On Fryszkowski’s problem

Abstract

Abstract. In this paper we give two partial answers to Fryszkowski's problem which can be stated as follows: given α ∈ (0, 1), an arbitrary non-empty set Ω and a set-valued mapping F : Ω → 2 Ω , find necessary and (or) sufficient conditions for the existence of a (complete) metric d on Ω having the property that F is a Nadler set-valued α-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 Ω having the property that F is a Nadler set-valued α-contraction with respect to d, in the case that α ∈ (0, 1 2 ) and there exists z ∈ Ω such that F (z) = {z} and, on the other hand, we give a sufficient condition for the existence of a complete metric d on Ω having the property that F is a Nadler set-valued α-contraction with respect to d, in the case that Ω is finite.Mathematics Subject Classification (2010): 54C60, 54H25.