Applied Mathematics
Vol. 3 No. 3 (2012) , Article ID: 18094 , 6 pages DOI:10.4236/am.2012.33036
Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Space Using R-Weakly Commuting Mappings
1School of Mathematics and Computer Applications, Thapar University, Patiala, India
2Deenbandhu Chhotu Ram, University of Science and Technology, Murthal, India
Email: *sauravmanro@hotmail.com
Received August 22, 2011; revised October 10, 2011; accepted October 18, 2011
Keywords: Intuitionistic Fuzzy Metric Space; Reciprocal Continuity; R-Weakly Commuting Mappings; Common Fixed Point Theorem
ABSTRACT
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.
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:

(3.2) 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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NOTES
*Corresponding author.