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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, $\left(X,d\right)$ a complete metric space and $G:S\to S$, $H:X\to X$ given functions. Let $f:S\to X$ satisfy the inequality

$d\left(H\left(f\left(G\left(x\right)\right)\right),f\left(x\right)\right)\le \delta \left(x\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x\in S$

for some function $\delta :S\to {ℝ}_{+}$. If $H$ is continuous and satisfies:

$d\left(H\left(u\right),H\left(v\right)\right)\le \varphi \left(d\left(u,v\right)\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u,v\in X,$

for a non-decreasing subadditive function $\varphi :{ℝ}_{+}\to {ℝ}_{+}$, and the series ${\sum }_{i=0}^{\infty }{\varphi }^{i}\left(\delta \left({G}^{i}\left(x\right)\right)\right)$ is convergent for every $x\in S$, then there exists a unique function $F:S\to X$ – a solution of the functional equation

$H\left(F\left(G\left(x\right)\right)\right)=F\left(x\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x\in S$

and satisfying

$d\left(F\left(x\right),f\left(x\right)\right)\le \sum _{i=0}^{\infty }{\varphi }^{i}\left(\delta \left({G}^{i}\left(x\right)\right)\right)·$

Moreover, an analogous result, for a mapping $f$ satisfying the inequality

$\left|\frac{H\left(f\left(G\left(x\right)\right)\right)}{f\left(x\right)}-1\right|\le \delta \left(x\right)$

is considered.

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