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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: 7 September 2012

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)
(2)
Department of Mathematics, Semnan University
(3)
Department of Mathematics, Payame Noor University

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.