Proceedings of the American Mathematical Society

Minimal displacement and retraction problems in infinite-dimensional Hilbert spaces

Author(s): Krzysztof Bolibok
Journal: Proc. Amer. Math. Soc. 132 (2004), 1103-1111.
MSC (2000): Primary 47H09, 47H10
Posted: September 18, 2003
Retrieve article in: PDF DVI PostScript

Abstract: We give the first constructive example of a Lipschitz mapping with positive minimal displacement in an infinite-dimensional Hilbert space

We use this construction to obtain an evaluation from below of the minimal displacement characteristic in the space

In the second part we present a simple and constructive proof of existence of a Lipschitz retraction from a unit ball

onto a unit sphere

in the space

, and we improve an evaluation from above of a retraction constant $k_{0}\left( H\right) .$

1.: Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439-445. MR 85h:46028
2.: K. Bolibok, Constructions of Lipschitzian mappings with non-zero minimal displacement in spaces $L^{1}\left( 0,1\right)$ and $L^{2}\left( 0,1\right),$ Ann. Univ. Marie Curie-Sklodowska Sect. A 50 (1996), 25-31. MR 98j:47123
3.: K. Bolibok, Construction of a Lipschitz retraction in the space $c_{0,}$ Ann. Univ. Marie Curie-Sklodowska Sect. A. 51 (1997), 43-46. MR 99m:46029
4.: K. Bolibok and K. Goebel, A note on minimal displacement and retraction problems, J. Math. Anal. Appl. 206 (1997), 308-314. MR 97m:47078
5.: C. Franchetti, Lipschitz maps and the geometry of the unit ball in normed spaces, Arch. Math. 46 (1986), 76-84. MR 87g:46025
6.: M. Furi and M. Martelli,On the minimal displacement of points under alpha-Lipschitz maps in normed spaces, Boll. Un. Mat. Ital. 9 (1974), 791-799. MR 51:6509
7.: K. Goebel, On the minimal displacement of points under Lipschitz mappings, Pacific J. Math. 45 (1973), 151-163. MR 48:7050
8.: K. Goebel, Metric environment of the topological fixed point theorems, ``Handbook of Metric Fixed Point Theory'', Kluwer, Dordrecht, 2001, 577-611. MR 2003e:47096
9.: K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990. MR 92c:47070
10.: K. Goebel and T. Komorowski, Retracting balls onto spheres and minimal displacement problems, Fixed Point Theory and Applications, Pitman Research Notes in Math., Longman Sci. Tech., Harlow, 1991, 155-172. MR 93b:47110
11.: T. Komorowski and J. Wosko, A remark on the retracting of a ball onto a sphere in an infinite-dimensional Hilbert space, Math. Scand. 67 (1990), 223-226. MR 92e:58012
12.: P. K. Lin and Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), 633-639. MR 86c:47074
13.: B. Nowak, On the Lipschitz retraction of the unit ball in infinite-dimensional Banach spaces onto its boundary, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 861-864. MR 82g:58008
14.: S. Reich, Minimal displacement of points under weakly inward pseudo-Lipschitzian mappings I, Atti. Accad. Naz. Linzei Rend. Cl. Sci. Fis. Mat. Natur. 59 (1975), 40-44. MR 56:9345
15.: S. Reich, Minimal displacement of points under weakly inward pseudo-Lipschitzian mappings II, Atti. Accad. Naz. Linzei Rend. Cl. Sci. Fis. Mat. Natur. 60 (1976), 95-96. MR 58:7264
16.: R. L. Thele, Some results on the radial projection in Banach spaces, Proc. Amer. Math. Soc. 42 (1974), 483-486. MR 48:6892

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H09, 47H10

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS