Oscillation criteria for second-order nonlinear difference equations of Euler type

The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for the second-order nonlinear difference equation Delta(2)x(n) + 1/n(n + 1) f(x(n)) = 0, where f (x) is continuous on R and satisfies the signum condition xf(x) > 0 if x not equal 0. The obtained results are best possible in a certain sense. Proof is given by means of the Riccati technique and phase plane analysis of a system. A discrete version of the Riemann-Weber generalization of Euler-Cauchy differential equation plays an important role in proving our results.