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)
On the stability of the monomial functional equation. (English) Zbl 1152.39023

Let X be a normed space, Y a Banach space and n a positive integer. For a mapping f:XY and x,yX we denote

D n f(x,y):= i=0 n n i(-1) n-i f(ix+y)-n!f(x)·

A solution f of the functional equation D n f(x,y)=0 (x,yX) is called a monomial function of degree n. The author proves the following stability results for the monomial equation. Suppose that for np0 a mapping f:XY satisfies

D n f(x,y)εx p +y p ,x,yX·(1)

Then there exists a unique monomial function F:XY of degree n such that

f(x)-F(x)Mε 2 n -2 p x p ,xX

with some explicitly given constant M depending on n and p. A similar stability result can be obtained if f satisfies the inequality (1) with some p<0 and for all x,yX{0}. However, the author proves that in this case the mapping f itself has to be a monomial function of degree n, i.e., surprisingly the superstability phenomenon appears. Applying the above results for n=1,2,3, one gets the (super)stability of additive, quadratic, cubic, ...mappings.

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