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Functional inequalities in non-Archimedean Banach spaces. (English) Zbl 1203.39015
The authors show that if \(f\) is a function between non-Archimedean spaces satisfying the functional inequality \(\|f(x)+f(y)+f(z)\| \leq \|k f((x+y+z)/k)\|\), where \(|k| < |3|\), then \(f\) is additive. They also prove the generalized Hyers-Ulam stability of the functional inequality above in non-Archimedean normed spaces.
Reviewer’s Comment: The authors assume that the domain of \(f\) is non-Archimedean, but it seems that they do not need this assumption.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
39B52 Functional equations for functions with more general domains and/or ranges
39B62 Functional inequalities, including subadditivity, convexity, etc.
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