Stability of Coincidence Points and Properties of Covering Mappings

Properties of closed set-valued covering mappings acting from one metric space into another are studied. Under quite general assumptions, it is proved that, if a given $\alpha$-covering mapping and a mapping satisfying the Lipschitz condition with constant $\beta<\alpha$ have a coincidence point, then this point is stable under small perturbations (with respect to the Hausdorff metric) of these mappings. This assertion is meaningful for single-valued mappings as well. The structure of the set of coincidence points of an $\alpha$-covering and a Lipschitzian mapping is studied. Conditions are obtained under which the limit of a sequence of $\alpha$-covering set-valued mappings is an $(\alpha-\varepsilon)$-covering for an arbitrary $\varepsilon>0$.