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1. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145
RESEARCH ARTICLE
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OPEN ACCESS
Fixed Point Theorem for Selfmaps On A Fuzzy 2-Metric Space
Under Occasionally Subcompatibility Condition
K.P.R. Sastry 1, M.V.R. Kameswari 2, Ch. Srinivasa Rao 3, and R. Venkata
Bhaskar 4
1
8-28-8/1, Tamil Street, Chinna Waltair, Visakhapatnam -530 017, India
Department of Engineering Mathematics, GIT, Gitam University, Visakhapatnam -530 045, India
3
Department of Mathematics, Mrs. A.V.N. College, Visakhapatnam -530 001, India
4
Department of Mathematics, Raghu Engineering College, Visakhapatnam -531 162, India.
2
Abstract
In this paper, we prove a common fixed point theorem for three selfmaps and extend it to four and six
continuous selfmaps on a fuzzy 2-metric space, using the notion of occasionally sub compatible map with
respect to another map. We show that the result of Surjeet Singh Chauhan and Kiran Utreja [7] follows as a
corollary.
Mathematical subject classification (2010): 47H10, 54H25
Key words:Fuzzy 2-metric space, coincidence point, weakly compatible maps, occasionally sub
compatiblemap w.r.t another map, sub compatible of type A.
For
1.Introduction
The concept of fuzzy sets was introduced by
(i)
is commutative and associative
L.A.Zadeh [9] in 1965 which became active field
(ii)
is continuous
of research for many researchers. In 1975,
(iii)
Karmosil and Michalek [5] introduced the concept
(iv)
of a fuzzy metric space based on fuzzy sets; this
notion was further modified by George and
Veermani [2] with the help of t-norms. Many
are examples of
authors made use of the definition of a fuzzy metric
continuous t-norms.
space due to George and Veermani [2] in proving
fixed point theorems in fuzzy metric spaces. Gahler
Definition 2.3: (Kramosil. I and Michelek. J [5])
[1] introduced and studied 2-metric spaces in a
A triple
is said to be a fuzzy metric space
series of his papers. Iseki, Sharma, and Sharma [3]
(FM space, briefly) if X is a nonempty set, is a
investigated, for the first time, contraction type
continuous t-norm and M is a fuzzy set on
mappings in 2-metric spaces. In this paper we
which satisfies the following
introduced the notion of occasionally sub
conditions:
compatible map with respect to another map. Using
this notion we prove a common fixed point theorem
for three selfmaps and extend it to four and six
For
and
continuous selfmaps on a fuzzy 2-metric space also
we show that the result of Surjeet Singh Chauhan
(i)
and Kiran Utreja [8] follows as a corollary.
(ii)
2. Preliminaries
We begin with some known definitions and results.
(iii)
(iv)
Definition 2.1 :( Zadeh. L.A [9]) A fuzzy set A in
a nonempty set X is a function with domain X and
values in
.
(v)
Definition 2.2: (Schweizer.B and Sklar. A [6] ) A
function
is said to be a
continuous t-norm if
satisfies the following
conditions:
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is
left
continuous.
Then
is called a fuzzy metric space on X.
The function
nearness between
and
denotes the degree of
with respect to t.
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2. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145
Example 2.4: (George and Veeramani [2]) Let
be a metric space. Define
and
for all
all
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(This corresponds to tetrahedron inequality in
2-metric space. The function value
may be interpreted as the
probability that the area of triangle is less than
t.)
is
left
continuous.
Example 2.8: Let
be a metric space. Define
and
and
,
Then
is a Fuzzy metric space. It is called
the Fuzzy metric space induced by
for all
Definition 2.5: (Gahler [1]) Let X be a nonempty
set. A real valued function
on
is said
to be a 2-metric on X if
given distinct elements
exists an element
such that
of
and all
Then
, there
Definition 2.9: (Jinkyu Han[4]) Let
fuzzy 2-metric space. Then,
,
.
is a 2-metric space.
Example 2.6: Let
and let
the
area of the triangle spanned by x, y and z which
may be given explicitly by the formula
 

 
d  x, y , z   x1 y2 z3  y3z2  x2 y1z3  y3z1  x3 y1z2  y2 z1
, where





x  x1 , x2 , x3 , y  y1 , y 2 , y3 , z  z1 , z 2 , z3
pair
 Then
is called a 2-metric space.
Definition 2.7: ( S.Sharma [7] ) A triple
is said to be a fuzzy metric 2-space, if X is a
nonempty set, is a continuous t-norm and M is a
fuzzy set on
which satisfies the
following conditions:
for
and
t1 , t2 , t3
,
for all
if and only if
at least two of the three points are equal,
,
( symmetry about first three variables)
be a
A sequence
in a fuzzy 2- metric
space is said to be convergent to a point
if
.
A sequence
in a fuzzy 2- metric
space
is called a Cauchy sequence, if for
any
and
there exists
such that, for all
and
,
when at least two of
are equal,
The pair
is a fuzzy 2- metric space.

A fuzzy 2- metric space in which every
Cauchy sequence is convergent is said to be
complete.
Definition 2.10: Let and be maps from a fuzzy
2-metric space
into itself. A point x in X
is called a coincidence point of A and B if
.
Definition 2.11: Two selfmaps and of a fuzzy
2-metric space
are said to be weakly
compatible if they commute at their coincidence
points, that is if
for some
, then
.
Definition 2.12: Two selfmaps and of a fuzzy
2-metric space
are said to be sub
compatible if there exists a sequence
in such
that
Definition 2.13: (Surjeet singh Chauhan and
Kiran Utreja [8]) Two selfmaps and of a
fuzzy 2-metric space
are said to be sub
compatible of type A, if there exists a sequence
in such that
, and
t1  t2  t3
,
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The following two results are proved in [4].
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3. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145
Lemma 2.14: For all
non-decreasing function.
is a
Lemma 2.15: Let
be a fuzzy 2-metric
If
there
k   0,1 such
with
exists
and
Definition 3.3: Let and be selfmaps of a fuzzy
2-metric space
.
is said to be sub
compatible with respect to
if there exists a
sequence
such that
that
for all
space.
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then
.
and
further
Surjeet Singh Chauhan and Kiran Utreja [6] proved
the following Theorem
Theorem 2.16: (Surjeet Singh Chauhan and
Kiran Utreja [8]) Let
and be six self
mappings of a fuzzy 2-metric space
with
continuous t-norm satisfying
.
Suppose the pairs
and
are sub
compatible of type A having the same coincidence
point and
,



 M Sx, PQy, z , t   M Sx, ABx, z , t   M PQy , Ty , z , t 
M  ABx, PQy , z , t   M  ABx, Ty , z , t 

M Sx , Ty , z , kt 
Now we state and prove our main results.
Theorem 3.4: Let
and be three continuous
self
mappings
of
a
fuzzy
2-metric
space
with
continuous
t-norm
satisfying
. Suppose
are
occasionally sub compatible with respect to and

  
Then
fixed point in
and
k   0,1 , t  0 .
have a unique common
.
for all x, y, z  X and some
Then
and
If further
then
and
in .
3. Main Results
In this section we first introduce the notions of
occasionally sub compatible map
weakly sub compatible map and
sub
compatible map with respect to another map.
We use these notions to prove some fixed point
theorems. Further we show that the result of Surjeet
Singh Chauhan and Kiran Utreja [7] follows as a
corollary.
Definition 3.1: Let and be selfmaps of a fuzzy
2-metric space
. is said to be occasionally
sub compatible with respect to , if there exists a
sequence
such that
 
(3.4.1)
(2.16.1)
for all x, y, z  X and some
 
M Ax, By , z , kt  M By , Sx , z , t  M Ax , Sy , z , t  M Ax , By , z , t
k   0,1 , t  0 .
have a coincidence point in .
and
are weakly compatible
have a unique common fixed point
Proof: We may suppose, without loss of generality,
that
has at least three elements. Since
is
occasionally sub compatible with respect to , there
exists a sequence
in such that
and
(since
A and S are continuous)
Thus a is coincidence point of
and
Also since is occasionally sub compatible with
respect to , there exists a sequence
in such
that
Definition 3.2: Let and be selfmaps of a fuzzy
2-metric space
.
is said to be weakly sub
compatible with respect to , if
and
.
Thus b is coincidence point of
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and
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

4. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145
Take
in

(3.4.1).
Then
www.ijera.com
We have

M  Aa , Bb, z , kt   M  Bb, Sa, z ,t  M  Aa, Sb, z, t  M  Aa, Bb, z, t 


 M  Aa , Bb, z , kt   M  Bb, Aa, z , t  M  Aa, Bb, z, t M  Aa, Bb, z, t
 M  Aa, Bb, z, kt   M  Aa, Bb, z, t  t  0
(since, by given condition,  is minimum t-norm)
Therefore
 M  Aa, Bb, z, t   1 t  0 and
For uniqueness, suppose
fixed point of
and

Thus
for all
Aa  Bb
Since
is fixed point of .
is common fixed point of
and
is another common
Then
and
 Bb, w, z , kt   M  ABb, Bw, z , kt 
M

 M  w, Bb , z ,t M  Bb , w, z ,t M  Bb , w , z ,t 

 M  Bw, SBb , z ,t M  ABb , Sw, z ,t M  ABb , Bw, z ,t 
Take
in

(3.4.1).
Then
 M  Bb, w, z , kt   M  Bb, w, z , kt  t  0

M  Aa , Bb , z ,kt   M  Bb , Sa , z ,t M  Aa ,Sb , z ,t M  Aa , Bb , z ,t 
 M  Bb, w, z , kt   1 t  0
 M  Aa, Ba, z , kt   M  Ba, Aa, z , t   M  Aa, Aa, z , t   M  Aa, Ba, z , t 
 M  Aa, Ba, z , kt   M  Ba, Aa, z , t  1  M  Aa, Ba, z , t 

 M  Aa, Ba, z , kt   M  Aa, Ba, z , t  t  0
Bb  w
 M  Aa, Ba, z , t   1 t  0

This completes the proof of Theorem.
Aa  Ba
Since
,
Therefore
is coincidence point of
Take
show that
Theorem 3.5: Let
and
mappings of a fuzzy 2-metric space
and .
in (3.4.1), then, as above we can
is coincidence point of
and .
with continuous t-norm satisfying
. Suppose
and are continuous and
and
are occasionally sub compatible with respect to
and respectively and
Since
and
are weakly compatible, they
commute at their coincidence points.
Thus we have,
M  Sx, PQy , z , t   M  Sx , ABx , z , t   


M  Sx , Ty , z , kt   M  PQy , Ty , z , t   M  ABx , PQy , z , t 
M  ABx, Ty , z , t 



and
Take
in (3.4.1). Then


M  Aa , BBb , z ,kt   M  BBb ,Sa , z ,t M  Aa ,SBb , z ,t M  Aa ,BBb , z ,t 
M
M
M  BBb , Aa , z , t   M  Aa , BSb , z , t 



 M  Aa , BBb , z , t 



 Aa , BBb, z , kt   
M  BBb , Aa , z , t   M  Aa , BBb , z , t 



 M  Aa , BBb , z , t 



 Aa , BBb, z , kt   
 Aa , BBb, z , kt   M  Aa , BBb, z , t  t  0
 M  Aa , BBb , z , t   1 t  0
M

Aa  BBb

Bb  BBb
Therefore
is fixed point of
be six self
(3.5.1)
for all x, y, z  X and some
Then
in .
and
k   0,1 , t  0 .
have a coincidence point
If further
and
compatible then
and
common fixed point in .
are weakly
have a unique
Proof: We may suppose, without loss of generality,
that
has at least three elements. Since
is
occasionally sub compatible with respect to ,
there exists a sequence
in such that
.
We have.
and
Therefore
is fixed point of
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.
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5. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145
Therefore
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is fixed point of
.
Thus a is coincidence point of
Take
Also since
is occasionally sub compatible with
respect to , there exists a sequence
in such
that
M  Sa , PQTb, z , t   M  Sa , ABa , z , t 
M  PQTb, TTb, z , t 

M  Sa , TTb , z , kt   

M  ABa , PQTb, z , t 

M  ABa , TTb, z , t 

M  Sa , TPQb , z , t   M  Sa , Sa , z , t  


 M  Sa , TTb , z , kt    M TPQb , TTb , z , t   M  Sa , TPQb , z , t 
 M  Sa , TPQb , z , t 



M  Sa , TTb , z , t   M  Sa , Sa , z , t  


 M  Sa , TTb , z , kt    M TTb , TTb , z , t   M  Sa , TTb , z , t 
 M  Sa , TTb , z , t 





M  Sa , TTb , z , t   1  1 

 M  Sa , TTb , z , kt   

M  Sa , TTb , z , t   M  Sa , TTb , z , t 


and
Thus b is coincidence point of
Take
and
in (3.5.1). Then


M Sa , Tb , z , kt 
M  Sa , PQb, z , t   M  Sa, ABa, z , t  


M  PQb, Tb, z , t   M  ABa, PQb, z , t 
M  ABa , Tb, z , t 



 M  Sa, Tb, z, t   M  Sa, Sa, z, t   M Tb, Tb, z , t  


 M  Sa, Tb, z, kt   

M  Sa, Tb, z, t   M  Sa, Tb, z, t 



in (3.5.1). Then
 Sa , TTb, z , kt   M  Sa , TTb, z , t  t  0
 M  Sa , TTb , z , kt   1 t  0
M

Sa  TTb
 M  Sa, Tb, z, kt   M  Sa, Tb, z, t  t  0

Sa  TSa
 M  Sa, Tb, z, kt   1 t  0

Sa  Tb
Therefore
Since
and
are weakly compatible,
they commute at their coincidence points.
We have
 M  Sa, Tb, z, kt   M  Sa, Tb, z, t  11 M  Sa, Tb, z, t   M  Sa, Tb, z, t 
is fixed point of
.
Thus we have,
Therefore
is fixed point of
.
and
Thus
Take
is
common
fixed
point
M  SSa , PQb, z , t   M  SSa , ABSa , z , t  


M  SSa , Tb , z , kt    M  PQb , Tb , z , t   M  ABSa , PQb , z , t 
M  ABSa , Tb, z , t 



M  SSa , Tb, z , t   M  SSa , SABa , z , t 


 M  SSa , Tb , z , kt    M Tb , Tb , z , t   M  SABa , Tb , z , t  
M  SABa , Tb, z , t 



M  SSa , Tb, z , t   M  SSa , SSa , z , t 


 M  SSa , Tb , z , kt    M Tb , Tb , z , t   M  SSa , Tb , z , t  
M  SSa , Tb, z , t 



For uniqueness, suppose
fixed point of
is another common
Then
 M  SSa, Tb, z, kt   M  SSa, Tb, z, t  1 1  M  SSa, Tb, z, t   M  SSa, Tb, z, t 
 M  SSa, Tb, z, kt   M  SSa, Tb, z, t  t  0
 M  SSa, Tb, z, kt   1 t  0


of
in (3.5.1). Then
SSa  Tb
SSa  Sa
Therefore
is fixed point of
.
We have.
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6. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145



M Sa , w, z , kt  M SSa , Tw, z , kt

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Also since
and commute, the pair
is weakly compatible.
M  SSa , PQw, z , t   M  SSa , ABSa , z , t  


Therefore , from Theorem 3.5,
is fixed point of
  M  PQw, Tw, z , t   M  ABSa , PQw, z , t 
.
M  ABSa , Tw, z , t 



Now we prove that
is fixed point of
and
 M  Sa, w, z , t   M  Sa, Sa, z , t   M  w, w, z , t  




M  Sa, w, z , t   M  Sa, w, z , t 
 Take


in (3.6.1). Then
 M  Sa, w, z , t  1 
M  SAa , PQb, z , t   M  SAa , ABAa , z , t  




 1 M  Sa, w, z , t  
M  SAa , Tb , z , kt   M  PQb , Tb , z , t   M  ABAa , PQb , z , t 


M  ABAa , Tb, z , t 



M  Sa, w, z , t  
M  Sa, w, z , kt   M  Sa, w, z , t  t  0
Since
M  Sa, w, z , kt   1t  0
Sa  w
Theorem 3.6: Let
and
be six
continuous self mappings of a fuzzy 2-metric
space
with
continuous
t-norm
satisfying
. Suppose
and
are occasionally sub compatible with respect to
and respectively

M  SAa , Tb, z , t   M  SAa , SAa , z , t 


M Tb, Tb, z , t   M  SAa , Tb, z , t  
M  SAa , Tb, z , t 



M  SAa , Tb, z , t   1  1

 M  SAa , Tb , z , kt   

M  SAa , Tb, z , t   M  SAa , Tb, z , t 
 M  SAa, Tb, z, kt   M  SAa, Tb, z, t  t  0

Now we extend Theorem 3.5 to six continuous
selfmaps.

M  Sx , PQy , z , t   M  Sx , ABx , z , t  


M  PQy , Ty , z , t   M  ABx , PQy , z , t 
M  ABx , Ty , z , t 



(3.6.1)
for all x, y, z  X and some

k   0,1 , t  0 .
 M  SAa, Tb, z, kt   1t  0


SAa  Tb
ASa  Sa
Therefore
is fixed point of
If
and
have a coincidence
further
, then
have a unique common fixed point in
and
Since
and commute, the pair
weakly compatible.
Therefore, from Theorem 3.5,
.
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is fixed point of
Therefore
Therefore
is fixed point of
.
in (3.6.1). Then
M  Sa , PQPb, z , t   M  Sa , ABa , z , t 



M  Sa , TPb , z , kt   M  PQPb , TPb , z , t   M  ABa , PQPb , z , t 
M  ABa , TPb, z , t 



.
Proof: From Theorem 3.5, we can prove the
following
a is coincidence point of
coincidence point of
and and
.
From Theorem 3.5, we have
Take
Then
point in .
commute,
 M SAa , Tb , z , kt 
This completes the proof of Theorem.
M Sx , Ty , z , kt 
and
Since
and
commute, so
b is
Sa  Tb .
is
is fixed point of
1144 | P a g e

7. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145
M  Sa , TPb, z , t   M  Sa , Sa , z , t   


 M  Sa , TPb , z , kt   M TPb , TPb , z , t   M  Sa , TPb , z , t 
M  Sa , TPb, z , t 





 M Sa , TPb , z , kt 
M  Sa , TPb, z , t   1  1 



M  Sa , TPb, z , t   M  Sa , TPb, z , t 

 
 M  Sa , TPb , z , kt   1t  0

 M Sa , TPb , z , kt  M Sa , TPb , z , t t  0

[1]
Sa  PTb

ACKNOWLEDGMENT
The fourth author (R.Venkata Bhaskar) is grateful
to
1. The authorities of Raghu Engineering College
for granting necessary permissions to carry on this
research and,
2. GITAM University for providing the necessary
facilities to carry on this research.
References
Sa  TPb

Sa  PSa
Therefore
[2]
is fixed point of
.
[3]
From Theorem 3.5, we have
is fixed point of
[4]
Therefore
Therefore
is fixed point of
.
and
[5]
For uniqueness, suppose
is another common
fixed point of
and in .
[6]
Then
[7]
Thus
is common fixed point of
in .



M Sa , w, z , kt  M SSa , Tw, z , kt

M  SSa , PQw, z , t   M  SSa , ABSa , z , t  


  M  PQw, Tw, z , t   M  ABSa , PQw, z , t 
M  ABSa , Tw, z , t 



 M  Sa, w, z, t   M  Sa, Sa, z, t  


  M  w, w, z, t   M  Sa, w, z, t   


 M  Sa, w, z, t 

 M  Sa, w, z, t  11 M  Sa, w, z, t  




 M  Sa, w, z, t 



 M  Sa, w, z, kt   M  Sa, w, z, t  t  0
www.ijera.com
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 M  Sa, w, z, kt   1t  0

Sa  w
This completes the proof of Theorem.
Note 3.7: It is evident that Theorem 2.16 becomes
a corollary of Theorem 3.6, because the pairs
and
are sub compatible of type A
implies that
and
are occasionally sub
compatible with respect to and respectively.
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