### Abstract

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u ε{lunate} K, and let Λ_{0} be the set of positive λ for which a positive solution-necessarily the minimum one-can be obtained by an iteration u_{n} = λAu_{n-1}, u_{0} = 0. We show that if K is normal, and if Λ is nonempty, then Λ_{0} is nonempty, and each set Λ_{0}, Λ is an interval with inf(Λ_{0}) = inf(Λ) = 0 and sup(Λ_{0}) = sup(Λ) (= λ*, say); but we may have λ* ∉ Λ_{0} and λ* ε{lunate} Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ_{0} is nonempty. Let u^{0}(λ) = lim_{n→∞}(λA)^{n}(0) be the minimum positive fixed point corresponding to λ ε{lunate} Λ_{0}. Then u^{0}(λ) is a continuous isotone convex function of λ on Λ_{0}.

Original language | English (US) |
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Pages (from-to) | 653-669 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 51 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1975 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics