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Generalized communication conditions and the eigenvalue problem for a monotone and homogeneous function. (English) Zbl 1208.47059
The author obtains a nonlinear version of the Perron-Frobenius theorem in the case of an eigenvalue problem involving a monotone and homogeneous self-mapping of a finite-dimensional positive cone. The approach relies on the study of the notions of projectively bounded and invariant sets.

MSC:
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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References:
[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 · doi:10.1016/S0362-546X(02)00170-0 · arxiv:math/0110108
[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 · doi:10.1017/S0305004105008832 · arxiv:math/0410084
[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 · doi:10.1016/j.na.2009.12.010
[4] Dellacherie, C.: Modèles simples de la théorie du potentiel non-linéaire. Lecture Notes in Math. 1426, pp. 52-104, Springer 1990.
[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 · doi:10.1090/S0002-9947-04-03470-1 · arxiv:math/0105091
[6] Gunawardena, J.: From max-plus algebra to nonexpansive maps: a nonlinear theory for discrete event systems. Theoret. Comput. Sci. 293 (2003), 141-167. · Zbl 1036.93045 · doi:10.1016/S0304-3975(02)00235-9
[7] Kolokoltsov, V. N.: Nonexpansive maps and option pricing theory. Kybernetika 34 (1998), 713-724. · Zbl 1274.91420 · www.kybernetika.cz
[8] Lemmens, B., Scheutzow, M.: On the dynamics of sup-norm nonexpansive maps. Ergodic Theory Dynam. Systems 25 (2005), 3, 861-871. · Zbl 1114.47044 · doi:10.1017/S0143385704000665
[9] Minc, H.: Nonnegative Matrices. Wiley, New York 1988. · Zbl 0638.15008
[10] Nussbaum, R. D.: Hilbert’s projective metric and iterated nonlinear maps. Memoirs of the AMS 75 (1988), 391. · Zbl 0666.47028 · doi:10.1090/memo/0391
[11] Nussbaum, R. D.: Iterated nonlinear maps and Hilbert’s projective metric. Memoirs of the AMS 79 (1989), 401. · Zbl 0669.47031 · doi:10.1090/memo/0401
[12] Seneta, E.: Non-negative Matrices and Markov Chains. Springer, New York 1980. · Zbl 1099.60004 · doi:10.1007/0-387-32792-4
[13] Zijm, W. H. M.: Generalized eigenvectors and sets of nonnegative matrices. inear Alg. Appl. 59 (1984), 91-113. · Zbl 0548.15015 · doi:10.1016/0024-3795(84)90161-7
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