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On a general Hyers-Ulam stability result. (English) Zbl 0826.39009
Let \(S\) be a set, \((X,d)\) be a complete metric space and \(k : \mathbb{R}^+ \to \mathbb{R}^+\), \(F : S \times S \to S\), \(R_i : S \times S \to S\) and \(H : X \times X \times X^p \to X\), \(\rho : S \times S \to \mathbb{R}^+\) be some given functions. Suppose that a function \(f : S \to X\) fulfills the inequality \[ d ( f ( F(x,y) ),\;H ( f(x), f(y), f ( R_1 (x,y) ), \ldots, f ( R_p (x,y) ) ) ) \leq \rho (x,y). \] The authors give some conditions under which there exists a solution \(g : S \to X\) of the functional equation \[ g ( F(x + y) ) = H ( f(x), f(y), f ( R_1 (x,y) ), \ldots, f ( R_p (x,y) )) \] such that \(d(g (x), g(y)) \leq \sum^\infty_{n = 1} k^{- n} [\Delta (G^{m - 1} (x))]\), where \(G(x) : = F(x,x)\) and
\(\Delta (x) : = d(f(G(x)), H(f(x), f(x), f(R_1 (x,x)), \ldots, f(R_p (x,x)))\). Numerous earlier results concerning stability in the sense of Hyers-Ulam are derivable from the theorems of this paper.

MSC:
39B52 Functional equations for functions with more general domains and/or ranges
39B42 Matrix and operator functional equations
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