On Fryszkowski's problem

Abstract

In this paper we give two partial answers to Fryszkowski's problem which can be stated as follows: given $\alpha \in (0,1)$, an arbitrary non-empty set $\Omega$ and a set-valued mapping $F:\Omega \rightarrow 2^{\Omega }$, find necessary and (or) sufficient conditions for the existence of a (complete) metric $d$ on $\Omega $ having the property that $F$ is a Nadler set-valued $\alpha $-contraction with respect to $d$. More precisely, on the one hand, we provide necessary and sufficient conditions for the existence of a complete and bounded metric $d$ on $\Omega $ having the property that $F$ is a Nadler set-valued $\alpha $-contraction with respect to $d$ , in the case that $\alpha \in (0,\frac{1}{2})$ and there exists $z\in \Omega $ such that $F(z)=\{z\}$ and, on the other hand, we give a sufficient condition for the existence of a complete metric $d$ on $\Omega $ having the property that $F$ is a Nadler set-valued $\alpha $-contraction with respect to $d$, in the case that $\Omega$ is finite.