Recurrent and almost recurrent multivalued maps and their selections.~II

L. I. Danilov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2012), pp. 3-21

Résumé

In the paper, we consider the problem of existence of recurrent and almost recurrent selections of multivalued mappings $\mathbb R\ni t\mapsto F(t)\in\operatorname{comp}U$ with nonempty compact sets $F(t)$ in a complete metric space $U$. The set $\operatorname{comp}U$ is equipped with the Hausdorff metric $\mathrm{dist}$. Recurrent and almost recurrent multivalued maps are defined as the functions with values in the metric space $(\operatorname{comp}U,\mathrm{dist})$. It is proved that there are recurrent (almost recurrent) selections of multivalued recurrent (almost recurrent) uniformly absolutely continuous maps. We also consider mappings $\mathbb R\ni t\mapsto F(t)$ with the sets $F(t)$ consisting of a finite number of points (the number depends on the $t\in\mathbb R$). We prove that if such a map is almost recurrent, then it has an almost recurrent selection. A multivalued recurrent mapping $t\mapsto F(t)$ with sets $F(t)$ consisting of at most $n$ points (where $n\in\mathbb N$) has a recurrent selection. If the sets $F(t)$ of a multivalued recurrent (almost recurrent) mapping $t\mapsto F(t)$ consist of $n$ points for all $t\in\mathbb R$, then all $n$ continuous selections of the map $F$ are recurrent (almost recurrent).

Détail

Keywords: recurrent function, selection, multivalued mapping.