Previously, Park et al. (2005) defined an intuitionistic fuzzy metric space and studied several fixedpoint theories in this space. This paper provides definitions and describe the properties of type(
β
) compatible mappings, and prove some common fixed points for four selfmappings that are compatible with type(
β
) in an intuitionistic fuzzy metric space. This paper also presents an example of a common fixed point that satisfies the conditions of Theorem 4.1 in an intuitionistic fuzzy metric space.
1. Introduction
Grabiec
[1]
demonstrated the Banach contraction theorem in the fuzzy metric spaces introduced by Kramosil and Michalek
[2]
. Park
[3

5]
, Park and Kim
[6]
also proved a fixedpoint theorem in a fuzzy metric space.
Recently, Park et al.
[7]
defined an intuitionistic fuzzy metric space while Park et al.
[8]
proved a fixedpoint Banach theorem for the contractive mapping of a complete intuitionistic fuzzy metric space. Park et al.
[9]
defined a type(
α
) compatible map and obtained results for five mappings using a type(
α
) compatibility map in intuitionistic fuzzy metric spaces. Furthermore, Park
[10]
introduced a type(
β
) compatible mapping and proved some of the properties of the type(
β
) compatibility mapping in an intuitionistic fuzzy metric space.
This paper proves some common fixed points for four selfmappings that satisfy type(
β
) compatibility mapping in intuitionistic fuzzy metric space, while it also provides an example in the given conditions for an intuitionistic fuzzy metric space.
2. Preliminaries
First, some definitions and properties of the intuitionistic fuzzy metric space
X
are provided, as follows.
Let us recall (
[11]
) that a continuous
t
–norm is a binary operation * : [0, 1]×[0, 1] → [0, 1], which satisfies the following conditions: (a) * is commutative and associative; (b) * is continuous; (c)
a
* 1 =
a
for all
a
∈ [0, 1]; (d)
a
*
b
≤
c
*
d
whenever
a
≤
c
and
b
≤
d
(
a
,
b
,
c
,
d
∈ [0, 1]).
Similarly, a continuous
t
–conorm is a binary operation ◊ : [0, 1] × [0, 1] → [0, 1], which satisfies the following conditions: (a) ◊ is commutative and associative; (b) ◊ is continuous; (c)
a
◊ 0 =
a
for all
a
∈ [0, 1]; (d)
a
◊
b
≤
c
◊
d
whenever
a
≤
c
and
b
≤
d
(
a
,
b
,
c
,
d
∈ [0, 1]).
 Definition 2.1.
[12]
The 5–tuple (
X
,
M
,
N
,*,◊) is said to be an intuitionistic fuzzy metric space if
X
is an arbitrary set, * is a continuous
t
–norm, ◊ is a continuous
t
–conorm, and
M
,
N
are fuzzy sets in
X
^{2}
× (0,∞), which satisfy the following conditions: for all
x
,
y
,
z
∈
X
, such that

(a)M(x,y,t) > 0,

(b)M(x,y,t) = 1 ⇔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 continuous,

(f)N(x,y,t) > 0,

(g)N(x,y,t) = 0 ⇔x=y,

(h)N(x,y,t) =N(y,x,t),

(i)N(x,y,t) ◊N(y,z,s) ≥N(x,z,t+s),

(j)N(x,y, •) : (0,∞) → (0, 1] is continuous.
Note that (
M
,
N
) is referred to as an intuitionistic fuzzy metric on
X
. The functions
M
(
x
,
y
,
t
) and
N
(
x
,
y
,
t
) denote the degree of proximity and the degree of nonproximity between
x
and
y
with respect to
t
, respectively.
 Example 2.2.
[13]
Let (
X
,
d
) be a metric space. Denote
a
*
b
=
ab
and
a
◊
b
= min{1,
a
+
b
} for all
a, b
∈ [0, 1] and let
M_{d}
,
N_{d}
be the fuzzy sets on
X
^{2}
× (0,∞), which are defined as follows :
for
k
,
m
,
n
∈
R
^{+}
(
m
≥ 1). Thus, (
X
,
M_{d}
,
N_{d}
,*,◊) is an intuitionistic fuzzy metric space, i.e., the intuitionistic fuzzy metric space induced by the metric
d
.
 Definition 2.3.
[13]
Let
X
be an intuitionistic fuzzy metric space.
(a) {
x_{n}
} is said to be convergent to a point
x
∈
X
by lim
_{n}
_{→∞}
x_{n}
=
x
if
for all
t
> 0.
(b) {
x_{n}
} is a Cauchy sequence if
for all
t
> 0 and
p
> 0.
(c)
X
is complete if every Cauchy sequence converges on
X
.
In this paper,
X
is considered to be the intuitionistic fuzzy metric space with the following condition:
for all
x
,
y
∈
X
and
t
> 0.
 Lemma 2.4.
[6]
Let {
x_{n}
} be a sequence in an intuitionistic fuzzy metric space
X
with the condition (1). If there exists a number
k
∈ (0,1) such that for all
x
,
y
∈
X
and
t
> 0,
for all
t
> 0 and
n
= 1, 2, 3 …, then {
x_{n}
} is a Cauchy sequence in
X
.
 Lemma 2.5.
[14]
Let
X
be an intuitionistic fuzzy metric space. If there exists a number
k
∈ (0,1) such that for all
x
,
y
∈
X
and
t
> 0,

M(x,y,kt) ≥M(x,y,t),

N(x,y,kt) ≤N(x,y,t),
then
x
=
y
.
3. Properties of type(β) compatible mappings and an example
This section introduces type(
α
) and type(
β
) compatible maps in an intuitionistic fuzzy metric space, and it also presents an example of the relations of type(
β
) compatible maps.
 Definition 3.1.
[14]
Let
A
,
B
be mappings from the intuitionistic fuzzy metric space
X
into itself. These mappings are said to be compatible if
for all
t
> 0, whenever {
x_{n}
} ⊂
X
such that lim
_{n}
_{→∞}
Ax_{n}
= lim
_{n}
_{→∞}
Bx_{n}
=
x
for some
x
∈
X
.
 Definition 3.2.
(
[10]
) Let
A
,
B
be mappings from the intuitionistic fuzzy metric space
X
into itself. The mappings are said to be type(
β
) compatible if
for all
t
> 0, whenever {
x_{n}
} ⊂
X
such that lim
_{n}
_{→∞}
Ax_{n}
= lim
_{n}
_{→∞}
Bx_{n}
=
x
for some
x
∈
X
.
 Proposition 3.3.
[10]
Let
X
be an intuitionistic fuzzy metric space and
A
,
B
be the continuous mappings from
X
into itself. Thus,
A
and
B
are compatible if they are type(
β
) compatible.
 Proposition 3.4.
[10]
Let
X
be an intuitionistic fuzzy metric space and
A
,
B
be mappings from
X
into itself. If
A
,
B
are type(
β
) compatible and
Ax
=
Bx
for some
x
∈
X
, then
ABx
=
BBx
=
BAx
=
AAx
.
 Proposition 3.5.
[10]
Let
X
be an intuitionistic fuzzy metric space and
A
,
B
be type(
β
) compatible mappings from
X
into itself. Let {
x_{n}
} ⊂
X
so lim
_{n}
_{→∞}
Ax_{n}
= lim
_{n}
_{→∞}
Bx_{n}
=
x
for some
x
∈
X
, then

(a)limn→∞BBxn=Axif A is continuous atx∈X,

(b)limn→∞AAxn=BxifBis continuous atx∈X,

(c)ABx=BAxandAx=BxifAandBare continuous atx∈X.
 Example 3.6.
Let
X
= [0,∞) with the metric
d
defined by
d
(
x
,
y
) = 
x
–
y
 and for each
t
> 0, let
M_{d}
,
N_{d}
be fuzzy sets on
X
^{2}
× [0,∞), which are defined as follows
for all
x
,
y
∈
X
. Clearly, (
X
,
M_{d}
,
N_{d}
, *, ◊) is an intuitionistic fuzzy metric space where *, ◊ are defined by
a
*
b
= min{
a, b
} and
a
◊
b
= max{
a, b
} for all
a, b
∈ [0, 1]. Let us define
A
,
B
:
X
→
X
as
Thus,
A
,
B
are discontinuous at
x
= 1. Let {
x_{n}
} ⊂
X
be defined by
Next, we have lim
_{n}
_{→∞}
Ax_{n}
= lim
_{n}
_{→∞}
Bx_{n}
= 1
Furthermore,
and
Therefore,
A
,
B
are type(
β
) compatible but they are not compatible.
4. Main Results and Example
This section proves the main theorem and presents an example using the given conditions in an intuitionistic fuzzy metric space.
 Theorem 4.1.
Let
X
be a complete intuitionistic fuzzy metric space where
t
*
t
≥
t
,
t
◊
t
≤
t
for all
t
∈ [0, 1]. Let
A
,
B
,
S
and
T
be mappings from
X
into itself so:

(a)AT(X) ∪BS(X) ⊂ST(X);

(b) there existsk∈ (0, 1) so for allx,y∈Xandt> 0,

M2(Ax,By,kt) * [M(Sx,Ax,kt)M(Ty,By,kt)]

*M2(Ty,By,kt) +aM(Ty,By,kt)M(Sx,By, 2kt)

≥ [pM(Sx,Ax,t) +qM(Sx,Ty,t)]M(Sx,By, 2kt),

N2(Ax,By,kt) ◊ [N(Sx,Ax,kt)N(Ty,By,kt)] ◊N2(Ty,By,kt) +aM(Ty,By,kt)N(Sx,By, 2kt) ≤ [pN(Sx,Ax,t) +qN(Sx,Ty,t)]N(Sx,By, 2kt),
where 0 <
p
,
q
< 1, 0 ≤
a
< 1 such that
p
+
q
–
a
= 1;

(c)SandTare continuous andST=TS;

(d) the pairs (A,S) and (B,T) are type(β) compatible.
Thus,
A
,
B
,
S
and
T
have a unique common fixed point in
X
.
Proof.
Let
x
_{0}
be an arbitrary point of
X
. Using (a), we can construct an {
x_{n}
} ⊂
X
as follows:
ATx
_{2}
_{n}
=
STx
_{2}
_{n}
_{+1}
,
BSx
_{2}
_{n}
_{+1}
=
STx
_{2}
_{n}
+2,
n
= 0, 1, 2,…. Next, let
z_{n}
=
STx_{n}
. Using (b), we obtain

M2(ATx2n,BSx2n+1,kt) * [M(STx2n,ATx2n,kt) ×M(TSx2n+1,BSx2n+1,kt)] *M2(TSx2n+1,BSx2n+1,kt) +aM(TSx2n+1,BSx2n+1,kt) ×M(STx2n,BSx2n+1, 2kt) ≥ [pM(STx2n+1,ATx2n,t) +qM(STx2n,TSx2n+1,t)] ×M(STx2n,BSx2n+1, 2kt),

N2(ATx2n,BSx2n+1,kt) ◊ [N(STx2n,ATx2n,kt) ×N(TSx2n+1,BSx2n+1,kt)] ◊N2(TSx2n+1,BSx2n+1,kt) +aN(TSx2n+1,BSx2n+1,kt) ×N(STx2n,BSx2n+1, 2kt) ≤ [pN(STx2n+1,ATx2n,t) +qN(STx2n,TSx2n+1,t)] ×N(STx2n,BSx2n+1, 2kt)
and

M2(STx2n+1,STx2n+2,kt) * [M(z2n,STx2n+1,kt) ×M(z2n+1,STx2n+2,kt)] *M2(z2n+1,STx2n+2,kt) +aM(z2n+1,STx2n+2,kt)M(z2n,STx2n+2, 2kt) ≥ [pM(z2n,STx2n+1,t) +qM(z2n,z2n+1,t)] ×M(z2n,STx2n+2, 2kt),

N2(STx2n+1,STx2n+2,kt) ◊ [N(z2n,STx2n+1,kt) ×N(z2n+1,STx2n+2,kt)] ◊N2(z2n+1,STx2n+2,kt) +aN(z2n+1,STx2n+2,kt)N(z2n,STx2n+2, 2kt) ≤ [pN(z2n,STx2n+1,t) +qN(z2n,z2n+1,t)] ×N(z2n,STx2n+2, 2kt).
Then,

M2(z2n+1,z2n+2,kt) *[M(z2n,z2n+1,kt)M(z2n+1,z2n+2,kt)] +aM(z2n+1,z2n+2,kt)M(z2n,z2n+2, 2kt) ≥ [p+q]M(z2n,z2n+1,t)M(z2n,z2n+2, 2kt),

N2(z2n+1,z2n+2,kt) ◊[N(z2n,z2n+1,kt)N(z2n+1,z2n+2,kt)] +aN(z2n+1,z2n+2,kt)N(z2n,z2n+2, 2kt) ≤ [p+q]N(z2n,z2n+1,t)N(z2n,z2n+2, 2kt),
and

M2(z2n+1,z2n+2,kt)M(z2n+1,z2n+2,kt)] +aM(z2n+1,z2n+2,kt)M(z2n,z2n+2, 2kt) ≥ [p+q]M(z2n,z2n+1,t)M(z2n,z2n+2, 2kt),

N2(z2n+1,z2n+2,kt)N(z2n+1,z2n+2,kt)] +aN(z2n+1,z2n+2,kt)N(z2n,z2n+2, 2kt) ≤ [p+q]N(z2n,z2n+1,t)N(z2n,z2n+2, 2kt).
Therefore, it follows that

z2n+1M(z2n+1,z2n+2,kt) ≥M(z2n,z2n+1,t),

N(z2n+1,z2n+2,kt) ≤N(z2n,z2n+1,t)
for all
t
> 0 and
k
∈ (0, 1). In general, for
m
= 1, 2, …, we have

M(zm+1,zm+2,kt) ≥M(zm,zm+1,t),

N(zm+1,zm+2,kt) ≤N(zm,zm+1,t)
Thus, {
z_{n}
} is a Cauchy sequence in
X
and, because
X
is complete, {
z_{n}
} converges to a point
z
∈
X
. Since {
ATx
_{2}
_{n}
}, {
BSx
_{2}
_{n}
_{+1}
} are subsequences of {
z_{n}
}, lim
_{n}
_{→∞}
ATx
_{2}
_{n}
=
z
= lim
_{n}
_{→∞}
BSx
_{2}
_{n}
_{+1}
.
Let
y_{n}
=
TX_{n}
,
u_{n}
=
Sx_{n}
for
n
= 1, 2,…. Thus, we have
Ay
_{2}
_{n}
→
z
,
Sy
_{2}
_{n}
→
z
,
Tu
_{2}
_{n}
_{+1}
→
z
and
Bu
_{2}
_{n}
_{+1}
→
z
. Furthermore,

M(AAy2n,SSy2n,t) → 1,

M(BBu2n+1,TT2n+1,t) → 1,

N(AAy2n,SSy2n,t) → 0,

N(BBu2n+1,TT2n+1,t) → 0
as
n
→∞. Based on the continuity of
T
and Proposition 3.4, we obtain
TBu
_{2}
_{n}
_{+1}
→
Tz
,
BBu
_{2}
_{n}
_{+1}
→
Tz
.
Next, by taking
x
=
y
_{2}
_{n}
,
y
=
Bu
_{2}
_{n}
_{+1}
in (b), for
n
→∞ we obtain,

M2(z,Tz,kt) * [M(z,z,kt)M(Tz,Tz,kt)] *M2(Tz,Tz,kt) +aM(Tz,Tz,kt)M(z,Tz, 2kt) ≥ [pM(z,z,t) +qM(z,Tz,t)]M(z,Tz, 2kt),

N2(z,Tz,kt) ◊ [N(z,z,kt)N(Tz,Tz,kt)] ◊N2(Tz,Tz,kt) +aN(Tz,Tz,kt)N(z,Tz, 2kt) ≤ [pN(z,z,t) +qN(z,Tz,t)]N(z,Tz, 2kt),
then

M2(z,Tz,kt) +aM(z,Tz, 2kt) ≥ [p+qM(z,Tz,t)]M(z,Tz, 2kt),N2(z,Tz,kt) ≤qN(z,Tz,t)N(z,Tz, 2kt).
Since
M
(
x
,
y
, •) is nondecreasing and
N
(
x
,
y
, •) is nonincreasing for all
x
,
y
∈
X
, we obtain

M(z,Tz,kt) +a≥p+qM(z,Tz,t),

N(z,Tz,kt) ≤qN(z,Tz,t)
and
Thus,
z
=
Tz
. Similarly, we have
z
=
Sz
.
Next, by taking
x
=
y
_{2}
_{n}
and
y
=
z
in condition (b), for
n
→∞ we obtain

M(z,Bz,kt) *M(z,Bz,kt) +aM(z,Bz,kt)M(z,Bz, 2kt) ≥ (p+q)M(z,Bz, 2kt),

N(z,Bz,kt) ◊N(z,Bz,kt) +aN(z,Bz,kt)N(z,Bz, 2kt) ≤ 0.
Thus,

M(z,Bz,kt) +aM(z,Bz,kt) ≥p+q,

N(z,Bz,kt) +aN(z,Bz,kt) ≤ 0.
Therefore,

M(z,Bz,kt) ≥ 1,

N(z,Bz,kt) ≤ 0
for all
t
> 0 and
k
∈ (0, 1). Thus,
z
=
Bz
. Similarly, we obtain
z
=
Az
. Therefore,
z
is a common fixed point of
A
,
B
,
S
and
T
.
Let
w
be another common fixed point of
A
,
B
,
S
and
T
.
Using condition (b), we have

M2(z,w,kt) * [M(z,z,kt)M(w,w,kt)] *M2(w,w,kt) +aM(w,w,kt)M(z,w, 2kt) ≥ [pM(z,z,t) +qM(z,w,t)]M(z,w, 2kt),

N2(z,w,kt) ◊ [N(z,z,kt)N(w,w,kt)] ◊N2(w,w,kt) +aN(w,w,kt)N(z,w, 2kt) ≤ [pN(z,z,t) +qN(z,w,t)]M(z,w, 2kt).
Thus,

M2(z,w,kt) +M(z,w, 2kt) ≥ (p+qM(z,w,t))M(z,w, 2kt),N2(z,w,kt) ≤qM(z,w,t)M(z,w, 2kt),
Therefore,

M(z,w,kt) ≤M(z,w, 2kt),

N(z,w,kt) ≥N(z,w, 2kt),
so
Thus,
z
=
w
. This means that
A
,
B
,
S
and
T
have a unique common fixed point.
 Corollary 4.2.
Let
X
be a complete intuitionistic fuzzy metric space where
t
*
t
≥
t
,
t
◊
t
≤
t
for all
t
∈ [0, 1] and let
A
,
B
be mappings from
X
into itself such that:

(e)A(X) ⊂S(X),

(f) there existsk∈ (0,1) so for allx,y∈Xandt> 0,

M2(Ax,Ay,kt) * [M(Sx,Ax,kt)M(Sy,Ay,kt)]M2(Sy,Ay,kt) +aM(Sy,Ay,kt)M(Sx,Ay, 2kt) ≥ [pM(Sx,Ax,t) +qM(Sx,Sy,t)]M(Sx,Ay, 2kt),

N2(Ax,Ay,kt) ◊ [N(Sx,Ax,kt)N(Sy,Ay,kt)kt)] ◊N2(Sy,Ay,kt) +aM(Sy,Ay,kt)N(Sx,Ay, 2kt) ≤ [pN(Sx,Ax,t) +qN(Sx,Sy,t)]N(Sx,Ay, 2kt),
where 0 <
p
,
q
< 1, 0 ≤
a
< 1 such that
p
+
q
–
a
= 1,

(g)Sis continuous,

(h)AandSare type(β) compatible.
Thus,
A
and
S
have a unique common fixed point in
X
.
Proof.
Therefore, if we enter
A
=
B
and
S
=
T
into Theorem 4.1, all of the conditions of Theorem 4.1 are satisfied. Thus, the proof of this corollary follows from Theorem 4.1.
 Example 4.3.
Let
with the metric
d
defined by
d
(
x
,
y
) = 
x
–
y
 and for each
t
> 0, let
M_{d}
,
N_{d}
be fuzzy sets on
X
^{2}
× [0,∞), which are defined as follows
for all
x
,
y
∈
X
. Clearly, (
X
,
M_{d}
,
N_{d}
, *, ◊) is a complete intuitionistic fuzzy metric space where *, ◊ are defined by
a
*
b
= min{
a, b
} and
a
◊
b
= max{
a, b
} for all
a, b
∈ [0, 1]. Let
A
,
B
,
S
and
T
be maps from
X
into itself, which are defined by
for all
x
∈
X
. Then,
Furthermore,
ST
=
TS
and
S
,
T
are continuous. If we take
and
t
= 1, the condition (b) of Theorem 4.1 is satisfied. Moreover,
A
,
S
are type(
β
) compatible if lim
_{n}
_{→∞}
x_{n}
= 0 where {
x_{n}
} ⊂
X
such that lim
_{n}
_{→∞}
Ax_{n}
= lim
_{n}
_{→∞}
S_{xn}
= 0 for some 0 ∈
X
.
Similarly,
B
,
T
are type(
β
) compatible. Thus,

M(0,B0,kt) +aM(0,B0,kt) ≥p+q,

N(0,B0,kt) +aN(0,B0,kt) ≤ 0.
Therefore,
M
(0,
B
0,
kt
) ≥ 1 and
N
(0,
B
0,
kt
) ≤ 0 for all t > 0 and
k
∈ (0, 1). Thus, 0 =
B
0. Similarly, we obtain 0 =
A
0. Therefore, 0 is a common fixed point of
A
,
B
,
S
and
T
.
Let w be another common fixed point of
A
,
B
,
S
and
T
. Then,

M2(0,w,kt) +M(0,w, 2kt) ≥ (p+qM(0,w,t))M(0,w,2kt),

M2(0,w,kt) ≤qM(0,w,t)M(0,w, 2kt).
Therefore, because

M(0,w,kt) ≤M(0,w, 2kt),

N(0,w,kt) ≥N(0,w, 2kt),
Thus,
Therefore, 0 =
w
. Thus,
A
,
B
,
S
and
T
have a unique common fixed point 0.
5. Conclusion
Park et al.
[7]
defined an intuitionistic fuzzy metric space and Park et al.
[8]
proved a fixedpoint Banach theorem for the contractive mapping of a complete intuitionistic fuzzy metric space. Park et al.
[9]
defined a type(
α
) compatible mapping and obtained results for five mappings using type(
α
) compatibility in intuitionistic fuzzy metric spaces. Furthermore, Park
[10]
introduced type(
β
) compatible mapping and proved some properties of type(
β
) compatibility in an intuitionistic fuzzy metric space. In this paper, we proved some common fixed points for four selfmappings that satisfy type(
β
) compatibility and we provided an example in the given conditions for an intuitionistic fuzzy metric space.
This paper attempted to develop a method to provide a proof based on the fundamental concepts and properties defined in the new space. I think that the results of this paper will be extended to the intuitionistic Mfuzzy metric space and other spaces. Further research should be conducted to determine how to combine the collaborative learning algorithm with our proof method in the future.
Acknowledgements
The author would like to thank the editors for their very helpful and detailed editorial comments. This paper was supported by the Chinju National University of Education Research Fund in 2012.
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