Extension of Some Common Fixed Point Theorems using Compatible Mappings in Fuzzy Metric Space
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This paper proves common fixed point theorems for four mappings within fuzzy metric spaces, utilizing the concept of compatibility. It introduces and explores definitions related to fuzzy metric spaces and compatible mappings, building upon previous works in the field. The main result establishes conditions under which mappings in a complete fuzzy metric space have a common fixed point, completing the proof through rigorous mathematical construction.
![International Journal of Research in Engineering and Science (IJRES)
ISSN (Online): 2320-9364, ISSN (Print): 2320-9356
www.ijres.org Volume 3 Issue 12 ǁ December. 2015 ǁ PP.59-61
www.ijres.org 59 | Page
Extension of Some Common Fixed Point Theorems using
Compatible Mappings in Fuzzy Metric Space
Vineeta Singh
(S.A.T.I., Vidisha) S. K. Malhotra (Govt. Benazir College, Bhopal) Subject Classification 54H25, 47H10
Abstract: In this paper we have proved some common Fixed Point theorems for four mappings using the
notion of compatibility.
Keywords: Fuzzy Metric Space, Compatible Mappings
I. Introduction
The concept OF Fuzzy sets was investigated by Zadeh [1]. Here we are dealing with the fuzzy metric space
defined by Kramosil and Michalek [ 2] and modified by George and Veeramani [3]. Grabiec[ 4] has also proved
fixed point results for fuzzy metric space with different mappings. Singh and Chauhan[5] gave the results using
the concept of compatible mappings in Fuzzy metric space. Jungck [ 6] introduced the concept of compatible
mapping of type (A) and type (B)In fuzzy metric space. Singh and jain [7] proved the fixed point theorems in
fuzzy metric space using the concept of compatibility and semicompatibily. Sharma[8] also done work on
compatible mappings.
II. FUZZY METRIC SPACE
Definition[2 ]: A 3-tuple (X,M, *) is said to be a fuzzy
metric space if X is an arbitrary set, * is a continuous t-norm
and M is a fuzzy set on X2
× [0, ] satisfying the following conditions
(f1) M(x, y, t) > 0
(f2) M(x, y, t) = 1 if and only if x = y
(f3) M(x, y, t) = M(y, x, t);
(f4) M(x, y, t) *M(y, z, s) ≤ M(x, z, t + s),
(f5) M(x, y, . ) : (0, ) → (0, 1] is continuous.
x,y,z X and t,s > 0
Then M is called a fuzzy metric on X. Then
M(x, y, t) denotes the degree i.e. of nearness between x and y
with respect to t.
Compatible and Non compatible mappings: Let A and S be mapping from a fuzzy metric space (X,M, *)
into itself.Then the mappings are said to be compatible if
n
Lim M(ASxn,SAxn,t) = 1, t >0,
whenever {xn} is a sequence in X such that
n
Lim Axn =
n
Lim x X
from the above definition it is inferred that A and S are non compatible maps from a fuzzy metric space (X,M,*)
into itself if
n
Lim Axn =
n
Lim Sxn = x X
but either
n
Lim M(ASxn,SAxn,t) ≠ 1, or the limit does not exist.
Main Results:
Theorem:-Let A,B,S,T be self maps of complete fuzzy metric space (X,M,*) such that a*b = min(a,b) for some
y in X.
(a) A(X) T(X),B(X) S(X),T(Y) A(Y)
(b) S and T are continuous.](https://image.slidesharecdn.com/i3125961-160226092324/85/Extension-of-Some-Common-Fixed-Point-Theorems-using-Compatible-Mappings-in-Fuzzy-Metric-Space-1-320.jpg)
![Extension of Some Common Fixed Point Theorems using Compatible Mappings in Fuzzy Metric
www.ijres.org 60 | Page
(c) [A,S],[B,T] are compatible pairs of maps
(d) For all x,y in X ,k (0,1) ,t> 0.
M(Ax,By,KT)≥ min { M(Sx,Ty,t),M(Ax,Sx,t),M(By,Ty,t),M(By,Sx,t),M(Ax,Ty,t),M(Ay,Tx,t)}
For all x,y X LIM n M(x,y,t) 1 then A,B,S,T have a common fixed point in X.
Proof:- Let x0 be an arbitrary point inX. Construct a sequence {yn} in X such that y2n-1= Tx2n-1=Ax2n-2
And y2n= Sx2n=Bx2n-1=Tx2n , for n= 0,1,2......
Put x= x2n, y= x2n+1
M(y2n+1,y2n+2,kt) = M (Ax2n,Bx2n+1,kt)
≥ min {(M(Sx2n, Tx2n+1,t),M((Ax2n,Sx2n,t),M(Bx2n+1,Tx2n+1,t),M(Tx2n,Ax2n+1,t),M(Ax2n,Tx2n+1,t),M(Bx2n+1,
Sx2n,t)}
≥ min{M(y2n,y2n+1,kt),M(y2n+1,y2n+2,t),1}
Which implies
{M(y2n+1,y2n+2,kt) ≥ M(y2n,y2n+1,t).
In general
{M(yn,yn+1,kt) ≥ M(yn-1,yn,t) (1)
To prove that {yn} is a Cauchy sequence we will prove (b) is true for all n≥n0 and every mN
{M(yn,yn+m,t) > 1 - (2)
Here we use induction method
From(1) we have
M(yn,yn+1,t) ≥ M(yn-1,yn,t /k) ≥ M((yn-2,yn-1,t/k2
) ≥ ........ ≥ M( y0,y1,t/kn
) 1 as n
i.e for t> 0, (0,1). We can choose n0 N, such that
{M(yn,yn+1,t) > 1 - (3)
Thus (2) is true for m=1.Suppose (2) is true for m then will show that it is true for m+1. By the definition of
fuzzy metric space, we have
M(yn,yn+m+1,t) ≥ min{M(yn,yn+m,t/2),M(yn+m,yn+m+1,t/2)} > 1 -
Hence(2) is true for m+1.Thus {yn} is a Cauchy sequence.By completeness of (X,M,*) ,{ yn} Converge (
Using (3), we have M(ASx2n,SAx2n ,t/2 ) 1
M(SAx2n,Sz,t) ≥ min{M(ASx2n,SAx2n ,t/2 ), M(SAx2n,Sz,t/2) }> 1 -
For all n ≥n0
Hence ASx2n Sz = TSx2n (4)
Similarly
BTx2n-1 Tz = ATx2n-1 (5)
Now put x= Sx2n and y = Tx2n-1
M(ASx2n, BTx2n-1,kt) ≥ min { M (S2
x2n, T2
x2n-1,t),M(ASx2n, S2
x2n,t),M(BTx2n-1, T2
x2n-1,t),M(BTx2n-1,
S2
x2n,t),M(ASx2n, T2
x2n-1,t),M(TSx2n,ATx2n-1,t)}
Taking limit as n and using (4) and (5)
We get M(Sz,Tz,kt) ≥M(Sz,Tz,t), which implies
Sz = Tz (6)
Now put x=y and y= Tx2n-1
M(Ay,BTx2n-1,kt) ≥ min{M(Sy, T2
x2n-1,t),M(Ay,Sy,t),M( BTx2n-1,Sy,t),M(Ay, T2
x2n-1,t),M(Ty,ATx2n-1,T)}
Taking the limit as n and using (5) and (6) we get
Az=Tz (7)
Now using (6) and (7)
M(Az,Bz,kt) ≥ min{M(Sz,Tz,t),M(Az, Sz,t),M(Bz,Tz,t),M(Bz,Sz,t),M(Az,Tz,t),M(Az,Tz,t)
= min{M(Tz,Tz,t),M(Az,Az,t),M(Az,Bz,t),M(Az,Bz,t),M(Az,Az,t),M(Az,Bz,t)}
≥M(Az,Bz,t)
Which implies Az=Bz
Using (6),(7) and (8)
We get
Az=Bz=Sz=Tz
Now
M(Ax2n,Bz,kt) ≥min{ M(Sx2n,Tz,t),M(Ax2n,Sx2n,t),(Bz,Tz,t),M(Bz,Sx2n,t),M(Ax2n,Tz,t),M(Tx2n,Az,t)}
Taking the limit as n and using (9) we get
Z =Bz
Thus z is common fixed point of A,B,S,T.
For uniqueness let w be another common fixed point then we have](https://image.slidesharecdn.com/i3125961-160226092324/85/Extension-of-Some-Common-Fixed-Point-Theorems-using-Compatible-Mappings-in-Fuzzy-Metric-Space-2-320.jpg)
![Extension of Some Common Fixed Point Theorems using Compatible Mappings in Fuzzy Metric
www.ijres.org 61 | Page
M(Az,Bw,kt) ≥min{M(Sz,Tw,t),M(Az,Sz,t),M(Bw,Tw,t),M(Bw,Sw,t),M(Az,Tw,t),M(Tz,Aw,t)}
i.e. M(z,w,kt) ≥M(z,w,t)
hence z =w this completes the proof.
III. Conclusion
Here we proved the theorem using the notion of compatibility without exploiting the condition of t-norm.
References
[1] L. A. Zadeh, Fuzzy sets, Infor. and Control, 8(1965), pg338-353.
[2] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11(1975), pg336-
344.
[3] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems,
64(1994),pg 395-399.
[4] M. Grabiec, Fixed points in fuzzy metric space, Fuzzy Sets and Systems, 27(1988),pg 385-389.
[5] B.Singh, M.S. Chauhan “Common fixed points of compatible maps in fuzzy metric space”,Fuzzy sets
and systems 115(2000)pg. 471-475.
[6] G.Jungck “Compatible mappings and common fixed points”International journal math sci 9(1986)pg
779-791
[7] Bijendra Singh, Shishir Jain “Semi compatibility, Compatibility and fixed point theorems in fuzzy
metric space Journal of the Chung Cheong mathematical society vol 18 no1 april (2005)
[8] Sushil Sharma” On fuzzy metric space” South east Asian Bulletin of Mathematics (2002)26,pg 133-
145.](https://image.slidesharecdn.com/i3125961-160226092324/85/Extension-of-Some-Common-Fixed-Point-Theorems-using-Compatible-Mappings-in-Fuzzy-Metric-Space-3-320.jpg)
Extension of Some Common Fixed Point Theorems using Compatible Mappings in Fuzzy Metric Space
- 1. International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 3 Issue 12 ǁ December. 2015 ǁ PP.59-61 www.ijres.org 59 | Page Extension of Some Common Fixed Point Theorems using Compatible Mappings in Fuzzy Metric Space Vineeta Singh (S.A.T.I., Vidisha) S. K. Malhotra (Govt. Benazir College, Bhopal) Subject Classification 54H25, 47H10 Abstract: In this paper we have proved some common Fixed Point theorems for four mappings using the notion of compatibility. Keywords: Fuzzy Metric Space, Compatible Mappings I. Introduction The concept OF Fuzzy sets was investigated by Zadeh [1]. Here we are dealing with the fuzzy metric space defined by Kramosil and Michalek [ 2] and modified by George and Veeramani [3]. Grabiec[ 4] has also proved fixed point results for fuzzy metric space with different mappings. Singh and Chauhan[5] gave the results using the concept of compatible mappings in Fuzzy metric space. Jungck [ 6] introduced the concept of compatible mapping of type (A) and type (B)In fuzzy metric space. Singh and jain [7] proved the fixed point theorems in fuzzy metric space using the concept of compatibility and semicompatibily. Sharma[8] also done work on compatible mappings. II. FUZZY METRIC SPACE Definition[2 ]: A 3-tuple (X,M, *) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on X2 × [0, ] satisfying the following conditions (f1) M(x, y, t) > 0 (f2) M(x, y, t) = 1 if and only if x = y (f3) M(x, y, t) = M(y, x, t); (f4) M(x, y, t) *M(y, z, s) ≤ M(x, z, t + s), (f5) M(x, y, . ) : (0, ) → (0, 1] is continuous. x,y,z X and t,s > 0 Then M is called a fuzzy metric on X. Then M(x, y, t) denotes the degree i.e. of nearness between x and y with respect to t. Compatible and Non compatible mappings: Let A and S be mapping from a fuzzy metric space (X,M, *) into itself.Then the mappings are said to be compatible if n Lim M(ASxn,SAxn,t) = 1, t >0, whenever {xn} is a sequence in X such that n Lim Axn = n Lim x X from the above definition it is inferred that A and S are non compatible maps from a fuzzy metric space (X,M,*) into itself if n Lim Axn = n Lim Sxn = x X but either n Lim M(ASxn,SAxn,t) ≠ 1, or the limit does not exist. Main Results: Theorem:-Let A,B,S,T be self maps of complete fuzzy metric space (X,M,*) such that a*b = min(a,b) for some y in X. (a) A(X) T(X),B(X) S(X),T(Y) A(Y) (b) S and T are continuous.
- 2. Extension of Some Common Fixed Point Theorems using Compatible Mappings in Fuzzy Metric www.ijres.org 60 | Page (c) [A,S],[B,T] are compatible pairs of maps (d) For all x,y in X ,k (0,1) ,t> 0. M(Ax,By,KT)≥ min { M(Sx,Ty,t),M(Ax,Sx,t),M(By,Ty,t),M(By,Sx,t),M(Ax,Ty,t),M(Ay,Tx,t)} For all x,y X LIM n M(x,y,t) 1 then A,B,S,T have a common fixed point in X. Proof:- Let x0 be an arbitrary point inX. Construct a sequence {yn} in X such that y2n-1= Tx2n-1=Ax2n-2 And y2n= Sx2n=Bx2n-1=Tx2n , for n= 0,1,2...... Put x= x2n, y= x2n+1 M(y2n+1,y2n+2,kt) = M (Ax2n,Bx2n+1,kt) ≥ min {(M(Sx2n, Tx2n+1,t),M((Ax2n,Sx2n,t),M(Bx2n+1,Tx2n+1,t),M(Tx2n,Ax2n+1,t),M(Ax2n,Tx2n+1,t),M(Bx2n+1, Sx2n,t)} ≥ min{M(y2n,y2n+1,kt),M(y2n+1,y2n+2,t),1} Which implies {M(y2n+1,y2n+2,kt) ≥ M(y2n,y2n+1,t). In general {M(yn,yn+1,kt) ≥ M(yn-1,yn,t) (1) To prove that {yn} is a Cauchy sequence we will prove (b) is true for all n≥n0 and every mN {M(yn,yn+m,t) > 1 - (2) Here we use induction method From(1) we have M(yn,yn+1,t) ≥ M(yn-1,yn,t /k) ≥ M((yn-2,yn-1,t/k2 ) ≥ ........ ≥ M( y0,y1,t/kn ) 1 as n i.e for t> 0, (0,1). We can choose n0 N, such that {M(yn,yn+1,t) > 1 - (3) Thus (2) is true for m=1.Suppose (2) is true for m then will show that it is true for m+1. By the definition of fuzzy metric space, we have M(yn,yn+m+1,t) ≥ min{M(yn,yn+m,t/2),M(yn+m,yn+m+1,t/2)} > 1 - Hence(2) is true for m+1.Thus {yn} is a Cauchy sequence.By completeness of (X,M,*) ,{ yn} Converge ( Using (3), we have M(ASx2n,SAx2n ,t/2 ) 1 M(SAx2n,Sz,t) ≥ min{M(ASx2n,SAx2n ,t/2 ), M(SAx2n,Sz,t/2) }> 1 - For all n ≥n0 Hence ASx2n Sz = TSx2n (4) Similarly BTx2n-1 Tz = ATx2n-1 (5) Now put x= Sx2n and y = Tx2n-1 M(ASx2n, BTx2n-1,kt) ≥ min { M (S2 x2n, T2 x2n-1,t),M(ASx2n, S2 x2n,t),M(BTx2n-1, T2 x2n-1,t),M(BTx2n-1, S2 x2n,t),M(ASx2n, T2 x2n-1,t),M(TSx2n,ATx2n-1,t)} Taking limit as n and using (4) and (5) We get M(Sz,Tz,kt) ≥M(Sz,Tz,t), which implies Sz = Tz (6) Now put x=y and y= Tx2n-1 M(Ay,BTx2n-1,kt) ≥ min{M(Sy, T2 x2n-1,t),M(Ay,Sy,t),M( BTx2n-1,Sy,t),M(Ay, T2 x2n-1,t),M(Ty,ATx2n-1,T)} Taking the limit as n and using (5) and (6) we get Az=Tz (7) Now using (6) and (7) M(Az,Bz,kt) ≥ min{M(Sz,Tz,t),M(Az, Sz,t),M(Bz,Tz,t),M(Bz,Sz,t),M(Az,Tz,t),M(Az,Tz,t) = min{M(Tz,Tz,t),M(Az,Az,t),M(Az,Bz,t),M(Az,Bz,t),M(Az,Az,t),M(Az,Bz,t)} ≥M(Az,Bz,t) Which implies Az=Bz Using (6),(7) and (8) We get Az=Bz=Sz=Tz Now M(Ax2n,Bz,kt) ≥min{ M(Sx2n,Tz,t),M(Ax2n,Sx2n,t),(Bz,Tz,t),M(Bz,Sx2n,t),M(Ax2n,Tz,t),M(Tx2n,Az,t)} Taking the limit as n and using (9) we get Z =Bz Thus z is common fixed point of A,B,S,T. For uniqueness let w be another common fixed point then we have
- 3. Extension of Some Common Fixed Point Theorems using Compatible Mappings in Fuzzy Metric www.ijres.org 61 | Page M(Az,Bw,kt) ≥min{M(Sz,Tw,t),M(Az,Sz,t),M(Bw,Tw,t),M(Bw,Sw,t),M(Az,Tw,t),M(Tz,Aw,t)} i.e. M(z,w,kt) ≥M(z,w,t) hence z =w this completes the proof. III. Conclusion Here we proved the theorem using the notion of compatibility without exploiting the condition of t-norm. References [1] L. A. Zadeh, Fuzzy sets, Infor. and Control, 8(1965), pg338-353. [2] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11(1975), pg336- 344. [3] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(1994),pg 395-399. [4] M. Grabiec, Fixed points in fuzzy metric space, Fuzzy Sets and Systems, 27(1988),pg 385-389. [5] B.Singh, M.S. Chauhan “Common fixed points of compatible maps in fuzzy metric space”,Fuzzy sets and systems 115(2000)pg. 471-475. [6] G.Jungck “Compatible mappings and common fixed points”International journal math sci 9(1986)pg 779-791 [7] Bijendra Singh, Shishir Jain “Semi compatibility, Compatibility and fixed point theorems in fuzzy metric space Journal of the Chung Cheong mathematical society vol 18 no1 april (2005) [8] Sushil Sharma” On fuzzy metric space” South east Asian Bulletin of Mathematics (2002)26,pg 133- 145.