On indexed families of multifunctions generated by families of functions (Q744036)

Let \((T,+)\) be an additive semigroup, \(X\) a nonempty set and \(\{F^t:t\in T\}\) a family of multifunctions \(F^t:X\to 2^X\). This family is collapsing (expanding) if \(F^{s+t}\subset F^t\circ F^s\) (resp., \(F^t\circ F^s\subset F^{s+t}\)) for \(s,t\in T\). The family is an iteration semigroup (iteration group) if both inclusions hold and \(T=(0,\infty )\) (resp., \(T=\mathbb{R}\)). The author studies the family of multifunctions \(H^t(x)=[f^t(x),g^t(x)]\), \(t\in T\), generated by functions \(f^t,g^t:X\to X\), where \(X\subset\mathbb{R}\). She gives necessary and sufficient conditions for \(\{H^t:t\in T\}\) to be collapsing, expanding or an iteration semigroup. The obtained results are applied to a multivalued iteration group of this type, investigated by \textit{M. C. Zdun} [J. Math. Anal. Appl. 398, No. 2, 638--648 (2013; Zbl 1261.39027)].