On externally complete subsets and common fixed points in partially ordered sets

Abu-Sbeih, Khamsi

Description

In this study, we introduce the concept of externally complete ordered sets. We discuss the properties of such sets and characterize them in ordered trees. We also prove some common fixed point results for order preserving mappings. In particular, we introduce for the first time the concept of Banach Operator pairs in partially ordered sets and prove a common fixed point result which generalizes the classical De Marr's common fixed point theorem. 2000 MSC : primary 06F30; 46B20; 47E10.

Sujets

• Partially ordered set
• Injective metric space
• Fixed point

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 9
• Langue: English

Extrait

AbuSbeih and KhamsiFixed Point Theory and Applications2011,2011:97 http://www.fixedpointtheoryandapplications.com/content/2011/1/97

R E S E A R C HOpen Access On externally complete subsets and common fixed points in partially ordered sets 1* 1,2 Mohammad Z AbuSbeihand Mohamed A Khamsi

* Correspondence: [e-mail] 1 Department of Mathematics & Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Full list of author information is available at the end of the article

Abstract In this study, we introduce the concept of externally complete ordered sets. We discuss the properties of such sets and characterize them in ordered trees. We also prove some common fixed point results for order preserving mappings. In particular, we introduce for the first time the concept of Banach Operator pairs in partially ordered sets and prove a common fixed point result which generalizes the classical De Marr’s common fixed point theorem. 2000 MSC: primary 06F30; 46B20; 47E10. Keywords:partially ordered sets, order preserving mappings, order trees, hypercon vex metric spaces, fixed point

1. Introduction This article focuses on the externally complete structure, a new concept that was initi ally introduced in metric spaces as externally hyperconvex sets by Aronszajn and Panitchpakdi in their fundamental article [1] on hyperconvexity. This idea developed from the original work of Quilliot [2] who introduced the concept of generalized metric structures to show that metric hyperconvexity is in fact similar to the complete lattice structure for ordered sets. In this fashion, Tarski’s fixed point theorem [3] becomes Sine and Soardi’s fixed point theorems for hyperconvex metric spaces [4,5]. For more on this, the reader may consult the references [68]. We begin by describing the relevant notation and terminology. Let (X,≺) be a par tially ordered set andM⊂Xa nonempty subset. Recall that an upper (resp. lower) bound forMis an elementpÎXwithm≺p(resp.p≺m) for eachmÎM; the least upper (resp. greatestlower) bound ofMwill be denoted supM(resp. infM). A none mpty subsetMof a partially ordered setXwill be called Dedekind complete if for any nonempty subsetA⊂M, supA(resp infA) exists inMprovidedAis bounded above (resp. bounded below) inX. Recall thatM⊂X issaid to be linearly ordered if for everym 1 ,m 2 ÎMwe havem 1 ≺m 2 orm 2 ≺m 1 . A linearly ordered subset ofXis called a chain. For anymÎXdefine ←,m] ={x∈X;x≺m}and [m,→={x∈X;m≺x} Recall that a connected partially ordered setXis called a tree ifXhas a lowest point e, and for everymÎX, the subset [e, m] is well ordered.

© 2011 AbuSbeih and Khamsi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.