## 1 Introduction  
In recent decades, many authors have studied problems of a number of different classes of advanced differential equations including the asymptotic and oscillatory behavior of their solutions, see [1; 2; 3; 4; 5; 6; 7; 8] and the references cited therein. For some more recent oscillation results, see [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. The interest in studying advanced differential equations is also caused by the fact that they appear in models of several areas in science. In [21; 22; 23], singular systems of differential equations are used to study the dynamics and stability properties of electrical power systems. Some additional mathematical background on this can be found in [24]. Systems of differential equations with delays are used to study additional properties of electrical power systems in [25; 26]. Non-linear advanced differential equations can be used to describe complex dynamical networks, see [27; 28; 29], and bring new insight to their stability. Furthermore, this type of equations can be also used in the modeling of dynamical networks of interacting free-bodies, see [30]. Finally, properties of advanced differential equations are used in the study of singular differential equations of fractional order, see [31; 32]. Several other examples in Physics can be found in [33]. In this paper, we consider an even-order non-linear advanced differential equation with a non-canonical operator of the following type:  
\[L_{y}+q\left(v\right)g\left(y\left(\eta\left(v\right)\right)\right)=0,\ \ \ \ \ L_{y}:=\left(a\left(v\right)\left(y^{\left(k-1\right)}\left(v\right)\right)^{ \beta}\right)^{\prime}, \tag{1}\]  
where \(\ v\geq v_{0},\kappa\) is even and \(\beta\) is a quotient of odd positive integers. The operator \(L_{y}\) is said to be in canonical form if \(\int_{v_{0}}^{\infty}a^{-1/\beta}\left(s\right)\mathrm{d}s=\infty\); otherwise, it is called noncanonical. Throughout this work, we suppose that:  
1. \(a\in C^{1}\left(\left[v_{0},\infty\right),\mathbb{R}\right),\,a\left(v\right) >0,\ a^{\prime}\left(v\right)\geq 0,\)
2. \(q,\eta\in C\left(\left[v_{0},\infty\right),\mathbb{R}\right),\,q\left(v\right) \geq 0,\)\(\eta\left(v\right)\geq v,\)\(\lim\limits_{v\to\infty}\eta\left(v\right)=\infty,\)
3. \(g\in C\left(\mathbb{R},\mathbb{R}\right)\) such that \(g\left(x\right)/x^{\beta}\geq k>0,\) for \(x\
eq 0\) and under the condition \[\zeta\left(v\right)=\int_{v_{0}}^{\infty}\frac{1}{a^{1/\beta}\left(s\right)} \mathrm{d}s<\infty.\] (2)  
**Definition 1**. : _The function \(y\in C^{\kappa-1}[v_{y},\infty)\), \(v_{y}\geq v_{0}\), is called a solution of (1), if \(\left(y^{\left(\kappa-1\right)}\left(v\right)\right)^{\beta}\in C^{1}[v_{y}, \infty)\), for \(a\in C^{1}\left([v_{0},\infty),\mathbb{R}\right),\ a\left(v\right)>0\) and \(y\left(v\right)\) satisfies (1) on \([v_{y},\infty)\)._  
**Definition 2**. : _Let_  
\[D=\left\{\left(v,s\right)\in\mathbb{R}^{2}:v\geq s\geq v_{0}\right\}\text{ and }D_{0}=\left\{\left(v,s\right)\in\mathbb{R}^{2}:v>s\geq v_{0}\right\}.\]  
_A kernel function \(H_{i}\in C\left(D,\mathbb{R}\right)\) is said to belong to the function class \(\Im\), written by \(H\in\Im\), if, for \(i=1,2\),_  
1. \(H_{i}\left(v,s\right)>0\)_, on_ \(D_{0}\) _and_ \(H_{i}\left(v,s\right)=0\) _for_ \(v\geq v_{0}\) _with_ \(\left(v,s\right)\
otin D_{0}\)_;_
2.