Fixed point theorems for increasing correspondences on lattices

Abstract For an ascending correspondence $$F:X\rightarrow 2^X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>X</mml:mi> </mml:msup> </mml:mrow> </mml:math> with chain-complete values on a complete lattice X , we prove that the set of fixed points is a complete lattice. This generalizes Zhou’s fixed point theorem. We provide an application to games with strategic complementarities.