- Research Article
- Open Access
Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces
© Xin-Qi Hu. 2011
- Received: 23 November 2010
- Accepted: 27 January 2011
- Published: 23 February 2011
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)
- Point Theorem
- Fixed Point Theorem
- Open Ball
- Cauchy Sequence
- Contractive Mapping
Since Zadeh  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 , Lakshmikantham and Ćirić  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.  gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang  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 , we give a new common fixed point theorem under weaker conditions than in  and give an example which shows that the result is a genuine generalization of the corresponding result in .
First we give some definitions.
Definition 1 (see ).
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 ).
The -norm is an example of -norm of H-type, but there are some other -norms of H-type .
Obviously, is a H-type norm if and only if for any , there exists such that for all , when .
Definition 3 (see ).
A 3-tuple 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 :
(FM-2) if and only if ;
(FM-5) is continuous.
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.
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 ).
Let be a fuzzy metric space, then
for all ;
for all and ;
(3)a fuzzy metric space is said to be complete if and only if every Cauchy sequence in is convergent.
For all , is nondecreasing.
In a fuzzy metric space , whenever for in , , , we can find a , such that .
For any , we can find an such that and for any we can find a such that ).
Definition 5 (see ).
whenever , and .
If not, since is nondecreasing and , there exists such that , then for , when as and we get , which is a contraction.
Condition (2.7) cannot guarantee the -property. See the following example.
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 ).
for all , then .
Definition 6 (see ).
Definition 7 (see ).
Definition 8 (see ).
Definition 9 (see ).
Definition 10 (see ).
for all are satisfied.
Definition 11 (see ).
for all .
It is easy to prove that, if and are commutative, then they are compatible.
for all .
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 .
The proof is divided into 4 steps.
Prove that and are Cauchy sequences.
for all .
for all with and . So is a Cauchy sequence.
Similarly, we can get that is also a Cauchy sequence.
Prove that and have a coupled coincidence point.
for all . Next we prove that and .
which implies that . Similarly, we can get .
Prove that and .
for all .
Since is continuous and for all , there exists such that and .
for all . We can get that and .
Prove 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 .
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.
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 ).
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.
for all . Then is a complete FM-space.
Then satisfies all the condition of Theorem 1, and there exists a point which is the unique common fixed point of and .
We consider the following cases.
Case 1 ( ).
it is easy to verified.
Case 2 ( ).
holds for all . So (3.36) holds for .
Case 3 ( ).
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 .
The author is grateful to the referees for their valuable comments and suggestions.
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