Common Fixed Point Theorem in Fuzzy Metric Spaces Under E. A. Like Property
Madhu Shrivastava1, K. Qureshi2, A. D. Singh3
1TIT Group of Instititution, Bhopal, India
2Retd. Additional Director, Bhopal, India
3Govt. M. V. M. College, Bhopal, India
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To cite this article:
Madhu Shrivastava, K. Qureshi, A. D. Singh. Common Fixed Point Theorem in Fuzzy Metric Spaces Under E. A. Like Property. Pure and Applied Mathematics Journal. Vol. 5, No. 4, 2016, pp. 141-144. doi: 10.11648/j.pamj.20160504.18
Received: July 4, 2016; Accepted: July 25, 2016; Published: August 21, 2016
Abstract: George and Veeramani [1] modify the concept of fuzzy metric spaces introduced by Kramosil and Michalek [4], Aamri and Moutawakil [8] generalized the notion of non-compatible mapping in metric space by E.A. property.Continuing the above conceptwe prove some commonfixed point theorem for a pair of weakly compatible maps under E.A. Like property.
Keywords: Fuzzy Metric Space, E. A. Property, E. A. Like Property, Weakly Compatible Maps
1. Introduction
Fuzzy set theory has various applications in different area. When the notion of fuzzy set was introduced, then it was the turning point in the development of mathematics. It was introduced by Zadeh [7]. Fuzzy set theory has various application in applied science such as neural network theory, stability theory, mathematical programming, modelling theory, engineering science, medical science etc.George and Veeramani [1] modify the concept of fuzzy metric spaces introduced by Kramosil and Michalek [4],with a view to obtain a Hausdorff topology on fuzzy metric spaces, continuously, many authors gives very important results a Sessa [14], Vasuki [12] etc. Aamri and Moutawakil [8] generalized the notion of non-compatible mapping in metric space by E.A. property.It was pointed out in [9] that property E.A. buys containment of ranges without any continuity requirement besides minimizes the commutatively at their point of coincidence.
In this paper, we establish some new results in common fixed point theorems in fuzzy metric spaces under E. A. Like [6].
2. Definition
Definition 2.1 [2] A binary operation is a continuous t-norms if ∗satisfying conditions:
(1) ∗is commutative and associative;
(2) ∗is continuous; if and only if
(3) for all
(4) whenever
and
and
Example 2.2
Definition 2.3 [1] A 3-tuple (X, M,∗) is said to be a fuzzy metric space if X is an arbitrary set,∗ is a continuous t-norm and M is a fuzzy set on satisfying the following conditions,
if and only if
is continuous.
Then M is called a fuzzy metric on X. Then denotes the degree of nearness between x and y with respect to t.
Example 2.4 [1] (Induced fuzzy metric) Let be a metric space. Denote
for all
and let
be fuzzy sets on
defined as follows:
Then
is a fuzzy metric space. We call this fuzzy metric induced by a metric d as the standard intuitionistic fuzzy metric.
Definition 2.5 Two self-mappings and
of a fuzzy metric space
are called compatible if
whenever
is a sequence in X such that
for some
Lemma 2.6 Let be fuzzy metric space. If there exists
such that
for all
and
then
Definition 2.7 Let be a set
and
self maps of
A point
is called a coincidence point of
and
iff
. We shall call
a point of coincidence of
and
Definition 2.8 [3] A pair of maps and
is called weakly compatible pair if they commute at coincidence points.
Definition 2.9 Let and
be two self-maps of a fuzzy metric space
we say that
and
satisfy the property E. A. if there exists a sequence
such that,
for some
Definition 2.10 Let and
be two self-maps of a fuzzy metric space
We say that
and
satisfy the property E. A. Like property if there exists a sequence
such that
for some
or
i. e,
Example 2.11 Let and
for all
then
is a fuzzy metric space.
Where
We define and
We have
And
Also
And
Definition 2.12 (Common E. A. Property) Let where
is a fuzzy metric space,then the pair
and
said to satisfy common E. A. property if there exist two sequences
and
in
such that
for some
Definition 2.13 (Common E. A. like Property) Let and
be self maps of a fuzzy metric space
then the pairs
and
said to satisfy common E. A. Like property if there exists two sequences
and
in X such that
,
Where or
Role of E.A. property in proving common fixed point theorems can be concluded by following,
(1) It buys containment of ranges without any continuity requirements.
(2) It minimizes the commutativity conditions of the maps to the commutativity at their points of coincidence.
(3) It allows replacing the completeness requirement of the space with a more natural condition of closeness of the range.
Of course, if two mappings satisfy E. A. like property then they satisfy E. A. property also, but, on the other hand, E. A. like property relaxes the condition of containment of ranges and closeness of the ranges to prove common fixed point theorems, which are necessary with E. A. property.
3. Main Results
Theorem-(3.1) - Let and
be self-maps of a fuzzy metric space
, satisfying
for all
in
and
such that following condition holds-(I)
for all
(II) and
Satisfy the E.A. Like property.
Where is a continuous function such that
for each
and
Then there exist a unique common fixed point of and
.
Proof – Since and
satisfy E. A. Like property. Therefore there exists a sequence
in
Such that
or
Suppose that
Therefore for some
Now we show that
from (I), we have
Taking ,we get.
This implies that i.e
is coincidence point of
and
Since and
are weakly compatible. Therefore
Now we show that If not from (I), we have
Taking ,we get.
Which is a contradiction. Hence . Hence
is a common fixed point of
and
Uniqueness – Let be another fixed point of and
, such that
, then from (I),we have
Which is a contradiction. Hence
Theorem–3.2 be self-maps of a fuzzy metric space
satisfying the following condition-
(I)
(II)-Pairs and
are weakly compatible.
(III)-Pairs and
satisfying common E. A. Like property.
for all in
and
where
and
(c and d) can not be simultaneous
and
Then A,B,S,T have a unique common fixed point.
Proof – Since (A,S) and (B,T) satisfy common E. A. Like property, therefore there exist two sequences and
in
such that
Where or
Suppose that Now We have
then
for some
Now we claim that ,from (I),We have
we get
Hence , Now We have
then
for some
Now we claim that , from (I),We have
we get
Hence
Since the pair is weakly compatible, therefore
Now we show that ,
we get
Hence
Since the pair is weakly compatible, therefore
Now we show that ,
we get
Hence .
Thus is common fixed pointof
and
Uniqueness – Suppose that is another common fixed point of
and
such that then from (I)
Hence
Theorem 3.3 Let and
be self maps of a fuzzy metric space
satisfying the following conditions:
and
for all
in
and
Pairs
or (
satisfy E. A. property
Pairs
and
are weakly compatible.
If the range of one of and
is a closed subset of
, then
and
have a common fixed point in
Proof- Proof as above.
References