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Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. (English) Zbl 1052.39031

The author formulates, in a general form, the method of proving the Hyers-Ulam stability for functional equations in several variables. This method appeared in his paper [Stochastica 4, No. 1, 23–30 (1980; Zbl 0442.39005)] and has been actually repeated in numerous papers of various authors.

The main result reads as follows: Assume that S is a set, (X,d) a complete metric space and G:SS, H:XX given functions. Let f:SX satisfy the inequality

d(H(f(G(x))),f(x))δ(x),xS

for some function δ:S + . If H is continuous and satisfies:

d(H(u),H(v))ϕ(d(u,v)),u,vX,

for a non-decreasing subadditive function ϕ: + + , and the series i=0 ϕ i (δ(G i (x))) is convergent for every xS, then there exists a unique function F:SX – a solution of the functional equation

H(F(G(x)))=F(x),xS

and satisfying

d(F(x),f(x)) i=0 ϕ i (δ(G i (x)))·

Moreover, an analogous result, for a mapping f satisfying the inequality

H(f(G(x))) f(x)-1δ(x)

is considered.


MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges