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Oscillation and nonoscillation theorems for second-order difference equations. (English) Zbl 0612.39002
A solution of the equation (1) $$\Delta^ 2_ ax_ n=F(n,x_ n,\Delta_ bx_ n)$$ is called nonoscillatory, if it is of constant sign, otherwise it is called oscillatory. Here, $$\Delta_ ax_ n=x_{n+1}-ax_ n,\Delta^ 2_ ax_ n=\Delta_ a(\Delta_ ax_ n)$$, and the function F is defined and finite on $$N\times R^ 2$$, $$N=\{n_ 0,n_ 0+1,...\}$$ with an integer value $$n_ 0$$. For u,v$$\in R$$ denote $$V=v+(b-a)u$$ and $$S=N\times \{(u,v):V=0\}$$. If $$a>0$$, $$F(n,u,v)=0$$ for (n,u,v)$$\in S$$ and $$F(n,u,v)(V+aV^ 2)>0$$ for $$(n,u,v)\in N\times R^ 2\setminus S$$, then every solution of (1) is nonoscillatory; If $$a<0$$, $$F(n,u,v)=0$$ for(n,u,v)$$\in S$$ and $$F(n,u,v)(V+aV^ 2)<0$$ for (n,u,v)$$\in N\times R^ 2\setminus S$$, then every solution of (1) is oscillatory. Some other similar Theorems are proved.
Reviewer: J.Gregor