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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: On externally complete subsets and common fixed points in partially ordered sets | Abu-Sbeih and Khamsi Fixed Point Theory and Applications 2011 2011 97 http www.fixedpointtheoryandapplications.eom content 2011 1 97 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access On externally complete subsets and common fixed points in partially ordered sets Mohammad Z Abu-Sbeih 1 and Mohamed A Khamsi1 2 Correspondence abusbeih@kfupm.edu.sa 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 Springer 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 initially 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 partially ordered set and M c X a non-empty subset. Recall that an upper resp. lower bound .
