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Zbl 1093.39024
On approximate derivations.
(English)
[J] Math. Inequal. Appl. 9, No. 1, 167-173 (2006). ISSN 1331-4343

The author of the present pleasant paper establishes that if $A$ is a subalgebra of a Banach algebra $B$ and $f: A \to B$ satisfies $\Vert f(x+y) - f(x) - f(y)\Vert \leq \delta$ and $\Vert f(xy) - xf(y) - f(x)y\Vert \leq \varepsilon$, for all $x, y \in A$ and for some $\delta, \varepsilon \geq 0$, then there exists a unique additive derivation $d:A \to B$ such that $\Vert f(x) - d(x)\Vert \leq \delta \quad (x \in A)$, and $x\left(f(y) - d(y)\right) = 0 \quad (x, y \in A)$. The result and its proof are still true for a more general case if we consider a normed algebra $A$ and replace $B$ by a Banach $A$-bimodule $X$. \par He also proves that if $B$ is a normed algebra with an identity belonging to $A$, then every mapping $f: A \to B$ satisfying $\Vert f(xy) - xf(y) - f(x)y\Vert \leq \varepsilon \quad (x, y \in A)$ must fulfil $f(xy) = xf(y) - f(x)y \quad (x, y \in A)$. This superstability result is nice, since there is not assumed any (approximately) additive condition on $f$. Some similar results in which one considers generalized derivations can be found in the reviewer's paper [Hyers-Ulam-Rassias stability of generalized derivations", Int. J. Math. Math. Sci. (to appear)].
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains
47B47 Derivations and linear operators defined by algebraic conditions

Keywords: derivation; Banach algebra; Hyers--Ulam--Rassias stability; superstability

Cited in: Zbl 1183.39023 Zbl 1104.39025

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