 Research Article
 Open access
 Published:
Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces
Fixed Point Theory and Applications volumeÂ 2011, ArticleÂ number:Â 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 Ä†iriÄ‡ [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 [8â€“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 â€‰â€‰).
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 .
References
Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338â€“353.
George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 1994,64(3):395â€“399. 10.1016/01650114(94)901627
George A, Veeramani P: On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems 1997,90(3):365â€“368. 10.1016/S01650114(96)002072
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis. Theory, Methods & Applications 2006,65(7):1379â€“1393. 10.1016/j.na.2005.10.017
Lakshmikantham V, Ä†iriÄ‡ L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,70(12):4341â€“4349. 10.1016/j.na.2008.09.020
Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2010,72(3â€“4):1298â€“1304. 10.1016/j.na.2009.08.018
Fang JX: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(5â€“6):1833â€“1843. 10.1016/j.na.2009.01.018
Ä†iriÄ‡ LB, MiheÅ£ D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topology and its Applications 2009,156(17):2838â€“2844. 10.1016/j.topol.2009.08.029
O'Regan D, Saadati R: Nonlinear contraction theorems in probabilistic spaces. Applied Mathematics and Computation 2008,195(1):86â€“93. 10.1016/j.amc.2007.04.070
Jain S, Jain S, Bahadur Jain L: Compatibility of type (P) in modified intuitionistic fuzzy metric space. Journal of Nonlinear Science and its Applications 2010,3(2):96â€“109.
Ä†iriÄ‡ LB, JeÅ¡iÄ‡ SN, Ume JS: The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 2008,37(3):781â€“791. 10.1016/j.chaos.2006.09.093
\'CiriÄ‡ L, Lakshmikantham V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stochastic Analysis and Applications 2009,27(6):1246â€“1259. 10.1080/07362990903259967
Ä†iriÄ‡ L, CakiÄ‡ N, RajoviÄ‡ M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008, 2008:11.
Aliouche A, Merghadi F, Djoudi A: A related fixed point theorem in two fuzzy metric spaces. Journal of Nonlinear Science and its Applications 2009,2(1):19â€“24.
Ä†iriÄ‡ L: Common fixed point theorems for a family of nonself mappings in convex metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(5â€“6):1662â€“1669. 10.1016/j.na.2009.01.002
Rao KPR, Aliouche A, Babu GR: Related fixed point theorems in fuzzy metric spaces. Journal of Nonlinear Science and its Applications 2008,1(3):194â€“202.
Ä†iriÄ‡ L, CakiÄ‡ N: On common fixed point theorems for nonself hybrid mappings in convex metric spaces. Applied Mathematics and Computation 2009,208(1):90â€“97. 10.1016/j.amc.2008.11.012
Ä†iriÄ‡ L: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos, Solitons and Fractals 2009,42(1):146â€“154. 10.1016/j.chaos.2008.11.010
Shakeri S, Ä†iriÄ‡ LJB, Saadati R: Common fixed point theorem in partially ordered fuzzy metric spaces. Fixed Point Theory and Applications 2010, 2010:13.
Ä†iriÄ‡ L, Samet B, Vetro C: Common fixed point theorems for families of occasionally weakly compatible mappings. Mathematical and Computer Modelling 2011,53(5â€“6):631â€“636. 10.1016/j.mcm.2010.09.015
Ä†iriÄ‡ L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Applied Mathematics and Computation 2011,217(12):5784â€“5789. 10.1016/j.amc.2010.12.060
Ä†iriÄ‡ L, Abbas M, DamjanoviÄ‡ B, Saadati R: Common fuzzy fixed point theorems in ordered metric spaces. Mathematical and Computer Modelling 2011,53(9â€“10):1737â€“1741. 10.1016/j.mcm.2010.12.050
Kamran T, CakiÄ‡ N: Hybrid tangential property and coincidence point theorems. Fixed Point Theory 2008,9(2):487â€“496.
HadÅ¾iÄ‡ O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and its Applications. Volume 536. Kluwer Academic, Dordrecht, The Netherlands; 2001:x+273.
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 1988,27(3):385â€“389. 10.1016/01650114(88)900644
Acknowledgment
The author is grateful to the referees for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/363716