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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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References:
[1] Yosida, Functional analysis 123 (1980) · Zbl 0830.46001 · doi:10.1007/978-3-642-61859-8
[2] DOI: 10.1073/pnas.27.4.222 · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] DOI: 10.1007/BF01830942 · Zbl 0652.39013 · doi:10.1007/BF01830942
[4] Holmes, Geometric functional analysis and its applications 24 (1975) · Zbl 0336.46001 · doi:10.1007/978-1-4684-9369-6
[5] Hewitt, Abstract harmonic analysis 115 (1963) · Zbl 0115.10603
[6] Ger, Proc. of the 4th International Conference on Functional Equations and Inequalities pp 5– (1994)
[7] Ger, Grazer Math. Ber. 316 pp 59– (1992)
[8] DOI: 10.1007/BF01831117 · Zbl 0836.39007 · doi:10.1007/BF01831117
[9] Day, Illinois J. Math. 1 pp 509– (1957)
[10] DOI: 10.1007/BF01833143 · Zbl 0706.39007 · doi:10.1007/BF01833143
[11] Badora, Ann. Polon. Math. 58 pp 147– (1993)
[12] Székelyhidi, C. R. Math. Rep. Acad. Sci. Canada 8 pp 127– (1986)
[13] Rao, Choquet-Deny type functional equations with applications to stochastic models (1994) · Zbl 0841.60005
[14] Ramachandran, Functional equations in probability theory (1991) · Zbl 0743.60022
[15] DOI: 10.1007/s000100050058 · Zbl 0922.39008 · doi:10.1007/s000100050058
[16] Nikodem, Ann. Polon. Math. 72 pp 207– (1999)
[17] Nikodem, Publ. Math. Debrecen 52 pp 575– (1998)
[18] Kuczma, An introduction to the theory of functional equations and inequalities (1985) · Zbl 0555.39004
[19] DOI: 10.1007/BF01830975 · Zbl 0806.47056 · doi:10.1007/BF01830975
[20] Hyers, Stability of functional equations in several variables (1998) · Zbl 0907.39025 · doi:10.1007/978-1-4612-1790-9
[21] DOI: 10.1007/BF01833149 · Zbl 0708.39005 · doi:10.1007/BF01833149
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