Asymptotic properties of solutions of second-order difference equations. (English) Zbl 1087.39003

Morchało, Jarosław

Arch. Math., Brno 38, No. 1, 15-26 (2002).

The author investigates the asymptotic relationship between the linear Sturm-Liouville difference equation \(\Delta (p_{n-1}\Delta x_{n-1})+q_nx_n=0\) and its “perturbation” \(\Delta (p_{n-1}\Delta x_{n-1})+q_nx_n= f(n,x_n,\Delta x_{n-1})\). It is shown that if the nonlinearity \(f\) in the second equation is small, in a certain sense, then both equations are asymptotically equivalent, again in a certain sense. The results of the paper are proved using the variation of parameters formula coupled with the Schauder fixed point theorem. The obtained asymptotic formulas are a discrete analogue of the results given in the paper J. Kuben [Czech. Math. J. 34, 189–202 (1984; Zbl 0555.34048)].

Reviewer: Ondřej Došlý (Brno)

MSC:

39A10

Additive difference equations

Keywords:

asymptotic equivalence; difference inequalities

Citations:

Zbl 0555.34048