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.
Similar content being viewed by others
References
Bauschke, H., Borwein, J.M.: Conical open mapping theorems and regularity. Proc. Cent. Math. Appl. 36, 1–10 (1999)
Cibulka, R.: Constrained open mapping theorem with applications. J. Math. Anal. Appl. 379, 205–215 (2011)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer, Dordrecht (2009)
Fabian, M., Hájek, P., Habala, P., Montesinos, V., Zizler, V.: Banach Space Theory: The Basis for Linear and Nonlinear Analysis. CMS Books in Mathematics. Springer, New York (2011)
Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)
Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspehi Mat. Nauk 55, 103–162 (2000)
Ioffe, A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math. Soc. 266, 1–56 (1981)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)
Mordukhovich, B.S., Wang, B.: Restrictive metric regularity and generalized differential calculus in Banach spaces. Int. J. Math. Math. Sci. 50, 2653–2680 (2004)
Páles, Z.: Linear selections for set-valued functions and extensions of bilinear forms. Arch. Math. 62, 427–432 (1994)
Páles, Z.: Inverse and implicit function theorems. J. Math. Anal. Appl. 209, 202–220 (1997)
Pataki, G.: On the closedness of the linear image of a closed convex cone. Math. Oper. Res. 32, 395–412 (2007)
Reif, J.: A characterization of (locally) uniformly convex spaces in terms of relative openness of quotient maps on the unit ball. J. Funct. Anal. 177, 1–15 (2000)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-013-0662-z