Bounded oscillation for second-order nonlinear neutral difference equations in critical and non-critical states

The sufficient and necessary conditions for the bounded oscillation of the second order nonlinear neutral delay difference equation Δ 2 [ x ( n ) - px ( n - τ ) ] = ∑ i = 1 m q i x ( n - σ i ) + f ( n , x ( n - η 1 ( n ) ) , … , x ( n - η l ( n ) ) ) are obtained in critical and non-critical cases.