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On a problem by K. Nikodem. (English) Zbl 1036.39020

A stability problem for the square-norm (quadratic) functional equation: \(g(x+y) + g(x-y)- 2 g(x) - 2 g(y) = 0\) is solved. Here \(g\) is defined in an Abelian group divisible by \(2\) and takes values in a Hilbert space. The problem is related to that raised by K. Nikodem, for the Cauchy functional equation, during the 38th International Symposium on Functional Equations (Noszvaj, 2000), cf. Report of Meeting [Aequationes Math. {61}, 301 (2001)].

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
39B62 Functional inequalities, including subadditivity, convexity, etc.
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