Hyers–Ulam stability and discrete dichotomy for difference periodic systems

Denote by Z₊ the set of all nonnegative integer numbers. Let A_n be an m×m invertible q-periodic complex matrix, for all n∈Z₊ and some positive integers m and q. First we prove that the discrete problem x_n+1=A_nx_n,x_n∈C^m is Hyers–Ulam stable if and only if the monodromy matrix T_q associated to the family A={A_n}_n∈Z+ possesses a discrete dichotomy. Let (a_n), (b_n) be complex valued 2-periodic sequences. Consider the non-autonomous recurrence z_n+2=a_nz_n+1+b_nz_n,n∈Z₊,z_n∈C and the matrixA_n=(11a_n+b_n−1a_n−1),n∈Z₊. We prove that the recurrence (a_n,b_n) is Hyers–Ulam stable if and only if the monodromy matrix T₂:=A₁A₀ has no eigenvalues on the unit circle.