Fixed Point Theorem of Compatible of Type (R) Using Implicit Relation in Fuzzy Metric Space

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072
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© 2016, IRJET ISO 9001:2008 Certified Journal Page 248
FIXED POINT THEOREM OF COMPATIBLE OF TYPE (R) USING
IMPLICIT RELATION IN FUZZY METRIC SPACE
Dr Mahendra Singh Bhadauriya1 , Ruchi Gangil2
1 Assistant Professor, ITM Group of Institutions, Gwalior (M.P), India
2 Assistant Professor, ITM Group of Institutions, Gwalior (M.P), India
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Abstract
In this paper authors prove a Common fixed print theorem
for compatible mapping of type (R) in Fuzzy metric space
by using implicit relation. Our result modifies as well as
generalize the results of M.Koireng et al [10]. In [1] Cho et
al introduced the concept of semi compatibility in the
abstract D-metric space. Recently B. Singh et al [15]
introduced the concept of semi compatible mapping in the
context of a fuzzy metric space. The earliest important
result of compatible mapping was obtained by jungck [7].
Pathak, Chang and Cho [11] introduced the concept of
compatible mapping of type (P). In 2004, R.Singh et al [12]
introduced the concept by idea of compatible mapping of
type (R) by combing the definition of compatible mapping
and compatible mapping of type (P).
Our aspire in this paper is to define compatible of type R in
fuzzy metric spaces and prove some common fixed point
theorem of compatible map of type (R) by generalized
some interesting result [9].
Key Words: 54H25, 54E50.
1. INTRODUCTION
The concept of fuzzy sets was introduced at first by Zadeh
[17] which laid the foundation of fuzzy mathematics.
George and Veeramani In [5] modified the notion of fuzzy
metric space introduced by Kramosil and Michalek [8].
They also obtained that every metric space induces a fuzzy
metric spaces. Sessa [16] proved a generalization of
commutatively which called weak commutatively. Further
Jungek [7] more generalized commutatively which is
called compatibility in metric space.
Sessa introduced the concept of weakly commuting
mappings and Jungck defined the notions of compatible
mappings in order to generalize the concept of weak
commutativity and showed that weak commuting
mappings are compatible but the converse is not true. In
recent years, a number of a fixed point theorems and
coincidence theorems have been obtained by various
authors utilizing this notion. Jungck further weakened the
notion of weak compatibility and Jungck and Rhoades
further extended weak compatibility.
2. PRELIMINARIES AND DEFINATIONS:
Definition 2.1 [5] The 3-tuple (X, M, *) is called a fuzzy
metric space if X is an arbitrary set, * is a continuous
t-norm and M is a fuzzy set on ),0(2
X which
satisfying the following conditions:
(1) 0),,( tyxM
(2) 1),,( tyxM If and only if yx 
(3) ),,(),,( txyMtyxM 
(4) ),,(),,(*),,( stzxMszyMtyxM 
(5) ]1,0[),0(:,.),( yxM is continuous, for all
Xzyx ,, and 0, st
Example 2.1.1 : Let ),( dX be a metric space and
abba * or },min{* baba  .
Let
),(
),,(
yxdt
t
tyxM

 for all Xyx , and
0t then ,*),( MX is a fuzzy metric space.
Definition 2.2 [13] A sequence }{ nx in a fuzzy metric
space ,*),( MX is said to be a Cauchy sequence if and
only if for each , 0t there exists Nx  such that
1),,( txxM mn , For all 0, xmn 
The sequence }{ nx is said to converge to a point x in X
if for each 0 , 0t there exists Nx 0 such that
1),,( txxM n for all 0xn 
A fuzzy metric space ,*),( MX is said to be complete if
every Cauchy sequence in it converges to a point in it.

_______________________________________________________________________________________
Definition 2.3 [15] A pair of self mappings (A, S) of fuzzy
metric space ,*),( MX is said to be compatible if
1),,( 

tSAxASxMLim nn
n
Whenever }{ nx is a sequence in X such that
xAxLimSxLim n
n
n
n


, for some Xx
Definition 2.4 [14] A pair (A, S) of self mappings of a
fuzzy metric space is said to be semi compatible if
SxASxLim n
n


whenever }{ nx is a sequence in X such
that xSxLimASxLim n
n
n
n


So (A, S) is semi compatible and SyAy  implies
SAyASy  by taking yxn }{ and SyAyx  .
Proposition 2.1 [2] In a fuzzy metric space ,*),( MX
limit of a sequence is unique.
Proof: Let xxn }{ and yxn }{ then
),,(1),,( tyxMLimtxxMLim n
n
n
n 

Now )2/,,(*)2/,,(),,( txyMtxxMtyxM nn
1*1),,( tyxM (By taking limit n )
1),,( tyxM for all 0t
Thus yx  and hence the limit is unique.
Proposition 2.2 [15] If (A, S) is a semi-compatible pair of
self maps of a fuzzy metric space ,*),( MX and S in
continuous then ),( SA is compatible.
Definition 2.5 [9] Self mappings T and S of a fuzzy
metric space (X, M, ∗) is said to be compatible of type (R) if
1),,( 

tTxSTxMLim nn
n
For all t > 0
1),,( 

tTTxSSxMLim nn
n
For all t > 0
Whenever { nx } is a sequence in X such that
tSxLimAxLim n
n
n
n


For some z ∈ X.
Lemma 2.1 [15] let (X, M, *) be a fuzzy metric space
If there exists k ϵ X such that
M(x, y, k t) ≥ M(x, y, t/kn) for positive integer n
taking limit as n → ∞,
M(x, y, k/t) ≥ 1 and hence x = y.
Lemma 2.2 [14] The only t-norm * satisfying r*r ≥ r for
all r∈[0,1] is the minimum t-norm, that is,
a* b = min {a , b} for all a, b ∈ [0,1]
Lemma 2.3 [13] :- Let (X, M, *) be a fuzzy metric space, if
Proposition 2.3 [9] Let (X, M, *) be a fuzzy metric space
and let T and S be Compatible mappings of type(R)
and T z = S z for some z ∈ X then TTz = TSz = STz = SSz.
Proposition 2.4 [9] Let (X , M , * ) be a fuzzy metric space
and let A and S be Compatible mappings of type(R) and let
T nx , S nx → z as n →∞ for some z∈ X.
Then
(I) 1),,( 

zTzSTxMLim n
n
for if S is continuous at t
(ii) 1),,( 

zSzSTxMLim n
n
for if T is continuous at t
(iii) TSz = STz and Tz=Sz if T and S are continuous at z.
A Class of Implicit Relation –
Let  be the set of all real and continuous from
R5
]1,0[: satisfying the following conditions:
(A-1)  Is non-increasing in second, third, fourth and fifth
argument
(A-2) 0),,,,( vvuvu  vu 
0),,,,( vvvvu  vu 
Example:- },,,.{),,,,( 5432154321 ttttMaxtttttt 
3. MAIN RESULT
THEOREM 3.1
Let A, B, S and T is self mappings of a complete fuzzy
metric space ,*),( MX with continuous t-norm defined
by ]1,0[},{}*min{*  bababa satisfying the
following conditions:
(i) )(XA )(XT , )(XB )(XS
(ii) One of A, B, S and T are continuous.
(iii) Pairs (A, S) and (B, T) are compatible of type (R)
(iv)  Some )1,0(k such that for all 0,,  tXyx
 ),,,(),,,(),,,( tAxSxMtTySxMktByAxM
),,(),,,( tAxTyMktByTyM ) 0
(v) 1),,(,,  tyxMXyx as t
Then A, B, S and T have a unique common fixed point.
Proof: Let Xx 0 be any point as )()( XTXA 
and )()( XBXS  , Xx  1 and Xx 2 such that
10 TxAx  and 21 SxBx  . Inductively we construct a
sequence }{ ny in X such that
12212   nnn TxAxy And
221222   nnn SxBxy ; )( 22 nn Sxy  n = 0, 1
With 122 ,  nn xyxx using contractive condition, we
get

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),,(),,,(),,,(( 22122122 tAxSxMtTxSxMktBxAxM nnnnnn 
 0),,(),,,(, 2121212  tAxTxMktBxTxM nnnn
),,(),,,(),,,(( 1221222212 tyyMtyyMktyyM nnnnnn 
),,,( 2212 ktyyM nn  0)),,( 1212  tyyM nn And
),,(),,,(),,,(( 1221222212 tyyMtyyMktyyM nnnnnn 
1),,,( 2212 ktyyM nn  ) 0
Since  is non-increasing and ψ is non decreasing in fifth
argument therefore,
 
 0).,(),.,(
),.,(),.,(),.,(
12122212
1221222212


tyyMktyyM
tyyMtyyMktyyM
nnnn
nnnnnn
Therefore by (A-2),
),,(),,( 212212 tyyMktyyM nnnn   and similarly
),,(),,( 122212 tyyMktyyM nnnn  
Hence ),,(),,( 11 tyyMktyyM nnnn   for all n.
Now we show that
1),,( 

tyyMLim npn
n
For all p and t > 0
Now ),,( 1 tyyM nn
 )/,,( 1 ktyyM nn 
 )/,,( 2
2 ktyyM nn 
 ..............
 1)/,,( 01 n
ktyyM As n
kt / as n
Thus the result holds for 1p . By induction hypothesis
suppose that the result hotels for p = r, now
11*1)2/,,(*)2/,,(
),,(
1
1




tyyMtyyM
tyyM
rnrnrnn
rnn
Thus the result holds for 1 rp
Hence }{ ny is a Cauchy sequence in X and as X is
complete we get Xzyn }{ .
Hence
zSxzAx nn  22 , ... (I)
zBxzTx nn   1212 , .... (II)
Let S be continuous and pairs (A, S) are compatible of
type(R) we get
AS Szx n 2 , SzSSx n 2
From contractive condition we get
),,(),,,(( 122122 tTxSSxMktBxASxM nnnn  ,
),,(),,,( 121222 ktBxTxMtASxSSxM nnnn 
0)),,( 212  tAxTxM nn
Taking limit as n we get
0)),,(),,,(
),,,(),,,(),,,((
tzzMktzzM
tSzSzMtzSzMktzSzM
 0)1,1,1),,,(),,,(( tzSzMktzSzM
Since  is non increasing and ψ is non decreasing in third,
fourth and fifth argument therefore we can write
0)),,(),,,(
),,,(),,,(),,,((
tzSzMtzSzM
tzSzMtzSzMktzSzM
),,(),,( tzSzMktzSzM  and
),,(),,( tzSzNktzSzN 
zSz 
Step II: Put 12,  nxyzx in condition (IV),
),,(),,,(),,,(( 1212 tAzSzMtTxSzMktBxAzM nn  ,
0),,(),,,( 121212  tAzTxMktBxTxM nnn
As n
0)),,(,
),,(),,,(),,,(),,,((
tAzzM
ktzzMtAzSzMtzSzMktzAzM
 zSz 
0)),,(),,,(
),,,(),,,(),,,((
tAzzMktzzM
tAzzMtzzMktzAzM
0)),,(,1),,,(,1),,,(( tzAzMtzAzMktzAzM
 Since  is non-increasing and ψ is non –decreasing in
second and fourth argument
0),,(),,,(
),,,(),,,(),,,((
tzAzMtzAzM
tzAzMtzAzMktzAzM
 ),,(),,( tzAzMktzAzM  And
),,(),,( tzAzNktzAzN 
 SzzAz 
Step III: As
XuXTXA  ),()(
zTuAz 
Put nxx 2 , uy  is condition (IV), we have
),,(),,,(),,,(( 2222 tAxSxMtTuSxMktBuAxM nnnn ,
 0),,(),,,( 2 tAxTuMktBuTuM n
Taking Limit n
 0),,(),,,(
),,,(),,,(),,,((
tzTuMktBuTuM
tzzMtTuzMktBuzM
 zTuAz 

0),,(),,,(
),,,(),,,(),,,((
tzzMktBuzM
tzzMtzzMktBuzM

0)1,,,(
,1,1),,,((
ktBuzM
ktBuzM
Since  is non increasing and ψ is non decreasing in
second, third and fifth argument then

_______________________________________________________________________________________
0),,(),,,(
),,,(),,,(),,,((
tBuzMktBuzM
tBuzMtBuzMktBuzM
 ),,(),,(( tBuzMktBuzM  And
),,(),,(( tBuzNktBuzN 
 zBu 
And we get zBuTu  and as (B, T) is compatible of
type R so we get
TBuBTu  i.e. TzBz 
Step IV: Take zx  and zy  in condition (IV), we get
),,,(),,,(),,,(( tAzSzMtTzSzMktBzAzM
0)),,(),,,( tAzTzMktBzTzM
0),,(),,,(
),,,(),,,(),,,((
tzBzMktBzBzM
tzzMtBzzMktBzzM
0),,(,1,1),,,(),,,(( tzBzMktBzzMktBzzM
Since  is non increasing in second, third and fifth
argument
0),,(),,,(
),,,(),,,(),,,((
tzBzMtzBzM
tzBzMtBzzMktBzzM
),,(),,( tzBzMktBzzM 
Bz = z
Therefore zTzBz 
And so we get
zTzSzBzAz 
Hence z is a common fixed point A, B, S, and T
Uniqueness: Let z and 'z be two common fixed points
of the maps A, B, S and T. Then
zSzTzBzAz  And
''''' zSzTzBzAz 
Using condition (IV), we get
),,,(),,',(),,',(( tAzSzMtTzSzMktBzAzM
0)),,'(),,','( tAzTzMktBzTzM
0)),,'(
),,','(),,,(),,',(),,',((
tzzM
ktzzMtzzMtzzMktzzM
0),,'(,1,1),,',(),,',(( tzzMtzzMktzzM
Since  is none increasing in third and fourth argument
so?
0)),',(
),,',(),,,(),,',(),,',(( '
tzzM
tzzMtzzMtzzMktzzM
0)),',(
),,',(),,,(),,',(),,',(( '
tzzM
tzzMtzzMtzzMktzzM
),',(),',( tzzMktzzM  and 'zz  .
Hence z is a unique common fixed point maps A, B, S, T.
COROLLARY 3.1
Let A, B, S and T be self mappings of a complete fuzzy
metric space ,*),( MX with continuous t-norm and co t-
norm defined by
]1,0[},{}*min{*  bababa
Satisfying I to III

0),,(),,,(
),,,(),,,(
tAyTxMtAxSxM
tTySxMktByAxM
COROLLARY 3.2
Let A, B, S and T be self mappings of a complete fuzzy
metric space ,*),( MX with
Continuous t-norm defined by
]1,0[},{}*min{*  bawherebaba
Satisfying i, ii, iii, v of theorem3.1 and there exist some
)1,0(k such that for all 0,,  tXyx

 0),,(),2,,(
),,,(),,,(),,,(
tAxTyMtByTyM
tAxSxMtTySxMktByAxM
ACKNOWLEDGEMENT
Authors are thankful to Dr. P. S. Bisen for their valuable
suggestion towards the improvement of this paper.
REFERENCES
.
[1] Y.J Cho, B.K. Sharma, and D.R. Sahu, Semi-
compatibility and fixed points, Math. Japonica
42(1), 91-98, 1995.
[2] Cho, on common fixed points in fuzzy metric
spaces, Inter. Math. Forum 1 (10), 471-479, 2006.
[3] Zi- Ke Deng, Fuzzy pseudo metric spaces, J. Math.
Anal. Appl.86, 74-95, 1982.
[4] M.A, Erceg., Metric spaces in Fuzzy set theory,
J. Math. Anal. Appl., 69, 205-230, 1979.
[5] A. George and P. Veeramani, On same results of
analysis for fuzzy metric spaces, Fuzzy sets and
systems 90, 365-368, 1997.
[6] M. Grabiec, Fixed points in fuzzy metric spaces,
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[7] G. Jungck, Compatible mappings and common
fixed points. Internat. J.Math. and Math. Sci., 9(4),
771-779, 1986.
[8] Kramosil and J. Machalek, Fuzzy metric and
statistical metric spaces, Kybernetika 11, 336-
344, 1975..
[9] M. Koireng et.al. , Common fixed point theorem of
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mappings in fixed point considerations, publ.inst.
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