# zbMATH — the first resource for mathematics

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
##### Keywords:
Hyers-Ulam stability; metric space; functional equation
Full Text: