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,Tiy)→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.