On common fixed point theorem in fuzzy metric space

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
118
On Common Fixed point Theorem in Fuzzy Metric space
Kamal Wadhwa
Govt. Narmada P.G. College, Hoshangabad, Madhya Pradesh, India
Jyoti Panthi
Govt. Narmada P.G. College, Hoshangabad, Madhya Pradesh, India
ABSTRACT
In this research article we are proving common fixed point theorem using Occasionally Weakly Compatible
Mapping in fuzzy metric space.
KEYWORDS:Common Fixed point, Fuzzy Metric space, Occasionally Weakly Compatible Mapping,
Continuous t-norm.
1. INTRODUCTION
It proved a turning point in the development of mathematics when the notion of fuzzy set was introduced
by Zadeh [24 ] which laid the foundation of fuzzy mathematics. Kramosil and Michalek [11] introduced the
notion of a fuzzy metric space by generalizing the concept of the probabilistic metric space to the fuzzy situation.
George and Veeramani [7] modified the concept of fuzzy metric spaces introduced by Kramosil and Michalek
[11]. There are many view points of the notion of the metric space in fuzzy topology for instance one can refer to
Kaleva and Seikkala [10], Kramosil and Michalek [11], George and Veeramani [7].
.
2. PRELIMINARIES:
Definition 2.1. [24] Let X be any non empty set. A fuzzy set M in X is a function with domain X and values in
[0, 1].
Definition 2.2. [19] A binary operation ∗ :[0,1]×[0,1]→[0,1] is a continuous t-norm if it satisfy the following
condition:
(i) ∗ is associative and commutative .
(ii) ∗ is continous function.
(iii) a∗1=a for all a∈ [0,1]
(iv) a∗b ≤ c∗d whenever a ≤ c and b ≤ d and a, b,c,d ∈[0,1]
Definition 2.3. [11] The 3 − tuple (X, M,∗) is called a fuzzy metric space in the sense of Kramosil and Michalek
if X is an arbitrary set, is a continuous t − norm and M is a fuzzy
set in X2
× [0,∞) satisfying the following conditions:
(a) M(x, y, t) > 0,
(b) M(x, y, t) = 1 for all t > 0 if and only if x = y,
(c) M(x, y, t) = M(y, x, t),
(d) M(x, y, t) M(y, z, s) ≤ M(x, z, t + s),
(e) M(x, y, .) : [0,∞) → [0, 1] is a continuous function, for all x, y, z ∈ X and t, s > 0.
Definition 2.4 [11] Let (X, M, ∗) be a fuzzy metric space . Then
(i) A sequence {xn} in X converges to x if and only if for each t >0 there exists n0 ∈N, such that,
lim n→∞ M (xn, x, t) = 1, for all n ≥ n0.
(ii) The sequence (xn) n∈N is called Cauchy sequence if lim → M (xn,xn+p, t) = 1, for all t > 0 and p ∈ N.
(iii) A fuzzy metric space X is called complete if every Cauchy sequence is convergent in X.
Definition 2.5. [23] Two self-mappings f and g of a fuzzy metric space (X, M, ∗) are said to be weakly
commuting if M(fgx, gfx, t) ≥ M(fx, gx, t), for each x ∈X and for each t > 0.
Definition 2.6 [5] Two self mappings f and g of a fuzzy metric space(X, M,∗) are called compatible if lim → M
(fgxn, gfxn, t) = 1 whenever {xn} is a sequence in X such thatlim → fx = lim → gx = x for some x in X.
Definition 2.7.[2] A pair of mappings f and g from a fuzzy metric space (X,M,∗) into itself are weakly
compatible if they commute at their coincidence points,i.e., fx = gx implies that fgx = gfx.
Definition 2.8 Let X be a set, f, g selfmaps of X. A point x in X is called a coincidence point of f and g iﬀ fx =
gx. We shall call w = fx = gx a point of coincidence of f and g.

119
Definition 2.9 [2] A pair of maps S and T is called weakly compatible pair if they commute at coincidence
points.
Definition 2.10.[4] Two self maps f and g of a set X are occasionally weakly compatible (owc) iff there is a
point x in X which is a coincidence point of f and g at which f and g commute.
A. Al-Thagafi and Naseer Shahzad [4] shown that occasionally weakly compatible is weakly compatible but
converse is not true.
Lemma 2.11 [4] Let X be a set, f, g owc self maps of X. If f and g have a unique point of coincidence, w = fx =
gx, then w is the unique common fixed point of f and g.
3. IMPLICIT RELATIONS:
(a) Let (Ф) be the set of all real continuous functions ∅ : ( )5
→ satisfying the condition ∅: (u, u, v, v, u,)
≥0 imply u ≥ v, for all u, v ∈ [0,1].
(b) Let (Ф) be the set of all real continuous functions ∅ : ( )4
→ satisfying the condition ∅: (u, v, u, u,) ≥0
imply u≥ v, for all u, v ∈ [0,1].
4. MAIN RESULTS
Theorem 4.1.: Let (X, M,∗) be a fuzzy metric space with ∗ continuous t-norm. Let A, B, S, T be self mappings
of X satisfying
(i) The pair (A, S) and (B, T) be owc.
(ii) For some ∅ ∈ Ф and for all x, y ∈ X and every t > 0,
∅{M (Ax, By, t), M (Sx, Ty, t), M (Sx, Ax, t), M (Ty, By, t), M (Sx, By, t)}≥ 0
then there exists a unique point w ∈X such that Aw = Sw = w and a unique point z ∈X such that Bz = T z = z.
Moreover, z = w, so that there is a unique common fixed point of A, B, S and T.
Proof: Let the pairs {A, S} and {B, T} be owc, so there are points x, y ∈X such that Ax = Sx and By = T y.
We claim that Ax = By. If not, by inequality (ii)
∅{M (Ax, By, t), M (Ax, By, t), M (Ax, Ax, t), M (By, By, t), M (Ax, By, t)}≥ 0
∅{M (Ax, By, t), M (Ax, By, t), 1, 1, M (Ax, By, t)}≥ 0
∅{M (Ax, By, t), M (Ax, By, t), 1, 1, M (Ax, By, t)}≥ 0
In view of Ф we get Ax = By i.e. Ax = Sx = By = T y
Suppose that there is a another point z such that Az = Sz then by (i) we have Az = Sz = By = T y, so Ax = Az
and w = Ax = Sx is the unique point of coincidence of A and S. By Lemma 2.11 w is the only common fixed
point of A and S. Similarly there is a unique point z ∈ X such that z = Bz = T z.
Assume that w ≠ z. We have
∅{M(Aw, Bz, t), M(Sw,Tz, t), M(Sw, Aw, t), M(Tz,Bz,t),M(Sw,Bz, t)}≥ 0
∅{M (w, z, t), M (w, z, t), M (w, w, t), M (z, z, t), M (w, z, t)}≥ 0
∅{M (w, z, t), M (w, z, t), 1, 1, M (w, z, t)}≥ 0
∅{M (w, z, t), M (w, z, t), 1, 1, M (w, z, t)}≥ 0
In view of Ф we get w = z. by Lemma 2.11 and z is a common fixed point of A, B, S and T. The uniqueness of
the fixed point holds from (ii)

120
Theorem 4.2.: Let (X, M, ∗ ) be a fuzzy metric space with ∗ continuous t-norm. Let A, B, S, T be self mappings
of X satisfying
(i) The pair (A, S) and (B, T) be owc.
(ii) For some ∅ ∈Ф and for all x, y ∈ X and every t > 0,
∅ M Sx, Ty, t , M Sx, Ax, t , M Sx, By, t , M Ty, Ax, t ≥ 0
then there exists a unique point w ∈ X such that Aw = Sw = w and a unique point z ∈X such that Bz = T z = z.
Moreover, z = w, so that there is a unique common fixed point of A, B, S and T.
Proof: Let the pairs {A, S} and {B, T} be owc, so there are points x, y ∈ X such that Ax = Sx and By = T y.
We claim that Ax = By. If not, by inequality (ii).
∅ , M Ax, By, t , M Ax, Ax, t , M Ax, By, t , M By, Ax, t ≥ 0
∅ M Ax, By, t , M Ax, Ax, t , M Ax, By, t , M Ax, By, t ≥ 0
∅ M Ax, By, t , 1, M Ax, By, t , M Ax, By, t ! ≥ 0
In view of Ф we get Ax = By i.e. Ax = Sx = By = T y
Suppose that there is a another point z such that Az = Sz then by (i) we have Az = Sz = By = T y, so Ax = Az
and w = Ax = Sx is the unique point of coincidence of A and S. By Lemma 2.12 w is the only common fixed
point of A and S. Similarly there is a unique point z ∈ X such that z = Bz = T z.
∅ M Sw, Tz, t , M Sw, Aw, t , M Sw, Bz, t , M Tz, Aw, t ≥ 0
∅ M w, z, t , M w, w, t , M w, z, t , M z, w, t ≥ 0
∅ M w, z, t ,1, M w, z, t , M w, z, t ≥ 0
In view of Ф we get w = z. by Lemma 2.11 and z is a common fixed point of A, B, S and T. The uniqueness of
the fixed point holds from (ii)
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On common fixed point theorem in fuzzy metric space

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