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Subadditive and subquadratic set-valued functions. (English) Zbl 0626.54019
Prace Naukowe Uniwersytetu Śląskiego w Katowicach, Nr. 889. Katowice: Uniwersytet Śląski. 73 p.; zł 135.00 (1987).
A set-valued function (svf) $$F: X\to Y$$ on abelian groups is additive [subadditive] if $$F(x+y)=F(x)+F(y)$$ $$[F(x+y)\subset F(x)+F(y)]$$. It is subquadratic if $$F(x+y)+F(x-y)\subset 2F(x)+2F(y)$$. The author deals with subadditive, quadratic and subquadratic svfs with nonempty compact and convex values in a topological vector space. She formulates a stability problem to ask whether any subadditive svf has an additive selection, and proves this theorem: if Y is a locally bounded vector space, $$F: {\mathbb{R}}\to cc(Y)=\{compact$$, convex, nonempty subsets of $$Y\}$$ is a subadditive svf, $$G: {\mathbb{R}}\to \{bounded$$, nonempty subsets of $$Y\}$$ is a measurable svf and there exists a set of positive Lebesgue measure $$W\subset {\mathbb{R}}$$ such that F(t)$$\subset G(t)$$ for all $$t\in W$$ then there exists $$y\in Y$$ such that ty$$\in F(t)$$ for all $$t\in {\mathbb{R}}$$. Numerous other theorems are proved on the properties of subadditive svfs on $${\mathbb{R}}$$ (or $${\mathbb{R}}_+)$$ and some of these are generalized to n- subadditive svfs on $${\mathbb{R}}^ n$$. Another chapter concerns boundedness of svfs and the connection between boundedness and connectivity. Finally there is a chapter on subquadratic and quadratic svfs in which it is proved, for example, that if Y is a locally convex vector space, $$W\subset {\mathbb{R}}$$ is a set of positive inner Lebesgue measure, and if $$F: {\mathbb{R}}\to cc(Y)$$ is a subquadratic svf that is bounded on W then there exists a set $$A\in cc(Y)$$ such that $$t^ 2A\subset F(t)$$ for all $$t\in {\mathbb{R}}$$. Conditions are given which ensure that a subquadratic svf contains a quadratic svf, and under which upper semi-continuity at a point implies the continuity of a quadratic svf everywhere.
Reviewer: J.P.Jerrard

##### MSC:
 54C60 Set-valued maps in general topology 26E25 Set-valued functions 54C65 Selections in general topology