Oscillation of a family of -difference equations

We obtain the complete classification of oscillation and nonoscillation for the q -difference equation x Δ Δ ( t ) + b ( − 1 ) n t c x ( q t ) = 0 , b ≠ 0 , where t = q n ∈ T = q N 0 , q > 1 , c , b ∈ R . In particular we prove that this q -difference equation is nonoscillatory, if c > 2 and is oscillatory, if c < 2 . In the critical case c = 2 we show that it is oscillatory, if | b | > 1 q ( q − 1 ) , and is nonoscillatory, if | b | ≤ 1 q ( q − 1 ) .