Fixed Point Theorem in Fuzzy Metric Space

Dr M Ramana Reddy. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 8, ( Part -2) August 2016, pp.46-50
www.ijera.com 46|P a g e
Fixed Point Theorem in Fuzzy Metric Space
Dr M Ramana Reddy
Assistant Professor of Mathematics, Dept. Of Mathematics, Sreenidhi Institute of Science and Technology,
Hyderabad-501301
ABSTRACT
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
Keywords: Fuzzy metric space , Common fixed point. Fixed point theorem
I. INTRODUCTION
Now a day’s some contractive condition is
a central area of research on Fixed point theorems
in fuzzy metric spaces satisfying. Zadeh[10] in
1965 was introduced fuzzy sets. After this
developed and a series of research were done by
several Mathematicians. Kramosil and Michlek [5]
Helpern [4] in 1981 introduced the concept of
fuzzy metric space in 1975 and fixed point
theorems for fuzzy metric space. Later in 1994,
A.George and P.Veeramani [3] modified the notion
of fuzzy metric space with the help of t-norm.
Fuzzy metric space, here we adopt the notion that,
the distance between objects is fuzzy, the objects
themselves may be fuzzy or not.
In this present papers Gahler [1],[2]
investigated the properties of 2-metric space, and
investigated contraction mappings in 2-metric
spaces. We know that 2-metric space is a real
valued function of a point triples on a set X, which
abstract properties were suggested by the area
function in the Eucledian space, The idea of fuzzy
2-metric space was used by Sushil Sharma [8] and
obtained some fruitful results. prove some
common fixed point theorem in fuzzy -metric space
by employing the notion of reciprocal continuity, of
which we can widen the scope of many interesting
fixed point theorems in fuzzy metric space.
II. PRELIMINARY NOTES
Definition 2.1. A tiangular norm ∗ (shortly t− norm) is a binary operation on the unit interval [0, 1] such that for
all a, b, c, d ∈ [0, 1] the following conditions are satisfied:
1. a ∗1 = a;
2. a ∗ b = b ∗ a;
3. a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d
4. a ∗ (b ∗ c) = (a ∗ b) ∗ c.
Example 2.2. Let (X, d) be a metric space. Define a ∗ b = ab (or a ∗ b = min{a, b}) and for all x, y ∈ X and t >
0,  
 yxdt
t
tyxM
,
,,

 . Then (X, M, ∗) is a fuzzy metric space and this metric d is the standard fuzzy
metric.
Definition 2.3. A sequence {xn} in a fuzzy metric space (X, M, ∗) is said
(i).To converge to x in X if and only if M(xn, x, t) = 1 for each t > 0.
(ii). Cauchy sequence if and only if M(xn+p, xn, t) = 1 for each p > 0, t > 0.
(iii).to be complete if and only if every Cauchy sequence in X is convergent in X.
Definition 2.4. A pair (f, g) or (A,S) of self maps of a fuzzy metric space (X, M, ∗) is said
(i). To be reciprocal continuous if fxfgx n
n


lim and gxgfx n
n


lim whenever there exist a sequence {xn}
such that xgxfx n
n
n
n


limlim for some x ∈ X.
RESEARCH ARTICLE OPEN ACCESS

(ii). semi-compatible if SxASx n
n


lim whenever there exists a sequence {xn} in X such that
xSxAx n
n
n
n


limlim for some x ∈ X.
Definition 2.5. Two self maps A and B of a fuzzy metric space (X, M, ∗) are said to be weak compatible if they
commute at their coincidence points, that is Ax = Bx implies ABx = BAx.
Definition 2.6. A pair (A, S) of self maps of a fuzzy metric space (X, M, ∗) is said to be Definition 2.7. A
binary operation ∗ : [0, 1] × [0, 1] × [0, 1] → [0, 1] is called a continuous t-norm if ([0, 1]), ∗) is an abelian
topological monoid with unit 1 such that a1∗b1∗c1≤a2∗b2∗c2 whenever a1 ≤ a2, b1 ≤ b2, c1 ≤ c2 for all a1, a2, b1, b2
and c1, c2 are in [0, 1].
Definition 2.8. A sequence {xn} in a fuzzy 2-metric space (X, M, ∗) is said
(i),To converge to x in X if and only if   1,,,lim 

taxxM n
n
for all a ∈ X and t > 0.
(ii).Cauchy sequence, if and only if   1,,,lim 

taxxM npn
n
for all a ∈ X and p > 0, t > 0.
(iii). To be complete if and only if every Cauchy sequence in X is convergent in X.
Theorem 3.1 – Let A, B, S, T, L and M be a complete ε-chainable fuzzy metric space (X, M *) with continuous
t-norm satisfying the conditions.
(1)    XSTXL  ,    XABXM 
(2) AB = BA, ST = TS, LB = BL, MT = TM;
;
(3) there exists k (0, 1) such that
           0,,,,,,,,,2,,,,min{ tMySTyMtLxABxMtSTyABxMtMyABxMktMyLxM
For every x, y X, α (0, 2) and t > 0. If L, AB is reciprocally continuous, semi-compatible maps. Then A, B,
S, T, L and M have a unique common fixed point in X.
Proof : Let x0 X then from (1) there exists x1, x2 X such that Lx0 = STx1 = y0 and Mx1 = ABx2 = y1. In
general we can find a sequence {xn} and {yn} in X such that Lx2n = STx2n+1 = y2n and Mx2n+1 y = x2n+1 for t > 0
and α = 1 – q with q (0, 1) in (4), we have
   ktMxLxMktyyM nnnn
,,,, 12222212 

     
   
0
,,,,,
,,,12,,
min
12122222
12221222






 


tMxSTxMtLxABxM
tSTxABxMtqMxABxM
nnnn
nnnn
     
   
0
,,,,,
,,,1,,
min
1222212
2121212






 


tyyMtyyM
tyyMtqyyM
nnnn
nnnn
    tyyMtyyM nnnn
,,,,,,1min 2212122 

   tyyMktyyM nnnn
,,min,, 2122212 

Again x = x2n+2 and y = x2n+3 with α = 1-q with q (0, 1) in (4), we have
   ktMxLxyMktyyM nnnn
,,,, 32223222 

    
   
0
,,,,,
,,,1,,
min
32322222
32223222






 


tMxSTxMtLxABxM
tSTxABxMtqMxABxM
nnnn
nnnn
     
   
0
,,,,,
,,,1,,
min
32222212
22123212






 


tyyMtyyM
tyyMtqyyM
nnnn
nnnn
     
   
0
,,,,,
,,,,,,,,,
min
32222212
221232222212










tyyMtyyM
tyyMqtyyMtyyM
nnnn
nnnnnn

As t-norm continuous, letting q → 1 we have,
      tyyMtyyMktyyM nnnnnn
,,,,,min,, 321222123222 

Hence,
   tyyMktyyM nnnn
,,,, 22123212 

Therefore for all n; we have
        1/,,...../,,/,,,, 1
2
111
 
n
nnnnnnnn
ktyyMktyyMktyyMtyyM as n → ∞.
For any t > 0. For each ε > 0 and each t > 0, we can choose n0 ε N such that M(yn, yn+1, t) > 1 – ε for all n > n0.
For m, n ε N, we suppose m ≥ n. Then we have that
     
           



11*.....*1*1*1/,,*
,*.....,,,,/,,,,
1
211
nmtyyM
nmtyyMnmtyyMtyyM
mm
nnnnmn
Hence {yn} is a Cauchy sequence in X; that is yn → z in X; so its subsequences Lx2n, STx2n+1, ABx2n, Mx2n+1
also converges to z. Since X is ε-chainable, there exists ε-chain from xn to xn+1, that is, there exists a finite
sequence xn = y1, y2, ......yt = xn+1 such that M(yi, yi-1, t)> 1-ε for all t > 0 and i = 1, 2, ...l. Thus we have M(xn,
xn+1, t) > M(y1, y2, t/l) * M(y2, y3, t/l)* ...*M(yi-1, yi, t/l) > (1 - ε) * (1 - ε) * (1 - ε) *... * (1 - ε) * ≥ (1 - ε), and so
{xn} is a Cauchy sequence in X and hence there exists z  X such that xn → z. Since the pair of (L, AB) is
reciprocal continuous; we have limn→∞L(AB)x2n → Lz and limn→∞AB(L)x2n → ABz and the semi compatibility
of (L, AB) which gives limn→∞AB(L)x2n → ABz, therefore Lz = ABz. We claim
Lz = ABz = z.
Step 1 : Putting x = z and y = x2n+1 with α = 1 in (4), we have
 
   
   









tMxSTxMtLzABzM
tSTxABzMtMxABzM
ktMxLzM
nn
nn
n
,,,,,
,,,,,,
min,,
1212
1212
12
Letting n → ∞; we have
          tzzMtLzLzMtzLzMtzLzMktzLzM ,,,,,,,,,,,min,, 
i.e.
z = Lz = ABz.
Step 2 : Putting x = Bz, y = x2n+1 with α = 1 in (4), we have
  
     
      









tMxSTxMtBzLBzABM
tSTxBzABMtMxBzABM
ktMxBzLM
nn
nn
n
,,,,,
,,,,,
min,,
1212
1212
12
Since LB = BL, AB = BA, so L(Bz) = B(Lz) = Bz and AB(Bz) = B(ABz) = Bz letting n → ∞; we have
          tzzMtzBzMtzBzMtzBzMktzBzM ,,,,,,,,,,,min,, 
i.e.
   tzBzMktzbzM ,,,, 
Therefore
   XSTXL  , there exists Xu  , such that z = Lz = Stu. Putting x = x2n, y = u with α = 1 in (4), we have
 
   
   
0
,,,,,
,,,,2,,
min,,,
22
22
2







tMuSTuMtLxABxM
tSTuABxMtMuABxM
ktMuLxM
nn
nn
n
           0,,,,,,,,,,,min,,, tMuzMtzzMtzzMtMuzMktMuzM
i.e.
   tMuzMktMuzM ,,,, 
Therefore
Z = Mu = STu.
Since M is ST-absorbing; then
    1/,,,,  RtMuSTuMktSTMuSTuM
i.e. STu = STMu => z = STz.
Step 4 : Putting x = x2n, y = z with α = 1 in (4), we have

 
   
   
0
,,,,,
,,,,,,
min,,,
22
22
2







tMzSTzMtLxABxM
tSTzABxMtMzABxM
ktMzLxM
nn
nn
n
           0,,,,,,,,,,,min,,, tMzzMtzzMtzzMtMzzMktMzzM
i.e.
   tMzzMktMzzM ,,,, 
Therefore
z = Mz = STz.
Step 5 : Putting x = x2n, y = Tz with α = 1 in (4), we have
  
     
      
0
,,,,,
,,,,,,
min,,,
22
22
2







tTzMTzSTMtLxABxM
tTzSTABxMtTzMABxM
ktTzMLxM
nn
nn
n
Since MT = TM, ST = TS therefore M(Tz) = T(Mz) = Tz, ST(Tz) = T(STz) = Tz;
           0,,,,,,,,,,,min,,, tTzTzMtzzMtzzMtTzzMktTzzM
i.e.
   tTzzMktTzzM ,,,, 
Therefore
z = Tz = Sz = Mz.
Hence
z = Az = Bz = Lz = Sz = Mz = Tz.
Uniqueness : Let w be another fixed point of A, B, L, S, M and T. Then putting x = u, y = w with α = 1 in (4)
we have
 
   
   
0
,,,,,
,,,,,,
min,,, 






tMwSTwMtLuABuM
tSTwABuMtMwABuM
ktMwLuM
         0,,,,,,,,,,,min twwMtuuMtwuMtwuM
Therefore
   twuMktwuM ,,,, 
Hence
z = w.
Corollary 3.2 : Let A, B, S, T, L and M be a complete ε-chainable fuzzy metric space (X,M,*) with continuous
t-norm satisfying the conditions (1) to (3) of theorem 3.1 and ; (5) there exists  1,0k such that
 
   
     
0}
2,,,,,,,,
,,,,2,,
,,,min{ 






tLxSTyMtMySTyMtLxABxM
tSTyABxMtMyABxM
ktMyLxM
For every x, y  X, α ε (0, 2) and t > 0. If L, AB is reciprocally continuous, semi-compatible maps. Then A, b,
S, T, L and M have a unique common fixed point in X.
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Fixed Point Theorem in Fuzzy Metric Space

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