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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 } .

International audienceLet $u_{\mu, x, s}(\cdot, 0)$ be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space $X$ $$\left\{\begin{array}{lc} \dot{u}(t) = A(t)u(t)+e^{i\mu t}x, \quad t>s \\ u(s) = 0. \end{array} \right.$$ Here, $x$ is a vector in $X,$ $\mu$ is a real number, $q$ is a positive real number and $A(\cdot)$ is a $q$-periodic linear operator valued function. Under some natu\-ral assumptions on the evolution family $\mathcal{U}=\{U(t, s): t\ge s\}$ gene\-rated by the family $\{A(t)\},$ we prove that if for each $\mu$, each $s\ge 0$ and every $x$ the solution $u_{\mu, x, s}(\cdot, 0)$ is bounded on ${\bf R}_+$ by a positive constant, depending only on $x,$ then the family $\mathcal{U}$ is uniformly exponentially stable. The approach is based on the theory of evolution semigroups

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) 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 ∈ Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ 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 ( q ) does not intersect the unit circle.

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