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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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