Unbounded solutions of quasi-linear difference equations

We study positive increasing solutions of the nonlinear difference equation δ(anφp(δχn))=bnf(χn+1, φp(u)=up-2u, p>1 where {an}, {bn} are positive real sequences for n ≥ 1, fR → R is continuous with uf(u) > 0 for u ≠ 0. A full characterization of limit behavior of all these solutions in terms of an, bn is established. Examples, showing the essential role of used hypotheses, are also included. The tools used are the Schauder fixed-point theorem and a comparison method based on the reciprocity principle.