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Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. (English) Zbl 1052.39031
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.
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 $$\delta:S\to\mathbb R_+$$. 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 $$\phi:\mathbb R_+\to\mathbb R_+$$, 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 $d(F(x),f(x))\leq\sum_{i=0}^{\infty}\phi^i(\delta(G^i(x))).$
Moreover, an analogous result, for a mapping $$f$$ satisfying the inequality $\left| \frac{H(f(G(x)))}{f(x)}-1\right| \leq\delta(x)$ is considered.

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