Continuous extension of order-preserving homogeneous maps

Andrew D. Burbanks; Colin T. Sparrow; Roger D. Nussbaum

Kybernetika (2003)

  • Volume: 39, Issue: 2, page [205]-215
  • ISSN: 0023-5954

Abstract

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Maps f defined on the interior of the standard non-negative cone K in 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 - { 0 } . In the case where the cycle time χ ( f ) of the original map does not exist, such eigenvectors must lie in ∂ K - { 0 } .

How to cite

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Burbanks, Andrew D., Sparrow, Colin T., and Nussbaum, Roger D.. "Continuous extension of order-preserving homogeneous maps." Kybernetika 39.2 (2003): [205]-215. <http://eudml.org/doc/33635>.

@article{Burbanks2003,
abstract = {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 $.},
author = {Burbanks, Andrew D., Sparrow, Colin T., Nussbaum, Roger D.},
journal = {Kybernetika},
keywords = {discrete event systems; order-preserving homogeneous maps; discrete event system; order-preserving homogeneous map},
language = {eng},
number = {2},
pages = {[205]-215},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Continuous extension of order-preserving homogeneous maps},
url = {http://eudml.org/doc/33635},
volume = {39},
year = {2003},
}

TY - JOUR
AU - Burbanks, Andrew D.
AU - Sparrow, Colin T.
AU - Nussbaum, Roger D.
TI - Continuous extension of order-preserving homogeneous maps
JO - Kybernetika
PY - 2003
PB - Institute of Information Theory and Automation AS CR
VL - 39
IS - 2
SP - [205]
EP - 215
AB - 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 $.
LA - eng
KW - discrete event systems; order-preserving homogeneous maps; discrete event system; order-preserving homogeneous map
UR - http://eudml.org/doc/33635
ER -

References

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  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 Zbl1048.47040MR1960046
  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 (1999) 
  3. Crandall M. G., Tartar L., 10.1090/S0002-9939-1980-0553381-X, Proc. Amer. Math. Soc. 78 (1980), 385–390 (1980) Zbl0449.47059MR0553381DOI10.1090/S0002-9939-1980-0553381-X
  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 
  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 
  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 (1981) MR0643014
  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 (1994) Zbl0844.58010MR1269677
  8. Riesz F., Sz.-Nagy B., Functional Analysis, Frederick Ungar Publishing Company, New York 1955 Zbl0732.47001MR0071727
  9. Thompson A. C., On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc. 14 (1963), 438–443 (1963) Zbl0147.34903MR0149237

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