Hyers-Ulam stability for linear differences with time dependent and periodic coefficients: The case when the monodromy matrix has simple eigenvalues

Let q ≥ 2 be a positive integer and let (a_j), (b_j) and (c_j) (with j nonnegative integer) be three given C-valued and q-periodic sequences. Let A(q) := A_q-1· · ·A₀, where A_j is defined below. Assume that the eigenvalues x, y, z of the "monodromy matrix" A(q) verify the condition (x -y)(y- z)(z - x) ≠ 0. We prove that the linear recurrence in C x_n+3 = a_nx_n+2 + b_nx_n+1 + c_nx_n, n ∈ Z₊ is Hyers-Ulam stable if and only if (|x| - 1)(|y| - 1)(|z| - 1) ≠ 0, i.e., the spectrum of A(q) does not intersect the unit circle Γ := (w ∈ C: |w| = 1).