Order-theoretical fixed point theorems for correspondences and application in game theory

For an ascending correspondence F:X→ 2X with chain-complete values on a complete lattice X, we prove that the set of fixed points is a complete lattice. This strengthens Zhou's fixed point theorem. For chain-complete posets that are not necessarily lattices, we generalize the Abian-Brown and the Markowsky fixed point theorems from single-valued maps to multivalued correspondences. We provide an application in game theory.