On sequences of iterations of increasing and continuous mappings on complete lattices

Abstract

We generalize the famous Tarski result by showing that: if X is a complete lattice, and f:X→X is an increasing and continuous mapping, then for all points x0∈X, the limits of sequences (fn(lim⁡supk⁡fk(x0)))n=1∞ and (fn(lim⁡infk⁡fk(x0)))n=1∞ are fixed points of f. These limits are the tight fixed-point bounds between which sufficiently large iterations fk(x0) are located. We provide an application of this result to studying best-response dynamics.