Coupled fixed point theorems on partially ordered G -metric spaces
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Coupled fixed point theorems on partially ordered G-metric spaces Erdal Karapınar1 , Poom Kumam2,3 and Inci M Erhan1* *
Correspondence: [email protected] Department of Mathematics, Atilim University, ˙Incek, Ankara 06836, Turkey Full list of author information is available at the end of the article 1
Abstract The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered G-metric spaces in a virtually different and more natural way. MSC: 46N40; 47H10; 54H25; 46T99 Keywords: coupled fixed point; coupled coincidence point; mixed g-monotone property; ordered set; G-metric space
1 Introduction and preliminaries The notion of metric space was introduced by Fréchet [] in . In almost all fields of quantitative sciences which require the use of analysis, metric spaces play a major role. Internet search engines, image classification, protein classification (see, e.g., []) can be listed as examples in which metric spaces have been extensively used to solve major problems. It is conceivable that metric spaces will be needed to explore new problems that will arise in quantitative sciences in the future. Therefore, it is necessary to consider various generalizations of metrics and metric spaces to broaden the scope of applied sciences. In this respect, cone metric spaces, fuzzy metric spaces, partial metric spaces, quasi-metric spaces and b-metric spaces can be given as the main examples. Applications of these different approaches to metrics and metric spaces make it evident that fixed point theorems are important not only for the branches of mainstream mathematics, but also for many divisions of applied sciences. Inspired by this motivation Mustafa and Sims [] introduced the notion of a G-metric space in (see also [–]). In their introductory paper, the authors investigated versions of the celebrated theorems of the fixed point theory such as the Banach contraction mapping principle [] from the point of view of G-metrics. Another fundamental aspect in the theory of existence and uniqueness of fixed points was considered by Ran and Reurings [] in partially ordered metric spaces. After Ran and Reurings’ pioneering work, several authors have focused on the fixed points in ordered metric spaces and have used the obtained results to discuss the existence and uniqueness of solutions of differential equations, more precisely, of boundary value problems (see, e.g., [–]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [], Gnana-Bhaskar and Lakshmikantham [] obtained interesting results related to differential equations with periodic boundary conditions by developing the mixed monotone property in the context of partially ordered metric spaces. As a continuation of this trend, © 2012 Karapınar et al.; 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, a
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