Positive decreasing solutions of quasi-linear difference equations

The second-order nonlinear difference equation Delta (a(n)Phi (p) (Deltax(n) )) = b(n)f(x(n)+1), Phi (p)(u) = \u \ (p-2)u, p > 1, where {a(n)}, {b(n)} are positive real sequences for n greater than or equa l to 1, f : R --> R is continuous with uf(u) > 0 for u not equal 0, is cons idered. A full characterization of limit behavior of all positive-decreasin g solutions in terms of a(n), b(n) is established. The obtained results ans wer some open problems formulated for p = 2. A comparison with the continuo us case jointly with similarities and discrepancies is given as well. (C) 2 001 Elsevier Science Ltd. All rights reserved.