On minimal (maximal) common fixed points of a commuting family of decreasing (increasing) maps

Abstract. We prove that in a complete partially ordered set, every commutative family of decreasing maps has a minimal common fixed point. Let (X,≤) be a partially ordered set. We call X complete if every linearly ordered subset of X has a greatest lower bound in X. A mapping f: X → X is called an decreasing (resp. increasing) if f(x) ≤ x (resp. f(x) ≥ x) for every x ∈ X. A family F of mappings of X into X is called commutative if f ◦ g = g ◦ f for every f, g ∈ F, where ◦ denotes the composition of maps. A subset S of (X,≤) is a directed set if every two, and hence finitely many, elements of S has a lower bound in S. For a set S, |S | denotes the cardinality of S. We shall use the following known fact whose proof can be found in [2]: Proposition 1 Let (X,≤) be a complete partially ordered set. Then every directed subset of X has a greatest lower bound in X. Now we prove the following theorem