On the Best Ulam Constant of the Linear Differential Operator with Constant Coefficients

Abstract

The linear differential operator with constant coefficients D(y)=y(n)+a1y(n−1)+…+any,y∈Cn(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 rk satisfying Rerk>0,1≤k≤n, then the best Ulam constant of D is KD=1|V|∫0∞|∑k=1n(−1)kVke−rkx|dx, where V=V(r1,r2,…,rn) and Vk=V(r1,…,rk−1,rk+1,…,rn),1≤k≤n, are Vandermonde determinants.

Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

“…Note that in (9) above with η = 2 and c = 3 and a homogeneous right-hand side, the Hyers-Ulam constant in this example is K = 1 c = 1 3 as well. Moreover, using different methods, the authors in [22] (Theorem 5) prove that (2) with a = 1 and b…”

“…A key motivation of this work is to compare the results on Hyers-Ulam stability (HUS) derived from integral transform methods with those derived using other techniques, such as in [21,22], often through carefully chosen examples. Another motivation is to bridge some of the gaps that may exist in the results between using one method versus another.…”

Best Constant in Ulam Stability for the Third Order Linear Differential Operator with Constant Coefficients

“…, where r k denote the roots of the characteristic equation. Important steps in finding the best Ulam constant of the same operator were made in [20], where the best Ulam constant was obtained only for the case of distinct roots in the characteristic equation.…”