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


39A10 Additive difference equations
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