## Abstract

This paper concerns the question whether the cone spectral radius r_{C}(f) of a continuous compact order-preserving homogenous map f: C → C on a closed cone C in Banach space X depends continuously on the map. Using the fixed point index we show that if there exists 0 < a_{C}1 < a_{2} < a_{3} <... not in the cone spectrum, σ_{C}(f), and limk→∞ ^{a}_{k} = r_{C}(f), then the cone spectral radius is continuous. An example is presented showing that if such a sequence (a_{k})_{k} does not exist, continuity may fail. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that if C is a polyhedral cone with m faces, then σ_{C}(f) contains at most m - 1 elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each nonpolyhedral cone there exist maps whose cone spectrum is infinite.

Original language | English (US) |
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Pages (from-to) | 2741-2754 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 141 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2013 |

## All Science Journal Classification (ASJC) codes

- Applied Mathematics
- Mathematics(all)

## Keywords

- Cone spectral radius
- Cone spectrum
- Continuity
- Fixed point index
- Nonlinear cone maps