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)
Some generalizations of Ulam-Hyers stability functional equations to Riesz algebras. (English) Zbl 1235.39030
Summary: {\it R. Badora} [J. Math. Anal. Appl. 276, No. 2, 589--597 (2002; Zbl 1014.39020)] proved the following stability result. Let $\epsilon$ and $\delta$ be nonnegative real numbers, then for every mapping $f$ of a ring $\Cal R$ onto a Banach algebra $\Cal B$ satisfying $||f(x + y) - f(x) - f(y)|| \leq \epsilon$ and $||f(x \cdot y) - f(x) f(y)|| \leq \delta$ for all $x, y \in \Cal R$, there exists a unique ring homomorphism $h : \Cal R \rightarrow \Cal B$ such that $||f(x) - h(x)|| \leq \epsilon, x \in \Cal R$. Moreover, $b \cdot (f(x) - h(x)) = 0, (f(x) - h(x)) \cdot b = 0$, for all $x \in \Cal R$ and all $b$ from the algebra generated by $h(\Cal R)$. In this paper, we generalize Badora’s stability result above on ring homomorphisms for Riesz algebras with extended norms.

MSC:
39B82Stability, separation, extension, and related topics
WorldCat.org
Full Text: DOI
References:
[1] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. · Zbl 0086.24101
[2] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] Z. Moszner, “On the stability of functional equations,” Aequationes Mathematicae, vol. 77, no. 1-2, pp. 33-88, 2009. · Zbl 1207.39044 · doi:10.1007/s00010-008-2945-7
[4] B. Paneah, “A new approach to the stability of linear functional operators,” Aequationes Mathematicae, vol. 78, no. 1-2, pp. 45-61, 2009. · Zbl 1207.39046 · doi:10.1007/s00010-009-2956-z
[5] D. G. Bourgin, “Approximately isometric and multiplicative transformations on continuous function rings,” Duke Mathematical Journal, vol. 16, pp. 385-397, 1949. · Zbl 0033.37702 · doi:10.1215/S0012-7094-49-01639-7
[6] R. Badora, “On approximate ring homomorphisms,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 589-597, 2002. · Zbl 1014.39020 · doi:10.1016/S0022-247X(02)00293-7
[7] W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, vol. 1, North-Holland, Amsterdam, The Netherlands, 1971.
[8] G.-L. Forti, “Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 127-133, 2004. · Zbl 1052.39031 · doi:10.1016/j.jmaa.2004.03.011
[9] J. Brzd\cek, “On a method of proving the Hyers-Ulam stability of functional equations on restricted domains,” The Australian Journal of Mathematical Analysis and Applications, vol. 6, pp. 1-10, 2009. · Zbl 1175.39014 · http://ajmaa.org/cgi-bin/paper.pl?string=v6n1/V6I1P4.tex