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

TL;DR

This paper characterizes Ulam stability for higher order linear difference equations in Banach spaces, establishing conditions based on roots of the characteristic equation and deriving an explicit formula for the best Ulam constant when roots are outside the unit circle.

Contribution

It provides a necessary and sufficient condition for Ulam stability and derives an explicit formula for the best Ulam constant in terms of roots and Vandermonde determinants.

Findings

Ulam stability occurs if roots are outside the unit circle.

Explicit formula for the best Ulam constant is derived.

The formula involves roots and Vandermonde determinants.

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.