Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Space Using R-Weakly Commuting Mappings ()
1. Introduction
Atanassove [1] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. In 2004, Park [2] defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norms and continuous t-conorms. Recently, in 2006, Alaca et al. [3] defined the notion of intuitionistic fuzzy metric space by making use of Intuitionistic fuzzy sets, with the help of continuous t-norm and continuous t-conorms as a generalization of fuzzy metric space due to Kramosil and Michalek [4]. In 2006, Turkoglu [5] et al. proved Jungck’s [6] common fixed point theorem in the setting of intuitionistic fuzzy metric spaces for commuting mappings. For more details on intuitionistic fuzzy metric space, one can refer to the papers [7-12].
The aim of this paper is to prove a common fixed point theorem in intuitionistic fuzzy metric space by using pointwise R-weak commutativity [5] and reciprocal continuity [9] of mappings satisfying contractive conditions.
2. Preliminaries
Definition 2.1 [13]. A binary operation 
is continuous t-norm if * satisfies the following conditions:
1) * is commutative and associative;
2) * is continuous;
3) 
for all
;
4) 
whenever 
and 
for all 
Definition 2.2 [13]. A binary operation 
is continuous t-conorm if ◊ satisfies the following conditions:
1) ◊ is commutative and associative;
2) ◊ is continuous;
3) 
for all
;
4) 
whenever 
and 
for all 
Alaca et al. [3] defined the notion of intuitionistic fuzzy metric space as:
Definition 2.3 [3]. A 5-tuple 
is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, ◊ is a continuous tconorm and 
are fuzzy sets on X2 × [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 [12]. Every fuzzy metric space 
is an intuitionistic fuzzy metric space of the form 
such that t-norm * and t-conorm 
are associated as 
for all
.
Remark 2.2 [12]. In intuitionistic fuzzy metric space
, 
is non-decreasing and 
is non-increasing for all
.
Definition 2.4 [3]. Let 
be an intuitionistic fuzzy metric space. Then
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 [3]. An intuitionistic fuzzy metric space 
is said to be complete if and only if every Cauchy sequence in X is convergent.
Example 2.1 [3]. Let 
and let * be the continuous t-norm and ◊ be the continuous tconorm defined by 
and 
respectively, for all
. For each 
and
, define M and N by
and
Clearly, 
is complete intuitionistic fuzzy metric space.
Definition 2.6 [3]. A pair of self mappings 
of a intuitionistic fuzzy metric space 
is said to be commuting if 
and 
for all
.
Definition 2.7 [3]. A pair of self mappings 
of a intuitionistic fuzzy metric space 
is said to be weakly commuting if 
and 
for all 
and
.
Definition 2.8 [12]. A pair of self mappings 
of a intuitionistic fuzzy metric space 
is said to be compatible if 
and 
for all
, whenever 
is a sequence in X such that 
for some 
Definition 2.9 [5]. A pair of self mappings 
of a intuitionistic fuzzy metric space 
is said to be pointwise R-weakly commuting, if given
, there exist 
such that for all 
and
Clearly, every pair of weakly commuting mappings is pointwise R-weakly commuting with
.
Definition 2.10 [9]. Two mappings A and S of a Intuitionistic fuzzy metric space 
are called reciprocally continuous if 
, whenever 
is a sequence such that
, 
for some z in X.
If A and S are both continuous, then they are obviously reciprocally continuous but converse is not true.
3. Lemmas
The proof of our result is based upon the following lemmas of which the first two are due to Alaca et al. [12]:
Lemma 3.1 [12]. Let 
is a sequence in a intuitionistic fuzzy metric space
. If there exists a constant 
such that
for all 
Then 
is a Cauchy sequence in X.
Lemma 3.2 [12]. Let 
be intuitionistic fuzzy metric space and for all
, 
and if for a number 
and
. Then x = y.
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:
(3.1)
(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.
4. Main Result
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.
5. Acknowledgements
We would like to thank the referee for the critical comments and suggestions for the improvement of my paper.
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