Common Fixed Point Theorem in Fuzzy Metric Spaces Under E. A. Like Property

Madhu Shrivastava¹, K. Qureshi², A. D. Singh³

¹TIT Group of Instititution, Bhopal, India

²Retd. Additional Director, Bhopal, India

³Govt. M. V. M. College, Bhopal, India

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To cite this article:

Madhu Shrivastava, K. Qureshi, A. D. Singh. Common Fixed Point Theorem in Fuzzy Metric Spaces Under E. A. Like Property. Pure and Applied Mathematics Journal. Vol. 5, No. 4, 2016, pp. 141-144. doi: 10.11648/j.pamj.20160504.18

Received: July 4, 2016; Accepted: July 25, 2016; Published: August 21, 2016

Abstract: George and Veeramani [1] modify the concept of fuzzy metric spaces introduced by Kramosil and Michalek [4], Aamri and Moutawakil [8] generalized the notion of non-compatible mapping in metric space by E.A. property.Continuing the above conceptwe prove some commonfixed point theorem for a pair of weakly compatible maps under E.A. Like property.

Keywords: Fuzzy Metric Space, E. A. Property, E. A. Like Property, Weakly Compatible Maps

1. Introduction

Fuzzy set theory has various applications in different area. When the notion of fuzzy set was introduced, then it was the turning point in the development of mathematics. It was introduced by Zadeh [7]. Fuzzy set theory has various application in applied science such as neural network theory, stability theory, mathematical programming, modelling theory, engineering science, medical science etc.George and Veeramani [1] modify the concept of fuzzy metric spaces introduced by Kramosil and Michalek [4],with a view to obtain a Hausdorff topology on fuzzy metric spaces, continuously, many authors gives very important results a Sessa [14], Vasuki [12] etc. Aamri and Moutawakil [8] generalized the notion of non-compatible mapping in metric space by E.A. property.It was pointed out in [9] that property E.A. buys containment of ranges without any continuity requirement besides minimizes the commutatively at their point of coincidence.

In this paper, we establish some new results in common fixed point theorems in fuzzy metric spaces under E. A. Like [6].

2. Definition

Definition 2.1 [2] A binary operation  is a continuous t-norms if ∗satisfying conditions:

(1) ∗is commutative and associative;

(2) ∗is continuous; if and only if

(3)  for all

(4)  whenever  and  and

Example 2.2

Definition 2.3 [1] A 3-tuple (X, M,∗) 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 on  satisfying the following conditions,

 if and only if

 is continuous.

Then M is called a fuzzy metric on X. Then  denotes the degree of nearness between x and y with respect to t.

Example 2.4 [1] (Induced fuzzy metric) Let be a metric space. Denote  for all  and let  be fuzzy sets on  defined as follows:  Then  is a fuzzy metric space. We call this fuzzy metric induced by a metric d as the standard intuitionistic fuzzy metric.

Definition 2.5 Two self-mappings and of a fuzzy metric space  are called compatible if  whenever is a sequence in X such that

for some

Lemma 2.6 Let be fuzzy metric space. If there exists such that

 for all  and then

Definition 2.7 Let be a setand  self maps of A point  is called a coincidence point of  and  iff . We shall call  a point of coincidence of and

Definition 2.8 [3] A pair of maps  and  is called weakly compatible pair if they commute at coincidence points.

Definition 2.9 Let  and  be two self-maps of a fuzzy metric spacewe say that and  satisfy the property E. A. if there exists a sequence  such that,  for some

Definition 2.10 Let and  be two self-maps of a fuzzy metric space We say that  and  satisfy the property E. A. Like property if there exists a sequence  such that  for some  or i. e,

Example 2.11 Let  and  for all  then  is a fuzzy metric space.

Where

We define  and

We have

And

Also

And

Definition 2.12 (Common E. A. Property) Let  where is a fuzzy metric space,then the pair  and  said to satisfy common E. A. property if there exist two sequencesand in such that

 for some

Definition 2.13 (Common E. A. like Property) Let  and  be self maps of a fuzzy metric space then the pairs  and  said to satisfy common E. A. Like property if there exists two sequences  and  in X such that

,

Where or

Role of E.A. property in proving common fixed point theorems can be concluded by following,

(1) It buys containment of ranges without any continuity requirements.

(2) It minimizes the commutativity conditions of the maps to the commutativity at their points of coincidence.

(3) It allows replacing the completeness requirement of the space with a more natural condition of closeness of the range.

Of course, if two mappings satisfy E. A. like property then they satisfy E. A. property also, but, on the other hand, E. A. like property relaxes the condition of containment of ranges and closeness of the ranges to prove common fixed point theorems, which are necessary with E. A. property.

3. Main Results

Theorem-(3.1) - Let  and  be self-maps of a fuzzy metric space , satisfying  for all in  and such that following condition holds-(I)  

for all

(II)  and  Satisfy the E.A. Like property.

Where  is a continuous function such that  for each  and

Then there exist a unique common fixed point of  and .

Proof – Since  and  satisfy E. A. Like property. Therefore there exists a sequence  in

Such that

or Suppose that

Therefore  for some  Now we show that  from (I), we have

Taking ,we get.

This implies that  i.e  is coincidence point of  and

Since  and  are weakly compatible. Therefore

Now we show that  If not from (I), we have

 

Taking ,we get.

Which is a contradiction. Hence . Hence  is a common fixed point of  and

Uniqueness – Let  be another fixed point of and , such that , then from (I),we have

Which is a contradiction. Hence

Theorem–3.2  be self-maps of a fuzzy metric space satisfying the following condition-

(I)

 

(II)-Pairs  and  are weakly compatible.

(III)-Pairs  and  satisfying common E. A. Like property.

for all  in and where  and (c and d) can not be simultaneous  and

Then A,B,S,T have a unique common fixed point.

Proof – Since (A,S) and (B,T) satisfy common E. A. Like property, therefore there exist two sequences  and  in  such that

Where  or

Suppose that Now We have  then  for some

Now we claim that ,from (I),We have

we get

Hence , Now We have  then  for some

Now we claim that , from (I),We have

 we get

Hence

Since the pair  is weakly compatible, therefore

Now we show that ,

we get

Hence

Since the pair  is weakly compatible, therefore

Now we show that ,

we get

Hence .

Thus  is common fixed pointof  and

Uniqueness – Suppose that  is another common fixed point of  and

such that  then from (I)

Hence

Theorem 3.3 Let  and  be self maps of a fuzzy metric space  satisfying the following conditions:

 and

 for all  in  and

 Pairs  or ( satisfy E. A. property

 Pairs  and  are weakly compatible.

If the range of one of  and  is a closed subset of , then  and have a common fixed point in

Proof- Proof as above.

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