A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible and Semi-Compatible Mappings

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 9 || November. 2016 || PP-27-31
www.ijmsi.org 27 | Page
A Common Fixed Point Theorem on Fuzzy Metric Space Using
Weakly Compatible and Semi-Compatible Mappings
V.Srinivas1
*, B.Vijayabasker Reddy2
1
Department of Mathematics, University College of Science, Saifabad, Osmania University, Hyderabad,
Telangana, India.
2
Department of Mathematics, Sreenidhi Institute of Science and Technology, Ghatkesar-501 301, Telangana,
India.
ABSTRACT: The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six
self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible
mappings in complete fuzzy metric space.
KEYWORDS: Fixed point, self maps, complete fuzzy metric space, semi-compatible mappings, weakly
compatible mappings.
AMS (2000) Mathematics Classification: 54H25, 47H10
I. INTRODUCTION AND PRELIMINARIES
Introduction: The concept of Fuzzy sets was introduced by Zadeh[14]. Following the concept of fuzzy sets,
fuzzy metric space initiated by Kramosil and Michalek.George and veeramani[5] modified the notion of fuzzy
metric spaces with the help of continuous-t norm. Recently, many others proved fixed points theorems involving
weaker forms of the compatible mappings in fuzzy metric space.Jungck and Rhoads[3] defined the concept of
weakly compatible mappings.Also B.Singh and S.Jain[2] introduced the notion of semi-compatible mappings in
fuzzy metric space.
Definition 1.1[1]: A binary operation * :[0,1][0,1] [0,1] is called continuous t-norm if * satisfies the
following conditions:
(i) * is commutative and associative
(ii) * is continuous
(iii) a*1=a for all a∈[0,1]
(iv) a*b ≤ c*d whenever a ≤c and b≤ d for all a, b,c,d [0,1]
Definition 1.2[1]: A 3-tuple (X, M,*) is said to be fuzzy metric space if X is an arbitrary set,* is continuous t-
norm and M is a fuzzy set on X2
(0,∞) satisfying the following conditions for all x,y,zX, s,t0
(FM-1) M(x,y,0)=0
(FM-2) M(x,y,t)=1 for all t>0 if and only if x=y
(FM-3) M(x,y,t)= M(y,x,t)
(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
(FM-6) lim ( , , ) 1
 

t
M x y t
Example 1.3 (Induced fuzzy metric space)[1]: Let (X, d) be a metric space defined a*b=min{a,b} for all x,y∈X
and t>0,
( , , ) ( )
( , )
t
M x y t a
t d x y
    

Then (X, M,*) is a fuzzy metric space. We call this fuzzy metric M induced by metric d is the standard fuzzy
metric. From the above example every metric induces a fuzzy metric but there exist no metric on X satisfying
(a).
Definition 1.4 [1]: Let (X, M,*) be a fuzzy metric space then a sequence <xn> in X is said to be convergent to
int lim ( , , ) 1 0.
 
  n
n
a po x X if M x x t for all t
Definition1.5 [1]: A sequence <xn> in X is called a Cauchy sequence if
lim ( , , ) 1 0 0.n p n
n
M x x t for all t and p
 
  

A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible and …
Definition 1.6 [1]: A fuzzy metric space (X, M,*) is said to be complete if every Cauchy sequence is
convergent to a point in X.
Lemma 1.7 [6] : For all x,yX, M(x,y,.) is non decreasing.
Lemma 1.8 [11] : Let (X,M,*)be a fuzzy metric space if there exists k(0,1) such that M(x,y,kt)≥M(x,y,t) then
x=y.
Proposition 1.9 [11] : In the fuzzy metric space (X, M,*) if a*a≥a for all a[0,1] then a*b= min{a,b}
Definition 1.10 [12]: Two self maps S and T of a fuzzy metric space (X,M,*) are said to be compatible
mappings if
n
lim M(STxn,TSxn,t )=1, whenever <xn> is a sequence in X such that
n
lim Sxn=
n
lim Txn= z for
some z∈X.
Definition 1.11[2]: Two self maps S and T of a fuzzy metric space (X,M,*) are said to be semi-compatible
mappings if
n
lim M(STxn,Tz,t )=1, whenever <xn> is a sequence in X such that
n
lim Sxn=
n
lim Txn= z for
some z∈X.
Definition 1.12 [3]: Two self maps S and T of a fuzzy metric space (X,M,*) are said to be weakly compatible if
they commute at their coincidence point. i.e if Su=Tu for some u∈X then STu=TSu.
II. MAIN RESULT
2.1 Theorem: Let A, B, P,Q ,S and T be self maps of a complete fuzzy metric space (X, M , *) satisfying the
conditions,
2.1.1 AP(X)  T(X) and BQ(X)  S(X).
2.1.2 The pairs (AP, S) is semi-compatible and (BQ,T) are weakly compatible.
2.1.3 M APx, BQy, kt 2
∗ M APx, BQy, kt M Ty, Sx, kt
≥
k1 M BQy, Sx, 2kt ∗ M APx, Ty, 2kt
+k2 M APx, Sx, kt ∗ M(BQy, Ty, kt
M(Ty, Sx, t)
for all x, y in X where k1, k2 ≥ 0, k1 + k2 ≥ 1
2.1.4 AP is continuous mapping.
then A,P, B,Q, S and T have a unique common fixed point in X.
Now we prove a Lemma
2.1.5 Lemma: Let A,P,B,Q, S and T be self mappings from a complete fuzzy metric space (X,M,*) into itself
satisfying the conditions 2.1.1 and 2.1.3 then the sequence yn defined by y2n=APx2n=Tx2n+1 and
y2n+1=BQx2n+1=Sx2n+2 for n0 relative to four self maps is a Cauchy sequence in X.
Proof: From the conditions 2.1.1 and 2.1.3 and from the definition of iterative sequence we have let x0 be
arbitrary point of X,AP(X)T(X) and BQ(X)S(X) there exists x1,x2X such that APx0=Tx1 and BQx1=Sx2.
Inductively construct a sequence xn and yn in X such that y2n=APx2n=Tx2n+1 and y2n+1=BQx2n+1=Sx2n+2 for
n0.
By taking x=x2n, y=x2n+1 in the inequality 2.1.3 then we get,
M APx2n, BQx2n+1, kt 2
∗ [M APx2n,BQx2n+1,kt M Tx2n+1, Sx2n, kt ]

k1 M BQx2n+1, Sx2n, 2kt ∗ M(APx2n, Tx2n+1, 2kt +
k2 M APx2n, Sx2n, kt ∗ M BQx2n+1, Tx2n+1, kt
M(Tx2n+1, Sx2n, t)
M y2n, y2n+1, kt 2
∗ [M y2n,y2n+1,kt M y2n, y2n−1, kt ]

k1 M y2n+1, y2n−1, 2kt ∗ M(y2n, y2n, 2kt +
k2 M y2n, y2n−1, kt ∗ M y2n+1, y2n, kt
M(y2n, y2n−1, t)
this implies
M(y2n, y2n+1,, kt) M y2n, y2n+1, kt ∗ M(y2n, y2n−1, kt)
≥
k1 M(y2n+1, y2n−1, 2kt) +
k2 M y2n, y2n−1, kt ∗ M y2n+1, y2n, kt
M(y2n, y2n−1, t) and
M y2n, y2n+1, kt M(y2n+1, y2n−1, 2kt
≥ k1 M(y2n+1, y2n−1, 2kt + k2 M(y2n−1, y2n+1, 2kt) M y2n, y2n−1, t
M y2n, y2n+1, kt M y2n+1, y2n−1, 2kt
≥ k1 + k2 M(y2n+1, y2n−1, 2kt M(y2n, y2n−1, t
This gives
M(y2n, y2n+1, kt) ≥ k1 + k2 M(y2n−1, y2n, t)
Since k1 + k2 ≥ 1
M(y2n, y2n+1, kt) ≥ M(y2n−1, y2n, t)
this implies M(yn, yn+1, kt) ≥ M(yn−1, yn, t)

Continuing in this process we get
M yn, yn+1,t ≥ M yn−1, yn,
t
k
≥ M yn−1, yn,
t
k2 ≥ M yn−1, yn,
t
k3 …
… . ≥ M yn−1, yn,
t
kn
And this implies M yn, yn+1, t → 1 as n → ∞.
Now for each >0 and t>0 we can choose n0N such that
M yn, yn+1, t > 1 − ε for m, n N. Suppose m  n
M(yn, ym , t) ≥
M yn, yn+1,
t
m−n
∗ M yn+1, yn+2,
t
m−n
∗ …
∗ M ym−1, ym ,
t
m−n
(1-)*(1-)*…..(1-)
(1-)
this shows that the sequence 𝑦𝑛 is a Cauchy sequence in complete fuzzy metric space X and hence it
converges to a limit, say zX
Proof of theorem 2.1:
From the Lemma APx2n → z , Tx2n+1 → z , BQx2n+1 → z, Sx2n+2 → z as n∞
Since AP is continuous AP(Sx2n+2) → 𝑧 and AP(APSx2n) → APz as n→∞
Also since the pair (AP,S) is semi-compatible implies AP(Sx2n+2) → Sz as n→∞ therefore APz=Sz.
Put x= z, y=x2n+1 in the inequality 2.1.3 then, we get
M APz, BQx2n+1, kt 2
∗ M APz, BQx2n+1, kt M Tx2n+1 , Sz, kt
≥
k1 M BQx2n+1, Sz, 2kt ∗ M APz, Tx2n+1, 2kt
+k2 M APz, Sz, kt ∗ M(BQx2n+1, Tx2n+1, kt)
M(Tx2n+1, Sz, t)
M APz, z, kt 2
∗ M APz, z, kt M z , APz, kt
≥
k1 M z, APz, 2kt ∗ M APz, z, 2kt
+k2 M APz, APz, kt ∗ M(z, z, kt)
M(z, APz, t)
M APz, z, kt 2
≥
k1 M z, APz, 2kt
+k2 1
M(z, APz, t)
M APz, z, kt ≥ k1M APz, z, kt +k2
1 − k1 M APz, z, kt ≥ k2
M APz, z, kt ≥
k2
1 − k1
≥ 1
Implies APz=Sz=z
Also from the condition AP(X)T(X) , there exists wX such that APz=Tw=z.
Now to prove BQw=z
Put x= z , y=w in the inequality 2.1.3 then, we get
M APz, BQw, kt 2
∗ M APz, BQw, kt M Tw , Sz, kt
≥
k1 M BQw, Sz, 2kt ∗ M APz, Tw, 2kt
+k2 M APz, Sz, kt ∗ M(BQw, Tw, kt)
M(Tw, Sz, t)
M z, BQw, kt 2
∗ M z, BQw, kt M z , z, kt
≥
k1 M BQw, z, 2kt ∗ M z, z, 2kt
+k2 M z, z, kt ∗ M(BQw, z, kt)
M(z, z, t)
this implies
M z, BQw, kt 2
≥
k1 M BQw, z, 2kt
+k2 M(BQw, z, kt)
M z, BQw, kt 2
≥
k1 M BQw, z, kt
+k2 M(BQw, z, kt)
M z, BQw, kt ≥ k1 + k1 ≥ 1 implies
BQw=z
Hence BQw=Tw=z.
Now the pair (BQ,T) is weakly compatible implies BQ(Tw)=T(BQ)w and this gives BQz=Tz.
Put x=z,y=z in the inequality 2.1.3 then, we get
M APz, BQz, kt 2
∗ M APz, BQz, kt M Tz , Sz, kt
≥
k1 M BQz, Sz, 2kt ∗ M APz, Tz, 2kt
+k2 M APz, Sz, kt ∗ M(BQz, Tz, kt)
M(Tz, Sz, t)
M z, BQz, kt 2
∗ M z, BQw, kt M BQz , z, kt

≥
k1 M BQz, z, 2kt ∗ M z, BQ, 2kt
+k2 M z, z, kt ∗ M(BQz, BQz, kt)
M(BQz, z, t)
this implies
M z, BQz, kt 2
≥
k1 M BQz, z, kt
+k2 1
M(BQz, z, t)
M z, BQz, kt (1 − k1) ≥ k2
M z, BQz, kt ≥
k2
1 − k1
≥ 1
Implies BQz=z.
Put x=Pz and y=z in the inequality 2.1.3 then, we get
M AP(Pz), BQz, kt 2
∗ M AP(Pz), BQz, kt M Tz , S(Pz), kt
≥
k1 M BQz, S Pz , 2kt ∗ M AP Pz , Tz, 2kt
+k2 M AP Pz , S Pz , kt ∗ M BQz, Tz, kt
M(Tz, S(Pz), t)
M Pz, z, kt 2
∗ M Pz, z, kt M z , Pz, kt
≥
k1 M z, Pz, 2kt ∗ M Pz, z, 2kt
+k2 M Pz, Pz, kt ∗ M(z, z, kt)
M(z, Pz, t)
this implies
M Pz, z, kt 2
≥
k1 M Pz, z, kt
+k2 1
M(z, Pz, t)
M Pz, z, kt (1 − k1) ≥ k2
M Pz, z, kt ≥
k2
1−k1
≥ 1 implies Pz=z then
APz=z gives A[P(z)]=z consequently Az=z.
Put x=z and y=Qz in the inequality 2.1.3 then, we get
Put x=z,y=Qz in the inequality 2.1.3 then, we get
M APz, BQ(Qz), kt 2
∗ M APz, BQ(Qz), kt M T(Qz) , Sz, kt
≥
k1 M BQ Qz , Sz, 2kt ∗ M APz, T Qz , 2kt
+k2 M APz, Sz, kt ∗ M BQ Qz , T Qz , kt
M(T(Qz), Sz, t)
M z, Qz, kt 2
∗ M z, Qz, kt M Qz , z, kt
≥
k1 M Qz, z, 2kt ∗ M z, Qz, 2kt
+k2 M z, z, kt ∗ M(Qz, Qz, kt)
M(Qz, z, t)
this implies
M z, Qz, kt 2
≥
k1 M Qz, z, kt
+k2 1
M(Qz, z, t)
M Qz, z, kt (1 − k1) ≥ k2
M Qz, z, kt ≥
k2
1−k1
≥ 1 this implies Qz=z then
BQz=z gives B[Q(z)]=z consequently Bz=z.
Therefore Az=Bz=Qz=Pz=Tz=Sz=z.
Uniqueness:Let z∗
≠ z be the another fixed point of the mappings A,B,P,Q,S and T then
Az*=Bz*=Pz*=Qz*=Sz*=Tz*=z*.
Put x=z and y=z* in the inequality 2.1.3 then, we get
Put x=z,y=z*
M APz, BQz∗
, kt 2
∗ M APz, BQz∗
, kt M Tz∗
, Sz, kt
≥
k1 M BQz∗
, Sz, 2kt ∗ M APz, Tz∗
, 2kt
+k2 M APz, Sz, kt ∗ M(BQz∗
, Tz∗
, kt)
M(Tz∗
, Sz, t)
M z, z∗
, kt 2
∗ M z, z∗
, kt M z∗
, z, kt
≥
k1 M z∗
, z, 2kt ∗ M z, z∗
, 2kt
+k2 M z, z, kt ∗ M(z∗
, z∗
, kt)
M(z∗
, z, t)
this implies
M z, z∗
, kt 2
≥
k1 M z∗
, z, kt
+k2 1
M(z∗
, z, t)
M z, z∗
, kt ≥
k2
1 − k1
≥ 1
Implies z*=z.
Hence the self mappings A,B,S,T,P, and Q has a unique common fixed point.

REFERENCES
[1] A George , P Veeramani , On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64, 1994, 395-399.
[2] B Singh , S Jain ,Semi –compatible and fixed point theorems in fuzzy metric space , Chungcheong Math. Soc. 18, 2005, 1-22.
[3] G Jungck , and B E Rhoads , Fixed point for set valued function without continuity ,Indian J.Pure and Appl.Math.29(3),1998,227-
238
[4] G Jungck , Compatible mappings and common fixed points , Internet.J.Math & Math. Sci. , 9(4), 1986, 771-779.
[5] Ivan Kramosil and Michalek, ”Fuzzy Metrics and Statistical Metric Spaces” Kybernetika, 11, 1975,336-334
[6] M Grabiec , Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27, 1988, 385-389.
[7] P.Balasubramaniam ,S. Murali Sankar S & R.P Pant , Common fixed points of four mappings in fuzzy metric space, J.Fuzzy Math.
10(2), 2002, 379-384
[8] R.P.Pant & K. Jha , A remark on, Common fixed points of four mappings in a fuzzy metric space, J.Fuzzy Math.12 (2),2004,433-
437.
[9] S Kutukcu , S Sharma & H A Tokgoz ,A fixed point theorem in fuzzy metric spaces, Int. J. Math. Analysis, 1(18), 2007 , 861-872.
[10] S N Mishra ,N Sharma &S L Singh , Common fixed points of mappings on fuzzy metric spaces, Internet .J.Math &Math
.Sci,17,1994, 253-258.
[11] S.Sharma , Common fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 125,2001, 1-8.
[12] Y J Cho, H K Pathak , S M Kang &J S Jung , Common fixed point of compatible maps of type β on fuzzy metric spaces, Fuzzy sets
and Systems, 93,1998, 99-111.
[13] V Srinivas,B.V.B.Reddy,R.Umamaheswarrao,A Common fixed point Theorem on Fuzzy metric Spaces,Kathmandu University
journal of Science , Engineering and Technology,8 (II),2012,77-82.
[14] L .A Zadeh L, Fuzzy Sets, Information and Control,8,1965,338-353

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