Monomial selections of set-valued maps
Let \((S,\bullet)\) be a semigroup and \(X\) be a locally convex space. Denote by \(WC(X)\) the family of nonempty weakly compact and convex subsets of \(X\). For functions \(f:S\to X\) and \(F:S^n\to X\) define \(\Delta_t f(s)= f(ts)-f(s)\) and \(_{(i)}\Delta_t F(s_1,\dots, s_n):= F(s_1,\dots, ts_i,\dots, s_n) -F(s_1, \dots,s_n)\). The main result of the paper is the following. Let \(\Phi: S^n\to WC(X)\) and assume that there exists a function \(F:S\to X\) such that \[ {1\over k_1! \dots k_n!} \Delta_{t_1}^{k_1} \dots \Delta_{t_n}^{k_n} f(s) \in \Phi(t_1, \dots,t_n), \quad s,t_1, \dots, t_n\in S. \] Then there exists a function \(F:S^n\to X\) such that \(F\) is a selection of \(\Phi\), i.e., \(F(t_1, \dots, t_n)\in \Phi(t_1, \dots,t_n)\), \(t_1,\dots, t_n\in S\) and, for all \(i=1, \dots,n\), \(F\) satisfies the functional equation \[ {1\over k_j!} {_{(i)} \Delta_u^{k_1}} F(t_1,\dots, t_n)=F(t_1, \dots,u, \dots,t_n),\quad u,\;t_1, \dots, t_n\in S. \]
scientific article; zbMATH DE number 1383750