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Additive selections and the stability of the Cauchy functional equation. (English) Zbl 1037.39008
The main result of the paper is the following Theorem:
Let $$(S,+)$$ be a left amenable semigroup and let $$X$$ be a Hausdorff locally convex linear space. Let $$F:S\to 2^X$$ be a set-valued map such that, for all $$s\in S$$, $$F(s)$$ is nonempty, convex and weakly compact. Then $$F$$ admits an additive selection $$A:S\to X$$ if and only if there exists a function $$f:S\to X$$ such that $$f(s+t)-f(t)\in F(s)$$, $$s,t\in S$$.
The main tool in the proof is the result stating that left (right) invariant means (defined on the space of real-valued bounded functions over a semigroup) can be extended to Hausdorff locally convex space-valued functions whose range has weakly compact convex closure. In the last part of the paper some functional inequalities related to quasi-additivity property are considered. Finally the complete characterization of quasi-additivity is given.

##### MSC:
 39B62 Functional inequalities, including subadditivity, convexity, etc. 39B52 Functional equations for functions with more general domains and/or ranges 54C60 Set-valued maps in general topology 54C65 Selections in general topology 39B72 Systems of functional equations and inequalities 39B82 Stability, separation, extension, and related topics for functional equations
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