Nonoscillation and oscillation of second order half-linear difference equations

This paper considers the oscillation problem for the second order half-linear difference equation Δ(∣Δxn−1∣r−1Δxn−1) + qn∣xn∣r−1xn = 0, where {qn}n=0∞ is a nonnegative sequence and r > 0. we establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When r = 1, our results are complete extensions of work by Zhang and Zhou [Oscillation and nonoscillation for second-order linear difference equations, Comp. Math. Appl. 39 (2000) 1-7], and reduce to the results in [Y.G. Sun, Oscillation and nonoscillation for second-order linear difference equations, Appl. Math. Comput. 170 (2005) 1095-1103]. When r > 0, our results generalize and complement Sun's results in [Y.G. Sun, Oscillation and nonoscillation for half-linear second order difference equations, Indian J. Pure Appl. Math. 35 (2004) 133-142].