =& \frac{{\tilde{L}}^{\prime }}{\tilde{L}}\Psi +\frac{{\acute{L}}_3\psi }{{\acute{L}}_2\psi }\Psi -\frac{\alpha {\sigma }^{\prime }{\psi }^{\prime }\left(\varrho \right)}{\psi \left(\varrho \right)}\Psi \hfill \\ & \le & \frac{{\tilde{L}}^{\prime }}{\tilde{L}}\Psi -\frac{\eta }{\nu }\Psi -k\delta \breve{O}\left(.,\zeta \right)-\frac{\alpha {\sigma }^{\prime }{\psi }^{\prime }\left(\varrho \right)}{\psi \left(\varrho \right)}\Psi \hfill \\ & \le & \left(\frac{{\tilde{L}}^{\prime }}{\tilde{L}}-\frac{\eta }{\nu }\right)\Psi -k\delta \breve{O}\left(.,\zeta \right)-\frac{\alpha {\sigma }^{\prime }{\psi }^{\prime }\left(\varrho \right)}{\psi \left(\varrho \right)}\Psi .\hfill \end{array}$$

(28)

From Equation (26), we find that

$${\Psi }^{\prime }\left(\iota \right)\le \left(\frac{{\tilde{L}}^{\prime }}{\tilde{L}}-\frac{\eta }{\nu }\right)\Psi -k\delta \breve{O}\left(.,\zeta \right)-\frac{\alpha {\sigma }^{\prime }}{{\nu }^{1/\alpha }\left(\varrho \left(.,\zeta \right)\right){\tilde{L}}^{1/\alpha }}{\Psi }^{\left(1/\alpha \right)+1}.$$

By Equation (27), we have

$${\Psi }^{\prime }\left(\iota \right)\le \left(\frac{{\tilde{L}}^{\prime }\left(\iota \right)}{\tilde{L}\left(\iota \right)}-\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\right)\Psi \left(\iota \right)-\left(\frac{\alpha {\sigma }^{\prime }\left(\iota \right){\tilde{V}}^{\alpha -1}\left(\varrho \left(\iota ,\zeta \right),{\iota }_2\right)}{\tilde{L}\left(\iota \right){\nu }^{1/\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}\right){\Psi }^2\left(\iota \right)-\frac{\beta }{\mu }\tilde{L}\left(\iota \right)\breve{O}\left(\iota ,\zeta \right).$$

(29)

We apply the inequality

$$Xu-B{u}^2\le \frac{X^2}{4B},$$

with

$$X\left(\iota \right)=\left(\frac{{\tilde{L}}^{\prime }\left(\iota \right)}{\tilde{L}\left(\iota \right)}-\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\right)\mathrm{and}Y=\frac{\alpha {\sigma }^{\prime }\left(\iota \right){\tilde{V}}^{\alpha -1}\left(\varrho \left(\iota ,\zeta \right),{\iota }_2\right)}{\tilde{L}\left(\iota \right){\nu }^{1/\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}.$$

(30)

Thus, we have

$${\Psi }^{\prime }\left(\iota \right)\le -\frac{\beta }{\mu }\tilde{L}\left(\iota \right)\breve{O}\left(\iota ,\zeta \right)+{\left(\frac{{\tilde{L}}^{\prime }\left(\iota \right)\nu \left(\iota \right)-\eta \left(\iota \right)\tilde{L}\left(\iota \right)}{\tilde{L}\left(\iota \right)\nu \left(\iota \right)}\right)}^2\times \frac{\tilde{L}\left(\iota \right){\nu }^{1/\alpha }\left(\varrho \left(\iota ,\zeta \right)\right)}{\alpha {\sigma }^{\prime }\left(\iota \right){\tilde{V}}^{\alpha -1}\left(\varrho \left(\iota ,\zeta \right),{\iota }_2\right)}.$$

That is to say, we find that

$${\Psi }^{\prime }\left(\iota \right)\le -\frac{\beta }{\mu }\tilde{L}\left(\iota \right)\breve{O}\left(\iota ,\zeta \right)+\frac{{\left({\tilde{L}}^{\prime }\left(\iota \right)\nu \left(\iota \right)-\tilde{L}\left(\iota \right)\eta \left(\iota \right)\right)}^2}{4\alpha \delta \left(\iota \right){\left(\nu \left(\iota \right)\right)}^2{\varrho }^{\prime }\left(\iota \right){\tilde{V}}^{\alpha -1}\left(\varrho \left(\iota ,\zeta \right),{\iota }_2\right)}.$$

(31)

By integrating Equation (31) from ${\iota }_3$ to $\iota$, we obtain

$${\int }_{{\iota }_3}^{\iota }\left[\frac{\beta }{\mu }\tilde{L}\left(\iota \right)\breve{O}\left(\iota ,\zeta \right)-\frac{{\left({\tilde{L}}^{\prime }\left(\xi \right)\nu \left(\xi \right)-\tilde{L}\left(\iota \right)\eta \left(\xi \right)\right)}^2}{4\alpha \delta \left(\xi \right){\left(\nu \left(\xi \right)\right)}^2{\varrho }^{\prime }\left(\xi \right){\tilde{V}}^{\alpha -1}\left(\varrho \left(\iota ,\zeta \right),{\iota }_2\right)}\right]\mathrm{d}\xi \le \Psi \left({\iota }_2\right),$$

which contradicts Equation (22), thus completing the proof.  □

Theorem 2.

Suppose that Equation (1) is nonoscillatory. If all solutions of the first delay equation

$${\theta }^{\prime }\left(\iota \right)+k\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)\theta \left(\varrho \left(\iota ,c\right)\right)=0,$$

(32)

where

$$\nu \left(\iota \right)=exp\left({\int }_{{\iota }_2}^{\iota }\frac{\eta \left(\xi \right)}{\nu \left(\xi \right)}\mathrm{d}\xi \right).$$

are oscillatory, then $N_1=\varnothing$ or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Proof. 

Assume that $\Omega >0$ is a solution to Equation (8). Then, $\psi \left(\iota \right)>0,$$\psi \left(\varsigma \left(\iota \right)\right)>0$ and $\psi \left(\varrho \left(\iota ,\zeta \right)\right)>0$ for $\iota \ge {\iota }_2$. Assume that $\psi$ satisfies Equation (9) on [${\iota }_2,\infty$). As in Theorem 1, we have ${\acute{L}}_2\psi >0$ on [${\iota }_2,\infty$). Using Equations (12) and (13) in Equation (16), we have

$${\acute{L}}_3\psi \left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_2\psi \left(\iota \right)+k\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\varrho \left(\iota ,c\right),{\iota }_2\right){\acute{L}}_2\psi \left(\varrho \left(\iota ,c\right)\right)\le 0.$$

Now, by setting $\varpi \left(\iota \right)={\acute{L}}_2\psi \left(\iota \right)$, we obtain

$${\varpi }^{\prime }\left(\iota \right)+\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\varpi \left(\iota \right)+k\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\varrho \left(\iota ,c\right),{\iota }_2\right)\varpi \left(\varrho \left(\iota ,c\right)\right)\le 0.$$

(33)

By multiplying Equation (33) by $\nu \left(\iota \right)$, we have

$${\left(\mu \varpi \right)}^{\prime }\left(\iota \right)+k\mu \left(\iota \right)\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)\varpi \left(\varrho \left(\iota ,c\right)\right)\le 0.$$

Since ${\nu }^{\prime }\left(\iota \right)\ge 0,$ we obtain

$${\theta }^{\prime }\left(\iota \right)+\frac{k\mu \left(\iota \right)}{\nu \left(\varrho \left(\iota ,\zeta \right)\right)}\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)\theta \left(\varrho \left(\iota ,c\right)\right)\le 0,$$

where $\theta =\mu \varpi .$ Hence, we have

$${\theta }^{\prime }\left(\iota \right)+k\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)\theta \left(\varrho \left(\iota ,c\right)\right)\le 0.$$

In light of Lemma 1 [41], we see that the first-order delay differential Equation (32) has a positive solution, which is a contradiction. The proof is complete.  □

Theorem 3.

Suppose that Equation (1) is nonoscillatory such that $\varrho \left(\iota ,\zeta \right)\le \varsigma \left(\iota \right)$ for $\iota \ge {\iota }_0$. Assume Equation (3) or (4) holds with

$$\left(\iota \right)=\left(k\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\varsigma \left(\iota \right),\varrho \left(\iota ,d\right)\right)-\left(\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}+\frac{{\rho }_0^{\alpha }\eta \left(\varsigma \left(\iota \right)\right)}{{\varsigma }_0\nu \left(\varsigma \left(\iota \right)\right)}\right)\right)\ge 0,$$

(34)

Then, $N_2=\varnothing$ or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Proof. 

Assume $\Omega >0$ is a solution to Equation (8). Then, $\psi \left(\iota \right)>0,$$\psi \left(\varsigma \left(\iota \right)\right)>0$ and $\psi \left(\varrho \left(\iota ,\zeta \right)\right)>0$ for $\iota \ge {\iota }_2$. Assume that Equation (10) is satisfied on [${\iota }_2,\infty$). We see that

$${\psi }^{\prime }\left(\iota \right)=\frac{{\acute{L}}_1^{1/\alpha }\psi \left(\iota \right)}{{\nu }^{1/\alpha }\left(\iota \right)}\le \frac{{\acute{L}}_1^{1/\alpha }\psi \left({\iota }_3\right)}{{\nu }^{1/\alpha }\left(\iota \right)},\iota \ge {\iota }_3,$$

(35)

Thus, it is impossible for ${\acute{L}}_2\psi \left(\iota \right)$ to be a non-positive function. By integration (twice) of Equation (35), we obtain that $\psi \left(\iota \right)<0$, and that is a contradiction to $\psi \left(\iota \right)>0$. Therefore, we assume ${\acute{L}}_2\psi \left(\iota \right)\ge 0$ for all large $\iota ,$ ${\iota }_2\le {\iota }_3\le \iota .$ However, since $u=\varrho \left(\iota ,d\right)$ and $v=\varsigma \left(\iota \right)$ in Equation (14), we obtain

$$\begin{array}{ccc}\hfill \psi \left(\varrho \left(\iota ,d\right)\right)& \ge & {\left(-{\acute{L}}_1\psi \left(\varsigma \left(\iota \right)\right)\right)}^{1/\alpha }\tilde{V}\left(\varsigma \left(\iota \right),\varrho \left(\iota ,d\right)\right)\hfill \\ & =& \tilde{V}\left(\varsigma \left(\iota \right),\varrho \left(\iota ,d\right)\right)g\left(\varsigma \left(\iota \right)\right),\hfill \end{array}$$

(36)

where $g\left(\iota \right)=-{\left({\acute{L}}_1\psi \left(\iota \right)\right)}^{1/\alpha }>0$ for ${\iota }_4\le \iota .$ Since ${\acute{L}}_2\psi \left(\iota \right)\ge 0$ and ${\acute{L}}_3\psi \left(\iota \right)\le 0$, we have

$$\begin{array}{ccc}\hfill \frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}{\acute{L}}_1\psi \left(\iota \right)+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}\frac{\eta \left(\varsigma \left(\iota \right)\right)}{\nu \left(\varsigma \left(\iota \right)\right)}{\acute{L}}_1\psi \left(\varsigma \left(\iota \right)\right)& \ge & {\acute{L}}_1\psi \left(\varsigma \left(\iota \right)\right)\left(\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}\frac{\eta \left(\varsigma \left(\iota \right)\right)}{\nu \left(\varsigma \left(\iota \right)\right)}\right)\hfill \\ & \ge & \left(\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}\frac{\eta \left(\varsigma \left(\iota \right)\right)}{\nu \left(\varsigma \left(\iota \right)\right)}\right){\acute{L}}_1\psi \left(\varsigma \left(\iota \right)\right)\hfill \end{array}$$

(37)

and

$${\acute{L}}_2\psi \left(\iota \right)+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}{\acute{L}}_2\psi \left(\varsigma \left(\iota \right)\right)\ge {\acute{L}}_2\psi \left(\iota \right)\left(\frac{{\varsigma }_0+{\rho }_0^{\alpha }}{{\varsigma }_0}\right).$$

(38)

By setting $\theta =g^{\alpha },$ from Equation (17), and since $g\left(\iota \right)$ is decreasing, with Equations (36)–(38), we have

$$\begin{array}{ccc}\hfill {\theta }^{″}\left(\iota \right)\left(\frac{{\varsigma }_0+{\rho }_0^{\alpha }}{{\varsigma }_0}\right)+\left(\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}+\frac{{\rho }_0^{\alpha }}{{\varsigma }_0}\frac{\eta \left(\varsigma \left(\iota \right)\right)}{\nu \left(\varsigma \left(\iota \right)\right)}\right)\theta \left(\varsigma \left(\iota \right)\right)& \ge & k{y}^{\alpha }\left(\varrho \left(\iota ,d\right)\right){\int }_c^d\tilde{\gamma }\left(\iota ,\zeta \right)\mathrm{d}\zeta \hfill \\ & \ge & k\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\varsigma \left(\iota \right),\varrho \left(\iota ,d\right)\right)\theta \left(\varsigma \left(\iota \right)\right).\hfill \end{array}$$

That is to say, we have

$${\theta }^{″}\left(\iota \right)\left(\frac{{\varsigma }_0+{\rho }_0^{\alpha }}{{\varsigma }_0}\right)\ge \left(k\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\varsigma \left(\iota \right),\varrho \left(\iota ,d\right)\right)-\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}+\frac{{\rho }_0^{\alpha }\eta \left(\varsigma \left(\iota \right)\right)}{{\varsigma }_0\nu \left(\varsigma \left(\iota \right)\right)}\right)\theta \left(\varsigma \left(\iota \right)\right).$$

(39)

Compared with Equation (1), we find that Equation (39) is oscillatory if Equation (3) or (4) is satisfied, thus completing the proof.  □

Theorem 4.

Suppose that Equation (1) is nonoscillatory and $\varrho \left(\iota ,\zeta \right)\le \varsigma \left(\iota \right)\le \iota$. If ∃ $\tilde{L}\left(\iota \right)\in C^1\left(\left[{\iota }_0,\infty \right),\left(0,\infty \right)\right),$ then for all sufficiently large ${\iota }_2\ge {\iota }_0$, ${\iota }_2\ge {\iota }_2$ such that Equations (22) and (3) or (4) satisfies Equation $\left(\iota \right)$ as in Theorem 3. Then, every solution to Equation (8) or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Theorem 5.

Suppose that Equation (1) is nonoscillatory and $\varrho \left(\iota ,\zeta \right)\le \varsigma \left(\iota \right)\le \iota$. If every solution of the first-order delay Equation (32) (where $\left(\iota \right)$ and $\nu \left(\iota \right)$ are defined as in Theorem 2 and Theorem 3, respectively) is oscillatory and Equation (3) or (4) is satisfied, then every solution to Equation (8) or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Corollary 1.

Suppose that Equation (1) is nonoscillatory and $\varrho \left(\iota ,\zeta \right)\le \varsigma \left(\iota \right)\le \iota$. If ∃ $\tilde{L}\left(\iota \right)\in C^1\left(\left[{\iota }_0,\infty \right),\left(0,\infty \right)\right)$ for all sufficiently large ${\iota }_2\ge {\iota }_0$ such that

$$\underset{\iota \to \infty }{limsup}{\int }_{{\iota }_3}^{\iota }\tilde{L}\left(\xi \right)\breve{O}\left(\iota ,\xi \right)\mathrm{d}\xi =\infty ,\mathrm{for}{\iota }_3\ge {\iota }_2$$

and Equation (3) or (4) holds as in Theorem 3 such that

$$\left(\frac{{\tilde{L}}^{\prime }\left(\iota \right)}{\tilde{L}\left(\iota \right)}-\frac{\eta \left(\iota \right)}{\nu \left(\iota \right)}\right)\le 0,$$

then every solution to Equation (8) or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Corollary 2.

Suppose that Equation (1) is nonoscillatory and $\varrho \left(\iota ,\zeta \right)\le \varsigma \left(\iota \right)\le \iota$. If

$$\underset{\iota \to \infty }{liminf}{\int }_{\varrho \left(\iota ,c\right)}^{\iota }\breve{O}\left(\iota ,\xi \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)\mathrm{d}\xi >\frac{1}{e},$$

and Equation (3) or (4) holds with $\left(\iota \right)$ as in Theorem 3, then every solution to Equation (8) or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Corollary 3.

Suppose that Equation (1) is nonoscillatory such that $\varrho \left(\iota ,\zeta \right)\le \varsigma \left(\iota \right)$ for $\iota \ge {\iota }_0$. If Equation (3) or (4) holds with $\left(\iota \right)$ as in Theorem 3, where

$${\int }_{\iota }^{\iota +\varrho \left(\iota ,c\right)}\breve{O}\left(\iota ,\xi \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)\mathrm{d}\xi >0\mathrm{for}\iota \ge {\iota }_0$$

and

$${\int }_{{\iota }_0}^{\infty }\breve{O}\left(\iota ,\zeta \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)ln\left(e{\int }_{\iota }^{\iota +\varrho \left(\iota ,c\right)}\breve{O}\left(\iota ,\xi \right){\tilde{V}}^{\alpha }\left(\iota ,{\iota }_2\right)\mathrm{d}\xi \right)\mathrm{d}\zeta =\infty ,$$

then every solution to Equation (8) or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

4. Further Results

In the following theorem, we obtain a new oscillation condition for the equation using the integral averaging technique. According to Philos in [42], we introduce the class of functions $\mathfrak{P}$. Let $\widehat{D}=\left\{\left(\iota ,\xi \right),{\iota }_0<\xi \le \iota \right\}$ and ${\widehat{D}}_0=\left\{\left(\iota ,\xi \right):{\iota }_0<\xi <\iota \right\}$. We say that a function $\widehat{H}\in \mathfrak{P}$ if the following are true:

(a)

$\widehat{H}\left(\iota ,\xi \right)$ is positive for all $\left(\iota ,\xi \right)\in {\widehat{D}}_0,$$\widehat{H}\left(\iota ,\iota \right)=0$;

(b)

$\widehat{H}$ has a nonpositive and continuous partial derivative on ${\widehat{D}}_0$ and for when it is continuous and $\overline{h}>0$:

$$\frac{\partial \widehat{H}\left(\iota ,\xi \right)}{\partial \xi }=\overline{h}(\iota ,\xi ){\left(\widehat{H}\left(\iota ,\xi \right))\right)}^{1/2}\mathrm{for}\mathrm{all}\left(\iota ,\xi \right)\in {\widehat{D}}_0.$$

Let us choose $\widehat{H}\left(\iota ,\xi \right)-{\left(\iota -\xi \right)}^n,$$n\in \mathbb{N}.$

Theorem 6.

Suppose that Equation (1) is nonoscillatory. There exist functions where $\tilde{L},$ $\widehat{h}\in C^1\left(\left[{\iota }_0,\infty \right),\left(0,\infty \right)\right),$ $\tilde{L}\left(\iota \right)$ is a positive function, and ${\widehat{h}}^{\prime }\left(\iota \right)\ge 0$ and $\varrho \left(\iota ,\zeta \right)\le \widehat{h}\left(\iota \right)\le \iota$ such that

$$\underset{\iota \to \infty }{limsup}{\int }_{{\iota }_3}^{\iota }\frac{1}{\widehat{H}\left(\iota ,{\iota }_1\right)}{\int }_{{\iota }_1}^{\iota }\left[\frac{\beta }{\mu }\tilde{L}\left(\xi \right)\breve{O}\left(\iota ,\xi \right)\widehat{H}\left(\iota ,\xi \right)-\frac{{\left(\overline{h}(\iota ,\xi )-X\left(\xi \right)\left(\widehat{H}\left(\iota ,\xi \right)\right)\right)}^2}{4B\left(\xi \right)}\right]\mathrm{d}\xi =\infty ,$$

(40)

for $\iota \ge {\iota }_0,$ where $\widehat{H}\in \mathfrak{P}$, with X and Y defined as in Equation (30). If Equation (3) or (4) holds (where $\left(\iota \right)$ is as in Theorem 2), then every solution to Equation (8) or ${\acute{L}}_2\psi \left(\iota \right)$ is oscillatory.

Proof. 

Let $\Omega$ be a positive solution to Equaiton (8). Then, $\psi \left(\iota \right)>0,$$\psi \left(\varsigma \left(\iota \right)\right)>0$ and $\psi \left(\varrho \left(\iota ,\zeta \right)\right)>0$ for $\iota \ge {\iota }_2$. As in Theorem 1, we arrive at Equaiton (29); that is we have

$$\begin{array}{ccc}\hfill {\int }_{{\iota }_1}^{\iota }\frac{\beta }{\mu }\tilde{L}\left(\xi \right)\breve{O}\left(\iota ,\xi \right)\widehat{H}\left(\iota ,\xi \right)\mathrm{d}\xi & \le & {\int }_{{\iota }_1}^{\iota }\widehat{H}\left(\iota ,\xi \right)\left[-{\Psi }^{\prime }\left(\xi \right)+X\left(\xi \right)\Psi \left(\xi \right)-Y\left(\xi \right){\Psi }^2\left(\xi \right)\right]\mathrm{d}\xi \hfill \\ & \le & -{\left.\widehat{H}\left(\iota ,\xi \right)\Psi \left(\xi \right)\right|}_{\xi -{\iota }_1}^{\xi -\iota }\hfill \\ & & +{\int }_{{\iota }_1}^{\iota }\frac{\partial \widehat{H}\left(\iota ,\xi \right)}{\partial \xi }\Psi \left(\xi \right)+\widehat{H}\left(\iota ,\xi \right)\left(X\left(\xi \right)\Psi \left(\xi \right)-Y\left(\xi \right){\Psi }^2\left(\xi \right)\right)\mathrm{d}\xi \hfill \\ & =& \widehat{H}\left(\iota ,{\iota }_1\right)\Psi \left({\iota }_1\right)\hfill \\ & & -{\int }_{{\iota }_1}^{\iota }\widehat{H}\left(\iota ,\xi \right)Y\left(\xi \right){\Psi }^2\left(\xi \right)+\overline{h}(\iota ,\xi ){\left(\widehat{H}\left(\iota ,\xi \right))\right)}^{1/2}\Psi \left(\xi \right)\mathrm{d}\xi \hfill \\ & & -{\int }_{{\iota }_1}^{\iota }\widehat{H}\left(\iota ,\xi \right)X\left(\xi \right)\Psi \left(\xi \right)\mathrm{d}\xi \hfill \\ & =& \widehat{H}\left(\iota ,{\iota }_1\right)\Psi \left({\iota }_1\right)\hfill \\ & & -{\int }_{{\iota }_1}^{\iota }\left[{\left({\left(\widehat{H}\left(\iota ,\xi \right)\right)}^{1/2}{Y}^{1/2}\left(\xi \right)\Psi \left(\xi \right)+\frac{\overline{h}(\iota ,\xi )-X\left(\xi \right)\widehat{H}\left(\iota ,\xi \right)}{2Y^{1/2}\left(\xi \right)}\right)}^2\mathrm{d}\xi \right]\hfill \\ & & +{\int }_{{\iota }_1}^{\iota }\frac{{\left(\overline{h}(\iota ,\xi )-X\left(\xi \right)\left(\widehat{H}\left(\iota ,\xi \right)\right)\right)}^2}{4Y\left(\xi \right)}\mathrm{d}\xi \hfill \\ & =& \widehat{H}\left(\iota ,{\iota }_1\right)\Psi \left({\iota }_1\right)+{\int }_{{\iota }_1}^{\iota }\frac{{\left(\overline{h}(\iota ,\xi )-X\left(\xi \right)\left(\widehat{H}\left(\iota ,\xi \right)\right)\right)}^2}{4Y\left(\xi \right)}\mathrm{d}\xi .\hfill \end{array}$$

Thus, we obtain

$$\frac{1}{\widehat{H}\left(\iota ,{\iota }_1\right)}{\int }_{{\iota }_1}^{\iota }\left[\frac{\beta }{\mu }\tilde{L}\left(\xi \right)\breve{O}\left(\iota ,\xi \right)\widehat{H}\left(\iota ,\xi \right)-\frac{{\left(\overline{h}(\iota ,\xi )-X\left(\xi \right)\left(\widehat{H}\left(\iota ,\xi \right)\right)\right)}^2}{4Y\left(\xi \right)}\right]\mathrm{d}\xi \le \Psi \left({\iota }_1\right),$$

which contradicts Equation (40). As proof of Theorem 3, we completed the rest of the proof, which was thus omitted.  □

Example 1.

Consider the equation

$$\frac{d^2}{d{\iota }^2}{\left(\frac{d}{d\iota }\psi \left(\iota \right)\right)}^3+9{\left(\frac{d}{d\iota }\psi \left(\iota \right)\right)}^3+6y\left(\frac{2t-3\pi }{2}\right)=0.$$

(41)

By Corollary 1, we see that its conditions with $\tilde{L}\left(\iota \right)=1$ hold, and thus Equation (41) is oscillatory. ($\psi \left(\iota \right)=sin\iota$ is one oscillatory solution.)

Example 2.

Consider the equation

$$\frac{d^3}{d{\iota }^3}\psi \left(\iota \right)+0.5\frac{d}{d\iota }\psi \left(\iota \right)+0.5y\left(\iota -\frac{3\pi }{2}\right)=0.$$

(42)

By Theorem 4, and by taking $\tilde{L}\left(\iota \right)=1$, $\varsigma \left(\iota \right)=\frac{2t-1}{2},$ we find that the condition in Equation (41) fails, while the rest of the conditions are satisfied. Therefore, Equation (42) has a nonoscillatory solution $\psi \left(\iota \right)=1/e^{\iota }.$

5. Conclusions

In this paper, we presented some new results about the oscillation behavior of Equation (8) when the related Equation (1) is nonoscillatory. By using the Riccati transformation technique and the principle of comparison with a first-order delay differential equation, we obtained criteria that guarantee the oscillation of all solutions of the studied equation. Our study is an improvement and an extension of the results found in the literature [13,23,25]:

Remark 3.

To obtain more results about the oscillation of Equation (8), the results presented in this work can be extended by the equation

$${\left(\nu \left(\iota \right){\left({\left(\Omega \left(\iota \right)+{\int }_c^d\rho \left(\iota ,\zeta \right)\Omega \left(\varsigma \left(\iota ,\zeta \right)\right)\mathrm{d}\zeta \right)}^{\prime }\right)}^{\alpha }\right)}^{″}+\eta \left(\iota \right){\left({\psi }^{\prime }\left(\iota \right)\right)}^{\alpha }+{\int }_c^d\gamma \left(\iota ,\zeta \right)ϝ\left(\Omega \left(\varrho \left(\iota ,\zeta \right)\right)\right)\mathrm{d}\zeta =0.$$

On the other hand, new conditions can be obtained without the restrictions

$$\eta \ge 0,\varrho \circ \varsigma =\varsigma \circ \varrho \mathit{and}{\varsigma }^{\prime }\ge {\varsigma }_0\ge 0,{\varrho }^{\prime }\left(\iota \right)\ge {\varrho }_0\ge 0.$$