Simple fixed point results for order-preserving self-maps and applications to nonlinear Markov operators

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(Tⁱx, T ⁱy) → 0 as i ↑ ∞ 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 Tⁱx ≤ x*for all i ∈ ℕ and x*≤ Tx*. 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.