Spectral characterizations for Hyers-Ulam stability

Abstract

First we prove that an $n\times n$ complex linear system is Hyers-Ulam stable if and only if it is dichotomic (i.e. its associated matrix has no eigenvalues on the imaginary axis $i\mathbb{R}$). Also we show that the scalar differential equation of order $n,$ \[\begin{split} x^{(n)}(t)=a_1x^{(n-1)}(t)+\ldots+a_{n-1}{x}'(t)+a_nx(t),\quad t\in\mathbb{R}_+:=[0, \infty), \end{split}\] is Hyers-Ulam stable if and only if the algebraic equation \[ \begin{split} z^n=a_1z^{n-1}+\cdots +a_{n-1}z+a_n, \end{split} \] has no roots on the imaginary axis.

Keywords

• diferential equations
• dichotomy
• hyers-ulam stability