Some Remarks on the Best Ulam Constant

The problem of Ulam stability for equations can be stated in terms of how much the mappings satisfying the equations approximately (in a sense) differ from the exact solutions of these equations. One of the best known results in this area is the following: Let g be a mapping from a normed space V into a Banach space B. Let ξ≥0 and t≠1 be fixed real numbers and g:V→B satisfy the inequality ∥g(u+v)−g(u)−g(v)∥ ≤ξ(∥u∥t+∥v∥t) for u,v∈V∖{0}. Then, there exists a unique additive f:V→B fulfilling the inequality ∥g(u)−f(u)∥ ≤ξ|1−2t−1|−1∥u∥t for u∈V∖{0}. There arises a natural problem if the constant, on the right hand side of the latter inequality, is the best possible. It is known as the problem of the best Ulam constant. We discuss this problem, as well as several related issues, show possible generalizations of the existing results, and indicate open problems. To make this publication more accessible to a wider audience, we limit the related information, avoid advanced generalizations, and mainly focus only on the additive Cauchy equation f(x+y)=f(x)+f(y) and on the general linear difference equation xn+p=a1xn+p−1+…+apxn+bn (considered for sequences in a Banach space). In particular, we show that there is a significant symmetry between Ulam constants of several functional equations and of their inhomogeneous or radical forms. We hope that in this way we will stimulate further research in this area.