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

Abstract

The linear differential operator with constant coefficients $$D(y)=y^{(n)}+a_1{y}^{(n-1)}+\dots +a_{n}y,y\in {\mathcal{C}}^n(\mathbb{R},X)$$ 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 $\Real r_k>0,$ $1\le k\le n,$ then the best Ulam constant of $D$ is $K_D=\frac{1}{|V|}{\int }_0^{\infty }\left|\sum _{k=1}^n(-1{)}^k{V}_k{e}^{-r_{k}x}\right|dx,$ where $V=V(r_1,r_2,\dots ,r_n)$ and $V_k=V(r_1,\dots ,r_{k-1},r_{k+1},\dots ,r_n),$ $1\le k\le n,$ are Vandermonde determinants.