Abstract
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.
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Acknowledgments
The first named author wants to express his thanks to the late Jiří Reif, his supervisor, who introduced him to the area of constrained open mapping theorems. We thank the referees for their valuable comments allowing us to improve the presentation of this note.
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In honor of Jonathan Borwein at the occasion of his 60.
The research of the first author was supported by the Research Plan MSM 4977751301. The second named author was supported in part by grant P 201/11/0345 and by Institutional Research Plan of the Academy of Sciences of Czech Republic No. AVOZ 101 905 03.
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Cibulka, R., Fabian, M. A note on Robinson–Ursescu and Lyusternik–Graves theorem. Math. Program. 139, 89–101 (2013). https://doi.org/10.1007/s10107-013-0662-z
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DOI: https://doi.org/10.1007/s10107-013-0662-z