A fixed point theorem for multifunctions and an application (Q1821800)

Let P and Q be ordered sets. A multifunction F of P into Q is a function that assigns to an element \(p\in P\) a non-empty subset F(p) in Q. Let \(F: P\to Q\) and \(G: Q\to R\) be two multifunctions, then the composition \(G\circ F: P\to R\) is defined as \((G\circ F)(p)=\{G(q)|\) \(q\in F(p)\}\). A fixed point of a multifunction \(F: P\to P\) is a point \(p\in P\) such that \(p\in F(p)\). The main result of the paper is a fixed point theorem for compositions of multifunctions that are chain-faithful. This result is then applied to get sufficient conditions for the fixed point property of the product of two ordered sets that have the fixed point property.