zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
The modified Hyers-Ulam-Rassias stability of a cubic type functional equation. (English) Zbl 1087.39027

The functional equation

f(x+y+2z)+f(x+y-2z)+f(2x)+f(2y)+7f(x)+7f(-x)=2f ( x + y ) + 2 f ( x + z ) + 2 f ( x - z ) + 2 f ( y + z ) + 2 f ( y - z )(1)

of a cubic type (fulfilled e.g. by f(x)=ax 3 +b) is considered for functions mapping a real vector space X into a Banach space Y. Its general solution is given and the stability in the sense of Hyers, Ulam, Rassias and Găvruta is proved. Instead of the classical method of the “Hyers sequence” the so-called fixed point alternative is used in the proof. The desired cubic function near the approximate solution of (1) is the fixed point of some operator acting on functions g:XY such that g(0)=0.

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