### 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) |
---|---|

Pages (from-to) | 653-669 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 51 |

Issue number | 3 |

DOIs | |

State | Published - 1975 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Nonlinear eigenvalue problems with positively convex operators.** / Laetsch, Theodore W.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 51, no. 3, pp. 653-669. https://doi.org/10.1016/0022-247X(75)90116-X

}

TY - JOUR

T1 - Nonlinear eigenvalue problems with positively convex operators

AU - Laetsch, Theodore W

PY - 1975

Y1 - 1975

N2 - 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 Λ0 be the set of positive λ for which a positive solution-necessarily the minimum one-can be obtained by an iteration un = λAun-1, u0 = 0. We show that if K is normal, and if Λ is nonempty, then Λ0 is nonempty, and each set Λ0, Λ is an interval with inf(Λ0) = inf(Λ) = 0 and sup(Λ0) = sup(Λ) (= λ*, say); but we may have λ* ∉ Λ0 and λ* ε{lunate} Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ0 is nonempty. Let u0(λ) = limn→∞(λA)n(0) be the minimum positive fixed point corresponding to λ ε{lunate} Λ0. Then u0(λ) is a continuous isotone convex function of λ on Λ0.

AB - 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 Λ0 be the set of positive λ for which a positive solution-necessarily the minimum one-can be obtained by an iteration un = λAun-1, u0 = 0. We show that if K is normal, and if Λ is nonempty, then Λ0 is nonempty, and each set Λ0, Λ is an interval with inf(Λ0) = inf(Λ) = 0 and sup(Λ0) = sup(Λ) (= λ*, say); but we may have λ* ∉ Λ0 and λ* ε{lunate} Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ0 is nonempty. Let u0(λ) = limn→∞(λA)n(0) be the minimum positive fixed point corresponding to λ ε{lunate} Λ0. Then u0(λ) is a continuous isotone convex function of λ on Λ0.

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U2 - 10.1016/0022-247X(75)90116-X

DO - 10.1016/0022-247X(75)90116-X

M3 - Article

VL - 51

SP - 653

EP - 669

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 3

ER -