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)
On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces. (English) Zbl 1117.39017

A quasi-norm is a real-valued function on a vector space X satisfying the following: (1) x=0 if and only if x=0; (2) λx=|λ|·x for all scalars λ and all xX; (3) There is a constant K1 such that x+yK(x+y) for all x,yX. Then the pair (X,·) is said to be a quasi-normed space. A quasi-norm · is called a p-norm (0<p1) if x+y p x p +y p (x,yX). By the Aoki-Rolewicz theorem [see S. Rolewicz, Metric linear spaces. 2nd ed. Mathematics and its applications (East European Series), 20. Dordrecht- Boston-Lancaster: D. Reidel Publishing Company, Warszawa: PWN-Polish Scientific Publishers. (1985; Zbl 0573.46001)]), each quasi-norm is equivalent to some p-norm. Since it is much easier to work with p-norms than quasi-norms, henceforth the authors restrict their attention mainly to p-norms.

The functional equation

f(ax+y)+f(x+ay)=(a+1)(a-1) 2 [f(x)+f(y)]+a(a+1)f(x+y)

is called the Euler-Lagrange type cubic functional equation. The authors prove the stability of this equation for fixed integers a with a0,±1 in the framework of quasi-Banach spaces by using the direct method.

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
46B03Isomorphic theory (including renorming) of Banach spaces
46B20Geometry and structure of normed linear spaces
39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)