Common Fixed Points of Commuting Monotone Mappings

Abstract

We are concerned here with the existence of fixed or common fixed points of commuting monotone self-mappings of a partially ordered set into itself. Let X be a partially ordered set. A self-mapping ƒ of X into itself is called an isotone mapping if x ⩾ y implies ƒ(x) ⩾ ƒ(y). Similarly, a self-mapping ƒ of X into itself is called an antitone mapping if x ⩾ y implies ƒ(x) ⩽ ƒ(y). An element X0 ∈ X is called well-ordered complete if every well-ordered subset with x0 as its first element has a supremum. An element x0 ∈ X is called chain-complete if every non-empty chain C ⊆ X such that x ⩾ x0 for all x ∈ C, has a supremum. X is called a well-ordered-complete semi-lattice if every non-empty well-ordered subset has a supremum. X is called a complete semi-lattice if every non-empty subset of X has a supremum.