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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 735217, 14 pages
http://dx.doi.org/10.1155/2013/735217
Research Article

Common Fixed Point Theorems in Fuzzy Metric Spaces Satisfying -Contractive Condition with Common Limit Range Property

1Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor, Uttar Pradesh 246701, India
2Department of Natural Resources Engineering and Management, University of Kurdistan, Hawler, Iraq
3Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Center, Pathum Thani 12121, Thailand

Received 18 June 2013; Accepted 28 July 2013

Academic Editor: Hassen Aydi

Copyright © 2013 Sunny Chauhan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The objective of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed point in fuzzy metric spaces. Some illustrative examples are furnished which demonstrate the validity of the hypotheses and degree of utility of our results. We derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. As an application to our main result, we prove an integral-type fixed point theorem in fuzzy metric space. Our results improve and extend a host of previously known results including the ones contained in Imdad et al. (2012).

1. Introduction

In 1965, Zadeh [1] studied the concept of a fuzzy set in his seminal paper. Thereafter, it was developed extensively by many researchers, which also include interesting applications of this theory in different fields. Fuzzy set theory has applications in applied sciences such as neural network theory, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory, and communication. In 1975, Kramosil and Michálek [2] introduced the concept of fuzzy metric space, which opened an avenue for further development of analysis in such spaces. Further, George and Veeramani [3] modified the concept of fuzzy metric space introduced by Kramosil and Michálek [2] and also have succeeded in inducing a Hausdorff topology on such a fuzzy metric space which is often used in current research these days. Most recently, Gregori et al. [4] showed several interesting examples of fuzzy metrics in the sense of George and Veeramani [3] and have also utilized such fuzzy metrics to color image processing.

On the other hand, Mishra et al. [5] extended the notion of compatible mappings to fuzzy metric spaces and proved common fixed point theorems in presence of continuity of at least one of the mappings, completeness of the underlying space, and containment of the ranges amongst involved mappings. Further, Singh and Jain [6] weakened the notion of compatibility by using the notion of weakly compatible mappings in fuzzy metric spaces and showed that every pair of compatible mappings is weakly compatible, but reverse is not true. Many mathematicians used different conditions on self-mappings and proved several fixed point theorems for contractions in fuzzy metric spaces (see [613]). However, the study of common fixed points of noncompatible maps is also of great interest according to Pant [14]. In 2002, Aamri and El Moutawakil [15] defined a property (E.A.) for self-mappings which contained the class of noncompatible mappings in metric spaces. In a paper of Ali and Imdad [16], it was pointed out that property (E.A.) allows replacing the completeness requirement of the space with a more natural condition of closedness of the range. Afterwards, Liu et al. [17] defined a new property which contains the property (E.A.) and proved some common fixed point theorems under hybrid contractive conditions. It was observed that the notion of common property (E.A.) relatively relaxes the required containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Subsequently, there are a number of results proved for contraction mappings satisfying property (E.A.) and common property (E.A.) in fuzzy metric spaces (see [1825]). In 2011, Sintunavarat and Kumam [26] coined the idea of “common limit range property” (also see [2733]) which relaxes the condition of closedness of the underlying subspace. Recently, Imdad et al. [34] extended the notion of common limit range property to two pairs of self-mappings which relaxes the requirement on closedness of the subspaces. Several common fixed point theorems have been proved by many researchers in the framework of fuzzy metric spaces via implicit relations (see [6, 22, 35]).

In this paper, we prove some common fixed point theorems for weakly compatible mappings with common limit range property in fuzzy metric spaces which include fuzzy metric spaces of two types, namely, Kramosil and Michálek fuzzy metric spaces and George and Veeramani fuzzy metric spaces. Some related results are also derived besides furnishing illustrative examples. We also present some integral-type common fixed point theorems in fuzzy metric spaces. Our results improve, extend, and generalize a host of previously known results existing in the literature.

2. Preliminaries

Definition 1 (see [36]). A binary operation is said to be continuous -norm if (1) is commutative and associative; (2) is continuous; (3) for all ; (4) whenever and for all .

Examples of continuous -norms are Lukasiewicz -norm, that is, , product -norm, that is, , and minimum -norm, that is, .

The fuzzy metric space of Kramosil and Michálek [2] is defined as follows.

Definition 2 (see [2]). The 3-tuple is said to be a KM-fuzzy metric space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions: for all and  (KM-1):;  (KM-2): if and only if ;  (KM-3):;  (KM-4):;  (KM-5): is left continuous.

Lemma 3 (see [37]). Let be a fuzzy metric space. Then is nondecreasing on for all .

The fuzzy metric space of George and Veeramani [3] is defined as follows.

Definition 4 (see [3]). The 3-tuple is said to be a GV-fuzzy metric space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions: for all and (GV-1): ; (GV-2): if and only if ; (GV-3): ; (GV-4): ; (GV-5): is continuous.

In view of (GV-1) and (GV-2), it is worth pointing out that (for all ) provided (see [24]).

Example 5 (see [3]). Let be a metric space. Define as for all and . Then is a GV-fuzzy metric space, where is the product -norm (or minimum -norm). Indeed, we call this fuzzy metric induced by metric the standard fuzzy metric. Hence every metric space is a fuzzy metric space.

Now we give some examples of fuzzy metric spaces according to Gregori et al. [4].

Example 6 (see [4]). Let be a nonempty set, a one-one function, and an increasing continuous function. For fixed , define as for all and . Then, is a fuzzy metric space on wherein is the product -norm.

Example 7 (see [4]). Let be a metric space and an increasing continuous function. Define as for all and . Then is a fuzzy metric space on wherein is the product -norm.

Example 8 (see [4]). Let be a bounded metric space with (for all , where is fixed constant in ) and an increasing continuous function. Define a function as for all and . Then is a fuzzy metric space on wherein is a Lukasiewicz -norm.

Definition 9 (see [24]). A sequence in a KM- (or GV-) fuzzy metric space is said to be convergent to some if for all there is some such that for all .

Lemma 10 (see [24]). If is a KM-fuzzy metric space and , are sequences in such that , , then for every continuity point of .

Definition 11 (see [5]). A pair of self-mappings of a KM- (or GV-) fuzzy metric space is said to be compatible if for all whenever is a sequence in such that for some .

Definition 12 (see [5]). A pair of self-mappings of a KM- (or GV-) fuzzy metric space is said to be noncompatible if there exists at least one sequence in such that for some but or nonexistent for at least one .

Definition 13 (see [38]). A pair of self-mappings of a nonempty set is said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points; that is, if for some , then .

Remark 14 (see [38]). Two compatible self-mappings are weakly compatible, but the converse is not true. Therefore the concept of weak compatibility is more general than that of compatibility.

Definition 15 (see [18]). A pair of self-mappings of a KM- (or GV-) fuzzy metric space is said to satisfy the property (E.A.) if there exists a sequence in such that for all for some .

Note that weak compatibility and property (E.A.) are independent of each other (see [39, Examples 2.1-2.2]).

Remark 16. In view of Definition 15, a pair of noncompatible mappings of a KM- (or GV-) fuzzy metric space satisfies the property (E.A.), but the converse need not be true (see [39, Remark 4.8 ]).

Definition 17 (see [18]). Two pairs and of self-mappings of a KM- (or GV-) fuzzy metric space are said to satisfy the common property (E.A.) if there exist two sequences , in such that for all for some .

Definition 18 (see [26]). A pair of self-mappings of a KM- (or GV-) fuzzy metric space is said to satisfy the common limit range property with respect to mapping (briefly, property) if there exists a sequence in such that for all where .

Definition 19 (see [27]). Two pairs and of self-mappings of a KM- (or GV-) fuzzy metric space are said to satisfy the common limit range property with respect to mappings and (briefly, property) if there exist two sequences , in such that for all where .

Remark 20. If and , then Definition 19 implies property (that is, Definition 18) according to Sintunavarat and Kumam [26].

Now we show that the property implies the common property (E.A.), but converse is not true. In this regard, see the following example.

Example 21. Let be a fuzzy metric space, where , with product -norm defined as for all and for all and . Define the self-mappings , , and by
Then we have , , , and .
Let us consider two sequences and in ; one can verify that but . Hence both pairs and do not satisfy the property while they satisfy the common property (E.A.).

Proposition 22. If the pairs and satisfy the common property (E.A.) and and are closed subsets of , then the pairs also share the property.

Definition 23 (see [40]). Let and be two families of self-mappings. The pair of families is said to be pairwise commuting if (1) for all ; (2) for all ; (3) for all and .

3. Main Results

Our results involve class of all mappings satisfying the following properties: : is continuous and nondecreasing on ; : for all .

We note that if , then , and that for all .

3.1. Fixed Point Theorems in KM-Fuzzy Metric Spaces

We begin with the following observation before proving our main result.

Lemma 24. Let , , , and be four self-mappings of a KM-fuzzy metric space . Suppose that (1)the pair or satisfies the (or ) property; (2) (or ); (3) (or ) is a closed subset of ; (4) converges for every sequence in whenever converges (or converges for every sequence in whenever converges); (5)for all , there exists : , for some ,
Then the pairs and satisfy the property.

Proof. If the pair enjoys the property, then there exists a sequence in such that where . By (2), , and for each sequence , there exists a sequence in such that . Therefore, due to the closedness of , so that and in all . Thus, we have , , and as . By (4), sequence converges and in all we need to show that as . Suppose that as , and then using inequality (14) with , , we have
Taking the limit as and using Lemma 10, we get or, equivalently,
As , we have for some . Then, in view of condition , we get , which is a contradiction, thereby implying which shows that the pairs and enjoy the property.

Remark 25. The converse of Lemma 24 is not true in general. For counterexamples, one can see Examples 27 and 30.

Theorem 26. Let , , , and be four self-mappings of a KM-fuzzy metric space satisfying inequality (14). Suppose that the pairs and enjoy the property. Then the pairs and have a coincidence point each. Moreover, , , , and have a unique common fixed point provided both pairs and are weakly compatible.

Proof. Since the pairs and satisfy the property, there exist two sequences and in such that where . Since , there exists a point such that . We show that . If not, then using inequality (14) with , , we get which, on making and using Lemma 10, reduces to and so
If , then for some . Then in view of condition we get , which is a contradiction. Therefore, so that which shows that is a coincidence point of the pair .
Also ; there exists a point such that . Now we assert that . Assume the contrary, and then using inequality (14) with , , we have which reduces to or, equivalently,
As implies for some , then in view of condition , we get , which is a contradiction. Therefore, so that which shows that is a coincidence point of the pair .
Since the pair is weakly compatible and , hence . Now we show that is a common fixed point of the pair . To prove this, we show that . If not, then using inequality (14) with , , we have and so Then on simplification, we obtain
Since , therefore for some . Then in view of condition , we get , which is a contradiction. Hence . Therefore, is a common fixed point of the pair .
Also the pair is weakly compatible and ; then . To accomplish this, we assert that . If not, then using inequality (14) with , , we have which reduces to and so
If , then for some . Then (in view of condition ) it follows that , which is a contradiction. Therefore, which shows that is a common fixed point of the pair . Uniqueness of common fixed point is an easy consequence of inequality (14) (in view of condition ).

Next, we give an example which is not applied by the results of Imdad et al. [21, Theorem 2.1] but can be applied to Theorem 26.

Example 27. Let be a fuzzy metric space, where , with product -norm defined as for all and for all and . Define the self-mappings , , , and by We obtain Hence and are not closed subsets of and so Theorem 2.1 of Imdad et al. [21] can not be applied to this example.
Next, we choose two sequences , (or , ), and then clearly which shows that both pairs and enjoy the property. By a routine calculation, one can verify inequality (14) (for all and ) wherein is defined by . Furthermore, we obtain that the pairs and are weakly compatible.
Therefore, all the conditions of Theorem 26 are satisfied and 3 is a unique common fixed point of , , , and which also remains a coincidence point as well.

Now we show that the result contained in Imdad et al. [21, Theorem 2.1] can be easily obtained by Theorem 26.

Theorem 28. Let , , , and be four self-mappings of a KM-fuzzy metric space satisfying inequality (14). Suppose that the following hypotheses hold: (1)the pairs and satisfy the common property (E.A.); (2) and are closed subsets of . Then the pairs and have a coincidence point each. Moreover, , , , and have a unique common fixed point provided both pairs and are weakly compatible.

Proof. Since the pairs and enjoy the common property (E.A.), there exist two sequences and in such that for some . Since and are closed subsets of , hence . Therefore, there exists a point such that . Similarly, . Therefore, there exists a point such that . The rest of the proof runs on the lines of the proof of Theorem 26.

Theorem 29. Let , , , and be four self-mappings of a KM-fuzzy metric space satisfying all the hypotheses of Lemma 24. Then , , , and have a unique common fixed point provided both pairs and are weakly compatible.

Proof. In view of Lemma 24, the pairs and enjoy the property; there exist two sequences and in such that where . The rest of the proof can be completed on the lines of the proof of Theorem 26. This completes the proof.

The following example demonstrates the utility of Theorem 29.

Example 30. In the setting of Example 27, replace the self-mappings , , , and by the following besides retaining the rest: Then we have and , whereas and are closed subsets of . Then, like the earlier example, the pairs satisfy the property and satisfy the property.
It easy to calculate that inequality (14) holds wherein is defined by . Moreover, the pairs and are weakly compatible.
Thus all the conditions of Theorem 29 are satisfied, and 3 is a unique common fixed point of the involved mappings , , , and .

By choosing , , , and suitably, we can derive a multitude of common fixed point theorems for a pair of mappings. As a sample, we deduce the following natural result for a pair of self-mappings.

Corollary 31. Let and be two self-mappings of a KM-fuzzy metric space satisfying the following conditions: (1)the pair enjoys the property; (2)for all , , there exists : , for some
Then and have a coincidence point. Moreover, if the pair is weakly compatible, then and have a unique common fixed point.

As an application of Theorem 26, we have the following result involving four finite families of self-mappings.

Theorem 32. Let , , , and be four finite families of self-mappings of a KM-fuzzy metric space such that , , , and which satisfy inequality (14). If the pairs and satisfy the property, then and have a point of coincidence each.
Moreover, , , , and have a unique common fixed point provided the pairs of families and are commute pairwise.

Proof. The proof of this theorem is similar to that of Theorem 3.1 contained in Imdad et al. [40]; hence the details are omitted.

Remark 33. Theorem 32 is a partial generalization of Theorem 26 as commutativity requirements in Theorem 32 are relatively stronger than weak compatibility used in Theorem 26.

Now, we indicate that Theorem 32 can be utilized to derive common fixed point theorems for any finite number of mappings. As a sample for five mappings, we can derive the following by setting one family of two members while the remaining families contain single members:

Corollary 34. Let , , , , and be five self-mappings of a KM-fuzzy metric space satisfying the following conditions: (1)the pairs and share the property; (2)for all , , there exists : , for some
Then the pairs and have a coincidence point each. Moreover, , , , , and have a unique common fixed point provided the pairs and commute pairwise (that is, , , , and ).

Similarly, we can derive a common fixed point theorem for six mappings by setting two families of two members while the remaining families contain single members:

Corollary 35. Let , , , , , and be six self-mappings of a KM-fuzzy metric space satisfying the following conditions: (1)the pairs and enjoy the property; (2)for all , , there exists : , for some
Then the pairs and have a coincidence point each. Moreover, , , , , , and have a unique common fixed point provided the pairs and commute pairwise (that is, , , , , , and ).

By setting , , , and in Theorem 32, we deduce the following.

Corollary 36. Let , , , and be four self-mappings of a KM-fuzzy metric space such that the pairs and satisfy the property. Suppose that for all , there exists : , for some where , , , and are fixed positive integers. Then the pairs and have a point of coincidence each. Further, , , , and have a unique common fixed point provided both pairs and commute pairwise.

Remark 37. The results similar to Theorem 28, Theorem 29, Corollary 31, Corollary 34, and Corollary 35 can be outlined in respect of Theorem 32 and Corollary 36.

3.2. Grabiec-Type Fixed Point Results

Inspired by the work of Grabiec [37], we state and prove some fixed point theorems for weakly compatible mappings with common limit range property.

Lemma 38 (see [37]). Let be a KM- (or GV-) fuzzy metric space. If there exists a constant such that for all , , then .

Theorem 39. Let , , , and be four self-mappings of a KM-fuzzy metric space . Suppose that (1)the pairs and enjoy the property; (2)for all , and for some
Then the pairs and have a coincidence point each. Moreover, , , , and have a unique common fixed point provided both pairs and are weakly compatible.

Proof. If the pairs and share the property, then there exist two sequences and in such that where . Since , there exists a point such that . Now we have to show that . On using inequality (45), we have
Letting and using Lemma 10,
Appealing to Lemma 38, we obtain and so which shows that is a coincidence point of the pair .
Also ; there exists a point such that . Now we have to assert that . On using inequality (45), we get or, equivalently,
In view of Lemma 38, we have ; that is, which shows that is a coincidence point of the pair .
As the pair is weakly compatible and , therefore . Now we show that is a common fixed point of the pair . To prove this, using inequality (45), we have which reduces to
Owing to Lemma 38, we get . Therefore, is a common fixed point of the pair .
Since pair is weakly compatible and , hence . On using inequality (45), we get Then on simplification, we have
By Lemma 38, we obtain which shows that is a common fixed point of the pair . Uniqueness of common fixed point is an easy consequence of the inequality (45) (in view of Lemma 38).

Remark 40. Theorem 39 improves and extends the results of Grabiec [37] and Imdad et al. [21, Theorem 2.5] and extends some relevant results contained in [16] to fuzzy metric spaces.

Remark 41. The results similar to Lemma 24, Theorem 28, Theorem 29, Theorem 32, Corollary 31, Corollary 34, Corollary 35, and Corollary 36 can be proved in view of contraction condition (45) which will generalize and extend several results from the literature. The listing of the possible corollaries are not included.

3.3. Fixed Point Theorems in GV-Fuzzy Metric Spaces

Lemma 42. Let , , , and be four self-mappings of a GV-fuzzy metric space satisfying conditions (1)–(4) of Lemma 24. Suppose that for all , for some , and for some
Then the pairs and satisfy the property.

Proof. As the pair enjoys the property, there exists a sequence in such that where . Since , each sequence there exists a sequence in such that . Therefore, due to the closedness of , so that . Thus in all we have , , and as . By (4) of Lemma 24, the sequence converges and in all we need to show that as . Suppose that as , and then using inequality (55) with , , we have in which, on making , we obtain
As implies , henceforth , which is a contradiction, thereby implying which shows that the pairs and enjoy the property.

Theorem 43. Let , , , and be four self-mappings of a GV-fuzzy metric space satisfying inequality (55). Suppose that the pairs and enjoy the property. Then the pairs and have a coincidence point each. Moreover, , , , and have a unique common fixed point provided both pairs and are weakly compatible.

Proof. If the pairs and satisfy the property, then there exist two sequences and in such that where . Since , there exists a point such that . We assert that . Assume the contrary, and then using inequality (55) with , , we get which, on making , reduces to
As implies , henceforth , which is a contradiction. Therefore, so that . Hence is a coincidence point of the pair .
Also there exists a point such that . Now we show that . If not, then using inequality (55) with , , we have which reduces to
As implies , henceforth , which is a contradiction. Therefore, so that which shows that is a coincidence point of the pair .
Since the pair is weakly compatible and , hence . Now we show that is a common fixed point of the pair . To prove this, we show that . Assume the contrary, and then using inequality (55) with , , we have Then on simplification, we obtain
As implies , henceforth , which is a contradiction. Hence . Therefore, is a common fixed point of the pair .
As the pair is weakly compatible and , then . To accomplish this, we assert that . If not, then using inequality (55) with , , we have