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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 142858, 7 pages
Comment on “Common Fixed Point Theorems for Commutating Mappings in Fuzzy Metric Spaces”
1School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China
2School of Information, Capital University of Economics and Business, Beijing 100070, China
Received 3 August 2012; Accepted 17 October 2012
Copyright © 2012 Yonghong Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the recent paper “common fixed point theorems for commutating mappings in fuzzy metric spaces,” the authors proved that a common fixed point theorem for commutating mappings in -complete fuzzy metric spaces and gave an example to illustrate the main result. In this note, we point out that the above example is incorrect because it does not satisfy the condition of -completeness, and then two appropriate examples are given. In addition, we prove that the theorem proposed by Zheng and Lian actually holds in an -complete fuzzy metric space. Our results improve and extend some existing results in the relevant literature.
In , Zheng and Lian extended Jungck's theorem in  to fuzzy metric spaces and obtained the following fixed point theorem for commutative mappings in fuzzy metric spaces in the sense of Kramosil and Michálek .
Theorem 1.1 (Zheng and Lian ). Let be a complete fuzzy metric space and let be a continuous map and a map. If(i),(ii) commutes with ,(iii)and for all and , where is an increasing and left-continuous function with for all .Then and have a unique common fixed point.
Remark 1.2. It can be seen from the proof of Theorem 1.1 that the fuzzy metric space is complete in the sense of -completeness.
Furthermore, the authors constructed the following example to illustrate the above theorem.
Example 1.3 (Zheng and Lian ). Let be endowed with the usual metric . For all and , define
Set and , , where is a constant.
In the Example 1.3, the authors claimed that is a complete fuzzy metric space in the sense of -completeness (now known as a -complete fuzzy metric space) with respect to -norm , and then checked all the conditions of Theorem 1.1. Therefore, they concluded that and have the unique common fixed point 0.
In fact, -completeness is a very strong kind of completeness. For instance, George and Veeramani  found that even is not -complete with respect to the standard fuzzy metric induced by Euclidean metric, and then proposed another kind of completeness (now known as an -complete fuzzy metric space) by modifying the definition of Cauchy sequence. For these two types of completeness, it is easy to see that every -complete fuzzy metric space is -complete. Therefore, the construction of fixed point theorems in -complete fuzzy metric spaces is more valuable and reasonable.
The main purpose of this note is to provide two appropriate examples for Theorem 1.1 and prove that this theorem does hold even if -completeness of the fuzzy metric space is replaced by -completeness. Our results not only improve and generalize Theorem 1.1, but also extend some main results of [2, 7].
For completeness and clarity, in this section, some related concepts and conclusions are summarized below. Let denote the set of all positive integers.
Definition 2.1 (Schweizer and Sklar ). A binary operation is called a continuous triangular norm (shortly, continuous -norm) if it satisfies the following conditions:(TN-1) is commutative and associative,(TN-2) is continuous,(TN-3) for every ,(TN-4) and whenever , and .
Definition 2.2 (Kramosil and Michálek ). The triple is called a fuzzy metric space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions: for all and ,(FM-1),(FM-2) if and only if ,(FM-3),(FM-4),(FM-5) and is left-continuous.
Remark 2.3. According to (FM-2) and (FM-4), it can easily be seen that is non-decreasing for all (see Lemma 4 in ).
Similar to the case in , in this note, we suppose that is a fuzzy metric space with the following additional condition:(FM-6), for all .
Definition 2.4 (Grabiec , George and Veeramani ). Let be a fuzzy metric space. Then(i)a sequence in is said to be convergent to a point , denoted by , if , for any ;(ii)a sequence in is called a -Cauchy sequence if and only if for any and ;(iii)a sequence in is called an -Cauchy sequence if and only if for each and , there exists such that , for any ;(iv)a fuzzy metric space is said to be -complete (M-complete) if every -Cauchy sequence (-Cauchy sequence) is convergent;(v)a map is said to be continuous at if converges to for each converging to .
The authors have proved the following conclusion (see the proof of Theorem 2.2 in ).
Lemma 2.5 (Zheng and Lian ). Let be an increasing and left-continuous function with for all . Then for any , where denotes the composition of with itself times.
3. Two Appropriate Examples
In this section, we will construct two appropriate examples for Theorem 1.1.
Example 3.1. Let be equipped with the usual metric , , for all , where is a constant. For all and , define
Clearly, is a -complete fuzzy metric space with regard to -norm .
Set and for all . It is obvious that . For any and , we have
Thus, all the conditions of Theorem 1.1 are satisfied and and have a unique fixed point, that is, .
Example 3.2. Let be the subset of defined by
where , , , , , and . , for all . For all and , define
where denotes the Euclidean distance of .
Clearly, is also a -complete fuzzy metric space with regard to -norm .
Let and be given by
Obviously, . Furthermore, it is easy to see that for any and . Hence, all the conditions of Theorem 1.1 are satisfied and is the unique common fixed point of and .
4. Main Results
Now, we will prove that Theorem 1.1 does hold even if -completeness is replaced by -completeness.
Theorem 4.1. Let be an -complete fuzzy metric space and let be a continuous map and a map. If(i),(ii) commutes with ,(iii)and for all and , where is an increasing and left-continuous function with for all .Then and have a unique common fixed point.
Proof. Let . From (i), we can find such that . By induction, we can find a sequence such that . For any , we have
As , for any , it follows by (FM-6) that . Hence, , for any .
Next, we claim that is an -Cauchy sequence. Suppose that it is not. Then there exist and two sequences , such that for every and , and then we can obtain that
Moreover, for every , we can choose the two smallest numbers and such that
For every , we can obtain
Clearly, this leads to a contradiction.
Here, we also consider another particular case. That is, for each , there exist and such that , for all . Then, for any , we know that . Since we can conclude that is a monotone and bounded sequence with respect to . Therefore, there exists such that . In addition, according to the foregoing inequality, we can obtain
By supposing that , it follows that , which is also a contradiction.
Hence, is an -Cauchy sequence in the -complete fuzzy metric space . Furthermore, we conclude that there exists a point such that . So .
By (iii), it can be seen that the continuity of implies that of . Consequently, we obtain that . According to the commutativity of and , we know that . Because of the uniqueness of limits, it follows immediately that . So . Thus, we have
Letting , we obtain that , for any . So . By (FM-2), we conclude that . Hence, , that is, is a common fixed point of and .
Furthermore, we show that is the unique common fixed point of and . Assume that and are two common fixed point of and , for any , we then obtain
As , we have . Thus for any . Furthermore, we can obtain . This completes the proof.
Remark 4.2. It should be pointed out that the foregoing three examples are suitable for Theorem 4.1.
Corollary 4.3. Let be an -complete fuzzy metric space and let be a continuous map and a map. If(i),(ii) commutes with ,(iii)and for all and , where .
Then and have a unique common fixed point.
This work was supported by “Qing Lan” Talent Engineering Funds by Tianshui Normal University, the Beijing Municipal Education Commission Foundation of China (no. KM201210038001), the National Natural Science Foundation (no. 71240002), and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the jurisdiction of Beijing Municipality (no. PHR201108333).
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