Nonlinear eigenvalue problems with positively convex operators

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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 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.

Original languageEnglish (US)
Pages (from-to)653-669
Number of pages17
JournalJournal of Mathematical Analysis and Applications
Volume51
Issue number3
DOIs
StatePublished - 1975

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Nonlinear Eigenvalue Problem
Operator
Cones
Positive Cone
Ordered Space
Normed Space
Convex function
Positive Solution
Continuous Function
Intersection
Fixed point
Iteration
Interval

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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

In: Journal of Mathematical Analysis and Applications, Vol. 51, No. 3, 1975, p. 653-669.

Research output: Contribution to journalArticle

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