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

The authors consider the difference equations Delta(a(n)Delta x(n)) = q(n)x(n+1) (*) and Delta(a(n)Delta x(n)) = q(n)f(x(n+1)), (**) where a(n) > 0, q(n) > 0, and f : R --> R is continuous with uf(u) > 0 for u not equal 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. (C) 1998 Elsevier Science Ltd. All rights reserved.