Hyers–Ulam stability and discrete dichotomy

Abstract

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.

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

“…The relationship between exponential stability and Hyers-Ulam stability has been studied in the articles [3,8,9,17,18], and this article continues these studies.…”

“…The history of the Ulam problem (concerning the stability of a functional equation) and of stability in the sense of Hyers-Ulam is well known. In particular, Hyers-Ulam stability for linear recurrences and for systems of linear recurrences is considered in [3][4][5][6][7][8][9][10][11][12][13][14][15][16], and the references therein.…”

A note on envelopes of homotopies

“…In the following theorems, A denotes a unital Banach algebra with unit 1 and K denotes an integer greater than 1. Recently, the reader can find the notion of (generalized) Hyers -Ulam stability in many papers [6,11,20,25,26].…”

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

“…Here, we continue the analysis started in [3] and, finally, we complete the discussion raised in [1] for periodic linear recurrences of order three. Thus, this article can be seen as a new link in the chain of articles [1][2][3][4][5] which address the Hyers-Ulam stability of linear scalar recurrences. The connections of this topic to those existing in the literature was already presented in [3], so we do not present them again here.…”