International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 683
ISSN 2229-5518
Abstract— Bijendra singh et al [11] proved a common fixed point theorem in fuzzy metric spaces for semi compatible and weak compatible mapping. In this paper we obtain a common fixed point theorem for six self maps for compatible of type (ß) with new implicit contractive condition in fuzzy metric space .
Index Terms— Common fixed points, Compatible maps of type( ß), Compatible Map,Continous maps, Fuzzy metric space, Implicit relation,Weak compatible maps,
—————————— ——————————
HE concept of fuzzy sets was introduced initially by
Zadeh [13]. Subsequently, several resercheral defined fuzzy metric space in various methods in various ab-
Whenever a ≤ c
a, b, c, d ∈ [ 0,1]
and b ≤ d for all
stract spaces. In this paper ,We deal with the fuzzy metric space defined by Kramosil and Michalek [6] and modified
by George and Veeramani [3]. They obtained that every
Definition 2.2 [12] : The 3-tuple (X, M, *) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-
2
metric space induces a fuzzy meric spaces. Also Grebic [4]
has proved fixed point theorem for fuzzy metric space. In fuzzy metric space, the notion of compatible maps was
norm and M is a fuzzy set on X
lowing conditions :
× ( 0, ∞) Satisfying the fol-
initiated by Mishra, Sharma and Singh [8]. Recently Jungek and Rhoades [5] introduced and worked on the concept of
weak compatible maps.
(FM 1)
For all x, y, z ∈ X
M ( x, y,0) = 0
and s, t > 0
In this paper we prove a fixed point theorem for A, B, S, T,
(FM 2) for all f
> 0 f if and only if x = y
P and Q self maps using implicit relation. Concept of weak compatibility and compatibility of type (ß )of self maps are
(FM 3)
M ( x, y, t ) = M ( y, x, t )
characterized to get common fixed points.
(FM 4) M ( x, y, t ) * M ( y, z, s) ≤ M ( x + z, t + s)
(FM 5)
M ( x, y,.) : ( 0, ∞) → [ 0,1]
is left
continuous
Definition 2.1 [13] : A binary operation *, [0, 1] × [0, 1]
(FM 6)
Lim M ( x, y, t ) = 1
n →∞
[0, 1] is called at norm if ([ 0,1],* )
monoid with unit 1 such that
a * b ≤ c * d
is an abelian topological
Note that every metric spaces induces a Fuzzy metric space.
Definition2.3 [11] : Let ( X , d ) be a Metric Space.
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Define a * b = min{
a, b} and
and x, y ∈ X
then x = y
M ( x, y, t ) =
and t > 0 .
t
t + d( x, j )
for all x, y ∈ X
Proof : Let there exists, k ∈ ( 0,1) such that
M ( x, y, kt ) ≥ M ( x, y, t )
Then ( X , M ,* ) is a fuzzy metric space. It is called
for all
x, y ∈ X
t > 0
the fuzzy metric space induced by d.
Definition 2.4 [3] : A sequence { x n }
in a fuzzy metric
then
M ( x, y, kt ) ≥ M ( x, y, t / k ) and so
M ( x, y, kt ) ≥ M ( x, y, t / k n ) for
space { X , M ,* ) is defined as Cauchy sequence if for each
positive integer n, taking limit n → ∞
∈> 0 and t > 0 , there exists x ∈ N
such that
M ( x, y, k ) ≥ 1 and hence x = y
Lemma 2.3 [12] : The only t-norm and satisfying
M ( x n , x m , t ) > 1− ∈ for all n, m ≥ x 0
r * r ≥ r
for all r ∈ [ 0,1] is the minimum t-norm,
The sequence { x n }
x in X iff
is said to be converge to a point
that is
a * b = min{
a, b}
for all a, b ∈ [ 0,1]
M ( x n , x, t ) > 1− ∈ for all n, m ≥ x 0
A fuzzy metric space ( X , M ,* ) is said to be complete if every Cauchy sequence in it converges to a point in it.
Definition 2.5 [1] : A pair of self mappings (A, S) of
fuzzy metric ( X , M ,* ) is said to be compatible if
Bijendra singh et al.[11] have proved the following result:
Let ( X , M ,* ) be a complete fuzzy metric space and
lim M ( ASx
n →∞
n , SAx n , t ) → 1∀t > 0 whenever { x n } is
let A, B, S, T, P and Q be mapping from X, into itself
a sequence in X such that
such that the following conditions are satisfied :
lim Sx n n →∞
= lim Ax n n →∞
= x for some x ∈ X
(a)
P( X ) ⊂ ST ( X ), Q( X ) ⊂ AB )( X )
If the self mapping A and B of a fuzzy metric space
( X , M ,* ) are compatible then they are weakly compatible but its converse is not true.
Lemma 2.1 [9] : Let ( X , M ,* ) be a fuzzy metric space,
(b) either AB or P is continuous
( c) (P, AB) is compatible and (Q, ST) is weakly compatible.
(d) AB = BA, ST = TS, PB = BP, QT = TQ
(e) There exist k ∈ ( 0,1) such that for every
then for all x, y ∈ X , M ( x, y,.)
is non-decreasing.
x, y ∈ X , t > 0
Lemma 2.2 [10] : Let ( X , M ,* ) be a fuzzy metric
space. If there exists k ∈ ( 0,1) such that
M ( x, y, kt ) ≥ M ( x, y, t )∀t > 0
M ( Px,Qy, qt) ≥ M ( ABx, STy, t) * M ( Px, ABx, t)
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* M ( Qy, STy , t ) * M ( Px , STy , t )
φ( M ( Px, Qy, kt ), M ( ABx, STy, t ), M ( Px, ABx, t ),
then A, B, S, T, P and Q have a unique common fixed
M (Qy, STy, kt ), M ( Px, STy, t )
) ≥ 0
point in X.
Now we are introducing an implicit-relation and we will prove the above result by using implicit relation in place of
Then A, B, S, T, P and Q have a unique common fixed
point in X.
Proof – Let x 0 ∈ X then there exist such that
contractive condition.
A Class of Implicit Relation:
Px 0
= STx 1 = y 0 and
Let φ be the set of all real continuous functions
Qx 1
= ABx 2
= y1
φ( t , t , t , t , t
) : ( R+) 5
→ R which is
Inductively, we can construct sequences { x n }
and
1 2 3 4 5
{ y n }
in X, such that
non-decreasing in the Ist argument satisfying the following conditions: (φ1) → for u, v ≥ 0
Px 2 n
= STx 2 n +1 = y2 n
and
φ( u, v, v, u,1) ≥ 0 implies u ≥ v
Qx 2 n +1
= ABx
2 n +2
= y2 n +1
(φ2) → φ( v, u,1,1, u ) ≥ 0 or
φ( u,1, u,1, u ) ≥ 0 or
for n = 0, 1, 2,……….
Step 1 : We put in contractive condition x = x 2 n , y = x 2 n +1
then we get
φ( u,1,1, u,1) ≥ 0 implies u ≥ 1
φ( M ( Px
2 n , Qx
2 n +1
, kt ), M ( ABx
2 n , STx
2 n +1
, t ),
Ex.
φ( t 1 , t 2 , t 3 , t 4 , t 5 ) = 13t 1 − 11t 2 − 3t 3 − 5t 4 + t 5 − 1
M ( Px2 n
M ( Px
, ABx2 n
, STx
, t), M (Qx
, t ) ≥ 0
2 n +1
, STx
2 n +1
, kt ),
Theorem 3.1 : Let ( X , M ,* ) be a complete fuzzy metric space and let A, B, S, T, P and Q be mapping from X into itself such
that the following conditions are satisfied :
2 n 2 n +1
φ( M ( y2 n , y2 n +1 , kt ), M ( y2 n −1 , y2 n , t ), M ( y2 n , y2 n −1 , t )
(a)
P( X ) ⊂ ST ( X ), Q( X ) ⊂ AB( X )
M ( y2 n +1 , y2 n , kt ), M ( y2 n , y2 n , t )
) ≥ 0
(b) Either A B or P is continuous.
(c) (P, AB) is compatible of type (ß) and (Q, ST) Is weakly compatible
(d) AB = BA, ST = TS, PB = BP, QT = TR
(e) there exist k ∈ ( 0,1) such that for every
x, y ∈ X and t > 0
⇒ From property of implicit relation
M ( y2 n , y2 n +1 , kt ) ≥ M ( y2 n −1 , y2 n , t ) Or M ( y2 n +1 , y2 n , kt ) ≥ M ( y2 n , y2 n −1 , t ) Similarly we have
M ( y2 n +2 , y2 n +1 , kt ) ≥ M ( y2 n +1 , y2 n , t )
Therefore, for all n, we have
M ( y n +1 , y n , kt ) ≥ M ( y n , y n −1 , t )
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To prove that { y n }
is a Cauchy sequence,
φ( M ( PABx2 n , ABABx2 n , t)
M ( PABx2 n , ABABx2 n , t)
M ( PABx2 n , ABABx2 n , t ), M (Qx2 n , STx2 n+1 , kt ),
M ( y
n+1
, yn
, t) ≥ M ( yn
, yn−1
, t / k ) ≥ M ( y
n−1
, yn−2
, t / k 2 )
M (( PABx2 n , STx2 n+1 , t )
) ≥ 0
≥ ………
≥ M ( y, y o
as n → ∞
, t / k n ) → 1
Letting n → ∞ , we get
φ( M ( ABz, z, kt ), M ( ABz, z, t ), M ( ABz, ABz, t ),
M ( z, z, t ), M ( ABz, z, t ) ≥ 0
Thus, the result hold for m = 1
By induction hypothesis suppose that result hold for m = p,
M ( yn , yn − p +1 , t ) ≥ M ( yn , yn − p , t / 2) * M ( yn +1 , yn − p +1 , t / 2)
φ( M ( ABz , z, kt ), M ( ABz , z, t ),1,1, M ( ABz , z, t )) ≥ 0
Since φ is non-decreasing in first arguments so,
φ( M ( ABz , z, t ), M ( ABz , z, t ),1,1, M ( ABz , z, t ) ≥ 0
m = p + 1 .
→ 1*1 = 1
Thus the result holds for
By the property of implicit relation
M ( ABz , z, t ) ≥ 1
Hence { y n }
is a Cauchy sequence in X. Which is
ABz = z
complete therefore { y n }
convergence to z i.e. y n
→ z ∈ X .
Step 3 : Putting x = z, y = x 2 n +1 , in contractive con-
Also its subsequences converges to the same point i.e. z ∈ X .
{ Qx 2 n +1 } → z and { STx 2 n +1 } → z
dition, we get
φ( M ( Pz , Qx 2 n +1 , kt ), M ( ABz , STx 2 n +1 , t ), M ( Pz , ABz , t ),
{ Px 2 n } → z and { ABx 2 n } → z
Case 1 : Suppose AB is continuous.
M ( Qx
2 n +1
, STx
2 n +1
, kt ), M ( Pz , STx
2 n +1
, t )) ≥ 0
As AB is continuous,
2
Letting Limit n → ∞ , we get
φ( M ( Pz, z, kt ), M ( z, z, t ), M ( Pz, z, t ),
( AB )
x 2 n → ABz
and
M ( z, z, t ), M ( Pz, z, t ) ≥ 0
( AB )Px 2 n → ABz
As (P, AB) is compatible pair, we have
P( AB )x 2 n → ABz
Step 2 : Putting x = ABx 2 n , y = x 2 n +1 in contrac- tive condition, we have
Since φ is non-decreasing in first argument, so
φ( M ( Pz , z, t ),1, M ( Pz , z, t ),1, M ( Pz , z, t )) ≥ 0
By the property of implicit relation
M ( Pz , z, t ) ≥ 1
Pz = z
Therefore ABz
= Pz = z
Step 4 : Put x = Bz
and y = x 2 n +1 in contractive
condition we get
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φ( M ( z, Qv , t ) ≥ 0
φ( M ( PBz, Qx2 n +1 , kt ), M ( ABBz, STx2 n +1 , t ),
⇒ Qv = z
M ( PBz, ABBz, t ), M (Qx2 n +1 , STx2 n +1 , kt ),
STv
= z = Qv
M ( PBz, SAx2 n +1 , t )
) ≥ 0
As ( Q, ST ) is weakly compatible, we have
[ As BP = PB, AB = BA so we have
STQv = QSTv
Thus STz = Qz .
P (Bz) = B (Pz) = Bz and
AB (Bz) = BA (Bz) = B (ABz) = Bz] Letting limit n → ∞ ,
φ( M ( Bz, z, kt ), M ( Bz, z, t ), M ( Bz, Bz, t ), M ( z, z, kt ),
Step 6 : Put x = x 2 n , y = z in contractive condition, we get
φ( M ( Px2 n , Qz, kt ), M ( ABx2 n , STz, t), M ( Px2 n , ABx2 n , t ),
M ( Bz, z, t )) ≥ 0
Since φ is non decreasing in first argument, so we can
M (Qz, STz, kt ), M ( Px2 n , STz, t )
) ≥ 0
write φ( M ( Bz , z, t ), M ( Bz , z, t ),1,1, M ( Bz , z, t )) ≥ 1
By property of implicit relation
Letting n → ∞ and using equation
{ Qx 2 n +1 } → z and { STx 2 n +1 } → z
φ( M ( z, Qz, kt ), M ( z, z, t ), M ( z, z, t ),
⇒ M ( Bz , z, t ) ≥ 1
⇒ Bz = z
Step5
M (Qz, z, kt ), M ( z, z, t ) ≥ 0
Now
Bz = z, ABz
= z ⇒ Az = z
φ( M ( z, Qz, kt ),1,1, M ( z, Qz, kt ),1) ≥ 0
M ( z, Qz, t ) ≥ 1
therefore Az
= Bz
= Pz = z
(Using property of implicit relation)
Step 5 : As P( X ) ⊂ ST ( X ) , therefore exist v ∈ X
Qz = z
. Such that z = Pz
= STv
. Putting x = x 2 n , y = v ,
Step 7 : Put x = x 2 n , y = Tz
in contractive
we get from contractive condition, we get,
φ( M ( Px2 n , Qv, kt ), M ( ABx2 n , STv, t ), M ( Px2 n , ABx2 n , t )
φ( M ( Px
Condition we get
, QTz, kt ), M ( ABx
, STTz, t), M ( Px
, ABx
, t ),
M (Qv, STv, kt ), M ( Px2 n , STv, t )
) ≥ 0
2 n
M (QTz, STTz, kt ), M ( Px
2 n
, STTz, t ) ≥ 0
2 n 2 n
Letting n → ∞ and using { Px 2 n } → z
and { ABx 2 n } → z we get
φ( M ( z, Qv, kt ), M ( z, z, t), M ( z, z, t ),
M (Qv, z, kt ), M ( z, z, t )) ≥ 0
φ( M ( z, Qv , t ),1,1, M ( z, Qv , kt ) ≥ 0
2 n
As QT = TQ and ST = TS, we have
QTz = TQz = Tz and ST (Tz) = T (STz) = Tz
Letting n → ∞ ,w e get
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φ( M ( z, Tz, kt ), M ( z, Tz, t ), M ( z, z, t ),
M (Tz, Tz, kt ), M ( z, Tz, t )) ≥ 0
Letting n → ∞ , we
φ( M ( Pz, z, kt ), M ( Pz, z, t ), M ( Pz, Pz, t ),
M ( z, z, kt ), M ( pz, z, t )) ≥ 0
φ( M ( z, Tz , kt ), M ( z, Tz , t ),1,1, M ( z, Tz , t )) ≥ 0
Since φ is non-decreasing in first argument so
φ( M ( z, Tz , t ), M ( z, Tz , t ),1,1, M ( z, Tz , t )) ≥ 0
φ( M ( Pz , z, kt ),1, M ( Pz , z, t ),1,1, M ( Pz , z, t ) ≥ 0
Since φ is non decreasing in the first argument so we can write
⇒ M ( z, Tz , t ) ≥ 1
(Using property of implicit relation)
⇒ Tz=z
φ( M ( Pz , z, t ), M ( Pz , z, t ),1,1, M ( Pz , z, t ) ≥ 0
M ( Pz , z, t ) ≥ 1 (By property of implicit relation)
Now
STz
= Tz
= z ⇒ Sz
= z hence
⇒ Pz = z
Sz = Tz
= Qz = z
Further using Step 5, 6, 7 we get
Combining result, we get
Qz = STz = Sz = Tz = z
Az = Bz
= Pz
= Qz = Tz
= Sz = z
Step 9 : As Q( X ) ⊂ AB( X ) , there exists w ∈ X
Case II : Suppose P is continuous
2
such that z = Qz = ABw
put x = w, y = x 2 n +1 in con-
As P is continuous, P
x 2 n → Pz
tractive condition we get
and P( AB )x 2 n → Pz
φ( M ( Pw, Qx
2 n +1
, kt ), M ( ABw, STx
2 n +1
, t), M ( Pw, ABw, t),
As (P, AB) is compatible of type (ß) M(PPx n ,(AB)(AB) x n ,t) → 1
2
M (Qx2 n +1 , STx2 n +1 , kt ), M ( Pw, STx2 n +1 , t ) ≥ 0
Letting n → ∞ we get
φ( M ( Pw, z, kt ), M ( z, z, t ), M ( Pw, z, t ),
M(Pz ,(AB)
x n , t ) → 1 when t > 0
2
M ( z, z, kt ), M ( Pw, z, t )) ≥ 0
we have (AB)
x2 n → Pz
Step 8 : Put x = Px 2 n , y = x 2 n +1 .
φ( M ( Pw , z, kt ),1, M ( Pw , z, t ),1, M ( Pw , z, t )) ≥ 0
φ( M ( PPx 2 n , Qx 2 n +1 , kt ), M ( ABPx 2 n , STx 2 n +1 , t ),
M ( PPx2 n , ABPx2 n , t), M (Qx2 n +1 , STx2 n +1 , kt ),
M ( PPx2 n , STx2 n +1 , t )) ≥ 0
Since φ is non decreasing in the first argument
φ( M ( Pw , z, t ),1, M ( Pw , z, t ),1, M ( Pw , z, t ) ≥ 0
M ( Pw , z, t ) ≥ 1
(Using property of implicit relation)
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⇒ Pw = z = ABw
(a)
P( X ) ⊂ S ( X ),Q( X ) ⊂ A( X )
As (P, AB) is compatible, we have
Pz = ABz
Also from Step 4, we get Bz = z
Thus Az = Bz = Pz = z and we due that z is the common fixed point of the six maps in the case also.
Uniqueness – Let u be another common fixed point of
A, B, S, T, P and Q
Then Au = Bu = Pu = Qu = Su = Tu = u
(b) Either A or P is continuous.
(c) (P, A) is compatible of type ß and (Q, S) is weakly compatible
(d) there exist k ∈ ( 0,1) such that for every
x, y ∈ X and t > 0
φ( M ( Px, Qy, kt ), M ( Ax, Sy, t ), M ( Px, Ax, t ),
Put x = z, y = u in contractive condition, we get
M (Qy, Sy, kt ), M (Qy, Sy, kt )
) ≥ 0
φ( M ( Pz , Qu , kt ), M ( ABz , STu , t ), M ( Pz , ABz , t ),
unique common fixed point in X.
Then A, S, P and Q have a
M ( Qu , STu , kt ), M ( Pz , STu , t ) ≥ 0
Taking n → ∞
φ( M ( z, u, kt ), M ( z, u, t ), M ( z, z, t ),
M (u, u, kt ), M ( z, u, t ) ≥ 0
φ( M ( z, u, kt ), M ( z, u, t ),1,1, M ( z, u, t ) ≥ 0
Since φ is non decreasing in first argument so
φ( M ( z, u, t ), M ( z, u, t ),1,1, M ( z, u, t ) ≥ 0
⇒ M ( z, u, t ) ≥ 1
(Using property of implicit relation)
⇒ z = u
Therefore z is the unique common fixed point of day maps A, B, S, T, P and Q.
Remark 3.1: If we take B = T = 1, the identify maps on x in the theorem 3.1 ,the condition (b) is satisfied trivially.
Corollary 3.1 : Let ( X , M ,* ) be a complete fuzzy metric space and let A, S, P and Q be mapping from X into itself such that
the following conditions are satisfied :
Remark 3.2 Corollary 3.1 is generalization of the result of cho [ ] in the sense that condion of compatibility of the pairs of self maps ,has been restricted to compatibility of type (ß) and wek compatibility and only one map of the pair is needed to contin- uous.
We establish A Common Fixed Point Theorem in Fuzzy Metric Spaces satisfying implicit relations for weakly compatible maps. There are some possible application in engineering, economics in dealing with problems arising in approximation theory, information system. In future scope we can obtain new implicit relation to relax conditions.
Acknowledgment
Our thanks to the Mr. Rakesh Rathore CEO Vikrant insti- tute of technology and management,Gwalior, who have contributed towards making of the paper.
References
[1] Bhardwaj R.K. et al, Some results on Fuzzy metric spaces, proceeding world congress on engineering 2011 Vol. 1, WCE 2011 London U.K.
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International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 690
ISSN 2229-5518
[2] Cho,Y.G ,Fixed point in fuzzy metric spaces, Journal of fuzzy Mathematics, 5 (1997), 949-962.
[3] George A and Veeramani P., on some results in fuzzy metric spaces, fuzzy sets and system, 64, 395-399 (1994)
[4] Grabice M., Fixed points in fuzzy metric spaces, fuzzy sets and systems 27 (1988), 385 389.
[5] Junck, G. and Rhoades, B.E., fixed points for set valanced functions without continuity Indian J. Pune Appli math 29,
227-238 (1998).
[6] Kramosil I and Michalek, J. Fuzzy metric and Statistical metric spaces, Kybernetica 11, 336-344 (1975)
[7] Khan, M.A.,Sub-compatible and sub-sequentialy continuous maps in fuzzy metric spaces. Appl. Math. Sci. Vol. 5. 2011, no.
29, 1421-1430.
[8] Mishra, S.N., Mishra N. and Singh S.L. Common fixed point of maps in fuzzy metric space, Int. J. Math. Math. Sci.
17, 253-258 (1994).
[9] Singh B. and Chauhan M.S., Common fixed points of com- patible maps in fuzzy metric space, fuzzy set and system 15 (2000), 471-475.
[10] Singh, B. and Jain, S,Semi-compatibility and fixed point theo- rems in fuzzy metric space using implicit Relations. Interna- tional Journal of mathematics and mathematical sciences, 16:
2005, 2617-2629.
[11] Singh B. ,R.K.Sharma,Arhind jain and mohit sharma, On
Fuzzy metric space, ultra scientist Vol. 21,393-398 (2009).
[12] Vasuki, R. Common fixed point theorem in a fuzzy metric space, fuzzy sets and system 97, 395-397 (1998)
[13] Zadeh L.A. (1965) : Fuzzy Sets, Inform. And Control, 89,
338
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