Covering mappings in metric spaces and fixed points

We consider metric spaces X and Y with metrics ρ X and ρ Y , respectively. Before stating the main results of this paper, we recall two known assertions. The first of them is the contraction mapping principle, which says that if a space X is complete, then any self-mapping of X satisfying the Lipschitz condition with Lipschitz constant less than 1 (i.e., a contraction) has a (unique) fixed point. The second assertion is Milyutin’s covering mapping theorem. To formulate it, we need the notion of a covering mapping. By B X ( r , x ) we denote the closed ball of radius r centered at x in the space X ; a similar notation is used in the space Y . For any M ⊂ Y , we set B Y ( r , M ) = ( r , y ) (this is the r -neighborhood of the set M ). Definition. Let α > 0. A mapping Ψ : X → Y is said to be α -covering if