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Stability of a mixed functional equation in several variables on Banach modules. (English) Zbl 1191.39027

The authors establish the general solution of the functional equation
\[ \sum_{i=1}^{n} f\bigg(x_i-\frac{1}{n}\sum_{j=1}^nx_j\bigg)=\sum_{i=1}^{n}f(x_i)- nf \bigg(\frac{1}{n}\sum_{i=1}^{n}x_i\bigg) \quad (n\geq 2) \]
and use the fixed point alternative of B. Margolis and J. B. Diaz [Bull. Am. Math. Soc. 74, 305–309 (1968; Zbl 0157.29904)] to prove its generalized Hyers-Ulam stability in Banach modules over a unital Banach algebra.

MSC:

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

Citations:

Zbl 0157.29904
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Full Text: DOI

References:

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