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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References:

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