We establish sufficient conditions for the existence and uniqueness of (N,λ)-periodic solutions for the following abstract model: Δαu(n) = Au(n + 1) + f(n, u(n)), n ∈ Z, where 0 < α ≤ 1, A is a closed linear operator defined in a Banach space X, Δα denotes the fractional difference operator in the Weyl-like sense, and f satisfies appropriate conditions.