RIEB, Kobe University

Consider a preordered metric space ( X , d , ⪯ ) . Suppose that d ( x , y ) ≤ d ( x ′ , y ′ ) if x ′ ⪯ x ⪯ y ⪯ y ′ . We say that a self-map T on X is asymptotically contractive if d ( T ⁱ x , T ⁱ y ) → 0 for all x , y ∈ X . We show that an order-preserving self-map T on X has a globally stable fixed point if and only if T is asymptotically contractive and there exist x , x ^∗ ∈ X such that T ⁱ x ⪯ x ^∗ for all i ∈ ℕ and x ^∗ ⪯ T x ^∗ . We establish this and other fixed point results for more general spaces where d consists of a collection of distance measures. We apply our results to order-preserving nonlinear Markov operators on the space of probability distribution functions on ℝ.