Letqbe a positive integer and let (an) and (bn) be two given ℂ-valued andq-periodic sequences. First we prove that the linear recurrence in ℂ0.1$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$is Hyers–Ulam stable if and only if the spectrum of the monodromy matrixTq: = Aq−1 · · · A0(i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = {z ∈ ℂ: |z| = 1}, i.e.Tqis hyperbolic. Here (and in as follows) we let0.2$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$Secondly we prove that the linear differential equation0.3$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$(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 system0.4$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$whereI2is the identity matrix of order 2 and0.5$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$
Let m be a given positive integer and let A be an m × m complex matrix. We prove that the discrete systemis Hyers-Ulam stable if and only if the matrix A possesses a discrete dichotomy. Also we prove that the scalar difference equation of order m x n+m = a 1 x n+m−1 + a 2 x n+m−2 + · · · + a m x n , n ∈ Z + , is Hyers-Ulam stable if and only if the algebraic equation z m = a 1 z m−1 + · · · + a m−1 z + a m , 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.
We study the local attractivity of mild solutions of equations in the formwhere A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at in nity.
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