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

Alina Ramona Baias; Dorian Popa

paper

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

arXiv:2007.04654

Abstract

In a Banach space $X$ the linear difference equation with constant coefficients $x_{n + p} = a_{1} x_{n + p - 1} + \dots + a_{p} x_{n},$ 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 $\frac{1}{∣ V ∣} s = 1 \sum \infty ∣ \frac{V _{1}}{r _{1}^{s}} - \frac{V _{2}}{r _{2}^{s}} + \dots + \frac{( - 1 ) ^{p + 1} V _{p}}{r _{p}^{s}} ∣,$ where $V = V (r_{1}, r_{2}, \dots, r_{p})$ and $V_{k} = V (r_{1}, \dots, r_{k - 1}, r_{k + 1}, \dots, r_{p}),$ $1 \leq k \leq p,$ are Vadermonde determinants.