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 stability of homomorphisms and amenability, with applications to functional equations. (English) Zbl 0619.39012
Following a well known result of D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.264)], the concept of stability for the homomorphisms from a group into a Banach space is defined. Some consequences of the stability are proved and then the connections between the stability of the homomorphisms and the amenability of the group are investigated. The results obtained are used for solving some alternative Cauchy functional equations.

MSC:
39B52Functional equations for functions with more general domains and/or ranges
39B99Functional equations
43A07Means on groups, semigroups, etc.; amenable groups
References:
[1]K. Baron, Functions with differences in subspaces, Manuscript.
[2]I. Fenyö, Osservazioni su alcuni teoremi di D. H. Hyers, Ist. Lombardo Accad. Sci. Lett. Rend. A 114 (1980), 235–242.
[3]G. L. Forti, On an alternative functional equation related to the Cauchy equation, Aequationes Math. 24 (1982), 195–206. · Zbl 0517.39007 · doi:10.1007/BF02193044
[4]G. L. Forti -L. Paganoni, A method for solving a conditional Cauchy equation on abelian groups, Ann. Mat. Pura Appl. (IV) 127 (1981), 79–99. · Zbl 0494.39005 · doi:10.1007/BF01811720
[5]Z. Gajda, On the stability of the Cauchy equation on semigroups, Manuscript.
[6]R. Ger, On a method of solving of conditional Cauchy equations, Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. No. 544–576 (1976), 159–165.
[7]F. P. Greenleaf, Invariant means on topological groups, Van Nostrand Mathematical Studies 16, New York, Toronto, London, Melbourne, 1969.
[8]D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. · doi:10.1073/pnas.27.4.222
[9]D. H. Hyers, The stability of homomorphisms and related topics, Global Analysis- Analysis on Manifold (Ed. T. M. Rassias), Teubner-Texte zur Mathematik, Band 57, Teubner Verlagsgesellschaft, Leipzig 1983, 140–153.
[10]M. Kuczma, Functional equations on restricted domains, Aequationes Math. 18 (1978), 1–34. · Zbl 0386.39002 · doi:10.1007/BF01844065
[11]Z. Moszner, Sur la stabilité de l’équation d’homomorphisme, Aequationes Math. 29 (1985), 290–306. · Zbl 0583.39012 · doi:10.1007/BF02189833
[12]L. Paganoni, Soluzione di una equazione funzionale su dominio ristretto, Boll. Un. Mat. Ital. (5) 17-B (1980), 979–993.
[13]L. Paganoni, On an alternative Cauchy equation, Aequationes Math. 29 (1985), 214–221. · Zbl 0583.39007 · doi:10.1007/BF02189830
[14]J. Ratz, On approximate additive mappings, General Inequalities 2 (Ed. E. F. Becken- bach), ISNM vol. 47, Birkhäuser, Basel, 1980, 233–251.
[15]L. Székelyhidi, Note on Hyers’ Theorem, C. R. Math. Rep. Acad. Sci. Canada, 8 (1986), 127–129.
[16]L. Székelyhidi, The Fréchet equation and Hyers theorem on noncommutative semigroups, Manuscript.