Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

Abstract

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 ⋯ A 0 , 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 n x n + 2 + b n x n + 1 + c n x 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 } .