The best Ulam constant of the n-th order linear differential operator with constant coefficients

Abstract

Let $X$ be a Banach space over the field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, and let $L:{\mathcal{C}}^n(\mathbb{R},X)\to \mathcal{C}(\mathbb{R},X)$ be the $n$-th order linear differential operator defined by $Ly:=y^{(n)}+a_1{y}^{(n-1)}+\cdots +a_{n}y$, where $a_1,\dots ,a_n\in \mathbb{K}$ are fixed scalars and $n$ is a fixed positive integer. It is well known that $L$ is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. In this paper, we determine the best Ulam constant of $L$ under this condition, for an arbitrary order $n$ and without any restriction on the multiplicities of the roots of the characteristic equation. This improves and completes several known Ulam stability results for this operator and provides the best estimate possible for the error between approximate and exact solutions of the equation $Ly=f$, where $f\in \mathcal{C}(\mathbb{R},X)$. Moreover, as an auxiliary result, we give an explicit formula for the general solution of this equation.