Selections of an iteration semigroup of linear set-valued functions
Let \(e_1,e_2,\dots, e_n\) be given linearly independent vectors in a real vector space \(X\) and let \[ C:= \{\lambda_1 e_1+ \lambda_2 e_2+\cdots+ \lambda_n e_n: \lambda_1\geq 0, \lambda_2\geq 0,\dots, \lambda_n\geq 0\}. \] A set-valued function \(F: C\to n(C)\), where \(n(C)\) is the set of all nonempty subsets of \(C\), is called linear iff \(F(x+ y)= F(x)+ F(y)\) for all \(x,y\in C\) and \(F(\lambda x)= \lambda F(x)\) for all \(x\in C\) and \(\lambda> 0\). A one-parameter family \(\{F^t: t\geq 0\}\) of set-valued functions \(F^t: C\to n(C)\) is said to be an iteration semigroup iff \[ F^{t+s}(x)= F^t[F^s(x)]= \bigcup \{F^t(y): y\in F^s(x)\} \] for all \(x\in C\) and \(t\geq 0\), \(s\geq 0\). Under additional assumptions there exists an iteration semigroup \(\{f^t: t\geq 0\}\) of continuous linear selections \(f^t\) of \(F^t\) for every \(t\geq 0\).
scientific article; zbMATH DE number 1221005