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Zbl 1235.39030
Polat, Faruk
Some generalizations of Ulam-Hyers stability functional equations to Riesz algebras. (English)
[J] Abstr. Appl. Anal. 2012, Article ID 653508, 9 p. (2012). ISSN 1085-3375; ISSN 1687-0409/e

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 2000:: *39B82 Stability, separation, extension, and related topics

Keywords: Ulam-Hyers stability

Citations: Zbl 1014.39020

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