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