zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.