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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 $f\left(\alpha x+\beta y\right)=\alpha f\left(x\right)+\beta f\left(y\right)$, where $\alpha$ and $\beta$ are given nonzero real numbers. Indeed, one of their main theorems states:

Let $A$ be a unital ${C}^{*}$-algebra with unitary group $U\left(A\right)$. Assume that $X$ and $Y$ are left Banach $A$-modules. Let $\varphi :{X}^{2}\to \left[0,\infty \right)$ be a function such that ${lim}_{n\to \infty }{2}^{n}\varphi \left(\frac{x}{{2}^{n}},\frac{y}{{2}^{n}}\right)=0$ for all $x,y\in X$ and there exists a constant $L<1$ with $2\psi \left(x\right)\le L\psi \left(2x\right)$ for all $x\in X$, where $\psi \left(x\right)=\varphi \left(\frac{x}{2\alpha },\frac{x}{2\beta }\right)+\varphi \left(\frac{x}{2\alpha },0\right)+\varphi \left(0,\frac{x}{2\beta }\right)$.

If a function $f:X\to Y$ satisfies $f\left(0\right)=0$ and

$\parallel f\left(\alpha x+\beta ay\right)-\alpha f\left(x\right)-\beta af\left(y\right)\parallel \le \varphi \left(x,y\right)$

for all $x,y\in X$ and for all $a\in U\left(A\right)$, then there exists a unique $A$-linear function $T:X\to Y$ such that $\parallel f\left(x\right)-T\left(x\right)\parallel \le \frac{1}{1-L}\psi \left(x\right)$ for all $x\in X$.

The readers may also refer to the following literature for more information on this subject: 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)].

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B22 Functional equations for real functions 39B52 Functional equations for functions with more general domains and/or ranges