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

Abstract

The linear differential operator with constant coefficients $$D(y)=y^{(n)}+a_1 y^{(n-1)}+\ldots+a_n y,\quad y\in \mathcal{C}^{n}(\mathbb{R}, X)$$ 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.