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On existence of positive solutions and bounded oscillations for neutral difference equations. (English) Zbl 0763.39002
The authors consider the linear homogeneous difference equation $$\Delta^ \alpha(x_ n-cx_{n-m})+p_ n x_{n-k}=0$$, where $$c,p_ n\geq 0$$. Here $$\Delta$$ is the forward difference operator, $$\Delta x_ n=x_{n+1}-x_ n$$, and $$\alpha=1,2$$. It is shown that positive solutions tending to zero or $$\infty$$ or remaining bounded, or oscillatory solutions, respectively, exist under appropriate conditions on $$c$$ and $$p_ n$$.
Reviewer’s remark: The right-hand sides of equations (2.17) and (2.20) should read $$(n+3)/(n^ 2(n+1))$$ and $$(n+3)/(n(n+1))$$, respectively, for these equations to have the indicated solutions.