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# Comment on ‘Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2011-93, 20 pages’

Fixed Point Theory and Applications20122012:144

https://doi.org/10.1186/1687-1812-2012-144

• Accepted: 22 August 2012
• Published:

## Abstract

In this paper, we provide an example to show that some results obtained in [Mongkolkeha et al. in Fixed Point Theory Appl. 2011, doi:10.1186/1687-1812-2011-93] are not valid.

MSC:47H09, 47H10.

## Keywords

• contraction mappings
• modular metric spaces
• metric space

We begin with the definition of a modular metric space.

Definition 1 [1]

Let X be a nonempty set. A function $\omega :\left(0,\mathrm{\infty }\right)×X×X\to \left[0,\mathrm{\infty }\right]$ is said to be metric modular on X if for all $x,y,z\in X$, the following conditions hold:

(i) ${\omega }_{\lambda }\left(x,y\right)=0$ for all $\lambda >0$ iff $x=y$;

(ii) ${\omega }_{\lambda }\left(x,y\right)={\omega }_{\lambda }\left(y,x\right)$ for all $\lambda >0$;

(iii) ${\omega }_{\lambda +\mu }\left(x,y\right)\le {\omega }_{\lambda }\left(x,z\right)+{\omega }_{\mu }\left(z,y\right)$ for all $\lambda ,\mu >0$.

Given ${x}_{\star }\in X$, the set ${X}_{\omega }\left({x}_{\star }\right)=\left\{x\in X:{lim}_{\lambda \to \mathrm{\infty }}{\omega }_{\lambda }\left(x,{x}_{\star }\right)=0\right\}$ is called a modular metric space generated by ${x}_{\star }$ and induced by ω. If its generator ${x}_{\star }$ does not play any role in the situation, we will write ${X}_{\omega }$ instead of ${X}_{\omega }\left({x}_{\star }\right)$.

We need the following theorems in the proof of the main result of this paper.

Theorem 2 [[1], Theorem 2.6]

If ω is metric (pseudo) modular on X, then the modular set ${X}_{\omega }$ is a (pseudo) metric space with (pseudo) metric given by
${d}_{\omega }^{\circ }\left(x,y\right)=inf\left\{\lambda >0:{\omega }_{\lambda }\left(x,y\right)\le \lambda \right\},\phantom{\rule{1em}{0ex}}x,y\in {X}_{\omega }.$

Theorem 3 [[1], Theorem 2.13]

Let ω be (pseudo) modular on a set X. Given a sequence $\left\{{x}_{n}\right\}\subset {X}_{\omega }$ and $x\in {X}_{\omega }$, we have ${d}_{\omega }^{\circ }\left({x}_{n},x\right)\to 0$ as $n\to \mathrm{\infty }$ if and only if ${\omega }_{\lambda }\left({x}_{n},x\right)\to 0$ as $n\to \mathrm{\infty }$ for all $\lambda >0$. A similar assertion holds for Cauchy sequences.

Let ω be modular on a set X. A mapping $T:{X}_{\omega }\to {X}_{\omega }$ is said to be contraction [[2], Definition 3.1] if there exists $k\in \left[0,1\right)$ such that
${\omega }_{\lambda }\left(Tx,Ty\right)\le k{\omega }_{\lambda }\left(x,y\right)$
(1)

for all $\lambda >0$ and $x,y\in {X}_{\omega }$.

Recently, Mongkolkeha et al. [2] proved the following theorems.

Theorem 4 [[2], Theorem 3.2]

Let ω be metric modular on X and ${X}_{\omega }$ be a modular metric space induced by ω. If ${X}_{\omega }$ is a complete modular metric space and $T:{X}_{\omega }\to {X}_{\omega }$ is a contraction mapping, then T has a unique fixed point in ${X}_{\omega }$. Moreover, for any $x\in {X}_{\omega }$, iterative sequence $\left\{{T}^{n}\left(x\right)\right\}$ converges to the fixed point.

Theorem 5 [[2], Theorem 3.4]

Let ω be metric modular on X and ${X}_{\omega }$ be a modular metric space induced by ω. If ${X}_{\omega }$ is a complete modular metric space and $T:{X}_{\omega }\to {X}_{\omega }$ is a mapping, which ${T}^{N}$ is a contraction mapping for some positive integer N. Then, T has a unique fixed point in ${X}_{\omega }$.

We show that Theorems 4 and 5 are not correct. To this end, we give the following example.

Example 6 Let $X=\mathbb{R}$ and define modular ω by ${\omega }_{\lambda }\left(x,y\right)=\mathrm{\infty }$ if $\lambda \le |x-y|$, and ${\omega }_{\lambda }\left(x,y\right)=0$ if $\lambda >|x-y|$. It is easy to verify that (see also [[1], Example 2.7]) ${X}_{\omega }=\mathbb{R}$ and ${d}_{\omega }^{\circ }\left(x,y\right)=|x-y|$. It follows from Theorem 3 that $\mathbb{R}$ is a complete modular metric space. Now, define $T:\mathbb{R}\to \mathbb{R}$ by $Tx=x+1$. We show that T is a contraction while it has no fixed point. Let $k\in \left[0,1\right)$ (for example, $k=1/2$) and $x,y\in \mathbb{R}$. If $\lambda \le |x-y|$, then ${\omega }_{\lambda }\left(x,y\right)=\mathrm{\infty }$ and (1) holds. If $|x-y|<\lambda$, then $|Tx-Ty|=|x-y|<\lambda$. Therefore, ${\omega }_{\lambda }\left(Tx,Ty\right)={\omega }_{\lambda }\left(x,y\right)=0$. Hence T is a contraction. On the other hand, by definition of T, it is easy to see that T has no fixed point. So, Theorems 4 and 5 are not correct.

Remark 7 In [[2], Example 3.7], the authors mentioned that ‘Thus, T is not a contraction mapping and then the Banach contraction mapping cannot be applied to this example.’ It is true that T is not contraction with the Euclidean metric, but one can easily verify that
${d}_{\omega }^{\circ }\left(Tx,Ty\right)\le \frac{\sqrt{3}}{2}{d}_{\omega }^{\circ }\left(x,y\right).$
Thus, the Banach contraction guarantees the existence of a fixed point. Note that
${d}_{\omega }^{\circ }\left(\left({a}_{1},0\right),\left({a}_{2},0\right)\right)=\sqrt{\frac{4|{a}_{1}-{a}_{2}|}{3}},\phantom{\rule{2em}{0ex}}{d}_{\omega }^{\circ }\left(\left(0,{b}_{1}\right),\left(0,{b}_{2}\right)\right)=\sqrt{|{b}_{1}-{b}_{2}|}$
and
${d}_{\omega }^{\circ }\left(\left(a,0\right),\left(0,b\right)\right)=\sqrt{\frac{4a}{3}+b}.$

## Authors’ Affiliations

(1)
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava Zang, Zanjan, 45137-66731, Iran
(2)
Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
(3)
Department of Mathematics, Payame Noor University, Tehran, Iran

## References

1. Chistyakov VV: Modular metric spaces, I: basic concepts. Nonlinear Anal. 2010, 72: 1–14. 10.1016/j.na.2009.04.057
2. Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011. doi:10.1186/1687–1812–2011–93Google Scholar