In this paper we provide a characterization theorem for set-valued solutions of the functional equation for the case of compact, convex valued maps. We prove that every solution of the functional equation is the sum of an additive single-valued function and a nonempty compact and convex set.

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