Oscillation theorems for second-order nonlinear delay difference equations

By means of Riccati transformation technique, we establish some new oscillation criteria for second-order nonlinear delay difference equation $$Delta (p_n (Delta x_n )^gamma ) + q_n f(x_{n - sigma } ) = 0,;;;;n = 0,1,2,...,$$ when $sumlimits_{n = 0}^infty {left( {frac{1}{{Pn}}} right)^{frac{1}{gamma }} = infty }$ . When $sumlimits_{n = 0}^infty {left( {frac{1}{{Pn}}} right)^{frac{1}{gamma }} < infty }$ we present some sufficient conditions which guarantee that, every solution oscillates or converges to zero. When $sumlimits_{n = 0}^infty {left( {frac{1}{{Pn}}} right)^{frac{1}{gamma }} = infty }$ holds, our results do not require the nonlinearity to be nondecreasing and are thus applicable to new classes of equations to which most previously known results are not.