Positive decreasing solutions of quasi-linear difference equations

The second-order nonlinear difference equation where {an}, {bn} are positive real sequences for n ≥ 1, f: → is continuous with uf(u) > 0 for u ≠ 0, is considered. A full characterization of limit behavior of all positive decreasing solutions in terms of an, bn is established. The obtained results answer some open problems formulated for p = 2. A comparison with the continuous case jointly with similarities and discrepancies is given as well.