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Asymptotic behavior of nonexpansive mappings in finite dimensional normed spaces

Author(s): Brian Lins
Journal: Proc. Amer. Math. Soc. 137 (2009), 2387-2392.
MSC (2000): Primary 47H09
Posted: December 23, 2008
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Abstract: If $ X$ is a finite dimensional real normed space, $ C$ is a closed convex subset of $ X$ and $ f:C \rightarrow C$ is nonexpansive with respect to the norm on $ X$, then we show that either $ f$ has a fixed point in $ C$ or there is a linear functional $ \varphi \in X^*$ such that $ \lim_{k \rightarrow \infty} \varphi(f^k(x)) = \infty$ for all $ x \in C$.

References:

1.
A. F. Beardon. Iteration of contractions and analytic maps, J. London Math. Soc. 41 (1990), 141-150. MR 1063551 (91i:51025)

2.
D. Burago, Y. Burago, and S. Ivanov. A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, Amer. Math. Soc., Providence, RI, 2001. MR 1835418 (2002e:53053)

3.
A. Całka. On conditions under which isometries have bounded orbits, Colloq. Math. 48 (1984), 219-227. MR 758530 (85m:54027)

4.
C. M. Dafermos and M. Slemrod. Asymptotic behavior of nonlinear contraction semigroups, J. Functional Anal. 13 (1973), 97-106. MR 0346611 (49:11336)

5.
M. Edelstein. On non-expansive mappings of Banach spaces, Proc. Camb. Phil. Soc. 60 (1964), 439-447. MR 0164222 (29:1521)

6.
J. Gunawardena and M. Keane. On the existence of cycle times for some nonexpansive maps, Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs, 1995.

7.
A. Karlsson. Non-expanding maps and Busemann functions, Ergodic Theory and Dynamical Systems 21 (2001), 1447-1457. MR 1855841 (2002f:37055)

8.
A. Karlsson, V. Metz and G. Noskov. Horoballs in simplices and Minkowski spaces, Int. J. Math. Math. Sci. (2006), Art. ID 23656, 20 pages. MR 2268510 (2007k:53047)

9.
E. Kohlberg. Invariant half-lines of nonexpansive piecewise-linear transformations, Math. Oper. Res. 5 (1980), 366-372. MR 594851 (82a:55005)

10.
E. Kohlberg and A. Neyman. Asymptotic behavior of nonexpansive mappings in normed linear spaces, Israel J. Math. 38 (1981), 269-275. MR 617673 (83g:47056)

11.
B. Lins. A Denjoy-Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains, Math. Proc. Camb. Phil. Soc. 143 (2007), 157-164. MR 2340981 (2008i:47102)

12.
R. D. Nussbaum. Fixed point theorems and Denjoy-Wolff theorems for Hilbert's projective metric in infinite dimensions, Topological Methods in Nonlinear Analysis 29 (2007), 199-249. MR 2345061

13.
A. T. Plant and S. Reich. The asymptotics of nonexpansive iterations, J. Functional Anal. 54 (1983), 308-319. MR 724526 (85a:47055)

14.
S. Reich. Fixed point iterations of nonexpansive mappings, Pacific J. Math. 60 (1975), 195-198. MR 0428130 (55:1159)

15.
S. Reich. Extension problems for accretive sets in Banach spaces, J. Functional Anal. 26 (1977) 378-395. MR 0477893 (57:17393)

16.
S. Reich. On the asymptotic behavior of nonlinear semigroups and the range of accretive operators, J. Math. Anal. Appl. 79 (1981), 113-126. MR 603380 (82c:47066)

17.
R. T. Rockafeller. Convex Analysis, Princeton University Press, Princeton, NJ, 1970. MR 0274683 (43:445)

18.
B. D. Rouhani and W. A. Kirk. Asymptotic properties of nonexpansive iterations in reflexive spaces, J. Math. Anal. Appl. 236 (1999), 281-289. MR 1704583 (2001a:47056)

19.
C. Walsh. The horofunction boundary of finite-dimensional normed spaces, Math. Proc. Camb. Phil. Soc. 142 (2007), 497-507. MR 2329698 (2008e:53150)

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Additional Information:

Brian Lins
Affiliation: Department of Mathematics and Computer Science, Hampden-Sydney College, Hampden-Sydney, Virginia 23943
Email: blins@hsc.edu

DOI: 10.1090/S0002-9939-08-09779-7
PII: S 0002-9939(08)09779-7
Received by editor(s): July 23, 2007,
Received by editor(s) in revised form: September 28, 2008
Posted: December 23, 2008
Communicated by: Marius Junge
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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