Hyers-Ulam stability for linear differences with time dependent and periodic coefficients

Let q ≥ 2 be a positive integer and let (a _j ), (b _j ), and (c _j ) (with j a non-negative integer) be three given C- valued and q-periodic sequences. Let A(q) := A _q-1 . . . A ₀ , where A _j is as is given below. Assuming that the "monodromy matrix" A(q) has at least one multiple eigenvalue, we prove that the linear scalar recurrence x _n+3 = a _n x _n+2 + b _n x _n+1 + cnxn, n ∈ Z ₊ is Hyers-Ulam stable if and only if the spectrum of A(q) does not intersect the unit circle G := w ∈ C: |w| = 1. Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A(q) does not intersect the unit circle.