Letqbe a positive integer and let (a_n) and (b_n) be two given ℂ-valued andq-periodic sequences. First we prove that the linear recurrence in ℂ 0.1 $$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$ is Hyers–Ulam stable if and only if the spectrum of the monodromy matrixT_q: =A_q−1· · ·A₀(i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = {z∈ ℂ: |z| = 1}, i.e.T_qis hyperbolic. Here (and in as follows) we let 0.2 $$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$ Secondly we prove that the linear differential equation 0.3 $${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$ (wherea(t) andb(t) are ℂ-valued continuous and 1-periodic functions defined on ℝ) is Hyers–Ulam stable if and only ifP(1) is hyperbolic; hereP(t) denotes the solution of the first-order matrix 2-dimensional differential system 0.4 $${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$ whereI₂is the identity matrix of order 2 and 0.5 $$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

Let $q\ge 2$ be a positive integer and let $\left(a_j\right)$ , $\left(b_j\right)$ , and $\left(c_j\right)$ (with j a non-negative integer) be three given $\mathbb{C}$ -valued and q-periodic sequences. Let $A\left(q\right):=A_{q-1}\cdots A_0$ , where $A_j$ is as is given below. Assuming that the “monodromy matrix” $A\left(q\right)$ has at least one multiple eigenvalue, we prove that the linear scalar recurrence $x_{n+3}=a_n{x}_{n+2}+b_n{x}_{n+1}+c_n{x}_n,n\in {\mathbb{Z}}_{+}$ is Hyers-Ulam stable if and only if the spectrum of $A\left(q\right)$ does not intersect the unit circle $\mathsf{\Gamma }:=\{w\in \mathbb{C}:|w|=1\}$ . 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 $A\left(q\right)$ does not intersect the unit circle.

Let $q\ge 2$ be a positive integer and let $\left(a_j\right),\left(b_j\right)$ and $\left(c_j\right)$ (with j nonnegative integer) be three given $\mathbb{C}$ -valued and q-periodic sequences. Let $A\left(q\right):=A_{q-1}\cdots A_0,$ where $A_j$ is defined below. Assume that the eigenvalues $x,y,z$ of the “monodromy matrix” $A\left(q\right)$ verify the condition $(x-y)(y-z)(z-x)\ne 0.$ We prove that the linear recurrence in $\mathbb{C}$ $x_{n+3}=a_n{x}_{n+2}+b_n{x}_{n+1}+c_n{x}_n,n\in {\mathbb{Z}}_{+}$ is Hyers–Ulam stable if and only if $\left(\right|x|-1)\left(\right|y|-1)\left(\right|z|-1)\ne 0,$ i.e., the spectrum of $A\left(q\right)$ does not intersect the unit circle $\mathsf{\Gamma }:=\{w\in \mathbb{C}:|w|=1\}.$

Let be a strongly continuous and -periodic evolution family acting on a complex Banach space . We prove that if and then the growth bound of the family is less than or equal to .