Best Ulam Constants for Higher-Order Linear Difference Equations in the Hyperbolic Case

For constant-coefficient higher-order linear difference equations on the nonnegative integers with values in a nonzero real or complex Banach space, the best Ulam constant is determined throughout the hyperbolic regime, where no characteristic root lies on the unit circle. Yuan, Xu and Brzdęk (2025) obtained a Green-function Ulam bound in this regime, but when there are at least two distinct characteristic roots and at least one lies inside the unit circle, neither the existence of a best constant nor the optimality of that bound was known. The Green bound is optimal for every hyperbolic root configuration. The key point is that bounded homogeneous corrections have no unstable component and hence decay at a uniform exponential rate. A finitely supported perturbation can then be translated far along the half-line and phase-aligned so that the corresponding Green correction asymptotically realizes the \( \ell^1\ \)-bound furnished by the Green kernel. Consequently, the best Ulam constant exists in every hyperbolic case and equals the \( \ell^1\ \)-norm of the Green kernel. The qualitative criterion that strong Hyers–Ulam stability is equivalent to the absence of characteristic roots on the unit circle was already known. Together with that criterion, the result gives the exact best constant whenever the equation is strongly Hyers–Ulam stable.