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 ![](https://www.scirp.org/html/6-7400579\95af4f93-e18d-4a92-865f-d5d36d83c6c1.jpg)
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 ![](https://www.scirp.org/html/6-7400579\3b3187f7-04e4-45a2-95ef-2825a3d266d8.jpg)
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
,
![](https://www.scirp.org/html/6-7400579\199ec450-f799-4451-b5bd-3aaab77d1a47.jpg)
and
![](https://www.scirp.org/html/6-7400579\c2efbe31-d2c6-4bb8-a3cd-69a813e49b94.jpg)
2) A sequence
in X is said to be convergent to a point
if, for all
,
![](https://www.scirp.org/html/6-7400579\4bdafe75-c950-4926-935c-a43786cd8bb3.jpg)
and
![](https://www.scirp.org/html/6-7400579\68f26bc0-9997-43b7-ad9f-9f0a847e6c6a.jpg)
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
![](https://www.scirp.org/html/6-7400579\f6e0ec13-6496-4f08-afb4-af3c7efe5c32.jpg)
and
![](https://www.scirp.org/html/6-7400579\0f151f2a-e8d9-4386-9abb-5c403e8ceedd.jpg)
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 ![](https://www.scirp.org/html/6-7400579\8ec5c77d-7c08-439c-934e-fc982d13ca2b.jpg)
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 ![](https://www.scirp.org/html/6-7400579\98856c62-8a41-4a55-b33a-c4fda3a2574d.jpg)
![](https://www.scirp.org/html/6-7400579\0c0c666b-f0fb-42a3-8a84-87d25e91d1d9.jpg)
and
![](https://www.scirp.org/html/6-7400579\ac0c0530-e951-4e94-aad8-bd93609ae078.jpg)
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
![](https://www.scirp.org/html/6-7400579\58587bbf-c1fd-4d4e-9e84-82b109be9945.jpg)
![](https://www.scirp.org/html/6-7400579\5dcb3321-4cf8-4589-91fe-e3a1c8730801.jpg)
for all ![](https://www.scirp.org/html/6-7400579\32047c0a-32b2-45d4-addc-2f1bcc11fea9.jpg)
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
![](https://www.scirp.org/html/6-7400579\9f89d186-fd6a-43c5-b4fa-6b12ad848001.jpg)
![](https://www.scirp.org/html/6-7400579\303c20e5-8529-492e-baa2-dc8d2595f1e6.jpg)
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
![](https://www.scirp.org/html/6-7400579\931b060f-c151-4f72-b7c5-5b4df8891dae.jpg)
That is
as
. By (3.1), for each n, there exists
such that
Thus, we have
,
,
and
as
whenever ![](https://www.scirp.org/html/6-7400579\4ffad9f7-f75c-420d-919d-217ffe0fcf41.jpg)
Now we claim that
as
.
Suppose not, then taking
in (3.2), we have
![](https://www.scirp.org/html/6-7400579\a7933014-d095-4abe-a045-ccf5034b6c9e.jpg)
![](https://www.scirp.org/html/6-7400579\9770659a-7295-4de2-9756-44861b96eade.jpg)
Taking
, we get
![](https://www.scirp.org/html/6-7400579\e889d804-2937-41ac-a5eb-d7b49f0eb495.jpg)
![](https://www.scirp.org/html/6-7400579\17b1c641-a842-4ae8-beb3-835d166941cf.jpg)
That is,
![](https://www.scirp.org/html/6-7400579\68c15d8f-bcaf-45af-8466-25b4e7388d61.jpg)
![](https://www.scirp.org/html/6-7400579\1fb8375b-99dc-486c-8b67-0b73c7f0ef64.jpg)
by the use of Lemma 3.2, we have
as
.
Now, we claim that
Again take
in (3.2), we have
![](https://www.scirp.org/html/6-7400579\d1d87db3-aa96-4ffd-af14-07a95cc487b4.jpg)
![](https://www.scirp.org/html/6-7400579\11b71c80-cede-46f2-8a42-6178063032f7.jpg)
![](https://www.scirp.org/html/6-7400579\170b8eb6-3dc1-446a-8f47-f44d84a27b35.jpg)
![](https://www.scirp.org/html/6-7400579\faaaac86-5482-46b5-bfe9-dad5871f9af2.jpg)
![](https://www.scirp.org/html/6-7400579\8abfe637-069c-491a-86b7-4bb9f202af19.jpg)
i.e.
![](https://www.scirp.org/html/6-7400579\179c4f30-f483-4290-836c-6a80927a8967.jpg)
![](https://www.scirp.org/html/6-7400579\4fdf484d-55f0-4083-b34b-e3ba6e7afe3f.jpg)
therefore, by use of Lemma 3.2, we have ![](https://www.scirp.org/html/6-7400579\c477dcdd-3e52-4453-92cf-3862d9d3b3fb.jpg)
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 ![](https://www.scirp.org/html/6-7400579\5f4cbd95-2902-4f3d-b274-8e165281d901.jpg)
![](https://www.scirp.org/html/6-7400579\5ae2bace-6fcd-4848-9b59-e0921cd13851.jpg)
We show that
is a Cauchy sequence in X. By (3.2) take
, we have
![](https://www.scirp.org/html/6-7400579\7c9f942e-6d58-4da8-bdcb-62c84ad67e71.jpg)
Now, taking
, we have
![](https://www.scirp.org/html/6-7400579\f444d152-b572-4461-a630-2634add8340c.jpg)
Similarly, we can show that
![](https://www.scirp.org/html/6-7400579\2e1942ee-e8f8-4c68-9709-7094578c1b8d.jpg)
Also,
![](https://www.scirp.org/html/6-7400579\cd626098-e0e2-461a-8090-e9074e662adc.jpg)
Taking
, we get
![](https://www.scirp.org/html/6-7400579\258ea4b6-617f-43c5-9f73-54871d9207c0.jpg)
Similarly, it can be shown that
![](https://www.scirp.org/html/6-7400579\4587168a-ae1a-433f-874a-88328ce8b272.jpg)
Therefore, for any n and t, we have
![](https://www.scirp.org/html/6-7400579\f2766f77-3aa2-4e1d-a425-35533c55d3be.jpg)
![](https://www.scirp.org/html/6-7400579\f0ef129b-6367-4482-b229-cf145d6e3b65.jpg)
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
![](https://www.scirp.org/html/6-7400579\f41e5087-cde8-4beb-92fa-3225cbab2868.jpg)
This gives
as
.
Hence,
.
Since
, therefore there exists a point
such that ![](https://www.scirp.org/html/6-7400579\32a1ac11-3cc4-4ded-acef-131e3b7c49be.jpg)
Now, again by taking
in (3.2), we have
![](https://www.scirp.org/html/6-7400579\38cf5385-a568-40f3-89e1-9efd9de97ddd.jpg)
![](https://www.scirp.org/html/6-7400579\b5852772-6a18-4acc-abcc-07ed558a509d.jpg)
and
![](https://www.scirp.org/html/6-7400579\24078292-4ac7-4825-b3d3-04f54f445f6f.jpg)
![](https://www.scirp.org/html/6-7400579\297d05b4-7b53-4c61-b22d-4df08b528e17.jpg)
![](https://www.scirp.org/html/6-7400579\018ac65a-92ce-4b56-b33e-2571cb475004.jpg)
![](https://www.scirp.org/html/6-7400579\4f1b1b57-df2e-49c6-af1a-21a5bb40b3fb.jpg)
Thus, by Lemma 3.2, we have ![](https://www.scirp.org/html/6-7400579\75f864a9-dac2-482a-b37b-6463459be0f7.jpg)
Thus, ![](https://www.scirp.org/html/6-7400579\6727b3dc-15eb-489b-9417-d141d2bdd890.jpg)
Since, A and S are pointwise R-weakly commuting mappings, therefore there exists
, such that
![](https://www.scirp.org/html/6-7400579\28853a33-fa48-4b61-afec-05410614952c.jpg)
and
![](https://www.scirp.org/html/6-7400579\5cf2c062-6afa-4547-854b-3cd8e67ae8c7.jpg)
Hence,
and ![](https://www.scirp.org/html/6-7400579\5cb5ff56-96bb-4102-b76a-d1a39f21a535.jpg)
Similarly, B and T are pointwise R-weakly commuting mappings, we have
Again, by taking
in (3.2),
![](https://www.scirp.org/html/6-7400579\b88e9963-58ba-4c64-90fb-af9206c71859.jpg)
![](https://www.scirp.org/html/6-7400579\43282331-2cb9-4e8a-9a50-b2adc40d2b83.jpg)
and
![](https://www.scirp.org/html/6-7400579\47fdf14d-f0ad-4f64-aa64-277f9d1a78f9.jpg)
![](https://www.scirp.org/html/6-7400579\e4c3d986-109e-4f15-b5c3-75a54f40be1f.jpg)
![](https://www.scirp.org/html/6-7400579\421f5780-9a70-4a44-955f-cb3974ca0767.jpg)
![](https://www.scirp.org/html/6-7400579\2a0aa636-d05e-41dc-a27c-9047ece6b2a6.jpg)
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 ![](https://www.scirp.org/html/6-7400579\7d353ce3-24f5-41ac-a291-21fc5f84682c.jpg)
![](https://www.scirp.org/html/6-7400579\3f26b570-eab7-43ac-9e38-799e332e6903.jpg)
and
![](https://www.scirp.org/html/6-7400579\44367a2c-04f6-44d8-95d5-ea084129c384.jpg)
![](https://www.scirp.org/html/6-7400579\be4aaf9e-78c5-4499-ac18-c42dd139e2c8.jpg)
This gives
and
![](https://www.scirp.org/html/6-7400579\377d2f4e-302c-47e7-81bf-96b2ac43ac46.jpg)
By Lemma 3.2, ![](https://www.scirp.org/html/6-7400579\ae2d05e8-6bd2-40af-af2d-c5667e41d6f5.jpg)
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
![](https://www.scirp.org/html/6-7400579\35b72e87-9c79-44bc-bbec-3ee5b79b36df.jpg)
![](https://www.scirp.org/html/6-7400579\aa6e6d24-7470-430a-90a2-d0b69a2df76d.jpg)
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 ![](https://www.scirp.org/html/6-7400579\c5964e76-ea18-466c-bfea-acffa332db7d.jpg)
and ![](https://www.scirp.org/html/6-7400579\662cb286-61b2-4c68-ab95-ee4a7ac97b35.jpg)
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.
NOTES