On a problem of Nirenberg concerning expanding maps in Hilbert space
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- by Janusz Szczepański PDF
- Proc. Amer. Math. Soc. 116 (1992), 1041-1044 Request permission
Abstract:
Let ${\mathbf {H}}$ be a Hilbert space and $f:{\mathbf {H}} \to {\mathbf {H}}$ a continuous map which is expanding (i.e., $||f({\mathbf {x}}) - f({\mathbf {y}})|| \geq ||{\mathbf {x}} - {\mathbf {y}}||$ for all ${\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}$) and such that $f({\mathbf {H}})$ has nonempty interior. Are these conditions sufficient to ensure that $f$ is onto? This question was stated by Nirenberg in 1974. In this paper we give a partial negative answer to this problem; namely, we present an example of a map $F:{\mathbf {H}} \to {\mathbf {H}}$ which is not onto, continuous, $F({\mathbf {H}})$ has nonempty interior, and for every ${\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}$ there is ${n_0} \in \mathbb {N}$ (depending on ${\mathbf {x}}$ and ${\mathbf {y}}$) such that for every $n \geq {n_0}$ \[ ||{F^n}({\mathbf {x}}) - {F^n}({\mathbf {y}})|| \geq {c^{n - m}}||{\mathbf {x}} - {\mathbf {y}}||\] where ${F^n}$ is the $n$th iterate of the map $F,c$ is a constant greater than 2, and $m$ is an integer depending on ${\mathbf {x}}$ and ${\mathbf {y}}$. Our example satisfies $||F({\mathbf {x}})|| = c||{\mathbf {x}}||$ for all ${\mathbf {x}} \in {\mathbf {H}}$. We show that no map with the above properties exists in the finite-dimensional case.References
-
L. E. J. Brouwer, Bewels der Invarianz des $n$-dimensionalen Gebiets, Math. Ann. 71 (1912), 305-313.
- L. E. J. Brouwer, Zur Invarianz des $n$-dimensionalen Gebiets, Math. Ann. 72 (1912), no. 1, 55–56 (German). MR 1511685, DOI 10.1007/BF01456889
- Felix E. Browder, The solvability of non-linear functional equations, Duke Math. J. 30 (1963), 557–566. MR 156204
- Kung Ching Chang and Shu Jie Li, A remark on expanding maps, Proc. Amer. Math. Soc. 85 (1982), no. 4, 583–586. MR 660608, DOI 10.1090/S0002-9939-1982-0660608-4
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835 J. Leray, Topologie des espaces abstraits de M. Banach, C. R. Acad. Sci. Paris Sér. I Math. 200 (1935), 1083-1085.
- George J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346. MR 169064
- 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, DOI 10.1016/0022-1236(84)90057-0
- 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
- Roger D. Nussbaum, Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972), 741–766. MR 306986, DOI 10.1016/0022-247X(72)90253-3
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 1041-1044
- MSC: Primary 47H99; Secondary 47H09
- DOI: https://doi.org/10.1090/S0002-9939-1992-1100665-0
- MathSciNet review: 1100665