Oscillation criteria for certain second order difference equations
The author considers second order difference equations of the form \[ \Delta^ 2y_{n-1}+\sum^ m_{i=1}q_{i,n}f_ i(y_ n)g_ i(\Delta y_{n-1})=0,\;n\in \mathbb{N}, \tag{*} \] and gives (in four theorems) sufficient conditions for all solutions of the equation \((*)\) to be oscillatory. In the reviewer's opinion some additional assumptions should be added, generally for the functions \(g_ i\), because it is easy to find equations of the type \((*)\) such that all assumptions are satisfied, nevertheless the equation possesses the nonoscillatory solution \(y_ n \equiv 1\) (on \(\mathbb{N})\), in fact a family of constant solutions. This refers to Lemma 2 and all theorems. Let us take (for Lemma 2) \(m=1\), \(q_{1,n} =n^ 2\), \(g(u)=u^ 2\), \(f(u)=u\). The sequence \(y_ n=1\), \(n \in \mathbb{N}\) is a positive solution of \((*)\) but the thesis \(y_{n+1}>y_ n\) and \(\Delta y_ n>0\) falls down (see also applications of this lemma in the proofs of the theorems). Similar results for the discrete Emden- Fowler equation were obtained by \textit{J. W. Hooker} and \textit{W. T. Patula} [J. Math. Anal. Appl. 91, 9-29 (1983; Zbl 0508.39005)].
scientific article; zbMATH DE number 428738