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Báo cáo hóa học: " A fixed point theorem for Meir-Keeler contractions in ordered metric spaces"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A fixed point theorem for Meir-Keeler contractions in ordered metric spaces | Harjani et al. Fixed Point Theory and Applications 2011 2011 83 http www.fixedpointtheoryandapplications.eom content 2011 1 83 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access A fixed point theorem for Meir-Keeler contractions in ordered metric spaces Jackie Harjani Belén Lopez and Kishin Sadarangani Correspondence ksadaran@dma. ulpgc.es Departamento de Matemáticas Universidad de Las Palmas de Gran Canaria Campus de Tafira Baja 35017 Las Palmas de Gran Canaria Spain Abstract The purpose of this paper is to present some fixed point theorems for Meir-Keeler contractions in a complete metric space endowed with a partial order. MSC 47H10. Keywords fixed point ordered metric spaces Meir-Keeler contraction V -J 1 Introduction and preliminaries The Banach contraction mapping principle is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Also its significance lies in its vast applicability in a number of branches of mathematics. Generalization of the above principle has been a heavily investigated branch of research. In particular Meir and Keeler 1 present the following fixed point theorem. Theorem 1.1. 1 Let X d be a complete metric space and T X X an operator. Suppose that for every E 0 there exists d e 0 such that for x y e X e d x y e 8 e d Tx Ty e. Then T admits a unique fixed point f e X and for any x e X the sequence Tnx converges to f. The purpose of this article is to present a version of Theorem 1.1 in the context of ordered metric spaces. Existence of fixed point in partially ordered sets has been recently studied in 2-20 . In the context of ordered metric spaces the usual contraction is weakened but at the expense that the operator is monotone. 2 Fixed point results nondecreasing case Our starting point is the following definition. Definition 2.1. Let X be a partially ordered set and T X X a mapping. We say that T is nondecreasing if for x y e X x y Tx Ty. This definition .