Nonlinear Analysis 75 (2012) 6148–6159 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Contractive mappings and existence of cycle times for a monotone and homogeneous function ✩ Rolando Cavazos-Cadena a , Daniel Hernández-Hernández b,∗ a Departamento de Estadística y Cálculo, Universidad Autónoma Agraria Antonio Narro, Buenavista, Saltillo COAH 25315, Mexico b Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato GTO 36000, Mexico article info Article history: Received 20 March 2012 Accepted 18 June 2012 Communicated by S. Carl MSC: primary 47J10 secondary 47H09 47H07 Keywords: Non-expansive map Topical functions Nonlinear Perron–Frobenius eigenvalue Contractive approximations Multiplicative ergodic behavior Hilbert’s distance abstract Given a monotone and homogeneous self-mapping f of the n-dimensional positive cone, a family of contractive mappings is used to define an equivalence relation in the index set, as well as a total order among the equivalence classes. Then, it is shown (i) that the cycle times are well-defined at each index belonging to the maximal and minimal classes, and (ii) that the cycle times of f exist at every index whenever a weak convexity condition is satisfied. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction This work concerns the long-term behavior of self-iterations of monotone and homogeneous functions of a finite- dimensional positive cone in itself. Given one of those mappings, say f , the paper establishes conditions ensuring the existence of the cycle times associated with f , which determine the grow rates of the components of the successive compositions of f , and in certain applications (within an additive context) can be thought of as an asymptotic average time to the next occurrence [1]. From an explicit example in [2], it is known that the cycle times of a monotone and homogeneous function f do not necessarily exist at every index. However, it was recently shown in [3] that there is always (at least) an index i at which the cycle time is well-defined, and the first objective of this work points in this direction: (i) To identify a subset E (f ) of the index set such that the cycle time of f is well-defined at each index belonging to E (f ). On the other hand, the aforementioned example in [2] shows that the existence of the cycle times at every index can occur only under special circumstances. For instance, if f admits an eigenvalue λ – that is, f (x) = λx for some x – then the cycle times are well-defined at every index and their common value is λ. However, the existence of an eigenvalue of f is a very strong property that can be ensured only under appropriate communication conditions [3,4], and the other major objective ✩ This work was supported by the PSF Organization under Grant No. 10-98-02, and in part by CONACYT under Grants 25357 and 61423. ∗ Corresponding author. E-mail addresses: rcavazos@uaaan.mx (R. Cavazos-Cadena), dher@cimat.mx (D. Hernández-Hernández). 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.06.026