| Title:
|
Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function (English) |
| Author:
|
Cavazos-Cadena, Rolando |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
665-683 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
projectively bounded and invariant sets |
| Keyword:
|
generalized Perron–Frobenius conditions |
| Keyword:
|
nonlinear eigenvalue |
| Keyword:
|
Collatz–Wielandt relations |
| MSC:
|
47H07 |
| MSC:
|
47H09 |
| MSC:
|
47J10 |
| idZBL:
|
Zbl 1208.47059 |
| idMR:
|
MR2722094 |
| . |
| Date available:
|
2010-10-22T05:25:38Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140777 |
| . |
| Reference:
|
[1] Akian, M., Gaubert, S.: Spectral theorem for convex monotone homogeneous maps, and ergodic control.Nonlinear Anal. 52 (2003), 2, 637–679. Zbl 1030.47048, MR 1938367, 10.1016/S0362-546X(02)00170-0 |
| Reference:
|
[2] Akian, M., Gaubert, S., Lemmens, B., Nussbaum, R.: Iteration of order preserving subhomogeneous maps on a cone.Math. Proc. Camb. Phil. Soc. 140 (2006), 157–176. Zbl 1101.37032, MR 2197581, 10.1017/S0305004105008832 |
| Reference:
|
[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. Zbl 1215.47052, MR 2587364 |
| Reference:
|
[4] Dellacherie, C.: Modèles simples de la théorie du potentiel non-linéaire.Lecture Notes in Math. 1426, pp. 52–104, Springer 1990. MR 1071533 |
| Reference:
|
[5] Gaubert, S., Gunawardena, J.: The Perron–Frobenius theorem for homogeneous, monotone functions.Trans. Amer. Math. Soc. 356 (2004), 12, 4931–4950. Zbl 1067.47064, MR 2084406, 10.1090/S0002-9947-04-03470-1 |
| Reference:
|
[6] Gunawardena, J.: From max-plus algebra to nonexpansive maps: a nonlinear theory for discrete event systems.Theoret. Comput. Sci. 293 (2003), 141–167. MR 1957616, 10.1016/S0304-3975(02)00235-9 |
| Reference:
|
[7] Kolokoltsov, V. N.: Nonexpansive maps and option pricing theory.Kybernetika 34 (1998), 713–724. MR 1695373 |
| Reference:
|
[8] Lemmens, B., Scheutzow, M.: On the dynamics of sup-norm nonexpansive maps.Ergodic Theory Dynam. Systems 25 (2005), 3, 861–871. MR 2142949 |
| Reference:
|
[9] Minc, H.: Nonnegative Matrices.Wiley, New York 1988. Zbl 0638.15008, MR 0932967 |
| Reference:
|
[10] Nussbaum, R. D.: Hilbert’s projective metric and iterated nonlinear maps.Memoirs of the AMS 75 (1988), 391. Zbl 0666.47028, MR 0961211, 10.1090/memo/0391 |
| Reference:
|
[11] Nussbaum, R. D.: Iterated nonlinear maps and Hilbert’s projective metric.Memoirs of the AMS 79 (1989), 401. Zbl 0669.47031, MR 0963567, 10.1090/memo/0401 |
| Reference:
|
[12] Seneta, E.: Non-negative Matrices and Markov Chains.Springer, New York 1980. MR 2209438 |
| Reference:
|
[13] Zijm, W. H. M.: Generalized eigenvectors and sets of nonnegative matrices.inear Alg. Appl. 59 (1984), 91–113. Zbl 0548.15015, MR 0743048 |
| . |
