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Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces
Fixed Point Theory and Applications volume 2011, Article number: 363716 (2011)
Abstract
We prove a common fixed point theorem for mappings under contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of S.Sedghi et al. (2010)
1. Introduction
Since Zadeh [1] introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space which have very important applications in quantum particle physics particularly in connection with both string and infinity theory.
Bhaskar and Lakshmikantham [4], Lakshmikantham and Ćirić [5] discussed the mixed monotone mappings and gave some coupled fixed point theorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al. [6] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang [7] gave some common fixed point theorems under contractions for compatible and weakly compatible mappings in Menger probabilistic metric spaces. Many authors [8–23] have proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.
In this paper, using similar proof as in [7], we give a new common fixed point theorem under weaker conditions than in [6] and give an example which shows that the result is a genuine generalization of the corresponding result in [6].
2. Preliminaries
First we give some definitions.
Definition 1 (see [2]).
A binary operation is continuous norm if is satisfying the following conditions:
(1)is commutative and associative;
(2) is continuous;
(3) for all ;
(4) whenever and for all .
Definition 2 (see [24]).
Let . A norm is said to be of Htype if the family of functions is equicontinuous at , where
The norm is an example of norm of Htype, but there are some other norms of Htype [24].
Obviously, is a Htype norm if and only if for any , there exists such that for all , when .
Definition 3 (see [2]).
A 3tuple is said to be 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 and :
(FM1);
(FM2) if and only if ;
(FM3);
(FM4);
(FM5) is continuous.
Let be a fuzzy metric space. For , the open ball with a center and a radius is defined by
A subset is called open if, for each , there exist and such that . Let denote the family of all open subsets of . Then is called the topology on induced by the fuzzy metric . This topology is Hausdorff and first countable.
Example 1.
Let be a metric space. Define norm and for all and , . Then is a fuzzy metric space. We call this fuzzy metric induced by the metric the standard fuzzy metric.
Definition 4 (see [2]).
Let be a fuzzy metric space, then
(1)a sequence in is said to be convergent to (denoted by ) if
for all ;
(2)a sequence in is said to be a Cauchy sequence if for any , there exists , such that
for all and ;
(3)a fuzzy metric space is said to be complete if and only if every Cauchy sequence in is convergent.
Remark 1 (see [25]).

(1)
For all , is nondecreasing.

(2)
It is easy to prove that if , , , then
(2.5)

(3)
In a fuzzy metric space , whenever for in , , , we can find a , such that .

(4)
For any , we can find an such that and for any we can find a such that ).
Definition 5 (see [6]).
Let be a fuzzy metric space. is said to satisfy the property on if
whenever , and .
Lemma 1.
Let be a fuzzy metric space and satisfies the property; then
Proof.
If not, since is nondecreasing and , there exists such that , then for , when as and we get , which is a contraction.
Remark 2.
Condition (2.7) cannot guarantee the property. See the following example.
Example 2.
Let be an ordinary metric space, for all , and be defined as following:
where . Then is continuous and increasing in , and . Let
then is a fuzzy metric space and
But for any , , , ,
Define , where and each satisfies the following conditions:
(1) is nondecreasing;
(2) is upper semicontinuous from the right;
(3) for all , where , .
It is easy to prove that, if , then for all .
Lemma 2 (see [7]).
Let be a fuzzy metric space, where is a continuous norm of Htype. If there exists such that if
for all , then .
Definition 6 (see [5]).
An element is called a coupled fixed point of the mapping if
Definition 7 (see [5]).
An element is called a coupled coincidence point of the mappings and if
Definition 8 (see [7]).
An element is called a common coupled fixed point of the mappings and if
Definition 9 (see [7]).
An element is called a common fixed point of the mappings and if
Definition 10 (see [7]).
The mappings and are said to be compatible if
for all whenever and are sequences in , such that
for all are satisfied.
Definition 11 (see [7]).
The mappings and are called commutative if
for all .
Remark 3.
It is easy to prove that, if and are commutative, then they are compatible.
3. Main Results
For convenience, we denote
for all .
Theorem 1.
Let be a complete FMspace, where is a continuous norm of Htype satisfying (2.7). Let and be two mappings and there exists such that
for all , .
Suppose that , and is continuous, and are compatible. Then there exist such that , that is, and have a unique common fixed point in .
Proof.
Let be two arbitrary points in . Since , we can choose such that and . Continuing in this way we can construct two sequences and in such that
The proof is divided into 4 steps.
Step 1.
Prove that and are Cauchy sequences.
Since is a norm of Htype, for any , there exists a such that
for all .
Since is continuous and for all , there exists such that
On the other hand, since , by condition () we have . Then for any , there exists such that
From condition (3.2), we have
Similarly, we can also get
Continuing in the same way we can get
So, from (3.5) and (3.6), for , we have
which implies that
for all with and . So is a Cauchy sequence.
Similarly, we can get that is also a Cauchy sequence.
Step 2.
Prove that and have a coupled coincidence point.
Since complete, there exist such that
Since and are compatible, we have by (3.12),
for all . Next we prove that and .
For all , by condition (3.2), we have
for all . Let , since and are compatible, with the continuity of , we get
which implies that . Similarly, we can get .
Step 3.
Prove that and .
Since is a norm of Htype, for any , there exists an such that
for all .
Since is continuous and for all , there exists such that and .
On the other hand, since , by condition we have . Then for any , there exists such that . Since
letting , we get
Similarly, we can get
From (3.18) and (3.19) we have
By this way, we can get for all ,
Then, we have
So for any we have
for all . We can get that and .
Step 4.
Prove that .
Since is a norm of Htype, for any , there exists an such that
for all .
Since is continuous and , there exists such that .
On the other hand, since , by condition we have . Then for any , there exists such that .
Since for ,
Letting yields
Thus we have
which implies that .
Thus we have proved that and have a unique common fixed point in .
This completes the proof of the Theorem 1.
Taking (the identity mapping) in Theorem 1, we get the following consequence.
Corollary 1.
Let be a complete FMspace, where is a continuous norm of Htype satisfying (2.7). Let and there exists such that
for all , .
Then there exist such that , that is, admits a unique fixed point in .
Let , where , the following by Lemma 1, we get the following.
Corollary 2 (see [6]).
Let for all and be a complete fuzzy metric space such that has property. Let and be two functions such that
for all , where , and is continuous and commutes with . Then there exists a unique such that .
Next we give an example to demonstrate Theorem 1.
Example 3.
Let , for all . is defined as (2.8). Let
for all . Then is a complete FMspace.
Let , and be defined as
Then satisfies all the condition of Theorem 1, and there exists a point which is the unique common fixed point of and .
In fact, it is easy to see that ,
For all and . (3.28) is equivalent to
Since , we can get
From (3.33), we only need to verify the following:
that is,
We consider the following cases.
Case 1 ().
Then (3.36) is equivalent to
it is easy to verified.
Case 2 ().
Then (3.36) is equivalent to
which is
since
that is
holds for all . So (3.36) holds for .
Case 3 ().
Then (3.36) is equivalent to
Let , we only need to verify
for all that . We can verify it holds.
Thus it is verified that the functions , , satisfy all the conditions of Theorem 1; is the common fixed point of and in .
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Hu, XQ. Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces. Fixed Point Theory Appl 2011, 363716 (2011). https://doi.org/10.1155/2011/363716
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DOI: https://doi.org/10.1155/2011/363716
Keywords
 Point Theorem
 Fixed Point Theorem
 Open Ball
 Cauchy Sequence
 Contractive Mapping