## Asymptotic behaviour of solutions of the second order difference equations.(English)Zbl 0799.39001

Under certain conditions, for the solutions of $$(*)$$ $$\Delta^ 2 y_ n = a_ n y_{n + 1} + f_ n (y_ n)$$ with $$\Delta y_ n = y_{n + 1} - y_ n$$ there are proved the representations $$y_ n = \alpha_ n u_ n + \beta_ n v_ n$$, where $$\alpha_ n \to \alpha$$, $$\beta_ n \to \beta$$ for $$n \to \infty$$, and $$u_ n$$, $$v_ n$$ are linearly independent solutions of $$(*)$$ with $$f_ n \equiv 0$$. In case of $$a_ n \equiv 0$$ in $$(*)$$, this result is sharpened to $$y_ n = \alpha n + \beta + o(1)$$ for $$n \to \infty$$.
Reviewer: L.Berg (Rostock)