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 + c n x n , n ∈ Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { 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.