Fixed point theorems for four mappings in fuzzy metric space using implicit relation

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
359
Fixed Point Theorems for Four Mappings in Fuzzy Metric Space
using Implicit Relation
Kamal Wadhwa1
, Rashmi Tiwari2
and Ved Prakash Bhardwaj3
1, 2, 3
Govt. Narmada Mahavidyalaya Hoshangabad (M.P.)
Abstract
The present paper deals with proving some common fixed point results in a fuzzy metric space using Implicit
Relation.
Index Terms: Fixed point, fuzzy metric space, occasionally weakly compatible mappings, t-norm, Implicit
Relation.
I. Introduction
It proved a turning point in the development of fuzzy mathematics when the notion of fuzzy set was introduced
by Zadeh [12] in 1965. Following the concept of fuzzy sets, fuzzy metric spaces have been introduced by
Kramosil and Michalek [15] and George and Veeramani [2] modified the notion of fuzzy metric spaces with the
help of continuous t-norm, which shows a new way for further development of analysis in such spaces.
Consequently in due course of time some metric fixed points results were generalized to fuzzy metric spaces by
various authors. Sessa [20] improved commutativity condition in fixed point theorem by introducing the notion
of weakly commuting maps in metric space. Vasuki [20] proved fixed point theorems for R-weakly commuting
mapping. Pant [18, 19] introduced the new concept of reciprocally continuous mappings and established some
common fixed point theorems. The concept of compatible maps by [15] and weakly compatible maps by [9] in
fuzzy metric space is generalized by A.Al Thagafi and Naseer Shahzad [1] by introducing the concept of
occasionally weakly compatible mappings.
In this paper we give a fixed point theorem on fuzzy metric space with an implicit relation defined in [14]. Our
results extend and generalize the result of Wadhwa and Dubey [11] using Chouhan, Khanduja and Singh [14].
We start with some preliminaries:
II. Preliminaries
Definition 2.1 [4]: A binary operation *: [0, 1] × [0, 1]→[0, 1] is continuous t-norm if satisfies the following
conditions:
(1) * is commutative and associative,
(2) * is continuous,
(3) a*1 = a for all a є [0, 1],
(4) a*b ≤ c*d whenever a ≤ c and b ≤ d for all a, b, c, d є [0, 1].
Examples of t- norm are a*b = min {a, b} and a*b = ab.
Definition 2.2 [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: (The functions M(x, y, t) denote
the degree of nearness between x and y with respect to t, respectively.)
1) M(x, y, 0) = 0 for all x, y ∈ X
2) M(x, y, t) = 1 for all x, y ∈ X and t>0 if and only if x = y
3) M(x, y, t) = M(y, x, t) for all x, y ∈ X and t>0
4) M(x, y, t)*M(y, z, s) ≤ M(x, z, t+s) for all x, y, z ∈ X and t, s > 0
5) For all x, y ∈ X, M(x, y, .) : [0, ∞) → [0, 1] is left continuous.
Remark 2.1: In a FM M(X, M, *), M(x, y, .) is non- decreasing for all x.
Example 2.3 (Induced fuzzy metric [2]): – Let (X, d) be a metric space. Denote a*b =ab for all a, b∈[0, 1] and
let
Md be fuzzy sets on X2
× [0, ∞) defined as follows:
Md (x, y, t) =
( , )
Then (X, Md,*) is a fuzzy metric space. We call this fuzzy metric induced by a metric d as the standard
intuitionistic fuzzy metric.
Definition 2.4: Let M(X, M, *) be a FM - space. Then
(i) A sequence{xn} in X is said to be Cauchy Sequence if for all t>0 and p>0
lim →∞ ( xn+p, xn, t) = 1
(ii) A sequence {xn} in X is said to be convergent to a point x ∈ X if for all t>0,
lim →∞ ( xn, x, t) = 1

360
Since * is continuous, the limit is uniquely determined from (5) and (11) respectively.
Definition 2.5: A FM-Space M(X, M, *) is said to be complete if and only if every Cauchy sequence in X is
convergent.
Definition 2.6: Let A and S be maps from a fuzzy metric M(X, M, *) into itself. The maps A and S are said to be
weakly commuting if M( ASz, SAz, t) ≥ M( Az, Sz, t) for all z ∈ X and t>0
Definition 2.7 [10]: Let f and g be maps from an FM-space M(X, M, *) into itself .The maps f and g are said to
be compatible if for all t>0, lim →∞ ( fgxn, gfxn, t) = 1 whenever {xn} is a sequence in X such that
lim →∞ fxn = lim →∞ gxn = z for some z ∈ X.
Definition 2.8: Two mappings A and S of a fuzzy metric space M(X, M, *) will be called reciprocally
continuous if ASun→Az and SAun→Sz, whenever {un} is a sequence such that for some Aun, Sun→z for some z
∈ X.
Definition 2.9: Let M(X, M, *) be a fuzzy metric space. f and g be self maps on X. A point x in X is called a
coincidence point of f and g iff fx =gx. In this case w = fx =g x is called a point of coincidence of f and g.
Definition 2.10[8]: A pair of maps S and T is called weakly compatible pair if they commute at the coincident
points i.e., if Su = Tu for some u in X then STu = TSu.
Definition 2.11[1]: Two self maps f and g of a set X are called occasionally weakly compatible (owc) iff there is
a point x in X which is coincidence point of f and g at which f and g commute.
Example 2.11.1[1]: Let R be the usual metric space. Define S, T: R→R by Sx=2x and Tx=x2
for all x∈R. Then
Sx=Tx for x=0, 2, but ST0=TS0, and ST2≠TS2. S and T are occasionally weakly compatible self maps but not
weakly compatible.
Definition 2.12 (Implicit Relation)[14]: Let φ5 be the set of all real and continuous function φ : (R+
)5
→R and
such that
2.12 (i) φ non increasing in 2nd
and 4th
argument and
2.12 (ii) for u, v ≥0, φ(u,v,v,v,v)≥0 ⇒ u ≥ v
Example: φ(t1, t2, t3, t4, t5) = t1 – max {t1, t2, t3, t4}.
Lemma 2.1: Let {un} be a sequence in a fuzzy metric space M(X, M, *). If there exist a constant k ∈ (0, 1) such
that
M( un, un+1, kt) ≥ M(un-1, un, t) for all t>0 and n = 1 , 2 , 3….. Then {un} is a Cauchy sequence in X.
Lemma 2.2: Let M(X, M, *) be a FM space and for all x, y ∈ X, t>0 and if for a number q ∈ (0, 1),
M( x, y, qt) ≥ M(x, y, t) then x = y.
Lemma 2.3[9]: Let X be a set, f and g be 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. Main Result
Theorem 3.1: Let M(X, M, *) be a complete fuzzy metric space and let A, B, S and T be self mappings of X.
Let the pairs {A, S} and {B, T} be owc. If for φ∈φ5 there exist q ∈ (0, 1) such that
φ
M(Ax, By, qt), M(Sx, Ax, t),
, !,") # ($!,% ,")&
#
,
' % , ,") ( ($!, !,")&
' (
,
)
* ( ,% ,")
* ($!, !,")
+ . )
( , !,") ($!,% ,")
-
+
. ≥ 0 ……………….. (1)
for all x, y ∈ X and t>0, and a,b,c,d≥0 with a&b and c&d cannot be simultaneously 0 , there exist a unique point
w ∈ X such that Aw = Sw = w and a unique point z ∈ X such that Bz = Tz = z. Morever 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 = Ty. We
claim that Ax = By. If not by inequality (1)
φ
, !,") # ($!,% ,")&
#
,
' % , ,") ( ($!, !,")&
' (
,
)
* ( ,% ,")
* ($!, !,")
+ . )
( , !,") ($!,% ,")
-
+
. ≥ 0
φ
M(Ax, By, qt), M(Ax, Ax, t),
,$!,") # ($!, ,")&
#
,
' , ,") ( ($!,$!,")&
' (
,
)
* ( , ,")
* ($!,$!,")
+ . )
( ,$!,") ($!, ,")
-
+
. ≥ 0
φ M(Ax, By, qt), 1, M(Ax, By, t), 1, M(Ax, By, t)& ≥ 0
φ M(Ax, By, qt), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t)& ≥ 0
Since, φ is non-increasing in 2nd
and 4th
argument therefore by 2.12 (i) and 2.12 (ii)
M(Ax, By, qt) ≥ M(Ax, By, t)
Therefore Ax = By i.e. Ax = Sx = By =Ty.
Suppose that there is a unique point z such that Az = Sz then by (1) we have

361
φ
M(Az, By, qt), M(Sz, Az, t),
2, !,") # ($!,%2,")&
#
,
' %2, 2,") ( ($!, !,")&
' (
,
)
* ( 2,%2,")
* ($!, !,")
+ . )
( 2, !,") ($!,%2,")
-
+
. ≥ 0
φ M(Az, By, qt), 1, M(Az, By, t), 1, M(Az, By, t)& ≥ 0
φ M(Az, By, qt), M(Az, By, t), M(Az, By, t), M(Az, By, t), M(Az, By, t)& ≥ 0
and 4th
argument therefore 2.12 (i) and 2.12 (ii)
M(Az, By, qt) ≥ M(Az, By, t)
Az = By = Sz =Ty, So Ax = Az and w = Ax = Sx the unique point of coincidence of A and S. By lemma (2.3) w
is the only common fixed point of A and S. Similarly there is a unique point z ∈ X such that z = Bz =Tz.
Assume that w ≠ z we have
φ
M(Aw, Bz, qt), M(Sw, Aw, t),
4, 2,") # ($2,%4,")&
#
,
' %4, 4,") ( ($2, 2,")&
' (
,
)
* ( 4,%4,")
* ($2, 2,")
+ . )
( 4, 2,") ($2,%4,")
-
+
. ≥ 0
φ
M(Aw, Bz, qt), M(w, w, t),
4,2,") # (2,4,")&
#
,
' 4,4,") ( (2,2,")&
' (
,
)
* (4,4,")
* (2,2,")
+ . )
(4,2,") (2,4,")
-
+
. ≥ 0
φ M(Aw, Bz, qt), M(w, w, t) , M(w, z, t), 1, M(w, z, t)& ≥ 0
φ M(Aw, Bz, qt), 1, M(w, z, t), 1, M(w, z, t)& ≥ 0
φ M(Aw, Bz, qt), M(w, z, t) , M(w, z, t), M(w, z, t), M(w, z, t)& ≥ 0
and 4th
argument
M(Aw, Bz, qt) ≥ M(w, z, t)
Since, w is the only common fixed point of A and we have z = Bz.
M(Aw, Bz, qt) = M(w, z, qt) ≥ M(w, z, t)
We have M(w, z, qt) ≥ M(w, z, t)
Hence z = w by Lemma (2.2) and z is a common fixed point of A, B, S and T. The uniqueness of the fixed point
holds from (1).
Definition 3.11 (Implicit Relation) [14]: Let φ6 be the set of all real and continuous function φ : (R+
)6
→R and
such that
3.11 (i) φ non increasing in 2nd
and 4th
argument and
3.11 (ii) for u, v ≥0, φ(u,v,v,v,v,v)≥0 ⇒ u ≥ v
Theorem 3.2: Let M(X, M, *) be a complete fuzzy metric space and let A, B, S and T be self mappings of X.
Let the pairs {A, S} and {B, T} be owc. If for φ∈φ6 there exist q ∈ (0, 1) such that
φ
, !,") # ($!,% ,")&
#
,
' % , ,") ( ($!, !,")&
' (
,
)
* ( ,% ,")
* ($!, !,")
+ . )
( , !,") ($!,% ,")
-
+ , M(By, Sx, t)
. ≥ 0 ……………….. (2)
for all x, y ∈ X and t>0, and a,b,c,d≥0 with a&b and c&d cannot be simultaneously 0 , there exist a unique point
w ∈ X such that Aw = Sw = w and a unique point z ∈ X such that Bz = Tz = z. Morever z = w, so that there is a
unique common fixed point of A, B, S and T.
Proof: Let the pairs {A, S} and {B, T} be owc, so there are points x, y ∈ X such that Ax = Sx and By = Ty. We
claim that Ax = By. If not by inequality (2)
φ
M(Ax, By, qt), M(Ax, Ax, t),
,$!,") # ($!, ,")&
#
,
' , ,") ( ($!,$!,")&
' (
,
)
* ( , ,")
* ($!,$!,")
+ . )
( ,$!,") ($!, ,")
-
+ , M(By, Ax, t)
. ≥ 0
φ(M(Ax, By, qt), 1, M(Ax, By, t), 1, M(Ax, By, t), M(By, Ax, t)) ≥ 0
φ(M(Ax, By, qt), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t), M(Ax, By, t), M(By, Ax, t)) ≥ 0
and 4th
argument therefore 3.11 (i) and 3.11 (ii)
M(Ax, By, qt) ≥ M(Ax, By, t)
Therefore Ax = By i.e. Ax = Sx = By = Ty. Suppose that there is another point z such that A z =S z then by (2)
we have A z = S z = Ty, So Ax = A z and w =Ax = Tx is the unique point of coincidence of A and T. By lemma
(2.2) w is a unique point z ∈ X such that w = Bz = Tz. Thus z is a common fixed point of A, B, S and T. The
uniqueness of fixed point holds by (2).

362
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Fixed point theorems for four mappings in fuzzy metric space using implicit relation

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