Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations Δ(anΔxn)=qnxn+1 and Δ(anΔxn)=qnf(xn+1) where an > 0, qn > 0, and f: R→R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.