Oscillations of difference equations with positive and negative coefficients (Q803440)

The author continues his studies of oscillatory behavior of solutions of difference equations, see e.g.: the author, \textit{Ch. G. Philos} and \textit{Y. G. Sficas} [J. Appl. Math. Simulation 2, No.2, 101-112 (1989; Zbl 0685.39004); Libertas Math. 9, 121-125 (1989; Zbl 0689.39002)]. Theorem 2: Let p,q: \(N\to {\mathbb{R}}^+\), \(k,r\in N:=\{0,1,2,...\}\), \(p'=\liminf_{n\to \infty}p_ n,\) \(q'=\limsup_{n\to \infty}q_ n,\) \(k>0\) or \(p'-q'>1\), \(q'(k-r)<1\); and every solution of the ``limiting'' equation \((1)\quad \Delta x_ n+p'x_{n-k}-q'x_{n-r}=0,\) \(n\in N\), oscillates. Then every solution of the equation \((2)\quad \Delta y_ n+p_ ny_{n-k}-q_ ny_{n-r}=0,\) \(n\in N\), also oscillates. Sufficient conditions for oscillation of all solutions of the equation (1) (Theorem 1), and for preservation of this property for the equation \(\Delta x_ n+(p'-\epsilon)x_{n-k}-(q'+\epsilon)x_{n-r}=0\) (Lemma 5) are given. See also: the reviewer and \textit{B. Szmanda} [Demonstr. Math. 17, 153-164 (1984; Zbl 0557.39004)].