Geodesic
Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function
Kybernetika, Tome 46 (2010) no. 4, pp. 665-683.

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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.
Classification : 47H07, 47H09, 47J10
Keywords: projectively bounded and invariant sets; generalized Perron–Frobenius conditions; nonlinear eigenvalue; Collatz–Wielandt relations 
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     author = {Cavazos-Cadena, Rolando},
     title = {Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function},
     journal = {Kybernetika},
     pages = {665--683},
     publisher = {mathdoc},
     volume = {46},
     number = {4},
     year = {2010},
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     zbl = {1208.47059},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010__46_4_a5/}
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Cavazos-Cadena, Rolando. Generalized communication conditions and the eigenvalue problem for a monotone and homogenous function. Kybernetika, Tome 46 (2010) no. 4, pp. 665-683. http://geodesic.mathdoc.fr/item/KYB_2010__46_4_a5/