Coincidence Points and Generalized Coincidence Points of Two Set-Valued Mappings

We consider set-valued mappings acting in metric spaces and show that, under natural general assumptions, the set of coincidence points of two such mappings one of which is covering and the other is Lipschitz continuous is dense in the set of generalized coincidence points of these mappings. We use this result to study the coincidence points and generalized coincidence points of a set-valued covering mapping and a set-valued Lipschitz mapping that depend on a parameter. In particular, we obtain conditions that guarantee the existence of a coincidence point for all values of the parameter under the assumption that a coincidence point exists for one value of the parameter.