Hyers–Ulam stability and discrete dichotomy for difference periodic systems

Denote by Z+Z+ the set of all nonnegative integer numbers. Let AnAn be an m×mm×m invertible q  -periodic complex matrix, for all n∈Z+n∈Z+ and some positive integers m and q. First we prove that the discrete problemequation(AnAn)xn+1=Anxn,xn∈Cm is Hyers–Ulam stable if and only if the monodromy matrix TqTq associated to the family A={An}n∈Z+A={An}n∈Z+ possesses a discrete dichotomy.Let (an)(an), (bn)(bn) be complex valued 2-periodic sequences. Consider the non-autonomous recurrenceequation(an,bnan,bn)zn+2=anzn+1+bnzn,n∈Z+,zn∈C and the matrixAn=(11an+bn−1an−1),n∈Z+. We prove that the recurrence (an,bn)(an,bn) is Hyers–Ulam stable if and only if the monodromy matrix T2:=A1A0T2:=A1A0 has no eigenvalues on the unit circle.