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A fixed point approach to the stability of a generalized Cauchy functional equation. (English) Zbl 1143.39016
Using a fixed point method, the authors prove the Hyers-Ulam-Rassias stability of a generalized Cauchy functional equation of the form $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$, where $\alpha$ and $\beta$ are given nonzero real numbers. Indeed, one of their main theorems states: Let $A$ be a unital $C^\ast$-algebra with unitary group $U(A)$. Assume that $X$ and $Y$ are left Banach $A$-modules. Let $\varphi : X^2 \to [0, \infty)$ be a function such that $\lim_{n \to \infty} 2^n \varphi(\frac{x}{2^n}, \frac{y}{2^n})=0$ for all $x, y \in X$ and there exists a constant $L < 1$ with $2\psi(x) \leq L\psi(2x)$ for all $x \in X$, where $\psi(x) = \varphi( \frac{x}{2\alpha}, \frac{x}{2\beta} ) + \varphi( \frac{x}{2\alpha}, 0 ) + \varphi( 0, \frac{x}{2\beta} )$. If a function $f : X \to Y$ satisfies $f(0) = 0$ and $$ \| f(\alpha x + \beta ay) - \alpha f(x) - \beta af(y) \| \leq \varphi(x,y) $$ for all $x, y \in X$ and for all $a \in U(A)$, then there exists a unique $A$-linear function $T : X \to Y$ such that $\| f(x) - T(x) \| \leq \frac{1}{1-L} \psi(x)$ for all $x \in X$. The readers may also refer to the following literature for more information on this subject: {\it S.-M. Jung} [J. Math. Anal. Appl. 329, No. 2, 879--890 (2007); Fixed Point Theory Appl. 2007, Article ID 57064, 9 p. (2007; Zbl 1155.45005)].

39B82Stability, separation, extension, and related topics
39B22Functional equations for real functions
39B52Functional equations for functions with more general domains and/or ranges