Research Article | Open Access

K. P. R. Rao, I. Altun, S. Hima Bindu, "Common Coupled Fixed-Point Theorems in Generalized Fuzzy Metric Spaces", *Advances in Fuzzy Systems*, vol. 2011, Article ID 986748, 6 pages, 2011. https://doi.org/10.1155/2011/986748

# Common Coupled Fixed-Point Theorems in Generalized Fuzzy Metric Spaces

**Academic Editor:**E. E. Kerre

#### Abstract

We prove two unique common coupled fixed-point theorems for self maps in symmetric *G*-fuzzy metric spaces.

#### 1. Introduction and Preliminaries

Mustafa and Sims [1–3] and Naidu et al. [4] demonstrated that most of the claims concerning the fundamental topological structure of -metric introduced by Dhage [5–8] and hence all theorems are incorrect. Alternatively, Mustafa and Sims [1, 2] introduced a -metric space and obtained some fixed-point theorems in it. Some interesting references in -metric spaces are [3, 9–15]. In this paper, we prove two unique common coupled fixed-point theorems for Jungck type and for three mappings in symmetric -fuzzy metric spaces.

Before giving our main results, we recall some of the basic concepts and results in -metric spaces and -fuzzy metric spaces.

*Definition 1 (see [2]). *Let be a nonempty set and let be a function satisfying the following properties:(*G _{1}*) if ,(

*G*) for all with ,(

_{2}*G*) for all with ,(

_{3}*G*), symmetry in all three variables,(

_{4}*G*) for all .

_{5}Then, the function is called a generalized metric or a -metric on and the pair is called a -metric space.

*Definition 2 (see [2]). *The -metric space is called symmetric if for all .

*Definition 3 (see [2]). *Let be a -metric space and let be a sequence in . A point is said to be limit of if and only if . In this case, the sequence is said to be -convergent to .

*Definition 4 (see [2]). *Let be a -metric space and let be a sequence in . is called -Cauchy if and only if . is called -complete if every -Cauchy sequence in is -convergent in .

Proposition 5 (see [2]). *In a -metric space , the following are equivalent.*(i)*The sequence is -Cauchy.*(ii)*For every there exists such that , for all .*

Proposition 6 (see [2]). *Let be a -metric space. Then, the function is jointly continuous in all three of its variables.*

Proposition 7 (see [2]). *Let be a -metric space. Then, for any , it follows that*(i)*if , then ,*(ii)*,*(iii)*,*(iv)*,*(v)*.*

Proposition 8 (see [2]). *Let be a -metric space. Then, for a sequence and a point , the following are equivalent:*(i)* is -convergent to ,*(ii)* as ,*(iii)* as ,*(iv)* as .*

Recently, Sun and Yang [16] introduced the concept of -fuzzy metric spaces and proved two common fixed-point theorems for four mappings.

*Definition 9 (see [16]). *A 3-tuple is called a -fuzzy metric space if is an arbitrary nonempty set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions for each :(i) for all with ,(ii) for all with ,(iii) if and only if ,(iv), where is a permutation function,(v) for all ,(vi) is continuous.

*Definition 10 (see [16]). *A -fuzzy metric space is said to be symmetric if for all and for each .

*Example 11. *Let be a nonempty set and let be a -metric on . Denote for all . For each , is a -fuzzy metric on .

Let be a -fuzzy metric space. For , and , the set is called an open ball with center and radius .

A subset of is called an open set if for each , there exist and such that .

A sequence in -fuzzy metric space is said to be -convergent to if as for each . It is called a -Cauchy sequence if as for each . is called -complete if every -Cauchy sequence in is -convergent in .

Lemma 12 (see [16]). *Let be a -fuzzy metric space. Then, is nondecreasing with respect to for all .*

Lemma 13 (see [16]). *Let be a -fuzzy metric space. Then, is a continuous function on .*

Now onwards, we assume the following condition: Using (P), one can prove the following lemma.

Lemma 14. *Let be a -fuzzy metric space. If there exists such that
**
for all and , then and .*

*Definition 15 (see [17]). *Let be a nonempty set. An element is called a coupled fixed point of the mapping if and .

*Definition 16 (see [18]). *Let be a nonempty set. An element is called(i)a coupled coincidence point of and if and ,(ii)a common coupled fixed point of and if and .

*Definition 17 (see [18]). *Let be a nonempty set. The mappings and are called -compatible if and whenever and for some .

Now, we give our main results.

#### 2. Main Results

Theorem 18. *Let be a -fuzzy metric space with for all and and let be mappings satisfying
**, where ,
**Then and have a unique common coupled fixed point of the form in .*

*Proof. *Let and denote . Let , . From (2), we have
Also,
Thus, . Hence,
For any positive integer and fixed positive integer , we have
Letting and using (P), we get
Hence, . Thus, is -Cauchy in . Similarly, we can show that is -Cauchy in . Since is -complete, and converge to some and in , respectively. Hence, there exist and in such that :
Letting , we get
Hence, . Similarly, it can be shown that . Since is -compatible, we have
Letting , we get
Similarly, we can show that
Thus,
From Lemma 14, we have and . Thus, and . Hence, is a common coupled fixed point of and .

Suppose is another common coupled fixed point of and :
Similarly,
Thus,
From Lemma 14, and . Thus, is the unique common coupled fixed point of and . Now, we will show that :
Thus,
From Lemma 14, we have . Thus, is a common fixed point of and , that is, . Suppose is another common fixed point of and :
Hence, . Thus, and have a unique common coupled fixed point of the form .

Finally, we prove a common coupled fixed-point theorem for three mappings in symmetric -fuzzy metric spaces.

Theorem 19. *Let be a symmetric -complete fuzzy metric space with for all and let be mappings satisfying
**, where . Then, there exists such that
**
Or
*

*Proof. *Let . Define the sequences and in as follows: ,; , ; , , . Suppose for some . Then, , where . Suppose . Then,
It is a contradiction. Hence, . From (25) and since is symmetric,
From Lemma 14, we have . Thus, . Similarly, if or , then also we can show that for some , in . Similarly, it can be shown that if or or then there exists such that
Now, assume that and for all . Write and :
Thus, . Similarly, we have .

Thus,
Similarly, we can show that
Thus,
Hence
Thus,
From , we have
As in Theorem 18, we can show that and are -Cauchy sequences in . Since is -complete, there exist such that and
Letting ,
From this, we have . As in the first part of proof, we can show that . Similarly, it can be shown that . Thus, is a common coupled fixed point of , , and . Suppose is another common coupled fixed point of , , and . Consider
Also,
Thus,
From Lemma 14, we have and . Thus, is the unique common coupled fixed point of , , and . Now, we will show that . Consider
Hence, . Thus, , , and have a unique common coupled fixed point of the form .

#### Acknowledgment

The authors are thankful to the referee for his valuable suggestions.

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#### Copyright

Copyright © 2011 K. P. R. Rao 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.