A note on Robinson-Ursescu and Lyusternik-Graves theorem

@article{Cibulka2013ANO,
  title={A note on Robinson-Ursescu and Lyusternik-Graves theorem},
  author={Radek Cibulka and Mari{\'a}n Fabian},
  journal={Math. Program.},
  year={2013},
  volume={139},
  pages={89-101}
}
The aim of this note is twofold. First, we prove an analogue of the wellknown 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.