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:(G1) if ,(G2) for all with ,(G3) for all with ,(G4), symmetry in all three variables,(G5) for all .
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.