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

Schmeidel, Ewa

Demonstr. Math. 26, No. 3-4, 811-819 (1993).

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)

MSC:

39A10

Additive difference equations

Keywords:

second order difference equations; asymptotic behaviour