Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalized Intuitionistic Fuzzy Metric Spaces

Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
DOI : 10.5121/mathsj.2015.2301 1
COMMON FIXED POINT THEOREMS IN
COMPATIBLE MAPPINGS OF TYPE (P*) OF
GENERALIZED INTUITIONISTIC FUZZY
METRIC SPACES
R.Muthuraj1
& R.Pandiselvi2
1
PG and Research Department of Mathematics, H.H.The Rajah’s College,
Pudukkottai – 622 001, India.
2
Department of Mathematics, The Madura college, Madurai – 625 011, India.
ABSTRACT
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in
intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
KEYWORDS
Intuitionistic fuzzy metric spaces, compatible mappings of type ( P ), type (P-1) and type (P-2) , common
fixed point .
1.INTRODUCTION
The Concept of fuzzy set was introduced by Zadeh [23] in 1965 .Following the concept of fuzzy
sets, Deng [6] Kaleva and Seikalla [12] and kramosil and Michalek [13] introduced the concept of
fuzzy metric space, George and Veeramani [7] modified the concept of fuzzy metric space
introduced by kramosil and Michalek [13] .
Further, Sedghi and Shobe [19] defined ℳ-fuzzy metric space and proved a common fixed point
theorem in it. Jong Seo Park [15] introduced the concept of semi compatible and Weak
Compatible maps in fuzzy metric space and proved some fixed point theorems satisfying certain
conditions in ℳ-fuzzy metric spaces.
As a generalization of fuzzy sets, Atanassov [1] introduced and studied the concept of
intuitionistic fuzzy sets. Using the idea of intuitionistic fuzzy sets Park [16] defined the notion of
intuitionistic fuzzy metric space with the help of continuous t- norm and continuous t- conorm as
a generalization of fuzzy metric space, George and Veeramani [8] had showed that every metric
induces an intuitionistic fuzzy metric and found a necessary and sufficient conditions for an
intuitionistic fuzzy metric space to be complete. Choudhary [4] introduced mutually contractive
sequence of self maps and proved a fixed point theorem. Kramaosil and Michalek [13] introduced
the notion of Cauchy sequences in an intuitionistic fuzzy metric space and proved the well known
fixed point theorem of Banach[2]. Turkoglu et al [22] gave the generalization of Jungck’s[11]
Common fixed point theorem to intuitionistic fuzzy metric spaces.

2
In this paper, we extend the result of common fixed point theorem for compatible mappings of
type (P-1) and type (P-2) in intuitionistic fuzzy metric space and prove common fixed point
theorem of type (P-1) and type (P-2) in intuitionistic fuzzy metric spaces, we also give an
example to validate our main theorem. Our results intuitionistically fuzzify the result of Muthuraj
and Pandiselvi [15].
2. PRELIMINARIES
We start with the following definitions.
Definition 2.1
A binary operation ∗ : [0,1] × [0,1] → [0,1] is said to be a continuous t-norm if * is satisfies the
following conditions.
(i) ∗ is commutative and associative,
(ii) ∗ is continuous,
(iii) a∗1 = a for all a∈ [0,1],
(iv) a∗b ≤ c∗d whenever a ≤ c and b ≤ d for all a,b,c,d ∈ [0,1].
Definition 2.2
A binary operation ◊ : [0,1] × [0,1] → [0,1] is said to be a continuous t-conorm if ◊ satisfies the
following conditions :
(i) ◊ is commutative and associative,
(ii) ◊ is continuous,
(iii) a ◊ 0 = a for all a ∈ [0,1],
(iv) a ◊ b ≤ c ◊ d whenever a ≤ c and b ≤ d for all a,b,c,d ∈ [0,1].
Definition 2.3
A 5-tuple (X, ℳ, , ∗, ◊) is called an intuitionistic fuzzy metric space if X is an arbitrary set, ∗
is a continuous t-norm, ◊ a continuous t-conorm and ℳ, are fuzzy sets on X3
× (0, ∞),
satisfying the following conditions, for each x, y, z, a∈X and
t, s 0,
a) ℳ( x, y, z, t ) + ( x, y, z, t ) ≤ 1.
b) ℳ( x, y, z, t ) 0.
c) ℳ( x, y, z, t ) = 1 if and only if x = y = z.
d) ℳ( x, y, z, t ) = ℳ ( p{ x, y, z}, t) where p is a permutation function,
e) ℳ( x, y, a, t ) ∗ ℳ( a, z, z, s ) ≤ ℳ( x, y, z, t + s )
f) ℳ( x, y, z ) : ( 0, ∞) → [0, 1] is continuous
g) ( x, y, z, t ) 0
h) ( x, y, z, t ) = 0, if and only if x = y = z,
i) ( x, y, z, t = ( p{ x, y, z}, t) where p is a permutation function,
j) ( x, y, a, t ) ◊ ( a, z, z, s ) ≥ ( x, y, z, t + s ),
k) ( x, y, z, ⋅) : ( 0, ∞) → [0, 1] is continuous.
Then (ℳ, ) is called an intuitionistic fuzzy metric on X.

3
Example 2.4
Let X = R, and ℳ(x, y, z, t ) =

| || || |
, ( x, y, z, t ) =
| || || |
| || || |
for every x,
y, z and t 0, let A and B defined as Ax = 2x + 1, Bx = x + 2, consider the sequence xn = +1, n
= 1 , 2,… Thus we have
lim
→∞
ℳ(Axn, 3, 3, t) = lim
→∞
ℳ(Bxn, 3, 3, t) =1 and
lim
→∞
( Axn, 3, 3, t) = lim
→∞
( Bxn, 3, 3, t) = 0, for every t 0.
Then A and B satisfying the property (E).
Definition 2.5
Let (X, ℳ, , ∗,◊ ) be an intuitionistic fuzzy metric space and {xn} be a sequence in X.
a) {xn} is said to be converges to a point x∈X, if lim
→∞
ℳ( x, x, xn, t ) = 1 and
lim
→∞
( x, x, xn, t ) = 0, for all t 0.
b) {xn} is called Cauchy sequence if lim
→∞
ℳ(xn+p, xn+p, xn, t) = 1 and
lim
→∞
(xn+p, xn+p, xn, t) = 0 for all t 0 and p 0.
c) An intuitionistic fuzzy metric space in which every Cauchy sequence is convergent is
said to be complete.
Lemma 2.6
Let (X, ℳ, , ∗, ◊) be an intuitionistic fuzzy metric space. Then ℳ(x, y, z, t) and (x, y, z, t)
are non-decreasing with respect to t, for all x, y, z in X.
Proof
By definition 2.3, for each x, y, z, a ∈X and t, s 0
we have ℳ(x, y, a, t ) ∗ ℳ(a, z, z, s ) ≤ ℳ(x, y, z, t + s ). If we set a = z,
we get ℳ(z, y, z, t ) ∗ ℳ(z, z, z, s ) ≤ ℳ(x, y, z, t + s ), that is
ℳ(x, y, z, t + s ) ≥ ℳ(x, y, z, t ).
Similarly, (x, y, a, t) ◊ (a, z, z, s ) ≥ (x, y, z, t + s ), for each x, y, z, a∈X and
t, s 0, by definition of (X, , ◊ ). If we set a = z, we get
(x, y, z, t ) ◊ (z, z, z, s ) ≥ (x, y, z, t + s )
that is (x, y, z, t + s ) ≤ (x, y, z, t) .Hence in IFMS (X, ℳ, , ∗, ◊ ),
ℳ(x, y, z, t ) and (x, y, z, t ) are non-decreasing with respect to t, for all x, y, z in X.

4
3.COMPATIBLE MAPPINGS OF TYPE
Definition 3.1
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗,◊) into itself.
Then the mappings are said to be compatible if
lim
→∞
ℳ(ASxn, SAxn, SAxn, t) = 1 and
lim
→∞
(ASxn, SAxn, SAxn, t) = 0, for all t 0 whenever {xn} is a sequence in X such that lim
→∞
Axn
= lim
→∞
Sxn = z for some z∈X.
Definition 3.2
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗, ◊ ) into
itself. Then the mappings are said to be compatible of type (P), if
lim
→∞
ℳ(AAxn, SSxn, SSxn, t ) = 1 and lim
→∞
(AAxn, SSxn, SSxn, t ) = 0 for all t 0, whenever
{xn} is a sequence in X such that lim
→∞
Axn = lim
→∞
Definition 3.3
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗, ◊) into itself.
Then the mappings are said to be R-Weakly commuting of type (P), if there exists some R 0,
such that ℳ(AAx, SSx, SSx, t) ≥ ℳ( Ax, Sx, Sx,

),
(AAx, SSx, SSx, t) ≤ (Ax, Sx, Sx,

), for all x in X and t 0.
Definition 3.4
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗, ◊) into
itself. Then the mappings are said to be compatible of type (P-1) if
lim
→∞
ℳ(SAxn, AAxn, AAxn, t ) = 1 and lim
→∞
( SAxn, AAxn, AAxn, t ) = 0 for all t 0, whenever
{xn} is a sequence in X such that lim
→∞
Axn = lim
→∞
S xn = z for some z∈X.
Definition 3.5
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗,◊ ) into itself.
Then the mappings are said to be compatible of type (P-2) if
lim
→∞
ℳ (AAxn, SSxn, SSxn, t) = 1 and lim
→∞
(AAxn, SSxn, SSxn, t) = 0 for all t 0 whenever {xn}
is a sequence in X such that lim
→∞
Axn = lim
→∞

5
Proposition 3.6
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗, ◊ ) into
itself.
a) If A is continuous map then the pair of mappings (A, S) is compatible of type (P-1) if and
only if A and S are compatible.
b) If S is a continuous map then the pair of mappings (A, S) is compatible of
type (P-2) if and only if A and S are compatible.
Proof
a) Let lim
→∞
Axn = lim
→∞
Sxn = z for some z ∈ X, and let the pair (A, S) be compatible of type
(P-1). Since A is continuous, we have lim
→∞
ASxn = Az and lim
→∞
AAxn = Az. Therefore it
follows that
ℳ SAx , ASx , ASx , t ≥ ℳ SAx , AAx , AAx ,

∗ ℳ AAx , ASx , ASx ,

and
SAx , ASx , ASx , t ≤ SAx , AAx , AAx ,

◊ AAx , ASx , ASx ,

yields lim
→∞
ℳ ( SAxn, ASxn, ASxn, t ) ≥ 1 ∗ 1 = 1 and
lim
→∞
( SAxn, ASxn, ASxn, t ) ≤ 0 ◊ 0 = 0 and so the mappings A and S are compatible.
Now, let A and S be compatible. Therefore it follows that
ℳSAx , AAx , AAx , t ≥ ℳ SAx , ASx , ASx ,
t
2

∗ ℳASx , AAx , AAx ,
t
2

SAx , AAx , AAx , t ≤ SAx , ASx , ASx ,
t
2

◊ ASx , AAx , AAx ,
t
2

yields lim
→∞
ℳ(SAxn, AAxn, AAxn, t ) ≥ 1 ∗ 1 = 1 and
lim
→∞
( SAxn, AAxn, AAxn, t ) ≤ 0 ◊ 0 = 0 and
so that pair of mappings (A,S) are compatible of type (P-1).

6
b) Let lim
→∞
Sxn = lim
→∞
Axn = z for some z in X and let the pair (A, S) be compatible of type
(P-2). Since S is continuous, we have lim
→∞
SAxn = Sz and
lim
→∞
SSxn = Sz. Therefore it follows that
ℳSAx , ASx , ASx , t ≥ ℳ SAx , SSx , SSx ,
t
2

∗ ℳ SSx , ASx , ASx ,

and
SAx , ASx , ASx , t ≤ SAx , SSx , SSx ,
t
2

◊ SSx , ASx , ASx ,
t
2

yields lim
→∞
ℳ( SAxn, ASxn, ASxn, t ) ≥ 1 ∗ 1 = 1 and
lim
→∞
( SAxn, ASxn, ASxn, t ) ≤ 0 ◊ 0 = 0 and so the mappings A and S are compatible.
Now let A and S be compatible. Then we have
ℳASx , SSx , SSx , t ≥ ℳ ASx , SAx , SAx ,
t
2

∗ ℳ SAx , SSx , SSx ,

and
ASx , SSx , SSx , t ≤ ASx , SAx , SAx ,
t
2

◊ SAx , SSx , SSx ,
t
2

yields lim
→∞
ℳ(ASxn, SSxn, SSxn, t ) ≥ 1 ∗ 1 = 1 and
lim
→∞
(ASxn, SSxn, SSxn, t) ≤ 0 ◊ 0 = 0 and so the pair of mappings (A, S) are
compatible of type (P-2).
Proposition 3.7
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗, ◊) into itself.
If the pair (A, S) is compatible of type (P-2) and Sz = Az for some z∈X. Then ASz = SSz.
Proof:
Let { xn} be a sequence in X defined by xn = z for n=1,2,… and let Az = Sz.
Then we have lim
→∞
Sxn = Sz and lim
→∞
Axn = Az. Since the pair (A, S) is compatible of type (P-2),
we have
ℳ( ASz, SSz, SSz, t ) = lim
→∞
ℳASxn, SSxn, SSxn, t ) = 1 and
( ASz, SSz, SSz, t ) = lim
→∞
(ASxn, SSxn, SSxn, t ) = 0.
Hence ASz = SSz.

7
Proposition 3.8
Let A and S self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗, ◊) with t ∗ t ≥
t and (1- t) ◊ (1- t) ≤ 1- t for all t ∈ [0, 1] if the pair (A, S) are compatible of type (p -1) and Axn,
Sxn → z for some z in X and a sequence {xn} in X.
Then AAxn → Sz, if S is continuous at z.
Proof
Since S is continuous at z, we have SAxn → Sz. Since the pair (A, S) are compatible of type (P-1),
we have ℳ(SAxn, AAxn, AAxn, t) → 1 as n → ∞. It follows that
ℳ( Sz, AAxn, AAxn, t) ≥ ℳ( Sz, SAxn, SAxn,

) ∗ ℳ( SAxn, AAxn, AAxn,

) and
(Sz, AAxn, AAxn, t ) ≤ (Sz, SAxn, SAxn,

) ∗ ( SAxn, AAxn, AAxn,

) yield
ℳ (Sz, AAxn, AAxn, t) ≥ 1 ∗1 = 1 and
(Sz, AAxn, AAxn, t) ≤ 0 ◊ 0 = 0 and so
we have AAxn → Sz as n → ∞.
Proposition 3.9
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, , ∗, ◊) with t ∗t ≥
t and (1- t) ◊ (1- t) ≤ 1- t for t ∈ [0, 1]. If the pair (A, S) are compatible of type (P - 2) and Axn,
Sxn→z for some z in X and sequence {xn} in X. Then SSxn → Az if A is continuous at z.
Proof
Since A is continuous at z, we have ASxn → Az. Since the pair (A, S) are compatible of type (P -
2), we have ℳ( ASxn, SSxn, SSxn, t ) →1 as n→∞, it follows that
ℳ(Az, SSxn, SSxn, t ) ≥ ℳ(Az, ASxn, ASxn,

) ∗ ℳ( ASxn, SSxn, SSxn,

) and
(Az, SSxn, SSxn, t ) ≤ (Az, ASxn, ASxn,

) ◊ (ASxn, SSxn, SSxn,

) yield
lim
→∞
ℳ(Az, SSxn, SSxn, t ) ≥ 1 ∗ 1 = 1 and
lim
→∞
( Az, SSxn, SSxn, t ) ≤ 0 ◊ 0 = 0 and so
we have SSxn → Az as n → ∞.

8
4. MAIN RESULTS
Theorem 4.1
Let (X, ℳ, , ∗,◊) be a complete generalized intuitionistic fuzzy metric space and let A, B, P,Q,
S and T be self mappings of X satisfying the following conditions.
(i) P(X) ⊆ ST (X), Q(X) ⊆ AB(X)
(ii) The pair (P, AB) and (Q, ST) are compatible mappings of type (P)
(iii) ST is continuous
(iv) ℳ( Px, Qz, Qz, qt) ≥ min {ℳ(ABx, Py, Qy, t), ℳ(ABx, Py, STz, t),
ℳ(Qy, STz, Py, t), ℳ(ABx, Qy, STz, t)} and
( Px, Qz, Qz, qt) ≤ max{ (ABx, Py,Qy, t), (ABx, Py, STz, t),
(Qy, STz, Py, t), (ABx, Qy, STz, t)}
then the mappings P, Q, AB and ST have a unique common fixed point in X.
Proof
Let x0 be any arbitrary point in X. Thus we construct a sequence {yn} in X such that
y2n-1 = STx2n-1 = Px2n-2 and y2n = ABx2n = Qx2n-1. Put x = x2n-1, y = x2n-1, z = x2n.
ℳ Px , Qx , Qx , qt ≥ min
$
%

ℳ ABx , Px , Qx , t ,
ℳABx , Px , STx , t ,
ℳ Qx , STx , Px , t ,
ℳ ABx , Qx , STx , t )
*
+
ℳ y , y , y , qt ≥ min
$
%

ℳ y , y , y , t ,
ℳ y , y , y , t ,
ℳ y , y , y , t ,
ℳy , y , y , t )
*
+
ℳ y , y , y , qt ≥ ℳy , y , y , t
This implies that ℳ y , y , y , tis an increasing sequence of positive real numbers.
Px , Qx , Qx , qt ≤ max
$
%

ABx , Px , Qx , t ,
ABx , Px , STx , t ,
Qx , STx , Px , t ,
ABx , Qx , STx , t )
*
+
y , y , y , qt ≤ max
$
%

y , y , y , t ,
y , y , y , t ,
y , y , y , t ,
y , y , y , t )
*
+
y , y , y , qt ≤ y , y , y , t
This implies that y , y , y , t is an decreasing sequence of positive real numbers.
Now to prove that ℳ y , y , y , t converges to 1 as n → ∞ and
y , y , y , t converges to 0 as n → ∞. By lemma 2.6,

9
ℳ y , y , y , t ≥ ℳ y , y , y ,
t
q
≥ ℳ y , y , y ,
t
q
. . . ≥ ℳ y0, y , y ,
t
q

Thus ℳ y , y , y , t ≥ ℳ y0, y , y ,

23 and
y , y , y , t ≤ y , y , y ,
t
q
≤ y , y , y ,
t
q
. . . ≤ y0, y , y ,
t
q

Then by the definition of IFMS,
ℳ( yn, yn+p, yn+p, t ) ≥ ℳ( yn, yn+1, yn+1,

4
) ∗… p times …∗ ℳ( yn+p-1, yn+p, yn+p,

4
)
≥ ℳ( y0, y1, y1,

23 ) ∗… p times …∗ ℳ( y0, y1, y1,

4235678 )
Thus by the definition of IFMS,
(yn, yn+p, yn+p, t) ≤ ( yn, yn+1, yn+1,

4
) ∗… p times …∗ ( yn+p-1, yn+p-1, yn+p,

4
)
≤ ( y0, y1, y1,

23 ) ∗ … p times … ∗ ( y0, y1, y1,

4235678 ).
lim
→∞
ℳ( yn, yn+p, yn+p, t ) ≥ 1 ∗ 1∗… p times …∗ 1. lim
→∞
ℳ( yn, yn+p, yn+p, t ) = 1 and
lim
→∞
( yn, yn+p, yn+p, t ) ≤ 0 ∗ 0∗…∗ p times …∗ 0. lim
→∞
(yn, yn+p, yn+p, t ) = 0.
Thus {yn} is a Cauchy sequence in intuitionistic fuzzy metric space X.
Since X is complete, there exists a point u∈X such that yn → u.
Thus {ABx2n}, {Qx2n-1}, {STx2n-1}, {Px2n-2} are Cauchy sequence converge to u.
Put x = ABx2n, y = u, z = STx2n-1 in (iv), we get
ℳ PABx , QSTx , QSTx , qt ≥ min
$
%

ℳABABx , Pu, Qu, t,
ℳABABx , Pu, STSTx , t ,
ℳQu, STSTx , Pu, t,
ℳABABx , Qu, STSTx , t ,)
*
+
and
PABx , QSTx , QSTx , qt ≤ max
$
%

ABABx , Pu, Qu, t,
ABABx , Pu, STSTx , t ,
Qu, STSTx , Pu, t,
ABABx , Qu, STSTx , t ,)
*
+
.

10
Now take the limit as n → ∞ and using (ii), we get,
ℳPu, Qu, Qu, qt ≥ min :
ℳ Pu, Pu, Qu, t, ℳ Pu, Pu, Qu, t
ℳ Qu, Qu, Pu, t, ℳ Pu, Qu, Qu, t
; and
Pu, Qu, Qu, qt ≤ max :
Pu, Pu, Qu, t, Pu, Pu, Qu, t
Qu, Qu, Pu, t, Pu, Qu, Qu, t
;.
Then by lemma 2.6, we get
ℳPu, Qu, Qu, qt ≥ ℳ Pu, Qu, Qu, t and
Pu, Qu, Qu, qt ≤ Pu, Qu, Qu, t.
Therefore Pu = Qu. Now put x = ABx2n, y = x2n-1, z = x2n-1, in (iv), we get
ℳ PABx , Qx , Qx , qt ≥ min
$
%

ℳ ABABx , Px , Qx , t,
ℳABABx , Px , STx , t ,
ℳ Qx , STx , Px , t ,
ℳABABx , Qx , STx , t )
*
+
and
PABx , Qx , Qx , qt ≤ max
$
%

ABABx , Px , Qx , t,
ABABx , Px , STx , t ,
Qx , STx , Px , t ,
ABABx , Qx , STx , t )
*
+
Thus we have ℳ Pu, u, u, qt ≥ ℳ Pu, u, u, t and
Pu, u, u, qt ≥ Pu, u, u, t.
Therefore Pu = u. This implies Pu = Qu = u.
Now put x = Px2n-2, y = Px2n-2, z = u in (iv), we get
ℳPPx , Qu, Qu, qt ≥ min
$
%

ℳ ABPx , PPx , QPx , t ,
ℳABPx , PPx , STu, t ,
ℳ QPx , STu, PPx , t ,
ℳABPx , QPx , STu, t )
*
+
and

11
PPx , Qu, Qu, qt ≤ max
$
%

ℳ ABPx , PPx , QPx , t ,
ℳABPx , PPx , STu, t ,
ℳ QPx , STu, PPx , t ,
ℳABPx , QPx , STu, t )
*
+
.
Now taking the limit as n → ∞ and on using (ii) and (iii), we get
ℳABu, u, u, qt ≥ min :
ℳABu, ABu, u, t , ℳABu, ABu, u, t ,
ℳ Qu, u, ABu, t , ℳABu, Qu, u, t
;
ABu, u, u, qt ≤ max :
ABu, ABu, u, t , ABu, ABu, u, t ,
Qu, u, ABu, t , ABu, Qu, u, t
;.
This implies
ℳABu, u, u, qt ≥ min :
ℳABu, ABu, u, t , ℳABu, ABu, u, t ,
ℳu, u, ABu, t , ℳABu, u, u, t
;
ABu, u, u, qt ≤ max :
ABu, ABu, u, t , ABu, ABu, u, t ,
u, u, ABu, t , ABu, u, u, t
; .
Therefore by lemma (2.6) we have ABu = u. Thus Pu = Qu = ABu = u.
Put x = u, y = u, z = Qx2n-1, in (iv) we get
ℳPu, QQx , QQx , qt ≥ min :
ℳu, u, u, t , ℳ u, u, STu, t ,
ℳu, STu, u, t , ℳ u, u, , STu, t
;
Pu, QQx , QQx , qt ≤ max :
u, u, u, t , u, u, STu, t ,
u, STu, u, t , u, u, , STu, t
;,
On using lemma, (2.6) we have
ℳSTu, STu, u, qt ≥ ℳ STu, STu, u, t and
ℳSTu, STu, u, qt ≥ ℳ STu, STu, u, t
(STu, STu, u, qt ) ≤ ( STu, STu, u, t ).
Thus STu = u. We get Pu = Qu = ABu = STu = u.
Uniqueness
Let w be another common fixed point of A, B, P, Q, S and T. Then
ℳ Pu, Qw, Qw, qt ≥ min :
ℳABu, Pw, Qw, t , ℳ ABu, Pw, STw, t ,
ℳ Qw, STw, Pw, t , ℳ ABu, Qw, STw, t
;

12
ℳ u, w, w, qt ≥ min :
ℳ u, w, w, t , ℳ u, w, w, t ,
ℳ w, w, w, t , ℳ u, w, w, t
;
ℳ u, w, w, qt ≥ ℳ u, w, w, t and
Pu, Qw, Qw, qt ≤ max :
ABu, Pw, Qw, t , ABu, Pw, STw, t ,
Qw, STw, Pw, t , ABu, Qw, STw, t
;
u, w, w, qt ≤ max :
u, w, w, t , u, w, w, t ,
w, w, w, t , u, w, w, t
; u, w, w, qt ≤ u, w, w, t ,
which is a contradiction. Therefore u = w.
Hence the common fixed point is unique.
Corollary 4.2
Let (X, ℳ, , ∗,◊) be a complete generalized intuitionistic fuzzy metric space and let A, P,Q and
S be self mappings of X satisfying the following conditions.
(i) P(X) ⊆ S(X), Q(X) ⊆ A(X)
(ii) The pair (P,A) and (Q,S) are compatible mappings of type (P)
(iii) S is continuous
(iv) ℳ( Px, Qz, Qz, qt ) ≥ min { ℳ( Ax, Py, Qy, t ), ℳ( Ax, Py, Sz, t ),
ℳ( Qy, Sz, Py, t ), ℳ( Ax, Qy, Sz, t )} and
( Px, Qz, Qz, qt ) ≤ max {( Ax, Py, Qy, t ), ( Ax, Py, Sz, t ),
( Qy, Sz, Py, t ), ( Ax, Qy, Sz, t )}.
Then the mappings P, Q, A and S have a unique common fixed point in X.
Corollary 4.3
Let (X, ℳ, , ∗,◊) be a complete generalized intuitionistic fuzzy metric space and let B,P,Q and
T be self mappings of X satisfying the conditions (i), (ii), (iii), (iv) with S = I and A = I;
Then the mappings B, P,Q and T have a unique common fixed point.
Corollary 4.4
Let ( X, ℳ, , ∗, ◊ ) be a complete generalized intuitionistic fuzzy metric space and let
A,B,P,Q,S and T be self mappings of X satisfying the following conditions:
(i) P(X) ⊆ ST(X), Q(X) ⊆ AB(X)
(ii) The pair (P, AB) and (Q, ST) are compatible mappings of type (P)
(iii) ST is continuous
(iv) ℳ( Px, Qz, Qz, qt) ≥ ℳ( ABx, Py, Qy, t) ∗ ℳ(ABx, Py, STz, t) ∗
ℳ( Qy, STz, Py, t) ∗ ℳ(ABx,Qy,STz,t) and
( Px, Qz, Qz, qt) ≤ ( ABx, Py, Qy, t) ◊ ( ABx, Py, STz, t) ◊
( Qy, STz, Py, t) ◊ ( ABx, Qy, STz, t)
Then the mappings P,Q,AB and ST have a unique common fixed point in X.

13
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Authors
Dr.R.Muthuraj received his Ph.D degree in Mathematics from Alagappa University
Karaikudi, Tamil nadu, India in April 2010. Presently he is an Assistant Professor ,
PG Research Department of Mathematics, H.H.The Rajah’s College,
Pudukkottai Tamilnadu ,India. He has published over 80 papers in refereed
National and International Journals. He is the reviewer and Editor of the reputed
International Journals.Eight members are doing research work under his guidance.
His research interests are Fuzzy Algebra, Lattice Theory, Discrete Mathematics,
Fuzzy Topology, Fixed point theory and Fuzzy Graph Theory.
R. Pandiselvi received her M.Phil degree from School of Mathematics, Madurai
Kamaraj University, Madurai, Tamilnadu, India. Now she is doing Ph.D at
Bharathidasan University Tiruchirappalli,Tamilnadu, India. Presently she is
working as an Associate Professor in Mathematics , The Madura college, Madurai,
Tamilnadu, India. She has published over 10 papers in reputed National and
International journals. Her research area is Fixed Point Theory.

Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalized Intuitionistic Fuzzy Metric Spaces

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