On the best Ulam constant of the linear differential operator with constant coefficients

TL;DR

This paper determines the exact Ulam stability constant for linear differential operators with constant coefficients in Banach spaces, linking stability to the roots of the characteristic equation and providing an explicit integral formula for the constant.

Contribution

It establishes a precise formula for the best Ulam constant based on the roots of the characteristic equation, extending stability analysis in Banach space differential operators.

Findings

Ulam stability occurs iff no roots on the imaginary axis

Explicit formula for the best Ulam constant involving Vandermonde determinants

Integral expression for the Ulam constant when roots have positive real parts

Abstract

The linear differential operator with constant coefficients $$D(y)=y^{(n)}+a_1 y^{(n-1)}+\ldots+a_n y,\quad y\in \mathcal{C}^{n}(\mathbb{R}, X)$$ acting in a Banach space $X$ is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of $D$ has distinct roots $r_k$ satisfying $\Real r_k>0,$ $1\leq k\le n,$ then the best Ulam constant of $D$ is $K_D=\frac{1}{|V|}\int_{0}^{\infty}\left|\sum\limits_{k=1}^n(-1)^kV_ke^{-r_k x}\right|dx,$ where $V=V(r_1,r_2,\ldots,r_n)$ and $V_k=V(r_1,\ldots,r_{k-1},r_{k+1}, \ldots, r_n),$ $1\leq k\leq n,$ are Vandermonde determinants.