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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 Applications volume 2012, Article number: 144 (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.
We begin with the definition of a modular metric space.
Definition 1 [1]
Let X be a nonempty set. A function is said to be metric modular on X if for all , the following conditions hold:
(i) for all iff ;
(ii) for all ;
(iii) for all .
Given , the set is called a modular metric space generated by and induced by ω. If its generator does not play any role in the situation, we will write instead of .
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 is a (pseudo) metric space with (pseudo) metric given by
Theorem 3 [[1], Theorem 2.13]
Let ω be (pseudo) modular on a set X. Given a sequence and , we have as if and only if as for all . A similar assertion holds for Cauchy sequences.
Let ω be modular on a set X. A mapping is said to be contraction [[2], Definition 3.1] if there exists such that
for all and .
Recently, Mongkolkeha et al. [2] proved the following theorems.
Theorem 4 [[2], Theorem 3.2]
Let ω be metric modular on X and be a modular metric space induced by ω. If is a complete modular metric space and is a contraction mapping, then T has a unique fixed point in . Moreover, for any , iterative sequence converges to the fixed point.
Theorem 5 [[2], Theorem 3.4]
Let ω be metric modular on X and be a modular metric space induced by ω. If is a complete modular metric space and is a mapping, which is a contraction mapping for some positive integer N. Then, T has a unique fixed point in .
We show that Theorems 4 and 5 are not correct. To this end, we give the following example.
Example 6 Let and define modular ω by if , and if . It is easy to verify that (see also [[1], Example 2.7]) and . It follows from Theorem 3 that is a complete modular metric space. Now, define by . We show that T is a contraction while it has no fixed point. Let (for example, ) and . If , then and (1) holds. If , then . Therefore, . 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
Thus, the Banach contraction guarantees the existence of a fixed point. Note that
and
References
Chistyakov VV: Modular metric spaces, I: basic concepts. Nonlinear Anal. 2010, 72: 1–14. 10.1016/j.na.2009.04.057
Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011. doi:10.1186/1687–1812–2011–93
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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
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Dehghan, H., Eshaghi Gordji, M. & Ebadian, A. 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 Appl 2012, 144 (2012). https://doi.org/10.1186/1687-1812-2012-144
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DOI: https://doi.org/10.1186/1687-1812-2012-144