Solution to a problem of Nirenberg concerning expanding maps
Authors:
Dean Ives and David Preiss
Journal:
Proc. Amer. Math. Soc. 149 (2021), 301-310
MSC (2010):
Primary 47H09
DOI:
https://doi.org/10.1090/proc/15214
Published electronically:
October 20, 2020
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: By constructing a map of a separable Hilbert space into itself that is continuous, expanding, nonsurjective, and equal to the identity on the unit ball we answer a problem stated by Louis Nirenberg.
- [1] Kung Ching Chang and Shu Jie Li, A remark on expanding maps, Proc. Amer. Math. Soc. 85 (1982), no. 4, 583–586. MR 660608, https://doi.org/10.1090/S0002-9939-1982-0660608-4
- [2] Jorge E. Hernández and M. Zuhair Nashed, Global invertibility of expanding maps, Proc. Amer. Math. Soc. 116 (1992), no. 1, 285–291. MR 1110543, https://doi.org/10.1090/S0002-9939-1992-1110543-9
- [3] Athanassios G. Kartsatos, On the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive Banach spaces, Trans. Amer. Math. Soc. 350 (1998), no. 10, 3967–3987. MR 1443880, https://doi.org/10.1090/S0002-9947-98-02033-9
- [4] Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093
- [5] Jean-Michel Morel and Heinrich Steinlein, On a problem of Nirenberg concerning expanding maps, J. Funct. Anal. 59 (1984), no. 1, 145–150. MR 763781, https://doi.org/10.1016/0022-1236(84)90057-0
- [6] Ronny Nahum, On the Lipschitz equivalence of unit balls and of spheres in normed spaces, preprint.
- [7] L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR 0488102
- [8] Janusz Szczepański, On a problem of Nirenberg concerning expanding maps in Hilbert space, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1041–1044. MR 1100665, https://doi.org/10.1090/S0002-9939-1992-1100665-0
- [9] Janusz Szczepański, A new result on the Nirenberg problem for expanding maps, Nonlinear Anal. 43 (2001), no. 1, 91–99. MR 1784447, https://doi.org/10.1016/S0362-546X(99)00180-7
- [10] Tian Xiang, Notes on expansive mappings and a partial answer to Nirenberg’s problem, Electron. J. Differential Equations (2013), No. 02, 16. MR 3020218
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47H09
Retrieve articles in all journals with MSC (2010): 47H09
Additional Information
Dean Ives
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
Email:
dean@math.ucl.ac.uk
David Preiss
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email:
d.preiss@warwick.ac.uk
DOI:
https://doi.org/10.1090/proc/15214
Received by editor(s):
March 1, 2020
Received by editor(s) in revised form:
May 28, 2020, and May 29, 2020
Published electronically:
October 20, 2020
Communicated by:
Stephen J. Dilworth
Article copyright:
© Copyright 2020
American Mathematical Society
