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Zbl 1052.39031
Forti, Gian-Luigi
Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations.
(English)
[J] J. Math. Anal. Appl. 295, No. 1, 127-133 (2004). ISSN 0022-247X

The author formulates, in a general form, the method of proving the Hyers-Ulam stability for functional equations in several variables. This method appeared in his paper [Stochastica 4, No. 1, 23--30 (1980; Zbl 0442.39005)] and has been actually repeated in numerous papers of various authors. \par The main result reads as follows: {Assume that $S$ is a set, $(X,d)$ a complete metric space and $G:S\to S$, $H:X\to X$ given functions. Let $f:S\to X$ satisfy the inequality $$d(H(f(G(x))),f(x))\leq\delta(x),\ \ \ x\in S$$ for some function $\delta:S\to\Bbb R_+$. If $H$ is continuous and satisfies: $$d(H(u),H(v))\leq\phi(d(u,v)),\ \ \ u,v\in X,$$ for a non-decreasing subadditive function $\phi:\Bbb R_+\to\Bbb R_+$, and the series $\sum_{i=0}^{\infty}\phi^i(\delta(G^i(x)))$ is convergent for every $x\in S$, then there exists a unique function $F:S\to X$ -- a solution of the functional equation $$H(F(G(x)))=F(x),\ \ \ x\in S$$ and satisfying $$d(F(x),f(x))\leq\sum_{i=0}^{\infty}\phi^i(\delta(G^i(x))).$$ } \par Moreover, an analogous result, for a mapping $f$ satisfying the inequality $$\left\vert \frac{H(f(G(x)))}{f(x)}-1\right\vert \leq\delta(x)$$ is considered.
[Jacek Chmielinski (Kraków)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: Hyers-Ulam stability; functional equations in several variables

Citations: Zbl 0442.39005

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