Oscillation and nonoscillation theorems for some mixed difference equations (Q1196609)

Let a be a positive real number, let \(\Delta_ a\) denote the difference operator defined by \(\Delta_ ax_ n:=x_{n+1}-ax_ n\) and put \(\Delta:=\Delta_ 1\). Difference equations of the following kind are considered: (1) \(\Delta^ i(\Delta_ ax_ n)+(-1)^ jb_ nx_ n=0\). Here \(i\in\{2,3\}\), \(j\in\{1,2\}\) and \((b_ n)\) is a sequence of positive real numbers with positive limit inferior. A sequence of real numbers is called nonoscillatory if it is eventually positive or negative, otherwise it is called oscillatory. The oscillatory (nonoscillatory) behaviour of the solutions of (1) is investigated.