Rapidly varying decreasing solutions of half-linear difference equations

In this paper a necessary and sufficient condition is derived for all positive decreasing solutions of a half-linear second order difference equation to be rapidly varying of index minus infinity. Relations with the standard classification of nonoscillatory solutions and with the notion of recessive solutions are also discussed. The results of this paper are complementary to those of a previous paper by the authors, and lead to a complete characterization of positive decreasing solutions with respect to their regularly or rapidly varying behavior.