| Title:
|
Continuous extension of order-preserving homogeneous maps (English) |
| Author:
|
Burbanks, Andrew D. |
| Author:
|
Sparrow, Colin T. |
| Author:
|
Nussbaum, Roger D. |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
39 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
[205]-215 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Maps $f$ defined on the interior of the standard non-negative cone $K$ in ${\mathbb{R}}^N$ which are both homogeneous of degree $1$ and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson’s part metric and continuous on the interior of the cone. It follows from more general results presented here that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that the extension must have at least one eigenvector in $K-\lbrace 0\rbrace $. In the case where the cycle time $\chi (f)$ of the original map does not exist, such eigenvectors must lie in $\partial {K}-\lbrace 0\rbrace $. (English) |
| Keyword:
|
discrete event systems |
| Keyword:
|
order-preserving homogeneous maps |
| MSC:
|
06F05 |
| MSC:
|
47H07 |
| MSC:
|
47N70 |
| MSC:
|
93B27 |
| MSC:
|
93B28 |
| MSC:
|
93C65 |
| idZBL:
|
Zbl 1249.93123 |
| idMR:
|
MR1996558 |
| . |
| Date available:
|
2009-09-24T19:52:52Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135522 |
| . |
| Reference:
|
[1] Burbanks A. D., Nussbaum R. D., Sparrow C. T.: Extension of order-preserving maps on a cone.Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 35–59 Zbl 1048.47040, MR 1960046 |
| Reference:
|
[2] Burbanks A. D., Sparrow C. T.: All Monotone Homogeneous Functions (on the Positive Cone) Admit Continuous Extension.Technical Report No. 1999-13, Statistical Laboratory, University of Cambridge 1999 |
| Reference:
|
[3] Crandall M. G., Tartar L.: Some relations between nonexpansive and order preserving mappings.Proc. Amer. Math. Soc. 78 (1980), 385–390 Zbl 0449.47059, MR 0553381, 10.1090/S0002-9939-1980-0553381-X |
| Reference:
|
[4] Gaubert S., Gunawardena J.: A Nonlinear Hierarchy for Discrete Event Systems.Technical Report No. HPL-BRIMS-98-20, BRIMS, Hewlett–Packard Laboratories, Bristol 1998 |
| Reference:
|
[5] Gunawardena J., Keane M.: On the Existence of Cycle Times for Some Nonexpansive Maps.Technical Report No. HPL-BRIMS-95-003, BRIMS, Hewlett–Packard Laboratories, Bristol 1995 |
| Reference:
|
[6] Nussbaum R. D.: Eigenvectors of Nonlinear Positive Operators and the Linear Krein–Rutman Theorem (Lecture Notes in Mathematics 886).Springer Verlag, Berlin 1981, pp. 309–331 MR 0643014 |
| Reference:
|
[7] Nussbaum R. D.: Finsler structures for the part-metric and Hilbert’s projective metric, and applications to ordinary differential equations.Differential and Integral Equations 7 (1994), 1649–1707 Zbl 0844.58010, MR 1269677 |
| Reference:
|
[8] Riesz F., Sz.-Nagy B.: Functional Analysis.Frederick Ungar Publishing Company, New York 1955 Zbl 0732.47001, MR 0071727 |
| Reference:
|
[9] Thompson A. C.: On certain contraction mappings in a partially ordered vector space.Proc. Amer. Math. Soc. 14 (1963), 438–443 Zbl 0147.34903, MR 0149237 |
| . |
