Polat, Faruk Some generalizations of Ulam-Hyers stability functional equations to Riesz algebras. (English) Zbl 1235.39030 Abstr. Appl. Anal. 2012, Article ID 653508, 9 p. (2012). Summary: 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 \(\mathcal R\) onto a Banach algebra \(\mathcal B\) satisfying \(||f(x + y) - f(x) - f(y)|| \leq \epsilon\) and \(||f(x \cdot y) - f(x) f(y)|| \leq \delta\) for all \(x, y \in \mathcal R\), there exists a unique ring homomorphism \(h : \mathcal R \rightarrow \mathcal B\) such that \(||f(x) - h(x)|| \leq \epsilon, x \in \mathcal R\). Moreover, \(b \cdot (f(x) - h(x)) = 0, (f(x) - h(x)) \cdot b = 0\), for all \(x \in \mathcal R\) and all \(b\) from the algebra generated by \(h(\mathcal R)\). In this paper, we generalize Badora’s stability result above on ring homomorphisms for Riesz algebras with extended norms. Cited in 3 Documents MSC: 39B82 Stability, separation, extension, and related topics for functional equations Keywords:Ulam-Hyers stability Citations:Zbl 1014.39020 PDFBibTeX XMLCite \textit{F. Polat}, Abstr. Appl. Anal. 2012, Article ID 653508, 9 p. (2012; Zbl 1235.39030) 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 [3] Z. Moszner, “On the stability of functional equations,” Aequationes Mathematicae, vol. 77, no. 1-2, pp. 33-88, 2009. · Zbl 1207.39044 [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 [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 [6] R. Badora, “On approximate ring homomorphisms,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 589-597, 2002. · Zbl 1014.39020 [7] W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, vol. 1, North-Holland, Amsterdam, The Netherlands, 1971. · Zbl 0231.46014 [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 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.