Common Fixed Point Theorem in Fuzzy Metric Spaces Under E. A. Like Property
Madhu Shrivastava^{1}, K. Qureshi^{2}, A. D. Singh^{3}
^{1}TIT Group of Instititution, Bhopal, India
^{2}Retd. Additional Director, Bhopal, India
^{3}Govt. 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 setand 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 spacewe 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 sequencesand 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