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Occasionally Weakly Compatible Mappings

Amit Kumar Govery*, Mamta Singh

School of Studies in Mathematics, Vikram University, Ujjain - 456010 (M.P.), India

Abstract: In this paper, the concept of compatible maps of type (A) and occasionally weakly compatible maps in fuzzy metric space have been applied to prove common fixed point theorem. A fixed point theorem for six self maps has been established using the concept of compatible maps of type (A) and occasionally weakly compatible maps, which generalizes the result of Cho .

Keywords: common fixed points, fuzzy metric space, compatible maps, compatible maps of type (A) and occasionally weakly compatible maps

Received July 08, 2015; Revised July 19, 2015; Accepted July 24, 2015

Citation: Amit Kumar Govery, Mamta Singh, Occasionally Weakly Compatible Mappings. Turkish Journal of Analysis and Number Theory, 3 (3). 78-82. January 2015. doi:10.12691/tjant-3-3-2

1. Introduction

The concept of Fuzzy sets was initially investigated by Zadeh [13] as a new way to represent vagueness in everyday life. Subsequently, it was developed by many authors and used in various fields. To use this concept in Topology and Analysis, several researchers have defined Fuzzy metric space in various ways. In this paper we deal with the Fuzzy metric space defined by Kramosil and Michalek [11] and modified by George and Veeramani [20]. Recently, Grebiec [1] has proved fixed point results for Fuzzy metric space. In the sequel, Singh and Chauhan [12] introduced the concept of compatible mappings of Fuzzy metric space and proved the common fixed point theorem. Jungck et. al. [2] introduced the concept of compatible maps of type (A) in metric space and proved fixed point theorems. Using the concept of compatible maps of type (A), Jain et. al. [18] proved a fixed point theorem for six self maps in a fuzzy metric space. Singh et. al. [7,8] proved fixed point theorems in a fuzzy metric space. Recently in 2012, Jain et. al. [4,5] and Sharma et. al. [6] proved various fixed point theorems using the concepts of semi-compatible mappings, property (E.A.) and absorbing mappings. The concept of occasionally weakly compatible mappings in metric spaces is introduced by Al-Thagafi and Shahzad [14] which is most general among all the commutativity concepts. Recently, Khan and Sumitra [15] extended the notion of occasionally weakly compatible maps to fuzzy metric space.

In this paper, a fixed point theorem for six self maps has been established using the concept of compatible maps of type (A) occasionally weakly compatible mappings, which generalizes the result of Cho [16].

For the sake of completeness, we recall some definitions and known results in Fuzzy metric space.

2. Definitions, lemmas, Remarks, Propositions

Definition 2.1. [10] A binary operation : is called a -norm if is an abelian topological monoid with unit 1 such that . Whenever and for . [0,1].

Examples of -norms are:

Definition 2.2. [10] The 3-tuple is said to be a Fuzzy metric space if X is an arbitrary set, is a continuous t-norm and M is a Fuzzy set in satisfying the following conditions:

For all and

(FM-1)

(FM-2)

(FM-3)

(FM-4)

(FM-5) is left

continuous,

(FM-6)

Note that can be considered as the degree of nearness between x and y with respect to t. We identify with for all . The following example shows that every metric space induces a Fuzzy metric space.

Example 2.1. [10] Let be a metric space. Define and for all . Then is a Fuzzy metric space. It is called the Fuzzy metric space induced by .

Definition 2.3. [10] A sequence in a Fuzzy metric space is said to be a Cauchy sequence if and only if for each , there exists such that for all .

The sequence is said to converge to a point in if and only if for each there exists such that for all .

A Fuzzy metric space is said to be complete if every Cauchy sequence in it converges to a point in it.

Definition 2.4. [12] Self mappings and of a Fuzzy metric space are said to be compatible if and only if for all , whenever is a sequence in such that for some in as .

Definition 2.5. [18] Self maps and of a Fuzzy metric space are said to be compatible maps of type if and for all , whenever is a sequence in such that for some in as .

Definition 2.6. [15] Two maps and from a Fuzzy metric space into itself are said to be Occasionally weakly compatible (owc) if and only if there is a point , which is coincidence point of and at which and commute.

Remark 2.1. [18] The concept of compatible maps of type and occasionally weakly compatibility is more general than the concept of compatible maps in a Fuzzy metric space.

Proposition 2.1. [18] In a Fuzzy metric space limit of a sequence is unique.

Lemma 2.1. [1] Let be a fuzzy metric space. Then for all , is a non-decreasing function.

Lemma 2.2. [16] Let be a fuzzy metric space. If there exists such that for all

Lemma 2.3. [18] Let be a sequence in a fuzzy metric space . If there exists a number such that and . Then is a Cauchy sequence in .

Proposition 2.2. [18] Let and be concept of compatible maps of type of a complete fuzzy metric space with continuous t-norm defined by for all and for some in . Then

Lemma 2.4. [3] The only -norm satisfying for all is the minimum -norm, that is for all .

3. Main Result

Theorem 3.1. Let be a complete fuzzy metric space and let and be mappings from into itself such that the following conditions are satisfied:

(a)

(b)

(c) either or is continuous;

(d) is compatible maps of type and is occasionally weakly compatible ;

(e) there exists such that for every and

Then and have a unique common fixed point in .

Proof: Let . From (a) there exist such that and .

Inductively, we can construct sequences and in such that and

Step 1. Put and in (e), we get

From lemma 2.1 and 2.2, we have

Similarly, we have

Thus, we have

and hence as for any .

For each and , we can choose such that

For , we suppose . Then we have

and hence is a Cauchy sequence in .

Since is complete, converges to some point . Also its subsequence’s converges to the same point i.e.

[picture] | (1)

[picture] | (2)

Case I. Suppose is continuous.

Since is continuous, we have

As is compatible pair of type , we have

Step 2. Put and in (e), we get

Taking , we get

Therefore, by using lemma 2.2, we get

[picture] | (3)

Step 3. Put and in (e), we have

Taking and using equation (1), we get

i.e.

Therefore, by using lemma 2.2, we get

Step 4. Putting and in condition (e), we get

As , so we have

Taking and using (1), we get

i.e.

Therefore, by using lemma 2.2, we get

and also we have

Therefore,

[picture] | (4)

Step 5. As , there exists such that .

Putting and in (e), we get

Taking and using (1) and (2), we get

i.e..

Therefore, by using lemma 2.2, we get . Hence . Since is occasionally weakly compatible, therefore, by proposition (2.2), we have

Step 6. Putting and in (e), we get

Taking and using (2) and step 5, we get

Therefore, by using lemma 2.2, we get

Step 7. Putting and in (e), we get

As and , we have

Taking , we get

i.e.

Therefore, by using lemma 2.2, we get

Now implies

Hence

[picture] | (5)

Combining (4) and (5), we get

Hence, is the common fixed point of and .

Case II. Suppose is continuous.

As is continuous,

As is compatible pair of type ,

Step 8. Putting and in condition (e), we have

Taking , we get

i.e.

Therefore by using lemma 2.2, we have

Further, using steps 5,6,7, we get

Step 9. As , there exists such that

Put and in (e), we get

Taking , we get

i.e.

Therefore, by using lemma 2.2, we get

Therefore,

As is compatible pair of type , then by proposition (2.2), we have

Also, from step 4, we get .

Further, using steps 5, 6, 7, we get

i.e. is the common fixed point of the six maps and in this case also.

Uniqueness: Let be another common fixed point of and .

Then . Put and in (e), we get

Taking , we get

i.e. .

Therefore by using lemma (2.2), we get .

Therefore is the unique common fixed point of self maps and .

Remark 3.1. If we take , the identity map on in theorem 3.1, then condition is satisfied trivially and we get

Corollary 3.1. Let be a complete fuzzy metric space and let and be mappings from into itself such that the following conditions are satisfied:

(a) ;

(b) either or is continuous;

(c) is compatible maps of type and is occasionally weakly compatible;

(d) there exists such that for every and

Then and have a unique common fixed point in .

Remark 3.2. In view of remark 3.1, corollary 3.1 is a generalization of the result of Cho [16] in the sense that condition of compatibility of the pairs of self maps has been restricted to compatibility of type occasionally weakly compatible and only one map of the first pair is needed to be continuous.

Acknowledgement

Authors are thankful to the referee for his valuable comments.

References

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