Additive selections of superadditive set-valued functions. (English) Zbl 0706.39006
A set-valued function F: \(C\to 2^ Y\) is said to be superadditive if \(F(x)+F(y)\subset F(x+y),\) x,y\(\in C\). The author proves that every superadditive set-valued function defined on a cone C with a cone basis in a topological vector space into the family of all compact convex subsets of a locally convex space admits an additive selection.
Reviewer: K.Nikodem
MSC:
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 39B72 | Systems of functional equations and inequalities |
| 46A03 | General theory of locally convex spaces |
| 54C60 | Set-valued maps in general topology |
Keywords:
superadditive set-valued function; cone; topological vector space; locally convex space; additive selectionReferences:
| [1] | Bernstein, F. andDoetsch, G.,Zur Theorie der konvexen Functionen. Math. Ann.76 (1915), 514–526. · JFM 45.0627.02 · doi:10.1007/BF01458222 |
| [2] | Boubaki, N.,Les structures fondamentales de l’analyse. Livre V: Espaces vectoriels topologiques. Hermann, Paris, 1953. |
| [3] | Gajda, Z. andGer, R.,Subadditive multifunctions and Hyers – Ulam stability. General inequalities 5 (Proc. Fifth Internat. Conf. on General Inequalities, Oberwolfach, 1986). (International Series of Numerical Mathematics, Vol. 80). Birkhäuser, Basel–Boston, 1987, pp. 281–291. |
| [4] | Ger, R. andKominek, Z.,Boundedness and continuity of additive and convex functionals. Aequationes Math.37 (1989), 252–258. · Zbl 0702.46004 · doi:10.1007/BF01836447 |
| [5] | Kranz, P.,Additive functionals on abelian semigroups. Comment. Math. Prace Mat.16 (1972), 239–246. · Zbl 0262.20087 |
| [6] | Mehdi, M. R.,On convex functions. J. London Math. Soc.39 (1964), 321–326. · Zbl 0125.06304 · doi:10.1112/jlms/s1-39.1.321 |
| [7] | Nikodem, K.,Additive set-valued functions in Hilbert spaces. Rev. Roumaine Math. Pures Appl.28 (1983), 239–242. · Zbl 0516.28014 |
| [8] | Nikodem, K.,Additive selections of additive set-valued functions. To appear. · Zbl 0686.46008 |
| [9] | Nikodem, K.,Midpoint convex functions majorized by midpoint concave functions. Aequationes Math.32 (1987), 45–51. · Zbl 0612.26007 · doi:10.1007/BF02311298 |
| [10] | Przesławski, K.,Linear and lipschitz continuous selectors for the family of convex sets in Euclidean vector space. Bull. Acad. Polon. Sci. Sér. Sci. Math.33 (1985), 31–33. · Zbl 0582.46046 |
| [11] | Rådström, H.,One-parameter semigroups of subsets of a real linear space. Ark. Mat.4 (1960), 87–97. · Zbl 0093.30401 · doi:10.1007/BF02591324 |
| [12] | Smajdor, W.,Subadditive and subquadratic set-valued functions. (Prace Nauk. Uniw. Śląsk. Katowic., No. 889). Uniw. Ślaski, Katowice, 1987. · Zbl 0626.54019 |
| [13] | Smajdor, W.,Superadditive set-valued functions and Banach–Steinhaus theorem. Radovi Mat.3 (1987), 203–214. · Zbl 0654.39007 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
