Nonlinear eigenvalue problems with positively convex operators

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 u_n = λAu_n-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) = lim_n→∞(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.