zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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


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


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


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.

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