A Tarski–Kantorovich theorem for correspondences

Abstract

For a weakly monotone (resp., strongly monotone) upper order hemicontinuous correspondence $F:A\rightrightarrows A$, where $A$ is a complete lattice (resp., a $\sigma$-complete lattice), we provide tight fixed-point bounds for sufficiently large iterations $F^k\left(a^0\right)$, starting from any point $a^0\in A$. Our results, hence, prove a generalization of the Tarski–Kantorovich theorem. We provide an application of our results to a class of social learning models on networks.