## 1 Introduction
\(H_{i}\left(v,s\right)\) _has a continuous and nonpositive partial derivative_ \(\partial H_{i}/\partial s\) _on_ \(D_{0}\) _and there exist functions_ \(\tau,\vartheta\in C^{1}\left(\left[v_{0},\infty\right),\left(0,\infty\right)\right)\) _and_ \(h_{i}\in C\left(D_{0},\mathbb{R}\right)\) _such that_ \[\frac{\partial}{\partial s}H_{1}\left(v,s\right)+\frac{\tau^{\prime}\left(s \right)}{\tau\left(s\right)}H_{1}\left(v,s\right)=h_{1}\left(v,s\right)H_{1}^{ \beta/\left(\beta+1\right)}\left(v,s\right)\] (3) _and_ \[\frac{\partial}{\partial s}H_{2}\left(v,s\right)+\frac{\vartheta^{\prime} \left(s\right)}{\vartheta\left(s\right)}H_{2}\left(v,s\right)=h_{2}\left(v,s \right)\sqrt{H_{2}\left(v,s\right)}.\] (4)  
Next we will discuss the results in [34; 35; 36]. Actually, our purpose in this article is to complement and improve these results. [PERSON] et al. in [34; 35] studied the even-order nonlinear advanced differential equations  
\[\left(\left(y^{\left(\kappa-1\right)}\left(v\right)\right)^{\beta}\right)^{ \prime}+q\left(v\right)y^{\beta}\left(\eta\left(v\right)\right)=0. \tag{5}\]  
By means of the Riccati transformation technique, the authors established some oscillation criteria of (5). [PERSON] and [PERSON] [36] investigated the second-order neutral Emden-Fowler delay dynamic equations  
\[y^{\left(\kappa\right)}\left(v\right)+q\left(v\right)y\left(\eta\left(v \right)\right)=0, \tag{6}\]  
and established some new oscillation for (5) under the condition  
\[\int_{v_{0}}^{\infty}\frac{1}{a^{1/\beta}\left(s\right)}\mathrm{d}s=\infty. \tag{7}\]  
To prove this, we apply the previous results to the equation  
\[y^{\left(\kappa\right)}\left(v\right)+\frac{q_{0}}{v^{\kappa}}y\left(\lambda v \right)=0,\ v\geq 1. \tag{8}\]  
if we set \(\kappa=4\) and \(\lambda=2\), then by applying conditions in [34; 35; 36] on Equation (8), we find the results in [35] improves those in [36]. Moreover, the those in [34] improves results in [35; 36]. Thus, the motivation in our paper is to complement and improve results in [34; 35; 36]. We will use the following methods:  
* Integral averaging technique. * Riccati transformations technique. * Method of comparison with second-order differential equations. We will also use the following lemmas from (1):  
**Lemma 1** ([3]). : _If \(y^{\left(\right)}\left(v\right)>0\), \(i=0,1,...,\kappa\), and \(y^{\left(\kappa+1\right)}\left(v\right)<0\), then_  
\[\frac{y\left(v\right)}{v^{\kappa}/\kappa!}\geq\frac{y^{\prime}\left(v\right) }{v^{\kappa-1}/\left(\kappa-1\right)!}.\]  
**Lemma 2** ([19]). : _Suppose that \(y\in C^{\kappa}\left(\left[v_{0},\infty\right),\left(0,\infty\right)\right),y^{ \left(x\right)}\) is of a fixed sign on \(\left[v_{0},\infty\right),y^{\left(x\right)}\) not identically zero and there exists a \(v_{1}\geq v_{0}\) such that_  
\[\zeta\left(v\right)=\int_{v_{0}}^{\infty}\frac{1}{a^{1/\beta}\left(s\right)} \mathrm{d}s<\infty.\]0  
_for all \(v\geq v_{1}\). If we have \(\lim_{v\rightarrow\infty}y\left(v\right)\
eq 0\), then there exists \(v_{\theta}\geq v_{1}\) such that_  
\[\zeta\left(v\right)=\int_{v_{0}}^{\infty}\frac{1}{a^{1/\beta}\left(s\right)} \mathrm{d}s<\infty.\]1  
_for every \(\theta\in\left(0,1\right)\) and \(v\geq v_{\theta}\)._  
**Lemma 3** ([2]). : _Let \(\beta\) be a ratio of two odd numbers, \(V>0\) and \(U\) are constants. Then_  
\[\zeta\left(v\right)=\int_{v_{0}}^{\infty}\frac{1}{a^{1/\beta}\left(s\right)} \mathrm{d}s<\infty.\]2  
**Lemma 4**. : _Suppose that \(y\) is an eventually positive solution of (1). Then, there exist three possible cases:_  
\[