A note on Robinson-Ursescu and Lyusternik-Graves theorem
The aim of this note is twofold. First, we prove an analogue of the well-known Robinson-Ursescu Theorem on the relative openness with a linear rate (restrictive metric regularity) of a multivalued mapping. Second, we prove a generalization of Graves Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a bunch of linear mappings. The approximated non-linear mapping is restricted to a closed convex subset of a Banach space.
robinson-ursescu theorem restrictive metric regularity open mapping graves theorem convex constraints