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 Drygas functional equation on groups. (English) Zbl 1130.39023

Suppose that G is a group. The system of functional equations

f(xy)+f(xy -1 )-2f(x)-f(y)-f(y -1 )=0f(yx)+f(y -1 x)-2f(x)-f(y)-f(y -1 )=0(*)

where x,yG is called stable if for any f:G satisfying the system of inequalities

|f(xy)+f(xy -1 )-2f(x)-f(y)-f(y -1 )|δ|f(yx)+f(y -1 x)-2f(x)-f(y)-f(y -1 )|δ

for some positive number δ, there is a solution φ of (*) and a positive number ε such that |f(x)-φ(x)|ε (xG).

In this paper the authors prove that the system (*) is not stable on an arbitrary group, in general; the system is stable on Heisenberg group

UT(3,K)1yt01x001:x,y,tK,

where K is a (commutative) field with characteristic different from two; the system is stable on certain class of n-Abelian groups; and finally that any group can be embedded into a group, where the system (*) is stable. See also V. A. Fauiziev and P. K. Sahoo [Stability of Drygas functional equation on T(3,), Int. J. Appl. Math. Stat. 7, No. Fe07, 70–81 (2007)].

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
39B72Systems of functional equations and inequalities