On coincidence points of mappings in partially ordered spaces (Q403891)

The authors prove coincidence theorems for two mappings in partially ordered spaces. Here we formulate just the first theorem: Theorem 1. Suppose that there exists an element \(x_0\in X\) such that \(\psi(x_0)\geq \varphi(x_0)\) and the following conditions hold: (a) \(\varphi\) is monotone; (b) \(\psi\) orderly covers the set \(W=\varphi(O_X(x_0))\); (c) for an arbitrary chain \(S\in{\mathfrak I}(\psi,\varphi,O_X(x_0),W)\), there is a lower bound \(u\in X\) for \(S\) such that \(\psi(u)\geq\varphi(u)\). Then \(\psi\) and \(\varphi\) have a coincidence point in \(\xi\in O_X(x_0)\) such that \(\xi\leq x_0\). Moreover the set of coincidence points has a minimal element. Theorem 1 implies the Knaster-Tarski fixed point theorem for monotone mappings and an assertion which generalizes the Birkhoff-Tarski theorem.