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(x0; y0) if x0 x y y0. We say that a self-map T on X is asymptotically contractive if d(Tix; Tiy) ! 0 for all x; y 2 X. We show that an order-preserving self-map T on X has a globally stable xed point if and only if T is asymptotically contractive and there exist x; x 2 X such that Tix x for all i 2 N and x Tx. We establish this and other xed 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. Keywords: xed point; order-preserving self-map; contraction; nonlin- ear Markov operator; global stability