Comparison theorems for second order nonlinear difference equations

New oscillation results are obtained for the second order nonlinear difference equation Δ rnf (Δxn−1) +g(n, xn) = 0, and its functional form Δ rnf (Δxn−1) +g(n, xτn ) = 0. The role played by the argument τn on the oscillation of the functional equation is explored. In particular, we characterize a class of sequences {τn} which have a harmless effect on the oscillation of this type of equations. Some of our results generalize, improve or unify known fundamental oscillation results for several particular cases of the above equations.  2004 Published by Elsevier Inc.