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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

  • Received: 8 January 2012
  • 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 ω : ( 0 , ) × X × X [ 0 , ] is said to be metric modular on X if for all x , y , z X , the following conditions hold:

(i) ω λ ( x , y ) = 0 for all λ > 0 iff x = y ;

(ii) ω λ ( x , y ) = ω λ ( y , x ) for all λ > 0 ;

(iii) ω λ + μ ( x , y ) ω λ ( x , z ) + ω μ ( z , y ) for all λ , μ > 0 .

Given x X , the set X ω ( x ) = { x X : lim λ ω λ ( x , x ) = 0 } is called a modular metric space generated by x and induced by ω. If its generator x does not play any role in the situation, we will write X ω instead of X ω ( x ) .

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 ω is a (pseudo) metric space with (pseudo) metric given by
d ω ( x , y ) = inf { λ > 0 : ω λ ( x , y ) λ } , x , y X ω .

Theorem 3 [[1], Theorem 2.13]

Let ω be (pseudo) modular on a set X. Given a sequence { x n } X ω and x X ω , we have d ω ( x n , x ) 0 as n if and only if ω λ ( x n , x ) 0 as n for all λ > 0 . A similar assertion holds for Cauchy sequences.

Let ω be modular on a set X. A mapping T : X ω X ω is said to be contraction [[2], Definition 3.1] if there exists k [ 0 , 1 ) such that
ω λ ( T x , T y ) k ω λ ( x , y )
(1)

for all λ > 0 and x , y X ω .

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

Theorem 4 [[2], Theorem 3.2]

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

Theorem 5 [[2], Theorem 3.4]

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

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

Example 6 Let X = R and define modular ω by ω λ ( x , y ) = if λ | x y | , and ω λ ( x , y ) = 0 if λ > | x y | . It is easy to verify that (see also [[1], Example 2.7]) X ω = R and d ω ( x , y ) = | x y | . It follows from Theorem 3 that R is a complete modular metric space. Now, define T : R R by T x = x + 1 . We show that T is a contraction while it has no fixed point. Let k [ 0 , 1 ) (for example, k = 1 / 2 ) and x , y R . If λ | x y | , then ω λ ( x , y ) = and (1) holds. If | x y | < λ , then | T x T y | = | x y | < λ . Therefore, ω λ ( T x , T y ) = ω λ ( x , y ) = 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 ω ( T x , T y ) 3 2 d ω ( x , y ) .
Thus, the Banach contraction guarantees the existence of a fixed point. Note that
d ω ( ( a 1 , 0 ) , ( a 2 , 0 ) ) = 4 | a 1 a 2 | 3 , d ω ( ( 0 , b 1 ) , ( 0 , b 2 ) ) = | b 1 b 2 |
and
d ω ( ( a , 0 ) , ( 0 , b ) ) = 4 a 3 + b .

Declarations

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.2009.04.057MathSciNetView ArticleGoogle Scholar
  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

Copyright

© Dehghan et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. 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.

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