Common coupled coincidence and coupled fixed point results in two generalized metric spaces
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Common coupled coincidence and coupled fixed point results in two generalized metric spaces Wasfi Shatanawi1*, Mujahid Abbas2 and Talat Nazir2 * Correspondence: [email protected]. jo 1 Department of Mathematics, The Hashemite University, Zarqa 13115, Jordan Full list of author information is available at the end of the article
Abstract In this article, we prove the existence of common coupled coincidence and coupled fixed point of generalized contractive type mappings in the context of two generalized metric spaces. These results generalize several comparable results from the current literature. We also provide illustrative examples in support of our new results. 2000 MSC: 47H10. Keywords: coupled coincidence point, common coupled fixed point, weakly compatible maps, generalized metric space
1 Introduction and preliminaries The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity [1-5]. Mustafa and Sims [4] generalized the concept of a metric space and call it a generalized metric space. Based on the notion of generalized metric spaces, Mustafa et al. [5-9] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [10] initiated the study of common fixed point theory in generalized metric spaces (see also [11]). Saadati et al. [12] proved some fixed point results for contractive mappings in partially ordered G-metric spaces. Abbas et al. [13] obtained some periodic point results in generalized metric spaces. Shatanawi [14] obtained some fixed point results for contractive mappings satisfying F-maps in G-metric spaces (see also [15]). Bhashkar and Lakshmikantham [16] introduced the concept of a coupled fixed point of a mapping F : X × X ® X (a nonempty set) and established some coupled fixed point theorems in partially ordered complete metric spaces. Later, Lakshmikantham and Ćirić [3] proved coupled coincidence and coupled common fixed point results for nonlinear mappings F : X × X ® X and g : X ® X satisfying certain contractive conditions in partially ordered complete metric spaces. Recently, Abbas et al. [17] obtained some coupled common fixed point results in two generalized metric spaces. Choudhury and Maity [18] also proved the existence of coupled fixed points in generalized metric spaces. Recently, Aydi et al. [19] generalized the results of Choudhury and Maity [18]. For other works on G-metric spaces, we refer the reader to [20,21]. The aim of this article is to prove some common coupled coincidence and coupled fixed points results for mappings defined on a set equipped with two generalized © 2011 Shatanawi 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, and reproduction in any medium, provided the original work is properly cited.
Shatanawi et al. Fixed Point Theory and Applicatio
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