On the best Ulam constant of a higher order linear difference equation

Abstract

In a Banach space $X$ the linear difference equation with constant coefficients $x_{n+p} = a_1x_{n+p-1} +\ldots + a_px_n,$ is Ulam stable if and only if the roots $r_k,$ $1\leq k\leq p,$ of its characteristic equation do not belong to the unit circle. If $|r_k| > 1,$ $1\leq k\leq p,$ we prove that the best Ulam constant of this equation is $ \frac{1}{|V|}\sum\limits_{s=1}^{\infty}|\frac{V_1}{r_1^s}-\frac{V_2}{r_2^s}+\ldots +\frac{(-1)^{p+1}V_p}{r_p^s}|,$ where $ V = V (r_1, r_2, \ldots, r_p)$ and $V_k =V (r_1,\ldots, r_{k-1}, r_{k+1},\ldots, r_p),$ $1\leq k\leq p,$ are Vadermonde determinants.