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In this paper, we prove a common fixed point theorem in Intuitionistic fuzzy metric space by using pointwise R-weak commutativity and reciprocal continuity of mappings satisfying contractive conditions.

Atanassove [

The aim of this paper is to prove a common fixed point theorem in intuitionistic fuzzy metric space by using pointwise R-weak commutativity [

Definition 2.1 [

1) * is commutative and associative;

2) * is continuous;

3) for all;

4) whenever and for all

Definition 2.2 [

1) ◊ is commutative and associative;

2) ◊ is continuous;

3) for all;

4) whenever and for all

Alaca et al. [

Definition 2.3 [^{2} × [0, ∞) satisfying the conditions:

1) for all and;

2) for all;

3) for all and if and only if;

4) for all and t > 0;

5) for all and;

6) is left continuous, for all;

7) for all and;

8) for all;

9) for all and if and only if;

10) for all and t > 0;

11) for all and;

12) is right continuous, for all;

13) for all.

The functions and denote the degree of nearness and the degree of non-nearness between x and y w.r.t. t respectively.

Remark 2.1 [

Remark 2.2 [

Definition 2.4 [

1) A sequence in X is said to be Cauchy sequence if, for all and,

and

2) A sequence in X is said to be convergent to a point if, for all,

and

Definition 2.5 [

Example 2.1 [

and

Clearly, is complete intuitionistic fuzzy metric space.

Definition 2.6 [

Definition 2.7 [

Definition 2.8 [

Definition 2.9 [

and

Clearly, every pair of weakly commuting mappings is pointwise R-weakly commuting with.

Definition 2.10 [

If A and S are both continuous, then they are obviously reciprocally continuous but converse is not true.

The proof of our result is based upon the following lemmas of which the first two are due to Alaca et al. [

Lemma 3.1 [

for all

Then is a Cauchy sequence in X.

Lemma 3.2 [

Lemma 3.3. Let be a complete intuitionistic fuzzy metric space with continuous t-norm * and continuous t-conorm ◊ defined by and for all Further, let and be pointwise R-weakly commuting pairs of self mappings of X satisfying:

(3.2) there exists a constant such that

for all, and. Then the continuity of one of the mappings in compatible pair or on implies their reciprocal continuity.

Proof. First, assume that A and S are compatible and S is continuous. We show that A and S are reciprocally continuous. Let be a sequence such that and for some as.

Since S is continuous, we have and as and since is compatible, we have

That is as. By (3.1), for each n, there exists such that Thus, we have, , and as whenever

Now we claim that as.

Suppose not, then taking in (3.2), we have

Taking, we get

That is,

by the use of Lemma 3.2, we have as.

Now, we claim that Again take in (3.2), we have

i.e.

therefore, by use of Lemma 3.2, we have

Hence, , as.

This proves that A and S are reciprocally continuous on X. Similarly, it can be proved that B and T are reciprocally continuous if the pair is assumed to be compatible and T is continuous.

The main result of this paper is the following theorem:

Theorem 4.1. Let be a complete intuitionistic fuzzy metric space with continuous t-norm * and continuous t-conorm defined by and for all

Further, let and be pointwise R-weakly commuting pairs of self mappings of X satisfying (3.1), (3.2). If one of the mappings in compatible pair or is continuous, then A, B, S and T have a unique common fixed point.

Proof. Let. By (3.1), we define the sequences and in X such that for all

We show that is a Cauchy sequence in X. By (3.2) take, we have

Now, taking, we have

Similarly, we can show that

Also,

Taking, we get

Similarly, it can be shown that

Therefore, for any n and t, we have

Hence, by Lemma 3.1, is a Cauchy sequence in X. Since X is complete, so converges to z in X. Its subsequences and also converge to z.

Now, suppose that is a compatible pair and S is continuous. Then by Lemma 3.2, A and S are reciprocally continuous, then, as.

As, is a compatible pair. This implies

This gives as.

Hence,.

Since, therefore there exists a point such that

Now, again by taking in (3.2), we have

and

Thus, by Lemma 3.2, we have

Thus,

Since, A and S are pointwise R-weakly commuting mappings, therefore there exists, such that

and

Hence, and

Similarly, B and T are pointwise R-weakly commuting mappings, we have

Again, by taking in (3.2),

and

By Lemma 3.2, we have Hence is common fixed point of A and S. Similarly by (3.2), is a common fixed point of B and T. Hence, is a common fixed point of A, B, S and T.

Uniqueness: Suppose that is another common fixed point of A, B, S and T.

Then by (3.2), take

and

This gives

and

By Lemma 3.2,

Thus, uniqueness follows.

Taking in above theorem, we get following result:

Corollary 4.1. Let be a complete intuitionistic fuzzy metric space with continuous t-norm * and continuous t-conorm defined by and for all Further, let A and B are reciprocally continuous mappings on X satisfying

for all, and then pair A and B has a unique common fixed point.

We give now example to illustrate the above theorem:

Example 4.1. Let and let and be defined by

and

Then is complete intuitionistic fuzzy metric space. Let A, B, S and T be self maps on X defined as:

and for all.

Clearly

1) either of pair (A, S) or (B, T) be continuous self-mappings on X;

2) ;

3) {A, S} and {B, T} are R-weakly commuting pairs as both pairs commute at coincidence points;

4) {A, S} and {B, T} satisfies inequality (3.2), for all, where.

Hence, all conditions of Theorem 4.1 are satisfied and x = 0 is a unique common fixed point of A, B, S and T.

We would like to thank the referee for the critical comments and suggestions for the improvement of my paper.