G -Metric spaces in any number of arguments and related fixed-point theorems
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RESEARCH
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G-Metric spaces in any number of arguments and related fixed-point theorems Antonio Roldán1* , Erdal Karapınar2 and Poom Kumam3 * Correspondence: [email protected] 1 University of Jaén, Campus las Lagunillas s/n, Jaén, 23071, Spain Full list of author information is available at the end of the article
Abstract Inspired by the notion of Mustafa and Sims’ G-metric space and the attention that this kind of metric has received in recent times, we introduce the concept of a G-metric space in any number of variables, and we study some of the basic properties. Then we prove that the family of this kind of metric is closed under finite products. Finally, we show some fixed-point theorems that improve and extend some well-known results in this field. MSC: 46T99; 47H10; 47H09; 54H25 Keywords: partially ordered set; fixed point; contractive mapping; G-metric space
1 Introduction In the s, Gähler [, ] tried to generalize the notion of metric and introduced the concept of -metric spaces inspired by the mapping that associated the area of a triangle to its three vertices. Later, Dhage [] changed the axioms and presented the concept of a D-metric. Unfortunately, both kinds of metrics appear not to have as good properties as their authors announced (see [–]). To overcome these drawbacks, Mustafa and Sims [] presented the notion of a G-metric space, which have received much attention since then. The literature on this topic, especially in related fixed point theory, has grown a lot in recent times (see, for instance, [–] and references therein). The main aim of the present paper is to introduce the notion of a G-metric space in any number of variables. To do that, we have been inspired by the perimeter of a triangle, as well as Dhage, which in the multidimensional case can be seen as the sum of all distances between any pair of points. In this sense, the axioms we present and the properties we deduce are very natural. We also prove two relevant facts: the product of metrics of this kind is also a metric in this sense, and there is no a trivially way to reduce the number of variables (which, for instance, permits us to reduce a Gn∗ -metric to the Mustafa and Sims’ spaces). Later, we demonstrate some fixed-point theorems distinguishing between the axioms that the metric verifies (Gn∗ -metrics and Gn -metrics). As a consequence, our main results are, obviously, valid in the context of G-metric spaces. 2 Preliminaries Let n be a positive integer such that n ≥ . Henceforth, X will denote a non-empty set and n X n will denote the product space X × X × · · · × X. Throughout this manuscript, m and k will denote non-negative integers. Unless otherwise stated, ‘for all m’ will mean ‘for all m ≥ ’. Let R+ = [, ∞) and let N = {, , , . . .}. ©2014 Roldán 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, p
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