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On the stability of homogeneous functional equations with degree \(t\) and \(n\)-variables. (English) Zbl 1051.39029
Let \(S\) be a nonempty set and let \(G\) be a multiplicative subsemi-group of the real (complex) field with the properties:
(1) \(1 \in G\);
(2) \(u^t, u^{1/t} \in G\) for all \(u \in G\) and \(t > 0\).
An operation \(\circ : S \times \cdots \times S \to S\) will be called \(n\)-times-symmetric if \(\circ\) satisfies \[ \circ \, [ \circ (x_1, x_2, \dots, x_n), \dots, \circ (x_1, x_2, \dots, x_n) ] = \circ \, [ \circ (x_1, \dots, x_1), \circ (x_2, \dots, x_2), \dots, \circ (x_n, \dots, x_n) ]. \] In this paper, the author investigates the Hyers-Ulam-Rassias stability of the functional equation \[ f(\circ (x_1, \dots, x_n)) = H\!\left( f(x_1)^{1/t}, \dots, f(x_n)^{1/t} \right), \] where \(H : G \times \cdots \times G \to G\) is a continuous \(G\)-homogeneous function of degree \(t\).

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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