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Semigroups of midpoint convex set-valued functions. (English) Zbl 0860.39038

A set-valued function \(F\) from a convex subset \(D\) of a real vector space \(X\) into the set \(n(Y)\) of all nonempty subsets of a real vector space \(Y\) is said to be midpoint convex if \[ \textstyle {1\over 2} \bigl[F(x) + F(y)\bigr] \subset F \bigl[\textstyle {1\over 2} (x+y) \bigr] \quad \text{for all} \quad x,y \in D. \] A set-valued function \(F:X \to n(Y)\) is said to be convex if \[ tF(x) + (1-t) F(y) \subset F \bigl(tx+ (1-t) y\bigr) \quad \text{for all} \quad x,y \in X \text{ and } t\in[0,1]. \] The author obtains the form of an midpoint convex set-valued function.
Theorem: Let \(X\) be a vector space and \(Y\) a locally convex space. If the set-valued function \(F:X \to c(Y)\) is midpoint convex then there exists exactly one additive function \(f:X \to Y\) such that \(F(x) = f(x) + F(o)\) for all \(x\in X\), where \(c(Y)\) denotes the family of all nonempty compact subsets of \(Y\).
One may assign to any family \(\{F^t:t \geq 0\}\) of convex set-valued functions \(F^t: X\to c(X)\), the family \(\{f^t: t\geq 0\}\), called the additive part, and the set-valued function \(t\mapsto F^t(o)\), called the translation part of \(\{F^t:t\geq 0\}\), by \(F^t(x) = f^t(x) + F^t(o)\). The following theorem is proved:
Let \(X\) be a locally convex space and let \(\{F^t:t \geq 0\}\) be a family of midpoint convex set-valued function \(F^t:X \to c(X)\) with an additive part \(\{f^t:t \geq 0\}\). Then \(\{F^t:t \geq 0\}\) is an iteration semigroup if and only if \(\{f^t:t \geq 0\}\) is an iteration semigroup and the condition \(F^{t+s} (o)= f^t[F^s(o) + F^t (o)]\) holds.
Some other similar results are also derived.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B12 Iteration theory, iterative and composite equations
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