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# An essential remark on fixed point results on multiplicative metric spaces

## 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.

## 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. [27] and related references therein).

### Definition 1.1

Let X be a non-empty set. A mapping $$d^{\ast} : X \times X \to[0, \infty)$$ is said to be a multiplicative metric if it satisfies the following conditions:

(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).

Also, $$(X, d^{\ast})$$ is called a multiplicative metric space.

For the sake of completeness, we shall present the definition of the (standard) metric.

### Definition 1.2

Let X be a non-empty set. A mapping $$d : X \times X \to[0, \infty)$$ is said to be a (standard) metric if it satisfies the following conditions:

(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).

Also, $$(X, d)$$ is called a (standard) metric space.

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.

## Main results

### Theorem 2.1

Let X be a non-empty 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

By using $$d(x,y)= \ln( d^{\ast}(x,y))$$, the first three assumptions of Definition 1.2 are obtained trivially. Since a logarithmic function is non-decreasing, (iv) yields

\begin{aligned} d(x,y)&=\ln\bigl(d^{\ast}(x, z)\bigr) \\ & \leq\ln\bigl( d^{\ast}(x,y)\cdot d^{\ast}(y, z)\bigr)= \ln \bigl(d^{\ast}(x,y)\bigr)+\ln \bigl(d^{\ast}(y, z)\bigr) \\ & =d(x,y)+d(y, z). \end{aligned}
(2.1)

□

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]

Let $$(X,d^{\ast})$$ be a complete multiplicative metric space and $$f : X \to X$$. Suppose that

$$\psi \bigl(d^{\ast}(fx,fy) \bigr) \leq\frac{\psi (M^{f}_{d^{\ast }}(x,y) )}{\varphi (M^{f}_{d^{\ast}}(x,y) )}$$
(2.2)

for any $$x,y \in X$$, where

$$M^{f}_{d^{\ast}}(x,y)=\bigl\{ d^{\ast}(x,y),d^{\ast}(fx,x),d^{\ast}(y,fy), \bigl(d^{\ast}(fx,y)d^{\ast}(x,fy)\bigr)^{\frac{1}{2}} \bigr\}$$
(2.3)

and $$\psi: [1,\infty)\rightarrow[1,\infty)$$ is continuous, non-decreasing, $$\psi^{-1}(\{1\})=\{1\}$$, and $$\varphi:[1,\infty)\rightarrow[1,\infty )$$ is lower semi-continuous and $$\varphi^{-1}(\{1\})=\{1\}$$. Then f has a unique fixed point.

Dorić [8] reported the following extension of the Banach contraction principle.

### Theorem 2.3

Let $$(X,d)$$ be a complete metric space and let $$f:X\rightarrow X$$ be a mapping such that for each pair of points $$x,y\in X$$,

$$\psi\bigl(d(fx,fy)\bigr)\leq\psi\bigl(M^{f}(x,y)\bigr)- \varphi\bigl(M^{f}(x,y)\bigr),$$
(2.4)

where

$$M^{f}(x,y)=\biggl\{ d(x,y),d(fx,x),d(y,fy), \frac{1}{2}\bigl[d(fx,y)+d(x,fy)\bigr] \biggr\}$$
(2.5)

and $$\psi: [0,\infty)\rightarrow[0,\infty)$$ is continuous, non-decreasing, $$\psi^{-1}(\{0\})=\{0\}$$, and $$\varphi:[0,\infty)\rightarrow[0,\infty )$$ is lower semi-continuous and $$\varphi^{-1}(\{0\})=\{0\}$$. Then F has a unique fixed point.

### Theorem 2.4

Theorem  2.2 is a consequence of 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 [27] from the relevant existing results in the literature. Regarding the analogy, we shall not list the other results.

## Conclusion

Some authors misuse the notion of the multiplicative calculus since they misunderstand the place and role of this calculus like other non-Newtonian calculuses. Indeed, it represents the same system of knowledge, only different by the presentation of them with respect to so-called 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.

## References

1. 1.

Bashirov, A, Kurpınar, E, Ozyapıcı, A: Multiplicative calculus and its applications. J. Math. Anal. Appl. 337(1), 36-48 (2008)

2. 2.

He, X, Song, M, Chen, D: Common fixed points for weak commutative mappings on a multiplicative metric space. Fixed Point Theory Appl. 2013, Article ID 48 (2013)

3. 3.

Abbas, M, Ali, B, Suleiman, YI: Common fixed points of locally contractive mappings in multiplicative metric spaces with application. Int. J. Math. Math. Sci. 2015, Article ID 218683 (2015). doi:10.1155/2015/218683

4. 4.

Mongkolkeha, C, Sintunavarat, W: Best proximity points for multiplicative proximal contraction mapping on multiplicative metric spaces. J. Nonlinear Sci. Appl. 8, 1134-1140 (2015)

5. 5.

Kang, SM, Kumar, P, Kumar, S, Nagpal, P, Garg, SK: Common fixed points for compatible mappings and its variants in multiplicative metric spaces. Int. J. Pure Appl. Math. 102(2), 383-406 (2015)

6. 6.

Yamaod, O, Sintunavarat, W: Some fixed point results for generalized contraction mappings with cyclic $$(\alpha, \beta)$$-admissible mappings in multiplicative metric space. J. Inequal. Appl. 2014, Article ID 488 (2014)

7. 7.

Abbas, M, De La Sen, M, Nazir, T: Common fixed points of generalized rational type cocyclic mappings in multiplicative metric spaces. Discrete Dyn. Nat. Soc. 2015, Article ID 532725 (2015)

8. 8.

Dorić, D: Common fixed point for generalized $$(\psi, \phi)$$-weak contractions. Appl. Math. Lett. 22, 1896-1900 (2009). doi:10.1016/j.aml.2009.08.001

## Acknowledgements

The third author is supported by Distinguished Scientist Fellowship Program (DSFP), King Saud University, Saudi Arabia.

## Author information

Authors

### Corresponding author

Correspondence to Bessem Samet.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Reprints and Permissions

Agarwal, R.P., Karapınar, E. & Samet, B. An essential remark on fixed point results on multiplicative metric spaces. Fixed Point Theory Appl 2016, 21 (2016). https://doi.org/10.1186/s13663-016-0506-7

• Accepted:

• Published:

• 47H10
• 26A33
• 45G10

### Keywords

• multiplicative metric
• fixed point