Ulam's Type Stability

Quite often (e.g., in applications) we have to do with functions that satisfy some equations only approximately. There arises a natural question what errors we commit when we replace such functions by the exact solutions to those equations. Some tools to evaluate them are provided within the theory of the Ulam (also Hyers-Ulam) type stability. The issue of Ulam's type stability of ( rst, functional, but next also di erence, di erential and integral) equations has been a very popular subject of investigations for the last nearly fty years (see, e.g., [3, 8, 9, 10]). The main motivation for it was given by S.M. Ulam in 1940. The following de nition somehow describes the main ideas of such stability notion for equations in n variables (R+ stands for the set of nonnegative reals). De nition 1. Let A be a nonempty set, (X, d) be a metric space, C ⊂ R+ n be nonempty, T map C into R+, and F1,F2 map a nonempty D ⊂ X into X n . We say that the equation F1φ(x1, . . . , xn) = F2φ(x1, . . . , xn) (1) is T stable provided for every ε ∈ C and φ0 ∈ D with