The Knaster–Tarski theorem versus monotone nonexpansive mappings

Abstract

Let X be a partially ordered set with the property that each family of order intervals of the form [a, b], [a, →) with the finite intersection property has a nonempty intersection. We show that every directed subset of X has a supremum. Then we apply the above result to prove that if X is a topological space with a partial order for which the order intervals are compact, F a nonempty commutative family of monotone maps from X into X and there exists c ∈ X such that c T c for every T ∈ F , then the set of common fixed points of F is nonempty and has a maximal element. The result, specialized to the case of Banach spaces gives a general fixed point theorem that drops almost all assumptions from the recent results in this area. An application to the theory of integral equations of Urysohn's type is also given.2010 Mathematics Subject Classification. Primary 06A45, 54F05; Secondary: 34A12, 46B20.

Notes on Knaster-Tarski Theorem versus Monotone Nonexpansive Mappings

“…Therefore, it was unknown whether an analogue to Kirk's fixed point theorem [26] for monotone nonexpansive mappings does exist. It was this problem that led Espínola and Wiśnicki [17] to discover the following results.…”

“…Lemma 5.1. [17] Let X be a partially ordered set for which each family of order intervals of the form [a, b], [a, →) with the finite intersection property has a nonempty intersection. Then every directed subset of X has a supremum.…”

“…Theorem 5.1. [17] Let X be a topological space with a partial order for which order intervals are compact and let T : X → X be monotone. If there exists c ∈ X such that c T (c), then the set of all fixed points of T is nonempty and has a maximal element.…”

Common Fixed-Point Theorems in Modular Function Spaces Endowed with Reflexive Digraph

“…eir starting point was to approach the fixed point by iterative techniques and successive approximations. Recently, Espínola and Wiśnicki [8] generalized the above results in Hausdorff topological spaces endowed with partial order. e key ingredient in such a generalization is the compactness of the order intervals mixed with Knaster-Tarski fixed point.…”

Remarks on asymptotic regularity and fixed points

“…For such mappings one of the fundamental results in fixed-point theory is the classical Knaster-Tarski theorem (also known as the Abian-Brown theorem), see [26], [38]. Recently, Espínola and Wiśnicki [15] studied the problem whether the classical Kirk's theorem for nonexpansive mappings (see [17]) still holds for monotone-nonexpansive mappings. They proved in some partially ordered sets a general theorem which guarantees the existence of a fixed point for monotone mappings (which need not be either monotone-nonexpansive nor continuous), and which does not impose any conditions on the Banach space.…”