An essential remark on fixed point results on multiplicative metric spaces
 Ravi P Agarwal^{1},
 Erdal Karapınar^{2, 3} and
 Bessem Samet^{4}Email author
https://doi.org/10.1186/s1366301605067
© Agarwal et al. 2016
Received: 19 September 2015
Accepted: 26 February 2016
Published: 5 March 2016
Abstract
In this short note, we announce that all the presented fixed point results in the setting of multiplicative metric spaces can be derived from the corresponding existing results in the context of standard metric spaces in the literature.
Keywords
MSC
1 Introduction and preliminaries
Recently, Bashirov et al. [1] announced multiplicative distance as a new distance notion. Following these initial papers, several authors have reported some fixed point results in the framework of multiplicative metric spaces (see e.g. [2–7] and related references therein).
Definition 1.1
 (i)^{∗} :

\(d^{\ast}(x,y) = 1\) for all \(x, y \in X \),
 (ii)^{∗} :

\(d^{\ast}(x,y) = 1\) if and only if \(x = y\),
 (iii)^{∗} :

\(d^{\ast}(x,y) = d^{\ast}(y, x)\) for all \(x, y \in X\),
 (iv)^{∗} :

\(d^{\ast}(x, z) \leq d^{\ast}(x,y)\cdot d^{\ast}(y, z)\) for all \(x,y,z \in X\) (multiplicative triangle inequality).
For the sake of completeness, we shall present the definition of the (standard) metric.
Definition 1.2
 (i):

\(d(x,y) = 1\) for all \(x, y \in X \),
 (ii):

\(d(x,y) = 1\) if and only if \(x = y\),
 (iii):

\(d(x,y) = d(y, x)\) for all \(x, y \in X\),
 (iv):

\(d(x, z) \leq d(x,y)+d(y, z)\) for all \(x,y,z \in X\) (standard triangle inequality).
Although the multiplicative metric was announced as a new distance notion, we note that composition of the multiplicative metric with a logarithmic function yields a standard metric. Hence, all fixed point results in the context of multiplicative metric spaces can easily be concluded from the corresponding existing famous fixed point results in the context of the standard metric.
2 Main results
Theorem 2.1
Let X be a nonempty set. A mapping \(d^{\ast} : X \times X \to[0, \infty)\) is said to be a multiplicative metric. Then the mapping \(d: X \times X \to[0, \infty)\) with \(d(x,y)= \ln( d^{\ast }(x,y))\) forms a metric.
Proof
It is clear that all topological notions (convergence, Cauchy, completeness) for multiplicative metric space are consequences of the standard topology of metric space.
Abbas et al. [7] published the following result.
Theorem 2.2
[7]
Dorić [8] reported the following extension of the Banach contraction principle.
Theorem 2.3
Proof
By using \(d(x,y)= \ln( d^{\ast}(x,y))\), we easily see that equation (2.3) yields (2.5). Hence, the inequalities (2.2) implies (2.4). Consequently, Theorem 2.3 provides the existence and uniqueness of the fixed point of f. □
It is clear that one can easily derive the other fixed results in [2–7] from the relevant existing results in the literature. Regarding the analogy, we shall not list the other results.
3 Conclusion
Some authors misuse the notion of the multiplicative calculus since they misunderstand the place and role of this calculus like other nonNewtonian calculuses. Indeed, it represents the same system of knowledge, only different by the presentation of them with respect to socalled reference function. Notice that in Newtonian calculus, the reference function is linear, whereas the reference function for multiplicative calculus is exponential. Consequently, every definition and also every theorem of Newtonian calculus has an analog in multiplicative calculus and vice versa. Therefore, ordinary and multiplicative fixed point theorems are applicable to the same class of functions. In this paper, we only underline these facts in the framework of fixed point theory. It would be possible to approach the problem globally by the use of the preceding discussion.
Declarations
Acknowledgements
The third author is supported by Distinguished Scientist Fellowship Program (DSFP), King Saud University, Saudi Arabia.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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