Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function

Rolando Cavazos-Cadena

Kybernetika (2010)

  • Volume: 46, Issue: 4, page 665-683
  • ISSN: 0023-5954

Abstract

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This work is concerned with the eigenvalue problem for a monotone and homogenous self-mapping f of a finite dimensional positive cone. Paralleling the classical analysis of the (linear) Perron–Frobenius theorem, a verifiable communication condition is formulated in terms of the successive compositions of f , and under such a condition it is shown that the upper eigenspaces of f are bounded in the projective sense, a property that yields the existence of a nonlinear eigenvalue as well as the projective boundedness of the corresponding eigenspace. The relation of the communication property studied in this note with the idea of indecomposability is briefly discussed.

How to cite

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Cavazos-Cadena, Rolando. "Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function." Kybernetika 46.4 (2010): 665-683. <http://eudml.org/doc/196772>.

@article{Cavazos2010,
abstract = {This work is concerned with the eigenvalue problem for a monotone and homogenous self-mapping $f$ of a finite dimensional positive cone. Paralleling the classical analysis of the (linear) Perron–Frobenius theorem, a verifiable communication condition is formulated in terms of the successive compositions of $f$, and under such a condition it is shown that the upper eigenspaces of $f$ are bounded in the projective sense, a property that yields the existence of a nonlinear eigenvalue as well as the projective boundedness of the corresponding eigenspace. The relation of the communication property studied in this note with the idea of indecomposability is briefly discussed.},
author = {Cavazos-Cadena, Rolando},
journal = {Kybernetika},
keywords = {projectively bounded and invariant sets; generalized Perron–Frobenius conditions; nonlinear eigenvalue; Collatz–Wielandt relations; projectively bounded and invariant sets; generalized Perron-Frobenius conditions; nonlinear eigenvalue; Collatz-Wielandt relations},
language = {eng},
number = {4},
pages = {665-683},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function},
url = {http://eudml.org/doc/196772},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Cavazos-Cadena, Rolando
TI - Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 4
SP - 665
EP - 683
AB - This work is concerned with the eigenvalue problem for a monotone and homogenous self-mapping $f$ of a finite dimensional positive cone. Paralleling the classical analysis of the (linear) Perron–Frobenius theorem, a verifiable communication condition is formulated in terms of the successive compositions of $f$, and under such a condition it is shown that the upper eigenspaces of $f$ are bounded in the projective sense, a property that yields the existence of a nonlinear eigenvalue as well as the projective boundedness of the corresponding eigenspace. The relation of the communication property studied in this note with the idea of indecomposability is briefly discussed.
LA - eng
KW - projectively bounded and invariant sets; generalized Perron–Frobenius conditions; nonlinear eigenvalue; Collatz–Wielandt relations; projectively bounded and invariant sets; generalized Perron-Frobenius conditions; nonlinear eigenvalue; Collatz-Wielandt relations
UR - http://eudml.org/doc/196772
ER -

References

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  1. Akian, M., Gaubert, S., 10.1016/S0362-546X(02)00170-0, Nonlinear Anal. 52 (2003), 2, 637–679. (2003) Zbl1030.47048MR1938367DOI10.1016/S0362-546X(02)00170-0
  2. Akian, M., Gaubert, S., Lemmens, B., Nussbaum, R., 10.1017/S0305004105008832, Math. Proc. Camb. Phil. Soc. 140 (2006), 157–176. (2006) Zbl1101.37032MR2197581DOI10.1017/S0305004105008832
  3. Cavazos–Cadena, R., Hernández–Hernández, D., Poisson equations associated with a homogeneous and monotone function: necessary and sufficient conditions for a solution in a weakly convex case, Nonlinear Anal.: Theory, Methods and Appl. 72 (2010), 3303–3313. (2010) Zbl1215.47052MR2587364
  4. Dellacherie, C., Modèles simples de la théorie du potentiel non-linéaire, Lecture Notes in Math. 1426, pp. 52–104, Springer 1990. (1426) MR1071533
  5. Gaubert, S., Gunawardena, J., 10.1090/S0002-9947-04-03470-1, Trans. Amer. Math. Soc. 356 (2004), 12, 4931–4950. (2004) Zbl1067.47064MR2084406DOI10.1090/S0002-9947-04-03470-1
  6. Gunawardena, J., 10.1016/S0304-3975(02)00235-9, Theoret. Comput. Sci. 293 (2003), 141–167. (2003) MR1957616DOI10.1016/S0304-3975(02)00235-9
  7. Kolokoltsov, V. N., Nonexpansive maps and option pricing theory, Kybernetika 34 (1998), 713–724. (1998) MR1695373
  8. Lemmens, B., Scheutzow, M., On the dynamics of sup-norm nonexpansive maps, Ergodic Theory Dynam. Systems 25 (2005), 3, 861–871. (2005) MR2142949
  9. Minc, H., Nonnegative Matrices, Wiley, New York 1988. (1988) Zbl0638.15008MR0932967
  10. Nussbaum, R. D., 10.1090/memo/0391, Memoirs of the AMS 75 (1988), 391. (1988) Zbl0666.47028MR0961211DOI10.1090/memo/0391
  11. Nussbaum, R. D., 10.1090/memo/0401, Memoirs of the AMS 79 (1989), 401. (1989) Zbl0669.47031MR0963567DOI10.1090/memo/0401
  12. Seneta, E., Non-negative Matrices and Markov Chains, Springer, New York 1980. (1980) MR2209438
  13. Zijm, W. H. M., Generalized eigenvectors and sets of nonnegative matrices, inear Alg. Appl. 59 (1984), 91–113. (1984) Zbl0548.15015MR0743048

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