We consider the difference equation Δ 2 x n + f ( n , x n + τ ) = 0 , τ = 0 , 1 , … , in the context of a Hilbert space. In this setting, we propose a concept of oscillation with respect to a direction and give sufficient conditions so that all its solutions be directionally oscillatory, as well as conditions which guarantee the existence of directionally positive monotone increasing solutions.