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Common fixed point theorems for nonlinear contractive mappings in fuzzy metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 191 (2013)
Abstract
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.
MSC:54E70, 47H25.
1 Introduction
The concept of fuzzy metric spaces was defined in different ways [1–3]. Grabiec [4] presented a fuzzy version of the Banach contraction principle in a fuzzy metric space in Kramosi and Michalek’s sense. Fang [5] proved some fixed point theorems in fuzzy metric spaces, which improved, generalized, unified and extended some main results of Edelstein [6], Istratescu [7], Sehgal and Bharucha-Reid [8].
In order to obtain a Hausdorff topology, George and Veeramani [9, 10] modified the concept of fuzzy metric space due to Kramosil and Michalek [11]. 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 [12] proved some common fixed point theorems for four mappings in GV fuzzy metric spaces. Gregori and Sapena [13] 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 [14] 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 [18], the authors gave the following results.
Theorem SAS [[18], Theorem 2.5]
Let for all and let be a complete fuzzy metric space such that M has an n-property. Let and be two functions such that
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 [17], Hu proved the following result.
Theorem of Hu [[17], Theorem 1]
Let be a complete fuzzy metric space, where ∗ is a continuous t-norm of H-type. Let and be two mappings and let there exist such that
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. [18] and Hu [17]. Finally, an example is presented to illustrate the main result in this paper.
2 Preliminaries
Definition 2.1 [9]
A binary operation is a continuous t-norm if ∗ satisfies the following conditions:
-
(1)
∗ is associative and commutative,
-
(2)
∗ is continuous,
-
(3)
for all ,
-
(4)
whenever and for all .
Two typical examples of the continuous t-norm are and for all .
Definition 2.2 [20]
A t-norm ∗ is said to be of Hadžić type (for short H-type) if the family of functions is equicontinuous at , where
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 [20].
Definition 2.3 (Kramosil and Michalek [11])
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 [9]:
(GV-1) for all and all ,
(GV-2) for some if and only if ,
(GV-4) is continuous.
Lemma 2.1 [4]
Let be a fuzzy metric space in the sense of GV. Then is nondecreasing for all .
Definition 2.4 (George and Veeramani [9])
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 [14]
An element is called a coupled coincidence point of the mappings and if
Here is called a coupled point of coincidence.
Definition 2.6 [15]
An element is called a common fixed point of the mappings and if
Definition 2.7 [17]
Let be a fuzzy metric space. The mappings F and g, where and , are said to be compatible if for all ,
whenever and are sequences in X such that and for some .
In [21], 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 [[22], Lemma 1], we get the following result.
Lemma 3.1 Let , let , and let . Assume that and
If each is nondecreasing, then for any .
Theorem 3.1 Let be a fuzzy metric space under a continuous t-norm ∗ of H-type. Let be a function satisfying that for any . Let and be two mappings with and assume that 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, .
Proof Since , there exist two sequences and in X such that
From (3.1) and (3.2) we have
and
It follows from (3.3) and (3.4) that
Let . Then
Since and , by Lemma 3.1 we have
Noting that , we get that
For any fixed , since , there exists such that . Next we show by induction that for any , there exists such that
It is obvious for since . Assume that (3.6) holds for some . Since , by (KM-5) we have
It follows from (3.1) and (3.6) that
Now from (3.7) and (3.8) we get
Similarly, we have
From (3.9) and (3.10) we conclude that
Since , this implies that (3.6) holds for . Therefore, there exists such that (3.6) holds for each .
Now we prove that and are Cauchy sequences in X. Let and . Since , there exists such that . Since is equicontinuous at 1 and , there is such that
By (3.5), one has . Since ∗ is continuous, there is such that for all ,
Hence, by (3.6) (replacing with ) and (3.11), we get
for any . Since
one has
By monotonicity of M, we have, for any ,
Thus and , i.e., and are Cauchy sequences in X. Since is complete and , there exist such that converges to and converges to .
Next we prove that and . Let ; since , there exists such that . By (KM-5) and (3.1), we have
Note that , and . Thus, letting in (3.12), we have
By induction we can get
By (GV-2) one has . Similarly, we can prove that .
Next we prove that if is another coupled coincidence point of g and F, then and . In fact, by (3.1) we have
It follows that
By induction we get
It follows from Lemma 3.1 and (GV-2) that and . This shows that g and F have the unique coupled point of coincidence.
Now we show that and . In fact, from (3.1) we get
and
Let . From (3.13) and (3.14) it follows that
By Lemma 3.1 we get , which implies that
Since converges to and converges to , we see that .
Now let . Then we have since . Since g and F are w-compatible, we have
which implies that is a coupled coincidence point of g and F. Since g and F have a unique coupled point of coincidence, we can conclude that , i.e., . Therefore, we have . Finally, we prove the uniqueness of a common fixed point of g and F. Let be such that . By (3.1) we have
which implies that
By Lemma 3.1 and (GV-2), we see that . This completes the proof. □
Theorem 3.2 Let be a fuzzy metric space under a continuous t-norm ∗ of H-type. Let be a function satisfying that for any . Suppose that and are two mappings such that , and assume that 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 in , that is, .
Proof Since , we can construct two sequences and in X such that
From (3.15) and (3.16) we have
and
Now, let . From (3.17) and (3.18) we get . It follows that
Since and for each , we have . By Lemma 3.1 we have
For any fixed , since , there exists such that . Similarly, since , there exists such that . By (3.17) we have
Next we show by induction that for any , there exists such that
This is obvious for since and . Assume that (3.22) holds for some . By (3.15), (3.22), (3.21) and (KM-5), we have
Similarly, we can prove that
Since , (3.22) holds for . Therefore, there exists such that (3.22) holds for all .
Let and . By hypothesis, is equicontinuous at 1 and , so there is such that
Since by (3.20) , there is such that for all , . Hence, it follows from (3.22) and (3.23) that
for all and any . Noting that (3.17) and (3.18), we have
This implies that 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 .
Next we prove that and . By (KM-5) and (3.15), we have, for any ,
Since and , letting in (3.24), we have . Noting that , we have . Similarly, we can prove that .
Let and . Since g and F are w-compatible, we have
This shows that is a coupled coincidence point of g and F. Now we prove that and . In fact, from (3.15) we have
Let . Then we have
Since and ∗ is continuous, we have
This shows that as , and so we have and . Therefore, we have and . Now, from (3.25) it follows that and .
Finally, we prove that . In fact, by (3.15) we have, for any ,
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.
Corollary 3.1 Let be a fuzzy metric space under a continuous t-norm ∗ of H-type. Let be a function satisfying that for any . Let be a mapping, and assume that for any ,
for all . Suppose that is complete. Then F has a unique fixed point , that is, .
Corollary 3.2 Let be a fuzzy metric space under a continuous t-norm ∗ of H-type. Let be a function satisfying that for any . Let be a mapping, and assume that for any ,
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.
Example 3.1 Let and for all . Define for all and . Then is a fuzzy metric space, but it is not complete. Define two mappings and by
and
It is easy to see that g and F are not commuting since , , and is complete.
Let by
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 1.1. Let and . Then we have
Case 1.2. Let and . Then
Case 1.3. Let and . This case is similar to Case 1.2.
Case 1.4. Let and . Then
Case 2. Let and . Then we have
Case 3. Let and . Then we have:
Case 3.1. If , then
Case 3.2. If , then
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 [[17], Theorem 1] cannot be applied to Example 3.1.
Remark 3.1 Our results improve the ones of Sedghi et al. [18] as follows:
-
(i)
from kt to ;
-
(ii)
the functions F and g are not required to be commutable.
Our results also improve the corresponding ones of Hu [17] as follows:
-
(a)
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);
-
(b)
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.
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Acknowledgements
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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Wang, S., Alsulami, S.M. & Ćirić, L. Common fixed point theorems for nonlinear contractive mappings in fuzzy metric spaces. Fixed Point Theory Appl 2013, 191 (2013). https://doi.org/10.1186/1687-1812-2013-191
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DOI: https://doi.org/10.1186/1687-1812-2013-191