Rasik M.Patel et. al. / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 2, Issue 10, 2015, pp.35-41
Common Fixed Point Theorem for Occasionally Weakly
Compatible Mapping in Q-Fuzzy Metric Space for Integral Type inequality
*
Rasik M.Patel1
The Research Scholar of
Sai Nath University, Ranchi (Jharkhand)
S.P.B.Patel Engineering College, Linch.
Email:[email protected]
Ramakant Bhardwaj2
TIT group of Institutes (TIT-Excellence)
Bhopal (M.P.),
India.
Email:[email protected]
Abstract
In this paper, the Fixed Point theory is an important and major topic of the nonlinear functional analysis that
deals with the investigation leading to the existence and approximation of a “Fixed Point”. The Common Fixed
Point Theorem in Q Fuzzy Metric Space, is established as a prime objective of this paper .The goal is achieved
by taking four self mappings on a Q Fuzzy Metric Space, satisfying the general contractive integral type
inequality along with the definition of occasionally weakly compatible.
Key words: Fixed point, Occasionally weakly Compatible mappings, 𝑄 -Fuzzy metric spaces.
2010 Mathematics Subject Classification: Primary 47H10, 54H25.
1. INTRODUCTION
The concept of fuzzy sets introduced by Zadeh [14] in 1965 plays an important role in topology and
analysis. Since then, there are many authors to study the fuzzy set with application. Especially,
Kramosil and Michalek [12] and George and Veeramani [8] modified the notion of fuzzy metric
spaces with the help of continuous t-norm, which shows a new way for further development of
analysis in such spaces. As a result of many fixed point theorem for various forms of mapping are
obtained in fuzzy metric spaces. Dhage [7] introduced the definition of D metric space and proved
many new fixed point theorem in D-metric spaces. Recently, Mustafa and Sims [15] presented a new
definition of G-metric space and made great contribution to the development of Dhage theory. On the
other hand, Lopez-Rodriguez and Romaguera [13] introduced the concept of Hausdorff fuzzy metric
in a more general space .The Q-fuzzy metrics spaces is introduced by Guangpeng Sun and kai
Yang[9] which can be consider as a Generalization of fuzzy metric spaces. R.Vasuki[16] proved fixed
point theorems for R-Weakly commutating mapping Pant [17,18,19] introduced the new concept of
reciprocal continuous mappings and established some common fixed points theorem.The concept of
compatible maps by [12] and weakly compatible maps by [10] in fuzzy metric space is generalized by
A.Al Thagafi and Naseer Shahzad [2] by introducing the concept of occasionally weakly compatible
mappings. Recent results on fixed point in Q-fuzzy metric space can be viewed in [9].The main
purpose of our paper is to prove common fixed point theorem in Q fuzzy metric space under general
contractive conditions satisfying the definition of occasionally weakly compatible map for integral
type inequality. This result generalizes and extends several known fixed point theorems for
occasionally weakly Compatible maps on G metric space.
2. PRELIMINARIES
DEFINITION 2.1 [1]: Let (X, d) be a complete metric space, c∈ (0, 1) and f: X→X be a mapping
𝑑(𝑓𝑥 ,𝑓𝑦 )
𝑑(𝑥,𝑦)
such that for each x, y ∈ X, 0
𝜑 𝑡 𝑑𝑡 ≤ 𝑐 0
𝜑 𝑡 𝑑𝑡 where 𝜑: [0,+∞) →[0,+∞) is a
Lebesgue integrable mapping which is summable on each compact subset of [0,+∞) , non negative,
𝜀
and such that for each ε > o, 0 𝜑 𝑡 𝑑𝑡, then f has a unique fixed point 𝑎 ∈ 𝑋 such that for each
𝑥 ∈ 𝑋, lim𝑛→∞ 𝑓 𝑛 𝑥 = 𝑎.
B.E.Rhoades [4], extending the result of Branciari by replacing the above condition by the following
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Rasik M.Patel et. al. / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 2, Issue 10, 2015, pp.35-41
DEFINITION 2.2 [3]: A binary operation *: [0, 1]
[0, 1]
[0, 1] is a continuous t-norm if it
satisfies the following conditions:
(1) * is associative and commutative,
(2) * is continuous,
(3) a * 1 = a for all a [0, 1],
(4) a * b c * d whenever a c and b d for all a, b, c, d [0, 1],
Two typical examples of continuous t-norm are a * b = ab and a * b = min (a, b).
DEFINITION 2.3 [3]: A 3-tuple (X, M,*) is called a fuzzy metric space if X is an arbitrary (Nonempty) set, * is a continuous t-norm and M is a fuzzy set on
satisfying the following
conditions: for all x, y, z X and t, s > 0,
(1) M(x, y, t)
.
(2) M(x, y, t)
if and only if x = y,
(3) M(x, y, t)
(4) M(x, y, t)
(5) M(x, y, .)
is continuous.
DEFINITION 2.4 [9]: A 3-tuple (X, Q,*) is called a Q-fuzzy metric space if X is an arbitrary (Nonempty) set, * is a continuous t-norm and M is a fuzzy set on
satisfying the following
conditions: for all
X and
0,
(1)
&
(2)
if and only if x = y = z.
(3)
(4)
(5)
is continuous.
A Q-fuzzy metric space is said to be symmetric if
for all
with
where p is a permutation function
for all
EXAMPLE 2.1: Let X = R and G is the G-metric on X. Denote
for any
define
.
for all x, y
.
for all
X. Then (X, Q,*) is a
-fuzzy
metric in X.
REMAK 2.1 Comparative study of Fuzzy metric space & Q-Fuzzy metric space:
1) In fuzzy metric space the fuzzy set M is defined on
where as in fuzzy metric space
the fuzzy set is defined on
.Thus it can be said that a Fuzzy Metric Space is the
extended version of the Fuzzy Metric Space in which Triangle Inequality is replaced by Rectangle
Inequality.
2) The concept of
Fuzzy Metric Space is on the G metric space which is a generalization of
ordinary metric space .Therefore the Fuzzy Metric Space is also called as the Generalized Fuzzy
Metric Space.
EXAMPLE 2.2: Let (X, M,*) be Fuzzy metric space. Define
for every x, y, z
by
X, then (X, Q,*) is a
-
fuzzy metric in X.
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Rasik M.Patel et. al. / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 2, Issue 10, 2015, pp.35-41
DEFINITION 2.5 [8] Let
be a
if and only if
as
each
&
-fuzzy metric space. A sequence {
for each
, there exist
} in X converges to
. It is called a Cauchy sequence if for
N such that
for each,
DEFINITION 2.6 [5] Let X be a set, f and g self maps of X. A point in X is called a coincidence
point of and iff
.We shall call
a point of coincidence of f and g.
DEFINITION 2.7 [9] Let f and g be self maps on a -fuzzy metric space
then the mappings
are said to be weakly compatible if they commute at their coincidence point, that is,
implies that
DEFINITION 2.8 [2] Let and
be self maps on a -fuzzy metric space
then the
mapping are occasionally weakly compatible iff there is a point x in X which is coincidence point of
and at which f and g commute.
Al-Thagafi and Naseer [2], (2008) shown that occasionally weakly is weakly compatible but converse
is not true.
EXAMPLE 2.3 Let R be the usual metric space. Define
by
for all
. Then
for
but
&
occasionally weakly compatible self maps but not weakly compatible.
LEMMA 2.1[9] If
respect to t for all
be a
in X .
LEMMA 2.2[9] Let
be a
for all
LEMMA 2.3[2] Let
coincidence,
-fuzzy metric space,then
and
. S and T are
is non decreasing with
-fuzzy metric space.,if there exists
and
, then
such that
.
be a set,
owc self maps of . If A and B have unique point of
, then w is the unique common fixed point of A and B.
PROOF. Since A and B are owc, there exists a point x in X such that
and
. Thus,
, which says that
is also a point of coincidence of
A and B. Since the point of coincidence
is unique by hypothesis,
, and
is a common fixed point of A and B. Moreover, if z is any common fixed point of A and B,
then
by the uniqueness of the point of coincidence point of A and B.
MAIN RESULT
THEOREM 3.1 Let
be a self mappings of the Symmetric
continuous t norm satisfying the following condition :
3.1(a) The pair {A,S} and {B,T} is Occasionally Weakly Compatible
3.1(b) There exist
such that
for all
in X and
. Then
- Fuzzy metric space with
have Unique Common Fixed Point in X.
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Rasik M.Patel et. al. / International Journal of Modern Sciences and Engineering Technology (IJMSET)
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Volume 2, Issue 10, 2015, pp.35-41
PROOF:
Since by hypothesis, the pair
and
points x, y in X such that
we have
By lemma 2.2 we have
and
is Occasionally weakly Compatible then there exists
. We claim that
. From Equation 3.1(b)
. So
Moreover, if there is another point z such that
, then, using 3.1(b) it follows that
or
and
is the unique point of coincidence of A
and S. Then by lemma 2.3, it follows that w is the unique common fixed point of A and S. By
symmetry, there is a unique common fixed point z in X such that
.
Now, we claim that
Suppose that
.Using equation 3.1(b)we have
By lemma 2.2 we have
Therefore w is a unique point of coincidence of
then by lemma 2.3
w is the unique common fixed point of
.
Example 3.1
Let
and G is the G Symmetric metric space on X such that
Denote
Then
for all
is a
in
and for each
Fuzzy Metric Space Define a mapping
define a fuzzy set
as
as
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Rasik M.Patel et. al. / International Journal of Modern Sciences and Engineering Technology (IJMSET)
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Volume 2, Issue 10, 2015, pp.35-41
We claim that the pair {A, S} and {B, T} is occasionally weakly Compatible
At
we have
and
also
and
thus the pair {A, S} is owc map. Similarly we can show the pair
{B, T} is occasionally weakly Compatible. For
and for all
and
, the
mappings satisfy equation 3.1(b)
Thus all the condition of theorem are verified. Hence 0 is the Common Fixed Point of
.
THEOREM 3.2 Let
be a self mappings of the Symmetric - Fuzzy metric space with
continuous t norm satisfying the following condition :
3.2(a) The pair {A,S} and {B,T} is Occasionally Weakly Compatible
3.2(b) There exist
such that
for all
in X and
. Then
Proof:
Since by hypothesis, the pair
points x, y in X such that
we have
By lemma 2.2 we have
have Unique Common Fixed Point in X.
and
and
is Occasionally weakly Compatible then there exists
. We claim that
. From Equation 3.2(b)
. So
Moreover, if there is another point z such that
, then, using 3.2(b) it follows that
or
and
is the unique point of coincidence of A
and S. Then by lemma 2.3, it follows that w is the unique common fixed point of A and S. By
symmetry, there is a unique common fixed point z in X such that
.
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Now, we claim that
we have
Suppose that
.Using equation 3.2(b),
By lemma 2.2 we have
Therefore w is a unique point of coincidence of
w is the unique common fixed point of
.
then by lemma 2.3
ACKNOWLEDGEMENT:
One of the Author (Dr. Ramakant Bhardwaj) is thankful to MPCOST Bhopal for the project No.2556.
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