Oscillation Properties for Advanced Difference Equations

Özkan Öcalan, Ömer Akin

In this paper, we provide some sufficient conditions for the oscillation of every solution of the difference equations \[ x_{n+1}-x_n+p_nx_{n-k}=0, \quad n=0,1,2,dots, \] whenever $k\in\{\dots,-3,-2\}$ and $p_n\leq0$; and also \[ x_{n+1}-x_n+um^m_{i=1}p_{in}x_{n-k_i}=0,\quad n=0,1,2,\dots, \] whenever $k_i\in\{\dots,-3,-2,-1\}$ and $p_{in}\leq0$ for $i=1,2,\dots,m$. We also obtain some alternative results for the oscillation of all solutions of these equations.