Research Journal of Applied Sciences, Engineering and Technology 2(5): 418-421, 2010
ISSN: 2040-7467
© M axwell Scientific Organization, 2010
Submitted Date: April 27, 2010
Accepted Date: May 17, 2010
Published Date: August 01, 2010
A Common Fixed Point Theorem in M-fuzzy Metric Spaces Satisfying
Integral Type Implicit Relations
Deo Brat Ojha and M anish Kumar M ishra
Departm ent of Mathematics, R.K.G .I.T., Delhi-M eerut Road G haziabad, 20100 3, UP, India
Abstract: In this note, a gen eral comm on fixe d point theorem o f integral type for two pair of w eakly
compatible mapping satisfying integral type implicit relations is obtained in M-fuzzy metric space.
Key w ords: Fixed point, M-fuzzy metric space, weakly compatible mappings
INTRODUCTION
In recent years, several common fixed point theorems
for contractive type, mappings have been established by
several authors like Jachymski (1995), Jungck (1993) and
Pant (1996). Using the concept of reciprocal continuity,
which is weaker from of continuity of mappings
Pant (1996) and Popa (1997, 1999) proved some fixedpoint theorems satisfying certain implicit relation.
Recently Aliouche and Djondi (2005) established a
general common fixed-point theorem for pair of
recipro cally continuous mappings satisfying an implicit
relation Pathak and Tiw ari (2007).
After introduction of fuzzy sets by Zadeh (1965) and
Kramosil and Michalek (1975), introduced the concept of
fuzzy metric space, many authors extended their views as
some George and V eeramani (1994), Grabiec (1988),
Subrahmanyam (1995) and Vasuki (1999). In recent
research Sedgh i and Sho be (2006 ), introduced M-fuzzy
metric space in D- metric space, Chauhan (2009), proved
some results using four weak co mpa tible ma pping s in M fuzzy m etric space and Chau han and Joshi (2009 ).
In this study, we established some fixed-point
theorems in M-fuzzy metric space using integral type
implicit relation w ith Integral type inequ ality.
MATERIALS AND METHODS
Definitions and preliminaries:
Definition: A binary operation *: [0,1]× [0,1]÷ [0,1] is a
continuous t - norm if it satisfies the following conditions:
C
C
C
C
* is communicative and associative
* is continuous
a * 1= a for all a , [0,1]
a * b # c * d whenever a # c and b # d foreach a, b,
c , [0,1]
Examples of continuous t-norm are:
C
C
a * b = ab and
a * b = min {a,b}
Definition: A 3-tuple (X, M, *) is called a M-fuzzy
metric space, if X is an arbitrary ( non-emp ty) set, * is a
continuous t-norm and M is a fuzzy set on X 3 ×(0, 4).
Satisfying the following conditions for each x, y, z, a , X
and t, s > 0:
C
C
C
C
C
M(x, y, z, t) > 0
M(x, y, z, t) = 1 iff x = y = z
M(x, y, z, t) = M(P{x, y, z}, t) (Symm etry) where P
is a permutation function.
M(x, y, a, t)* M(a, z, z, s) # M(x , y, z, t+s)
M(x, y, z): (0, 4) ÷ [0,1] is continuous
Remark: Let (X , M,*) be a M- fuzzy metric space. Then
for every t > 0 and for every x, y , X. We have M (x, x,
y, t) = M(x, y, y, t).
Definition: Let (X , M ,*) be a M -fuzzy metric space, for
t > 0, the open ball B M (x, r, t) with centre x , X and the
radius 0 < r < 1 is defined by BM (x, r, t) = {y , X: M(x,
y, y, t) > 1-r}. A subset A of X is called open set if for
each x , A there exist t > 0 and 0 < r < 1, such that
B M (x, r, t) # A.
Definition: A sequence {x n } in X converges to X iff M(x,
x, x n , t) ÷ 1 as n ÷ 4, for each t > 0. It is called a Cauchy
sequence if for each 0 < , < 1 and t > 0, there exist n 0 , N
such that M(x n, x n, x m , t) > 1 - g for each n, m $ n 0 . The Mfuzzy metric space (X, M,*) is said to be complete if
every Cau chy sequence is convergent.
Lemm a: Let (X, M,*) be a M-fuzzy metric space. Then
M(x, x, y, t) is non-decreasing with respect to t for all x,
y, z in X.
Corresponding Author: Manish Kumar Mishra, Department of Mathematics, R.K.G.I.T., Delhi-Meerut Road Ghaziabad, 201003,
U.P, India
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Res. J. Appl. Sci. Eng. Technol., 2(5): 418-421, 2010
t-norm * define by a*b = min(a, b), a, b , [0,1], and R: R +
÷ R + is a Lebesgue - integrable mapping w hich is
summable.
Lemm a: Let (X , M,*) be a M-fuzzy metric space. Then
M is continuous function on X 3 ×(0, 4).
Definition: Let F and G be two self maps of (X, M,*)
then F and G are said weakly com patible if there exists v
in X with F v = G v imp lies FG v = G Fv.
Definition: Implicit relation:
Let n be the set of all real continuous function n: (R + 4 ) ÷
R, non-decreasing and satisfying the following conditions:
C
such that:
C
C
for all
implies that u $ v
AB (X) d G(X ), ST(X) d J(X)
AB, G) and (ST, J) are weakly compatible for some
N , L. x, y, z , X, t > 0
Then AB, ST, J and G have unique common fixed point.
C
implies that u $ 1
C
implies that u $ 1
C
implies that u $ 1
Proof: Let x 0 , X be any arbitrary point. Since A B(X ) d
G(X) and ST(X ) d J(X), there exit a point x 1 , x 2 , X such
that ABx 0 = Gx 1 and STx 1 = Jx 2 . Induc tively we get a
sequence {y n } as y 2 n -1 = Gx 2 n -1 = ABx 2 n -2 , y 2 n = Jx 2 n =
STx 2 n -1 n = 1,2,...
Let M n = M (y n , y n + 1 , y n + 2 , t) < 1 for all n. Put x = x 2 n + 1 ,
y = y 2n-2 , z = x 2 n in (iii) we g et:
where; R: R + ÷ R + is a Lebesg ue-integrable mapping,
which is summ able, nonnegative and such that:
for each g > 0
Example define n(x, y, z, w) 5x-6y+2z-w then n , N
and N (t) = (i) Cos w
(ii)
Thus we have M 2 n > M 2n-1
(1)
Thus {M 2 n , n $ 0} is an increasing sequence of positive
real numbers in [0,1] and therefore tends to limit m # 1.
W e claim m = 1, for m < 1 taking limit in Eq. (1), we
get m < m, which is a contradiction. Therefore m = 1.
For any +ve intege r r:
Then
so, Sin(u-1) $ 0, which implies u $ 1.
M (y n , y n , y n + r, t) $
M (y n , y n , y n + 1 , t/r)*M (y n + 1 , y n + 1 , y n + 2 , t/r)*...*
M (y n+r-1 , y n+r-1 , y n + r, t/r) > (1-g)*(1-g)*.... r times = (1-g)
Y M (y n , y n , y n + r, t) > (1-g)
The main object of this study is to prove a common fixed
point theorem for a mapping satisfying certain integral
type implicit relations which are viable, productive and
powerful tool in finding the existence of common fixed
point in M-fuzzy metric space.
Thus, M (y n , y n , y n + r, t) > (1-g) for all n, s $ n 0 , where;
n 0 , N. Thus {y n }is a C auch y sequence in X . Since X is
com plete there is a point p , X s.t., y n ÷ p. Thus
subsequences {ABx 2 n }, {Gx 2n-1 }, {Jx 2 n }, {STx 2 n }, also
converges to p. Since A B(X ) d G(X) and ST(X ) d J(X),
RESULTS
Theorem: Let A, B, S, T, J and G be self maps of a
com plete M-fuzzy metric space (X, M,*) with continuous
419
Res. J. Appl. Sci. Eng. Technol., 2(5): 418-421, 2010
then there must exist u, v , X such that, p = Gv, p =
Ju, put x = v, y = x 2n-1 , z = x 2 n in condition (iii):
Uniqueness: Let w be a fixed point of AB, ST, J and G
other then p. Then ABw = STw = Jw = Gw = w , put
x = p, y = p, z = w in (iii) we g et:
Therefore, ABv = p. Hence, ABv = p = Jv. Put x = v,
y = x 2 n - + 1 and z = u in condition (iii), w e get:
Therefore, p = w.
Hence p is a unique fixed point of AB, ST, J and G.
Corollary: Let A, S, I, G be self maps of a c omplete Mfuzzy metric space (X, M,*) with continuous t-norm *
defined by a*b = min(a,b), a,b , [0,1] satisfying:
C
C
A(X) CG(X), S(X) CJ(X)
(A, G) and (S, J) are weakly compatible for some n
, N. x, y, z , X, t > 0
C
and
R: R + : R + is a Lebesgue - integrable mapping which
is sum mab le
Therefore, STu = p. Hence Ju = STu = p. Consequently,
ABv = Gv = Ju = STu. Since (AB, G) and (ST, J) are weak
compatible. Therefore, ABGv = GABv implies Abp = Gp.
Also STJu = JSTu implies STp = Jp. Therefore, p is a
coincidence point of AB , ST, J and G . Put x = v, y = x 2n-1 ,
z = p in c ondition (iii) we get:
Then A, S, J and G have comm on fixe d point.
Proof: Taking B = T = Identity mapping in above
theore m, w e get the requ ired result.
C O N C L U SI O N
W e establish A Comm on Fixed Point Theorem in Mfuzzy Me tric Spaces satisfying integ ral type im plicit
relations.
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