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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.

39B82Stability, separation, extension, and related topics
Full Text: DOI
[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