Hyers-Ulam stability and discrete dichotomy

Let m be a given positive integer and let A be an m×m complex matrix. We prove that the discrete systemXn+1=AXn,n∈Z+ is Hyers-Ulam stable if and only if the matrix A possesses a discrete dichotomy. Also we prove that the scalar difference equation of order mxn+m=a1xn+m−1+a2xn+m−2+⋯+amxn,n∈Z+, is Hyers-Ulam stable if and only if the algebraic equationzm=a1zm−1+⋯+am−1z+am,z∈C has no roots on the unit circle. This latter result is essentially known, for further details see for example [24] and [2]. However, our proofs are completely different and moreover, it seems that our approach opens the way to obtain many other results in this topic.