Common fixed point theorems for nonlinear contractive mappings in fuzzy metric spaces
© Wang et al.; licensee Springer 2013
Received: 11 May 2013
Accepted: 5 July 2013
Published: 22 July 2013
In this paper, we prove several common fixed point theorems for nonlinear mappings with a function ϕ in fuzzy metric spaces. In these fixed point theorems, very simple conditions are imposed on the function ϕ. Our results improve some recent ones in the literature. Finally, an example is presented to illustrate the main result of this paper.
The concept of fuzzy metric spaces was defined in different ways [1–3]. Grabiec  presented a fuzzy version of the Banach contraction principle in a fuzzy metric space in Kramosi and Michalek’s sense. Fang  proved some fixed point theorems in fuzzy metric spaces, which improved, generalized, unified and extended some main results of Edelstein , Istratescu , Sehgal and Bharucha-Reid .
In order to obtain a Hausdorff topology, George and Veeramani [9, 10] modified the concept of fuzzy metric space due to Kramosil and Michalek . Many fixed point theorems in complete fuzzy metric spaces in the sense of George and Veeramani (GV) [9, 10] have been obtained. For example, Singh and Chauhan  proved some common fixed point theorems for four mappings in GV fuzzy metric spaces. Gregori and Sapena  proved that each fuzzy contractive mapping has a unique fixed point in a complete GV fuzzy metric space, in which fuzzy contractive sequences are Cauchy.
In 2006, Bhaskar and Lakshmikantham  introduced the concept of coupled fixed point in metric spaces and obtained some coupled fixed point theorems with the application to a bounded value problem. Based on Bhaskar and Lakshmikantham’s work, many researchers have obtained more coupled fixed point theorems in metric spaces and cone metric spaces; see [14, 15]. Recently, the investigation of coupled fixed point theorems has been extended from metric spaces to probabilistic metric spaces and fuzzy metric spaces; see [16–19]. In , the authors gave the following results.
Theorem SAS [, Theorem 2.5]
for all , where , and g is continuous and commutes with F. Then there exists a unique such that .
Let , where and each satisfies the following conditions:
(ϕ-1) ϕ is nondecreasing,
(ϕ-2) ϕ is upper semicontinuous from the right,
(ϕ-3) for all , where , .
In , Hu proved the following result.
Theorem of Hu [, Theorem 1]
for all , . Suppose that and that g is continuous, F and g are compatible. Then there exists such that , that is, F and g have a unique common fixed point in X.
In this paper, inspired by Sedghi et al. and Hu’s work mentioned above, we prove some common fixed point theorems for ϕ-contractive mappings in fuzzy metric spaces, in which a very simple condition is imposed on the function ϕ. Our results improve the corresponding ones of Sedghi et al.  and Hu . Finally, an example is presented to illustrate the main result in this paper.
Definition 2.1 
∗ is associative and commutative,
∗ is continuous,
for all ,
whenever and for all .
Two typical examples of the continuous t-norm are and for all .
Definition 2.2 
The t-norm ∗2 is an example of t-norm of H-type, but t-norm ∗1 is not of H-type. Some other t-norm of H-type can be found in .
Definition 2.3 (Kramosil and Michalek )
A fuzzy metric space (in the sense of Kramosil and Michalek) is a triple , where X is a nonempty set, ∗ is a continuous t-norm and is a mapping, satisfying the following:
(KM-1) for all ,
(KM-2) for all if and only if ,
(KM-3) for all and all ,
(KM-4) is left continuous for all ,
(KM-5) for all and all .
In Definition 2.3, if M is a fuzzy set on and (KM-1), (KM-2), (KM-4) are replaced with the following (GV-1), (GV-2), (GV-4), respectively, then is called a fuzzy metric space in the sense of George and Veeramani :
(GV-1) for all and all ,
(GV-2) for some if and only if ,
(GV-4) is continuous.
Lemma 2.1 
Let be a fuzzy metric space in the sense of GV. Then is nondecreasing for all .
Definition 2.4 (George and Veeramani )
Let be a fuzzy metric space. A sequence in X is called an M-Cauchy sequence if for each and , there is such that for all . The fuzzy metric space is called M-complete if every M-Cauchy sequence is convergent.
Definition 2.5 
Here is called a coupled point of coincidence.
Definition 2.6 
Definition 2.7 
whenever and are sequences in X such that and for some .
In , Abbas et al. introduced the concept of w-compatible mappings. Here we give a similar concept in fuzzy metric spaces as follows.
Definition 2.8 Let be a fuzzy metric space, and let and be two mappings. F and g are said to be weakly compatible (or w-compatible) if they commute at their coupled coincidence points, i.e., if is a coupled coincidence point of g and F, then .
3 Main results
In this section, the fuzzy metric space is in the sense of GV and the fuzzy metric M is assumed to satisfy the condition for all .
By using the continuity of ∗ and [, Lemma 1], we get the following result.
If each is nondecreasing, then for any .
for all . Suppose that is complete and that g and F are w-compatible, then g and F have a unique common fixed point , that is, .
Since , this implies that (3.6) holds for . Therefore, there exists such that (3.6) holds for each .
Thus and , i.e., and are Cauchy sequences in X. Since is complete and , there exist such that converges to and converges to .
By (GV-2) one has . Similarly, we can prove that .
It follows from Lemma 3.1 and (GV-2) that and . This shows that g and F have the unique coupled point of coincidence.
Since converges to and converges to , we see that .
By Lemma 3.1 and (GV-2), we see that . This completes the proof. □
for all . Suppose that is complete and that g and F are w-compatible, then g and F have a unique common fixed point in , that is, .
Since , (3.22) holds for . Therefore, there exists such that (3.22) holds for all .
where . Thus and , i.e., and are the Cauchy sequences. Since is complete and , there exists such that converges to and converges to .
Since and , letting in (3.24), we have . Noting that , we have . Similarly, we can prove that .
This shows that as , and so we have and . Therefore, we have and . Now, from (3.25) it follows that and .
By induction we can get . Letting and noting that as , we have for any , i.e., . Therefore, u is a common fixed point of g and F. The uniqueness of u is similar to the final proof line of Theorem 3.1. This completes the proof. □
In Theorem 3.1 and Theorem 3.2, if we let for all , we get the following result.
for all . Suppose that is complete. Then F has a unique fixed point , that is, .
for all . Suppose that is complete. Then F has a unique fixed point , that is, .
Now, we illustrate Theorem 3.1 by the following example.
It is easy to see that g and F are not commuting since , , and is complete.
Then for any .
Now, we verify (3.1) for . We shall consider the following four cases.
Case 1. Let and . In this case there are four possibilities:
Case 1.3. Let and . This case is similar to Case 1.2.
Case 3. Let and . Then we have:
Case 4. and . This case is similar to Case 3.
For , since , by Cases 1-4 above, we can see that for all . It is easy to see that is a coupled coincidence point of g and F. Also, g and F are w-compatible at . By Theorem 3.1, we conclude that g and F have a unique common fixed point in X. Obviously, in this example, 0 is the unique common fixed point of g and F.
Since is not continuous at and is not complete, Hu’s Theorem 3.1 [, Theorem 1] cannot be applied to Example 3.1.
from kt to ;
the functions F and g are not required to be commutable.
in our Theorem 3.1, the function is only required to satisfy the condition for any . However, the function in Hu’s result is required to satisfy the conditions (ϕ-1)-(ϕ-3);
in our results, the mappings F and g are required to be weakly compatible, but in Hu’s result the mappings F and g are required to be compatible.
Also, in our results the mapping g is not required to be continuous, but the condition is imposed on the mapping g in the results of Sedghi et al. and Hu.
This work is supported by the Fundamental Research Funds for the Central Universities (Grant Number: 9161013002). The author S. Alsulami thanks the Deanship of Scientific Research (DSR), King Abdulaziz University, for financial support under grant No. (130-037-D1433).
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